**Extending Cosmology: The Metric Approach**

**Extending Cosmology: The Metric Approach**

Sergio Mendoza Additional information is available at the end of the chapter

10.5772/53878

Sergio Mendoza

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53878

## **1. Introduction**

In this chapter it is reviewed a possible physical scenario for which the introduction of a fundamental constant of nature with dimensions of acceleration into the theory of gravity makes it possible to extend gravity in a very consistent manner. In the non-relativistic regime a MOND-like theory with a modification in the force sector is obtained. This description turns out to be the the weak-field limit of a more general metric relativistic theory of gravity. The mass and length scales involved in the dynamics of the whole universe require small accelerations which are of the order of Milgrom's acceleration constant and so, it turns out that this relativistic theory of gravity can be used to explain the expansion of the universe. In this work it is explained how to build that relativistic theory of gravity in such a way that the overall large-scale dynamics of the universe can be treated in a pure metric approach without the need to introduce dark matter and/or dark energy components.

Cosmological and astrophysical observations are generally explained introducing two unknown mysterious dark components, namely dark matter and dark energy. These ad hoc hypothesis represent a big cosmological paradigm, since they arise due to the fact that Einstein's field equations are forced to remain unchanged under certain observed astrophysical phenomenology.

A natural alternative scenario would be to see whether viable cosmological solutions can be found if dark unknown entities are assumed non-existent. The price to pay with this assumption is that the field equations of the theory of gravity need to be extended and so, new Friedmann-like equations will arise. The most natural approach to extend gravity arises when a metric extension *f*(*R*) is introduced into the theory [see e.g. 9, and references therein].

In a series of recent articles, Bernal, Capozziello, Cristofano & de Laurentis [4], Bernal, Capozziello, Hidalgo & Mendoza [5], Carranza et al. [10], Hernandez et al. [17, 18], Mendoza et al. [22, 23] have shown how relevant the introduction of a new fundamental physical constant <sup>a</sup><sup>0</sup> <sup>≈</sup> <sup>10</sup><sup>−</sup>10m/s2 with dimensions of acceleration is in excellent agreement with

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Mendoza; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Mendoza, licensee InTech. This is an open access chapter distributed under the terms of the Creative

different phenomenology at many astrophysical mass and length sizes, from solar-system to extragalactic and cosmological scales. The introduction of the so called Milgrom's acceleration constant a<sup>0</sup> in a description of gravity means that any gravitational field produced by a certain distribution of mass (and hence energy) needs to incorporate the acceleration a<sup>0</sup> together with Newton's gravitational constant *G* and the speed of light *c* in the description of gravity.

where

*<sup>µ</sup>*(*y*) =

where the dimensionless quantity

and a mass-length scale

*function*) *g*(*x*) is such that:

*a* a0

Newtonian to the MONDian regime occurs.

correct MONDian and Newtonian limits, yielding:

= *g*(*x*) :=

gravitational acceleration experienced by a test particle is given by

1, for *y* ≫ 1, *y*, for *y* ≪ 1,

The usual approach to MOND as expressed by equation (2) means that Newton's 2nd law of mechanics needs to be modified [see e.g. 1]. As explained by Mendoza et al. [23], a better physical approach can be constructed if the modification is made in the force (gravitational) sector. Indeed, by the use of Buckingham's theorem of dimensional analysis [cf. 35], the

*<sup>x</sup>* :<sup>=</sup> *lM*

 *GM* a0

The length *lM* plays an important role in the description of the theory and is such that when *lM* ≫ *<sup>r</sup>*, the strong Newtonian regime of gravity is recovered and when *lM* ≪ *<sup>r</sup>* the weak MONDian regime of gravity appears. As such, the dimensionless acceleration (or *transition*

In general terms, a mass distribution whose length is much greater than its associated mass-length *<sup>l</sup>*<sup>M</sup> is in the MONDian regime (since *<sup>x</sup>* ≪ 1) and a mass distribution whose length is much smaller than its mass-length scale is in the Newtonian regime (since *x* ≫ 1). The case *x* = 1 can roughly be thought of as the point where the transition from the

A general transition function *g*(*x*) was built by [23] taking Taylor expansion series about the

1/2

*x*2, when *x* ≫ 1,

*lM* :=

and *<sup>y</sup>* :<sup>=</sup> *<sup>a</sup>*

a0 .

*<sup>a</sup>* = <sup>a</sup>0*g*(*x*), (3)

*<sup>r</sup>* , (4)

*<sup>x</sup>*, when *<sup>x</sup>* <sup>≪</sup> 1. (6)

. (5)

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In section 2 it is shown, through a description of an extended Newtonian gravity scenario, the advantages of working with a modification of gravity dependent on the mass and lengths associated with the dimensions and masses of the sources that generate the gravitational field, and not with the dynamical acceleration they produce on test particles. Section 3 describes how it is possible to build a metric theory of gravity which generalises the extended Newtonian description mentioned in section 2 and section 4 interconnects this extended relativistic description of gravity with a metric description of gravity for which the energy-momentum tensor appears in the gravitational field's action. On section 5 we use the developed theory of gravity for cosmological applications in a dust universe and see how it is a coherent representation of gravity at cosmological scales. Finally on section 6, we discuss the consequences of the developed approach of gravity and some of the future developments of the theory.

## **2. Extended Newtonian gravity**

Milgrom [26, 27, 29] constructed a MOdified Newtonian Dynamics (MOND) theory, based on the introduction of a fundamental constant of nature <sup>a</sup><sup>0</sup> <sup>=</sup> 1.2 <sup>×</sup> <sup>10</sup><sup>−</sup>10m s<sup>−</sup><sup>2</sup> in such a way that the acceleration experienced by a test particle on a gravitational field produced by a point mass source *M* is such that:

$$a = \begin{cases} -\frac{GM}{r^2}, & \text{for} \\ -\frac{\sqrt{\mathbf{a}\_0 GM}}{r}, & \text{for} \end{cases} \qquad a \gg \mathbf{a}\_0. \tag{1}$$

where *r* is the radial distance to the central mass. In other words, for accelerations *a* ≫ a0, Newtonian gravity is recovered and new MONDian effects are expected to appear for accelerations *<sup>a</sup>* <sup>a</sup>0. The strong *<sup>a</sup>* ≪ <sup>a</sup><sup>0</sup> MONDian regime means that Kepler's third law is not valid since for a circular orbit about the central mass *M*, the acceleration *a* = *v*/*r*, where *<sup>v</sup>* is velocity of the test mass, and so *<sup>v</sup>* = (a0*GM*) 1/4 ∝ *M*1/4, which is the Tully-Fisher relation [see e.g. 33] for the case of a spiral galaxy and is the same relation experienced by wide-open binaries [17] and by the tail of the "rotation curve" in globular clusters [15, 16].

In order to interpolate from the strong *<sup>a</sup>* ≫ <sup>a</sup><sup>0</sup> Newtonian regime to the weak *<sup>a</sup>* ≪ <sup>a</sup><sup>0</sup> one, the traditional MONDian approach is to construct a somewhat built-by-hand interpolation function *µ*(*y*) in such a way that

$$a\mu(y) = -\frac{GM}{r^2},\tag{2}$$

where

2 Open Questions in Cosmology

the description of gravity.

of the theory.

**2. Extended Newtonian gravity**

a point mass source *M* is such that:

function *µ*(*y*) in such a way that

*a* =

where *<sup>v</sup>* is velocity of the test mass, and so *<sup>v</sup>* = (a0*GM*)

− *GM*

− <sup>√</sup>a0*GM*

different phenomenology at many astrophysical mass and length sizes, from solar-system to extragalactic and cosmological scales. The introduction of the so called Milgrom's acceleration constant a<sup>0</sup> in a description of gravity means that any gravitational field produced by a certain distribution of mass (and hence energy) needs to incorporate the acceleration a<sup>0</sup> together with Newton's gravitational constant *G* and the speed of light *c* in

In section 2 it is shown, through a description of an extended Newtonian gravity scenario, the advantages of working with a modification of gravity dependent on the mass and lengths associated with the dimensions and masses of the sources that generate the gravitational field, and not with the dynamical acceleration they produce on test particles. Section 3 describes how it is possible to build a metric theory of gravity which generalises the extended Newtonian description mentioned in section 2 and section 4 interconnects this extended relativistic description of gravity with a metric description of gravity for which the energy-momentum tensor appears in the gravitational field's action. On section 5 we use the developed theory of gravity for cosmological applications in a dust universe and see how it is a coherent representation of gravity at cosmological scales. Finally on section 6, we discuss the consequences of the developed approach of gravity and some of the future developments

Milgrom [26, 27, 29] constructed a MOdified Newtonian Dynamics (MOND) theory, based on the introduction of a fundamental constant of nature <sup>a</sup><sup>0</sup> <sup>=</sup> 1.2 <sup>×</sup> <sup>10</sup><sup>−</sup>10m s<sup>−</sup><sup>2</sup> in such a way that the acceleration experienced by a test particle on a gravitational field produced by

where *r* is the radial distance to the central mass. In other words, for accelerations *a* ≫ a0, Newtonian gravity is recovered and new MONDian effects are expected to appear for accelerations *<sup>a</sup>* <sup>a</sup>0. The strong *<sup>a</sup>* ≪ <sup>a</sup><sup>0</sup> MONDian regime means that Kepler's third law is not valid since for a circular orbit about the central mass *M*, the acceleration *a* = *v*/*r*,

relation [see e.g. 33] for the case of a spiral galaxy and is the same relation experienced by wide-open binaries [17] and by the tail of the "rotation curve" in globular clusters [15, 16]. In order to interpolate from the strong *<sup>a</sup>* ≫ <sup>a</sup><sup>0</sup> Newtonian regime to the weak *<sup>a</sup>* ≪ <sup>a</sup><sup>0</sup> one, the traditional MONDian approach is to construct a somewhat built-by-hand interpolation

*<sup>a</sup>µ*(*y*) = <sup>−</sup> *GM*

*<sup>r</sup>*<sup>2</sup> , for *<sup>a</sup>* <sup>≫</sup> <sup>a</sup>0,

*<sup>r</sup>* , for *<sup>a</sup>* <sup>≪</sup> <sup>a</sup>0, (1)

1/4 ∝ *M*1/4, which is the Tully-Fisher

*<sup>r</sup>*<sup>2</sup> , (2)

$$
\mu(y) = \begin{cases} 1, & \text{for} \\ y, & \text{for} \end{cases} \quad y \gg 1, \quad \text{and} \quad y := \frac{a}{\mathfrak{a}\_0}.
$$

The usual approach to MOND as expressed by equation (2) means that Newton's 2nd law of mechanics needs to be modified [see e.g. 1]. As explained by Mendoza et al. [23], a better physical approach can be constructed if the modification is made in the force (gravitational) sector. Indeed, by the use of Buckingham's theorem of dimensional analysis [cf. 35], the gravitational acceleration experienced by a test particle is given by

$$a = \mathfrak{a}\_0 \mathfrak{g}(\mathfrak{x}),\tag{3}$$

where the dimensionless quantity

$$\mathbf{x} := \frac{l\_M}{r} \mathbf{y} \tag{4}$$

and a mass-length scale

$$l\_M := \left(\frac{GM}{\mathfrak{a}\_0}\right)^{1/2}.\tag{5}$$

The length *lM* plays an important role in the description of the theory and is such that when *lM* ≫ *<sup>r</sup>*, the strong Newtonian regime of gravity is recovered and when *lM* ≪ *<sup>r</sup>* the weak MONDian regime of gravity appears. As such, the dimensionless acceleration (or *transition function*) *g*(*x*) is such that:

$$\frac{a}{\mathbf{a}\_0} = g(\mathbf{x}) := \begin{cases} \mathbf{x}^2, & \text{when} \quad \mathbf{x} \gg 1, \\ \mathbf{x}, & \text{when} \quad \mathbf{x} \ll 1. \end{cases} \tag{6}$$

In general terms, a mass distribution whose length is much greater than its associated mass-length *<sup>l</sup>*<sup>M</sup> is in the MONDian regime (since *<sup>x</sup>* ≪ 1) and a mass distribution whose length is much smaller than its mass-length scale is in the Newtonian regime (since *x* ≫ 1). The case *x* = 1 can roughly be thought of as the point where the transition from the Newtonian to the MONDian regime occurs.

A general transition function *g*(*x*) was built by [23] taking Taylor expansion series about the correct MONDian and Newtonian limits, yielding:

**Figure 1.** The figure taken from Mendoza et al. [23] shows the acceleration function *a* in units of Milgrom's acceleration constant a<sup>0</sup> as a function of the parameter *x*. The thick dash-dot curve is the extreme limiting value *n* → ∞, i.e. *a*/a<sup>0</sup> = *x* for *x* ≤ 1 and *a*/a<sup>0</sup> = *x*<sup>2</sup> for *x* ≥ 1. The curves above and below this extreme acceleration line represent values of *n* = 4, 3, 2, 1, for the minus and plus signs of equation (7) respectively. The extreme limiting curve has a kink at *x* = 1.

$$\log(\mathbf{x}) = \mathbf{x} \, \frac{1 \pm \mathbf{x}^{n+1}}{1 \pm \mathbf{x}^n} \,. \tag{7}$$

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Extending Cosmology: The Metric Approach http://dx.doi.org/10.5772/53878

Care must be taken when the introduction of a new fundamental constant of nature with dimensions of acceleration a<sup>0</sup> is made. In fact, the introduction of a<sup>0</sup> does not impose any causality arguments such as the ones given by the velocity of light *c*. In fact, one may think of a<sup>0</sup> as a fundamental constant needed to transit from one gravity regime to another. In this

as the new fundamental constant of nature. The constant Σ0, with dimensions of surface mass density, enters in the description of the gravitational theory in such a way that equations (3)

and the acceleration in the full MONDian regime and the corresponding Tully-Fisher relation

Also, a more manageable extended fundamental quantity, directly measurable through the

*<sup>G</sup> <sup>g</sup>* (*lM*/*r*), *lM* :<sup>=</sup>

1/2

With this, the acceleration of a test particle in the full MOND regime and the Tully-Fisher

The choice of a new fundamental constant of nature has many ways in which it can be introduced into the theory [35]. In this work, the use of a<sup>0</sup> is kept as it is traditionally

*<sup>r</sup>* , *<sup>v</sup>* <sup>=</sup> (*ǫ<sup>M</sup> <sup>M</sup>*)

*<sup>r</sup>* , *<sup>v</sup>* <sup>=</sup> *<sup>G</sup>*1/2Σ1/4

1/2

*<sup>a</sup>* <sup>=</sup> <sup>−</sup>*<sup>G</sup>* (Σ0*M*)

with dimensions of velocity to the fourth over mass, for which

*<sup>a</sup>* <sup>=</sup> <sup>−</sup>*ǫ<sup>M</sup>*

*<sup>a</sup>* <sup>=</sup> <sup>−</sup>(*ǫ<sup>M</sup> <sup>M</sup>*)

<sup>Σ</sup><sup>0</sup> :<sup>=</sup> <sup>a</sup>0/*<sup>G</sup>* <sup>=</sup> 1.8 kg m<sup>−</sup>2, (9)

*<sup>a</sup>* = −*G*Σ0*<sup>g</sup>* (*lM*/*r*), *lM* := (*M*/Σ0). (10)

*<sup>ǫ</sup><sup>M</sup>* :<sup>=</sup> <sup>a</sup>0*<sup>G</sup>* <sup>=</sup> 8.004 <sup>×</sup> <sup>10</sup><sup>−</sup>21m4 <sup>s</sup>−<sup>4</sup> kg<sup>−</sup>1, (12)

*G*<sup>2</sup> *M*/*ǫ*<sup>0</sup>

<sup>0</sup> *<sup>M</sup>*1/4. (11)

. (13)

1/4 . (14)

respect for example, instead of using a<sup>0</sup> as a fundamental constant, one may define

and (5) are given by:

Tully-Fisher relation, can be defined:

are

relation are:

This non-singular function converges to the correct expected limits of equation (6) for any value of the parameter *n* ≥ 0. As shown in Figure 1, the transition function *g*(*x*) rapidly converges to the limit "step function"

$$g(\mathbf{x})\Big|\_{\mathbf{x}\to\infty} = \begin{cases} \mathbf{x}, & \text{for } 0 \le \mathbf{x} \le 1, \\ \mathbf{x}^2, & \text{for } \mathbf{x} \ge 1, \end{cases} \tag{8}$$

when *n* 3. The parameter *n* needs to be found empirically by astronomical observations. The value found by Mendoza et al. [23] for the rotation curve of our galaxy is *n* 3 and the one found by Hernandez & Jiménez [15], Hernandez et al. [16, 17] is *n* 8, with a minus sign selection on the numerator and denominator on the right hand side of equation (7). These authors have shown that a large value of *n* is coherent with solar system motion of planets, rotation curves of spiral galaxies, equilibrium relations of dwarf spheroidal galaxies and their correspondent relations in globular clusters, the Faber-Jackson relation and the fundamental plane of elliptical galaxies as well as with the orbits of wide binary stars. The *n* = 3 model in which a small, but measurable transition is obtained, has also been tested on earth and moon-like experiments by Meyer et al. [25] and Exirifard [12] respectively, showing that it is coherent with such precise measurements. In fact, these experiments also validate all *n* ≥ 3 models.

Care must be taken when the introduction of a new fundamental constant of nature with dimensions of acceleration a<sup>0</sup> is made. In fact, the introduction of a<sup>0</sup> does not impose any causality arguments such as the ones given by the velocity of light *c*. In fact, one may think of a<sup>0</sup> as a fundamental constant needed to transit from one gravity regime to another. In this respect for example, instead of using a<sup>0</sup> as a fundamental constant, one may define

4 Open Questions in Cosmology

(a/a0)

10

8

6

4

2

0

converges to the limit "step function"

models.

*g*(*x*) *n*→∞ x

**Figure 1.** The figure taken from Mendoza et al. [23] shows the acceleration function *a* in units of Milgrom's acceleration constant a<sup>0</sup> as a function of the parameter *x*. The thick dash-dot curve is the extreme limiting value *n* → ∞, i.e. *a*/a<sup>0</sup> = *x* for *x* ≤ 1 and *a*/a<sup>0</sup> = *x*<sup>2</sup> for *x* ≥ 1. The curves above and below this extreme acceleration line represent values of *n* = 4, 3, 2, 1,

This non-singular function converges to the correct expected limits of equation (6) for any value of the parameter *n* ≥ 0. As shown in Figure 1, the transition function *g*(*x*) rapidly

when *n* 3. The parameter *n* needs to be found empirically by astronomical observations. The value found by Mendoza et al. [23] for the rotation curve of our galaxy is *n* 3 and the one found by Hernandez & Jiménez [15], Hernandez et al. [16, 17] is *n* 8, with a minus sign selection on the numerator and denominator on the right hand side of equation (7). These authors have shown that a large value of *n* is coherent with solar system motion of planets, rotation curves of spiral galaxies, equilibrium relations of dwarf spheroidal galaxies and their correspondent relations in globular clusters, the Faber-Jackson relation and the fundamental plane of elliptical galaxies as well as with the orbits of wide binary stars. The *n* = 3 model in which a small, but measurable transition is obtained, has also been tested on earth and moon-like experiments by Meyer et al. [25] and Exirifard [12] respectively, showing that it is coherent with such precise measurements. In fact, these experiments also validate all *n* ≥ 3

for the minus and plus signs of equation (7) respectively. The extreme limiting curve has a kink at *x* = 1.

= 

*g*(*x*) = *x*

x

<sup>1</sup> ± *<sup>x</sup>n*+<sup>1</sup>

*x*, for 0 ≤ *x* ≤ 1,

<sup>1</sup> <sup>±</sup> *<sup>x</sup><sup>n</sup>* . (7)

*<sup>x</sup>*2, for *<sup>x</sup>* <sup>≥</sup> 1, (8)

0 0.5 1 1.5 2 2.5 3 3.5

$$
\Sigma\_0 := \mathbf{a}\_0 / G = 1.8 \,\mathrm{kg} \,\mathrm{m}^{-2} \,, \tag{9}
$$

as the new fundamental constant of nature. The constant Σ0, with dimensions of surface mass density, enters in the description of the gravitational theory in such a way that equations (3) and (5) are given by:

$$a = -G\Sigma\_0 \lg \left( l\_M / r \right), \qquad l\_M := \left( M / \Sigma\_0 \right). \tag{10}$$

and the acceleration in the full MONDian regime and the corresponding Tully-Fisher relation are

$$a = -G\frac{\left(\Sigma\_0 M\right)^{1/2}}{r}, \qquad v = G^{1/2} \Sigma\_0^{1/4} M^{1/4}.\tag{11}$$

Also, a more manageable extended fundamental quantity, directly measurable through the Tully-Fisher relation, can be defined:

$$\mathfrak{a}\mathfrak{e}\_{M} \coloneqq \mathfrak{a}\_{0}\mathbf{G} = 8.004 \times 10^{-21} \mathbf{m}^{4} \mathbf{s}^{-4} \mathbf{k} \mathbf{g}^{-1} \tag{12}$$

with dimensions of velocity to the fourth over mass, for which

$$a = -\frac{\varepsilon\_M}{G} \lg \left( l\_M / r \right), \qquad l\_M := \left( G^2 \, M / \epsilon\_0 \right). \tag{13}$$

With this, the acceleration of a test particle in the full MOND regime and the Tully-Fisher relation are:

$$a = -\frac{\left(\varepsilon\_M M\right)^{1/2}}{r}, \qquad v = \left(\varepsilon\_M M\right)^{1/4}.\tag{14}$$

The choice of a new fundamental constant of nature has many ways in which it can be introduced into the theory [35]. In this work, the use of a<sup>0</sup> is kept as it is traditionally done, but we note the fact that *ǫ*<sup>0</sup> is the best fundamental constant to use since it is directly measured through the flattened rotation curves of spiral galaxies.

The extended Newtonian model of gravity presented in this section is equivalent with MOND on spherical and cylindrical symmetry but deviates considerable from it for systems away from this symmetry [23]. As we have already shown, there are however many advantages of using this approach, the most objective meaning that the modification is made on the force sector and not a modification on the dynamics.

## **3. Relativistic metric extension**

Finding a relativistic theory of gravity for which one of its non-relativistic limits converges to MOND yields usually strange assumptions and/or complicated ideas [see e.g. 3, 6, 30]. A good first approach was provided by a slight modification of Einstein's field equations by Sobouti [36], but the attempt is not complete.

In order to find an elegant and simple theory of gravity for which a MONDian solution is found, Bernal, Capozziello, Hidalgo & Mendoza [5] used a correct dimensional metric interpretation of Hilbert's gravitational action *S*<sup>f</sup> for a point mass source M generating the gravitational field, in such a way that:

$$S\_{\rm f} = -\frac{c^3}{16\pi G L\_M^2} \int f(\chi) \sqrt{-g} \,\mathrm{d}^4 \mathbf{x}\_{\prime} \tag{15}$$

which slightly differs from its traditional form (see e.g. [7, 9, 37]) since the following dimensionless quantity has been introduced:

$$\chi := L\_M^2 R \,\!\!\!\/ \, : \tag{16}$$

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(18)

139

*f* ′

(*χ*) *χµν* <sup>−</sup> <sup>1</sup>

where the dimensionless Ricci tensor *χµν* is given by:

Einstein's summation convention over repeated indices.

(*χ*) *χ* − 2 *f*(*χ*) + 3*L*<sup>2</sup>

*f* ′

The trace of equation (18) is:

*α* .

where *T* := *T<sup>α</sup>*

<sup>2</sup> *<sup>f</sup>*(*χ*)*gµν* <sup>−</sup> *<sup>L</sup>*<sup>2</sup>

*χµν* := *<sup>L</sup>*<sup>2</sup>

and *<sup>R</sup>µν* is the standard Ricci tensor. The Laplace-Beltrami operator has been written as <sup>∆</sup> := ∇*α*∇*<sup>α</sup>* and the prime denotes derivative with respect to its argument. The energy-momentum tensor *<sup>T</sup>µν* is defined through the following standard relation: *<sup>δ</sup>S*<sup>m</sup> <sup>=</sup> <sup>−</sup> (1/2*c*) *<sup>T</sup>αβ <sup>δ</sup>gαβ*. In here and in what follows, we choose a (+, −, −, −) signature for the metric *<sup>g</sup>µν* and use

*<sup>M</sup>* <sup>∆</sup>*<sup>f</sup>* ′

In order to search for a MONDian solution, Bernal, Capozziello, Hidalgo & Mendoza [5] analysed the problem in two ways. First by performing an order of magnitude approach to the problem, and second, by doing a full perturbation analysis. Since the second technique is merely to fix constants of proportionality of the problem, their order of magnitude approach and its consequences are discussed in the remain of this section. Also, since we are interested at the moment on a point mass distribution generating a stationary spherically symmetric space-time, the trace equation (20) contains all the relevant information relating the field equations. At this point it is also useful to assume a power law form for the function

An order of magnitude approach to the problem means that d/d*χ* ≈ 1/*χ*, ∆ ≈ −1/*r*<sup>2</sup> and

*<sup>χ</sup>*(*b*−1)

Note that the second term on the left-hand side of equation (22) is much greater than the first

*<sup>r</sup>*<sup>2</sup> <sup>≈</sup> <sup>8</sup>*πGML*<sup>2</sup>

*M*

the mass density *ρ* ≈ *M*/*r*3. With this, the trace (20) takes the following form:

*χ<sup>b</sup>* (*b* − 2) − 3*bL*<sup>2</sup>

term when the following condition is satisfied:

(*χ*) = <sup>8</sup>*πGL*<sup>2</sup>

*M*

*f*(*χ*) = *χb*. (21)

*M*

*<sup>c</sup>*2*r*<sup>3</sup> . (22)

*<sup>c</sup>*<sup>4</sup> *<sup>T</sup>*, (20)

*M* 

∇*µ*∇*<sup>ν</sup>* − *<sup>g</sup>µν*<sup>∆</sup>

*M <sup>c</sup>*<sup>4</sup> *<sup>T</sup>µν*,

<sup>=</sup> <sup>8</sup>*πGL*<sup>2</sup>

 *f* ′ (*χ*)

*<sup>M</sup>Rµν*, (19)

where *R* is Ricci's scalar and *LM* defines a length fixed by the parameters of the model: The explicit form of the length *L* has to be obtained once a certain known limit of the theory is taken, usually a non-relativistic limit. Note that the definition of *χ* gives a correct dimensional character to the action (15), something that is not completely clear in all previous works dealing with a metric description of the gravitational field. For *f*(*χ*) = *χ* the standard Einstein-Hilbert action of general relativity is obtained.

On the other hand, the matter action has its usual form,

$$S\_{\rm m} = -\frac{1}{2c} \int \mathcal{L}\_{\rm m} \sqrt{-g} \,\mathrm{d}^4 \mathbf{x},\tag{17}$$

with L<sup>m</sup> the matter Lagrangian density of the system. The null variations of the complete action, i.e. *<sup>δ</sup>* (*S*<sup>H</sup> + *<sup>S</sup>*m) = 0, yield the following field equations:

$$\begin{split} f'(\boldsymbol{\chi})\,\boldsymbol{\chi}\_{\mu\nu} - \frac{1}{2}f(\boldsymbol{\chi})\boldsymbol{g}\_{\mu\nu} - \boldsymbol{L}\_{\boldsymbol{M}}^{2} \left( \nabla\_{\mu}\nabla\_{\nu} - \boldsymbol{g}\_{\mu\nu}\Delta \right) f'(\boldsymbol{\chi}) \\ = \frac{8\pi G L\_{\boldsymbol{M}}^{2}}{c^{4}} T\_{\mu\nu} \end{split} \tag{18}$$

where the dimensionless Ricci tensor *χµν* is given by:

$$\chi\_{\mu\nu} := L\_M^2 \mathbb{R}\_{\mu\nu\nu} \tag{19}$$

and *<sup>R</sup>µν* is the standard Ricci tensor. The Laplace-Beltrami operator has been written as <sup>∆</sup> := ∇*α*∇*<sup>α</sup>* and the prime denotes derivative with respect to its argument. The energy-momentum tensor *<sup>T</sup>µν* is defined through the following standard relation: *<sup>δ</sup>S*<sup>m</sup> <sup>=</sup> <sup>−</sup> (1/2*c*) *<sup>T</sup>αβ <sup>δ</sup>gαβ*. In here and in what follows, we choose a (+, −, −, −) signature for the metric *<sup>g</sup>µν* and use Einstein's summation convention over repeated indices.

The trace of equation (18) is:

$$f'(\chi)\,\chi-2f(\chi)+3L\_M^2\,\Delta f'(\chi)=\frac{8\pi GL\_M^2}{c^4}T,\tag{20}$$

where *T* := *T<sup>α</sup> α* .

6 Open Questions in Cosmology

done, but we note the fact that *ǫ*<sup>0</sup> is the best fundamental constant to use since it is directly

The extended Newtonian model of gravity presented in this section is equivalent with MOND on spherical and cylindrical symmetry but deviates considerable from it for systems away from this symmetry [23]. As we have already shown, there are however many advantages of using this approach, the most objective meaning that the modification is made

Finding a relativistic theory of gravity for which one of its non-relativistic limits converges to MOND yields usually strange assumptions and/or complicated ideas [see e.g. 3, 6, 30]. A good first approach was provided by a slight modification of Einstein's field equations by

In order to find an elegant and simple theory of gravity for which a MONDian solution is found, Bernal, Capozziello, Hidalgo & Mendoza [5] used a correct dimensional metric interpretation of Hilbert's gravitational action *S*<sup>f</sup> for a point mass source M generating the

which slightly differs from its traditional form (see e.g. [7, 9, 37]) since the following

*χ* := *L*<sup>2</sup>

where *R* is Ricci's scalar and *LM* defines a length fixed by the parameters of the model: The explicit form of the length *L* has to be obtained once a certain known limit of the theory is taken, usually a non-relativistic limit. Note that the definition of *χ* gives a correct dimensional character to the action (15), something that is not completely clear in all previous works dealing with a metric description of the gravitational field. For *f*(*χ*) = *χ* the standard

−*g* d4*x*, (15)

*<sup>M</sup>R*, (16)

−*g* d4*x*, (17)

measured through the flattened rotation curves of spiral galaxies.

on the force sector and not a modification on the dynamics.

*<sup>S</sup>*<sup>f</sup> <sup>=</sup> <sup>−</sup> *<sup>c</sup>*<sup>3</sup>

16*πGL*<sup>2</sup> *M f*(*χ*)

**3. Relativistic metric extension**

gravitational field, in such a way that:

Sobouti [36], but the attempt is not complete.

dimensionless quantity has been introduced:

Einstein-Hilbert action of general relativity is obtained. On the other hand, the matter action has its usual form,

*<sup>S</sup>*<sup>m</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup>

action, i.e. *<sup>δ</sup>* (*S*<sup>H</sup> + *<sup>S</sup>*m) = 0, yield the following field equations:

2*c* L<sup>m</sup>

with L<sup>m</sup> the matter Lagrangian density of the system. The null variations of the complete

In order to search for a MONDian solution, Bernal, Capozziello, Hidalgo & Mendoza [5] analysed the problem in two ways. First by performing an order of magnitude approach to the problem, and second, by doing a full perturbation analysis. Since the second technique is merely to fix constants of proportionality of the problem, their order of magnitude approach and its consequences are discussed in the remain of this section. Also, since we are interested at the moment on a point mass distribution generating a stationary spherically symmetric space-time, the trace equation (20) contains all the relevant information relating the field equations. At this point it is also useful to assume a power law form for the function

$$f(\chi) = \chi^b. \tag{21}$$

An order of magnitude approach to the problem means that d/d*χ* ≈ 1/*χ*, ∆ ≈ −1/*r*<sup>2</sup> and the mass density *ρ* ≈ *M*/*r*3. With this, the trace (20) takes the following form:

$$
\chi^b \left( b - 2 \right) - 3bL\_M^2 \frac{\chi^{(b-1)}}{r^2} \approx \frac{8\pi G M L\_M^2}{c^2 r^3}.\tag{22}
$$

Note that the second term on the left-hand side of equation (22) is much greater than the first term when the following condition is satisfied:

$$
\mathcal{R}r^2 \lesssim \frac{3b}{2-b}.\tag{23}
$$

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limit, the velocity of light *c* should not appear on equation (28) and so, the only way this

As discussed by Bernal, Capozziello, Hidalgo & Mendoza [5], the length *LM* must be constructed by fundamental parameters describing the theory of gravity and since the only two characteristic lengths of the problem are the mass-length *lM* and the gravitational radius

*<sup>r</sup>*<sup>g</sup> <sup>=</sup> *GM*

where the constant of proportionality *ζ* is a dimensionless number that can be found by a

√2

*<sup>ζ</sup>* <sup>=</sup> <sup>2</sup>

Substituting equation (31) and the value *b* = 3/2 into relation (29), it then follows that

*<sup>α</sup>* = *<sup>β</sup>* = 1/2, i.e. *LM* ≈ *<sup>r</sup>*

*<sup>a</sup>* ≈ −(a0*GM*)

*<sup>R</sup>* <sup>≈</sup> *<sup>r</sup>*<sup>g</sup> *lM* 1

which is the traditional form of MOND for a point mass source (see e.g. [2, 28, 29] and

1/2

If we now substitute this last result and the value *b* = 3/2 in equation (28) we get:

references therein). Also, the results of equation (34) in (27) mean that

and so, inequality (24) is equivalent to

*<sup>M</sup>* <sup>∝</sup> *<sup>c</sup>*(4−2*b*)/(*b*−1) <sup>=</sup> *<sup>c</sup>*2, and so, *LM* <sup>∝</sup> *<sup>c</sup>*<sup>−</sup>1. (29)

*<sup>c</sup>*<sup>2</sup> , (30)

<sup>9</sup> , (32)

*<sup>r</sup>* , (34)

*<sup>r</sup>*<sup>2</sup> , (35)

*<sup>M</sup>* . (33)

*<sup>M</sup>*, with *<sup>α</sup>* <sup>+</sup> *<sup>β</sup>* <sup>=</sup> 1, (31)

1/2 <sup>g</sup> *l* 1/2

condition is fulfilled is that *LM* depends on a power of *c*, i.e.

then the correct dimensional form of the length *LM* is given by

*LM* = *<sup>ζ</sup> <sup>r</sup><sup>α</sup>*

full perturbation analysis technique and is given by [5]:

g*l β*

*<sup>L</sup>*−<sup>2</sup>

At the same order of approximation, Ricci's scalar *<sup>R</sup>* <sup>≈</sup> *<sup>κ</sup>* <sup>=</sup> *<sup>R</sup>*−<sup>2</sup> <sup>c</sup> , where *<sup>κ</sup>* is the Gaussian curvature of space and *R*c its radius of curvature and so, relation (23) essentially means that

$$R\_{\mathbb{C}} \gg r.\tag{24}$$

In other words, the second term on the left-hand side of equation (22) dominates the first one when the local radius of curvature of space is much grater than the characteristic length *r*. This should occur in the weak-field regime, where MONDian effects are expected. For a metric description of gravity, this limit must correspond to the relativistic regime of MOND.

Under assumption (24), equation (22) takes the following form:

$$R^{(b-1)} \approx -\frac{8\pi GM}{3bc^2rL\_M^{2(b-1)}}.\tag{25}$$

We now recall the well known relation followed by the Ricci scalar at second order of approximation at the non-relativistic level [19]:

$$R = -\frac{2}{c^2} \nabla^2 \phi = +\frac{2}{c^2} \nabla \cdot a,\tag{26}$$

where the negative gradients of the gravitational potential *φ* provide the acceleration a := −∇*φ* felt by a test particle on a non-relativistic gravitational field. At order of magnitude, equation (26) can be approximated as

$$R \approx -\frac{2\phi}{c^2 r^2} \approx \frac{2a}{c^2 r}.\tag{27}$$

Substitution of this last equation on relation (25) gives

$$\begin{split} a &\approx -\frac{c^2 r}{2L\_M^2} \left(\frac{8\pi GM}{3bc^2 r}\right)^{1/(b-1)}, \\ &\approx -c^{(2b-4)/(b-1)} r^{(b-2)/(b-1)} L\_M^{-2} \left(GM\right)^{1/(b-1)}.\end{split} \tag{28}$$

This last equation converges to a MOND-like acceleration *a* ∝ 1/*r* if *b* − 2 = − (*b* − 1), i.e. when *b* = 3/2. Also, at the lowest order of approximation, in the extreme non-relativistic limit, the velocity of light *c* should not appear on equation (28) and so, the only way this condition is fulfilled is that *LM* depends on a power of *c*, i.e.

$$L\_M^{-2} \propto c^{(4-2b)/(b-1)} = c^2, \quad \text{and so,} \qquad L\_M \propto c^{-1}.\tag{29}$$

As discussed by Bernal, Capozziello, Hidalgo & Mendoza [5], the length *LM* must be constructed by fundamental parameters describing the theory of gravity and since the only two characteristic lengths of the problem are the mass-length *lM* and the gravitational radius

$$r\_{\rm g} = \frac{GM}{c^2},\tag{30}$$

then the correct dimensional form of the length *LM* is given by

8 Open Questions in Cosmology

*Rr*<sup>2</sup> <sup>3</sup>*<sup>b</sup>* 2 − *b*

At the same order of approximation, Ricci's scalar *<sup>R</sup>* <sup>≈</sup> *<sup>κ</sup>* <sup>=</sup> *<sup>R</sup>*−<sup>2</sup> <sup>c</sup> , where *<sup>κ</sup>* is the Gaussian curvature of space and *R*c its radius of curvature and so, relation (23) essentially means that

In other words, the second term on the left-hand side of equation (22) dominates the first one when the local radius of curvature of space is much grater than the characteristic length *r*. This should occur in the weak-field regime, where MONDian effects are expected. For a metric description of gravity, this limit must correspond to the relativistic regime of MOND.

*<sup>R</sup>*(*b*−1) ≈ − <sup>8</sup>*πGM*

We now recall the well known relation followed by the Ricci scalar at second order of

*<sup>c</sup>*<sup>2</sup> <sup>∇</sup>2*<sup>φ</sup>* = +

where the negative gradients of the gravitational potential *φ* provide the acceleration a := −∇*φ* felt by a test particle on a non-relativistic gravitational field. At order of magnitude,

> *<sup>c</sup>*2*r*<sup>2</sup> <sup>≈</sup> <sup>2</sup>*<sup>a</sup> c*2*r*

1/(*b*−1)

*<sup>r</sup>*(*b*−2)/(*b*−1)*L*−<sup>2</sup>

This last equation converges to a MOND-like acceleration *a* ∝ 1/*r* if *b* − 2 = − (*b* − 1), i.e. when *b* = 3/2. Also, at the lowest order of approximation, in the extreme non-relativistic

,

*<sup>M</sup>* (*GM*)

*<sup>R</sup>* ≈ − <sup>2</sup>*<sup>φ</sup>*

8*πGM* 3*bc*2*r*

<sup>3</sup>*bc*2*rL*2(*b*−1) *M*

2

Under assumption (24), equation (22) takes the following form:

*<sup>R</sup>* <sup>=</sup> <sup>−</sup> <sup>2</sup>

approximation at the non-relativistic level [19]:

Substitution of this last equation on relation (25) gives

*<sup>a</sup>* ≈ − *<sup>c</sup>*2*<sup>r</sup>* 2*L*<sup>2</sup> *M*

≈ −*c*(2*b*−4)/(*b*−1)

equation (26) can be approximated as

. (23)

. (25)

*<sup>c</sup>*<sup>2</sup> ∇ · <sup>a</sup>, (26)

. (27)

1/(*b*−1) . (28)

*Rc* ≫ *r*. (24)

$$L\_M = \zeta \, r\_\mathcal{g}^a l\_{M'}^\mathcal{P} \qquad \text{with} \qquad a + \beta = 1,\tag{31}$$

where the constant of proportionality *ζ* is a dimensionless number that can be found by a full perturbation analysis technique and is given by [5]:

$$
\zeta = \frac{2\sqrt{2}}{9},
\tag{32}
$$

Substituting equation (31) and the value *b* = 3/2 into relation (29), it then follows that

$$
\alpha = \beta = 1/2, \qquad \text{i.e.} \qquad \mathcal{L}\_M \approx r\_{\text{g}}^{1/2} l\_M^{1/2} \,. \tag{33}
$$

If we now substitute this last result and the value *b* = 3/2 in equation (28) we get:

$$a \approx -\frac{\left(\mathbf{a}\_0 GM\right)^{1/2}}{r},\tag{34}$$

which is the traditional form of MOND for a point mass source (see e.g. [2, 28, 29] and references therein). Also, the results of equation (34) in (27) mean that

*<sup>R</sup>* <sup>≈</sup> *<sup>r</sup>*<sup>g</sup> *lM* 1 *<sup>r</sup>*<sup>2</sup> , (35)

and so, inequality (24) is equivalent to

$$l\_M \gg r\_{\mathfrak{g}}.\tag{36}$$

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Extending Cosmology: The Metric Approach http://dx.doi.org/10.5772/53878

χ

**Figure 2.** The figure shows the transition function *f*(*χ*), as a function of the dimensionless Ricci scalar *χ*, for different regimes of gravity, converging to *f*(*χ*) = *χ* for *χ* ≫ 1 (general relativity) and to *f*(*χ*) = *χ*3/2 for *χ* ≪ 1 (a relativistic regime with MOND as its weak field limit) -see equation (37). The thick dash-dot curve is the extreme limiting value *p* → ∞, i.e. *f*(*χ*) = *χ*3/2 for *χ* ≤ 1 and *f*(*χ*) = *χ* for *χ* ≥ 1. The curves above and below this extreme function represent values of *p* = 3, 2, 1, 0 for the

it coincides with the mass of the central object generating the gravitational field as expressed in equations (31) and (33). Generally speaking what the meaning of *M* would be for a particular distribution of mass and energy needs further research, beyond the scope of this book chapter. Nevertheless one expects that for dust systems with spherically symmetric distributions, the function *M* would be given by the standard mass-energy relation [see e.g.

In very general terms, the definition of *M* in this last equation means that *M* would not be invariant. However, in some particular systems with high degree of symmetry it is possible to make this quantity invariant. For example, in the case of a spherically symmetric spacetime produced by a point mass that quantity is simply the "Schwarzschild" mass of the point mass generating the gravitational field. In the cosmological case it is also possible to define

The field equations produced by the null variations of the addition of the field's action *<sup>S</sup>*<sup>f</sup> + *S*<sup>m</sup> can be constructed in the following form. Harko et al. [14] have built an *F*(*R*, *T*) theory

*<sup>F</sup>*(*R*, *<sup>T</sup>*) :<sup>=</sup> *<sup>f</sup>*(*χ*)

*L*2 *M*

*T r*<sup>2</sup> d*r*, (39)

, (40)

*<sup>M</sup>* :<sup>=</sup> <sup>4</sup>*<sup>π</sup> c*2 

minus and plus signs of equation (7) respectively. The extreme limiting curve has a kink at *χ* = 1.

0 0.5 1 1.5 2

(a/a0)

31]:

2

1.5

1

0.5

0

it as an invariant quantity as discussed in section 5.

of gravity, so making the natural identification:

The regime imposed by equation (36) is precisely the one for which MONDian effects should appear in a relativistic theory of gravity. This is an expected generalisation of the results presented in section 2. Note that in the weak field limit regime for which *lM* ≪ *<sup>r</sup>* together with equation (36) yields *<sup>r</sup>* ≫ *lM* ≫ *<sup>r</sup>*g. In this connection, we also note that Newton's theory of gravity is recovered in the limit *lM* ≫ *<sup>r</sup>* ≫ *<sup>r</sup>*g.

In exactly the same way as it was done to build the transition function for the case of extended Newtonian gravity in section 2, a general function *f*(*χ*) can be constructed:

$$f(\chi) = \chi^{3/2} \frac{1 \pm \chi^{p+1}}{1 \pm \chi^{3/2+p}} \to \begin{cases} \chi^{3/2}, & \text{for } \chi \ll 1, \\ \chi, & \text{for } \chi \gg 1. \end{cases} \tag{37}$$

In other words, general relativity is recovered when *χ* ≫ 1 in the strong field regime and the relativistic version of MOND with *χ*3/2 is recovered for the weak field regime of gravity when *χ* ≪ 1 (see Figure 2). The unknown parameter *p* ≥ −1 needs to be calibrated with astronomical observations, in an analogous form as the calibration of the parameter *n* in equation (7) was done. This is a much harder task and a matter of future research. However, since the non-relativistic approach to gravity explained in section 2 means that the transition from the Newtonian to the MONDian regimes of gravity is very sharp, it most probably means that the function *f*(*χ*) = *χ* for *χ* ≥ 1 and that *f*(*χ*) = *χ*3/2 for *χ* ≤ 1, but this has to be tested by some astronomical observations.

The mass dependence of *χ* and *LM* mean that Hilbert's action (15) is a function of the mass *M*. This is usually not assumed, since that action is thought to be purely a function of the geometry of space-time due to the presence of mass and energy sources. However, it was Sobouti [36] who first encountered this peculiarity in the Hilbert action when dealing with a metric generalisation of MOND. Following the remarks by Sobouti [36] and Mendoza & Rosas-Guevara [24] one should not be surprised if some of the commonly accepted notions, even at the fundamental level of the action, require generalisations and re-thinking. An extended metric theory of gravity goes beyond the traditional general relativity ideas and in this way, we need to change our standard view of its fundamental principles.

## **4.** *F*(*R*, *T*) **connection**

For the description of gravity shown in section 3 it follows that an adequate way of writing up the gravitational field's action is given by:

$$S\_{\rm f} = -\frac{c^3}{16\pi G} \int \frac{f(\chi)}{L\_M^2} \sqrt{-g} \,\mathrm{d}^4 \mathrm{x}.\tag{38}$$

The function *LM* is a function of the mass of the system and in general terms it is a function of the space-time coordinates. For the particular case of a spherically symmetric space-time

10 Open Questions in Cosmology

of gravity is recovered in the limit *lM* ≫ *<sup>r</sup>* ≫ *<sup>r</sup>*g.

be tested by some astronomical observations.

up the gravitational field's action is given by:

**4.** *F*(*R*, *T*) **connection**

*lM* ≫ *<sup>r</sup>*g. (36)

*<sup>χ</sup>*, for *<sup>χ</sup>* <sup>≫</sup> 1. (37)

−*g* d4*x*. (38)

The regime imposed by equation (36) is precisely the one for which MONDian effects should appear in a relativistic theory of gravity. This is an expected generalisation of the results presented in section 2. Note that in the weak field limit regime for which *lM* ≪ *<sup>r</sup>* together with equation (36) yields *<sup>r</sup>* ≫ *lM* ≫ *<sup>r</sup>*g. In this connection, we also note that Newton's theory

In exactly the same way as it was done to build the transition function for the case of extended

→ 

In other words, general relativity is recovered when *χ* ≫ 1 in the strong field regime and the relativistic version of MOND with *χ*3/2 is recovered for the weak field regime of gravity when *χ* ≪ 1 (see Figure 2). The unknown parameter *p* ≥ −1 needs to be calibrated with astronomical observations, in an analogous form as the calibration of the parameter *n* in equation (7) was done. This is a much harder task and a matter of future research. However, since the non-relativistic approach to gravity explained in section 2 means that the transition from the Newtonian to the MONDian regimes of gravity is very sharp, it most probably means that the function *f*(*χ*) = *χ* for *χ* ≥ 1 and that *f*(*χ*) = *χ*3/2 for *χ* ≤ 1, but this has to

The mass dependence of *χ* and *LM* mean that Hilbert's action (15) is a function of the mass *M*. This is usually not assumed, since that action is thought to be purely a function of the geometry of space-time due to the presence of mass and energy sources. However, it was Sobouti [36] who first encountered this peculiarity in the Hilbert action when dealing with a metric generalisation of MOND. Following the remarks by Sobouti [36] and Mendoza & Rosas-Guevara [24] one should not be surprised if some of the commonly accepted notions, even at the fundamental level of the action, require generalisations and re-thinking. An extended metric theory of gravity goes beyond the traditional general relativity ideas and in

For the description of gravity shown in section 3 it follows that an adequate way of writing

 *f*(*χ*) *L*2 *M*

The function *LM* is a function of the mass of the system and in general terms it is a function of the space-time coordinates. For the particular case of a spherically symmetric space-time

*χ*3/2, for *χ* ≪ 1,

Newtonian gravity in section 2, a general function *f*(*χ*) can be constructed:

<sup>1</sup> <sup>±</sup> *<sup>χ</sup>*3/2+*<sup>p</sup>*

this way, we need to change our standard view of its fundamental principles.

*<sup>S</sup>*<sup>f</sup> <sup>=</sup> <sup>−</sup> *<sup>c</sup>*<sup>3</sup>

16*πG*

*<sup>f</sup>*(*χ*) = *<sup>χ</sup>*3/2 <sup>1</sup> <sup>±</sup> *<sup>χ</sup>p*+<sup>1</sup>

**Figure 2.** The figure shows the transition function *f*(*χ*), as a function of the dimensionless Ricci scalar *χ*, for different regimes of gravity, converging to *f*(*χ*) = *χ* for *χ* ≫ 1 (general relativity) and to *f*(*χ*) = *χ*3/2 for *χ* ≪ 1 (a relativistic regime with MOND as its weak field limit) -see equation (37). The thick dash-dot curve is the extreme limiting value *p* → ∞, i.e. *f*(*χ*) = *χ*3/2 for *χ* ≤ 1 and *f*(*χ*) = *χ* for *χ* ≥ 1. The curves above and below this extreme function represent values of *p* = 3, 2, 1, 0 for the minus and plus signs of equation (7) respectively. The extreme limiting curve has a kink at *χ* = 1.

it coincides with the mass of the central object generating the gravitational field as expressed in equations (31) and (33). Generally speaking what the meaning of *M* would be for a particular distribution of mass and energy needs further research, beyond the scope of this book chapter. Nevertheless one expects that for dust systems with spherically symmetric distributions, the function *M* would be given by the standard mass-energy relation [see e.g. 31]:

$$M \coloneqq \frac{4\pi}{c^2} \int T r^2 \,\mathrm{d}r,\tag{39}$$

In very general terms, the definition of *M* in this last equation means that *M* would not be invariant. However, in some particular systems with high degree of symmetry it is possible to make this quantity invariant. For example, in the case of a spherically symmetric spacetime produced by a point mass that quantity is simply the "Schwarzschild" mass of the point mass generating the gravitational field. In the cosmological case it is also possible to define it as an invariant quantity as discussed in section 5.

The field equations produced by the null variations of the addition of the field's action *<sup>S</sup>*<sup>f</sup> + *S*<sup>m</sup> can be constructed in the following form. Harko et al. [14] have built an *F*(*R*, *T*) theory of gravity, so making the natural identification:

$$F(\mathbf{R}, T) := \frac{f(\chi)}{L\_{\mathbf{M}}^2},\tag{40}$$

it is possible to use all their results for our particular case expressed in equation (40). For example, the null variations of the complete action *<sup>S</sup>*<sup>f</sup> + *<sup>S</sup>*<sup>m</sup> for the particular case of equation (40) is given by Harko et al. [14]:

$$\begin{aligned} \left(\frac{f\_R}{L\_M^2}\right) R\_{\mu\nu} - \frac{1}{2L\_M^2} f \, g\_{\mu\nu} + \left[g\_{\mu\nu}\Delta - \nabla\_\mu \nabla\_\nu\right] \left(\frac{f\_R}{L\_M^2}\right) &= \\ \frac{8\pi G}{c^4} T\_{\mu\nu} - \left(\frac{f}{L\_M^2}\right)\_T \left[T\_{\mu\nu} + \Theta\_{\mu\nu}\right], \end{aligned} \tag{41}$$

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(47)

*νλuνu<sup>λ</sup>* <sup>=</sup> *<sup>λ</sup>µ*, (46)

Extending Cosmology: The Metric Approach http://dx.doi.org/10.5772/53878

and as such the geodesic equation has a force term:

*<sup>λ</sup><sup>µ</sup>* :=8*π<sup>G</sup> c*4 *ρc*<sup>2</sup> + *p*

*<sup>p</sup>* = 0 (dust) fluid and (ii) for the cases in which *FT*(*R*, *<sup>T</sup>*) = 0.

where the four-force

the universe at any epoch.

**5. Cosmological applications**

the interval d*s* is given by [21]:

d*s*

<sup>2</sup> = *c*2d*t*

<sup>2</sup> − *a*2(*t*)

*d*2*x<sup>µ</sup> ds*<sup>2</sup> <sup>+</sup> <sup>Γ</sup>*<sup>µ</sup>*

(*gµν* − *<sup>u</sup>µuν*) ∇*<sup>ν</sup> <sup>p</sup>*,

<sup>−</sup><sup>1</sup> 8*πG <sup>c</sup>*<sup>4</sup> <sup>+</sup>

is perpendicular to the four velocity d*xα*/d*s*. As discussed by Harko et al. [14], the motion of test particles is geodesic, i.e. *<sup>λ</sup><sup>µ</sup>* <sup>=</sup> 0 and/or <sup>∇</sup>*αTαβ* <sup>=</sup> 0, (i) for the case of a pressureless

In what follows we will see how all the previous ideas can be applied to a Friedmann-Lemaître-Robertson-Walker dust universe and so, the divergence of the energy momentum tensor in equation (45) is null. It is worth noting that this condition on the energy-momentum tensor for many applications needs to be zero, including applications to

There are many good and interesting attempts to explain many cosmological observations using modified theories of gravity [see e.g. 32, and references therein], however these theories are not generally fully consistent with the gravitational anomalies shown at galactic and extragalactic scales discussed in sections 2 and 3. To see whether the gravitational *f*(*χ*) theory developed in the previous sections can deal with cosmological data, let us now apply the results obtained in those sections to an isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) universe following the procedures first explored by Carranza et al. [10]. In this case,

> d*r*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>κ</sup>r*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*

where *<sup>a</sup>*(*t*) is the scale factor of the universe normalised to unity, i.e. *<sup>a</sup>*<sup>0</sup> = 1, at the present epoch *<sup>t</sup>*0, and the angular displacement dΩ<sup>2</sup> := <sup>d</sup>*θ*<sup>2</sup> + sin<sup>2</sup> *<sup>θ</sup>* <sup>d</sup>*ϕ*<sup>2</sup> for the polar d*<sup>θ</sup>* and azimuthal d*ϕ* angular displacements with a comoving distance coordinate *r*. In what follows we assume a null space curvature *κ* = 0 at the present epoch in accordance with observations and deal with the expansion of the universe dictated by the field equations (41), avoiding any form of dark unknown component. Since we are interested on the compatibility of this cosmological model with SNIa observations, in what follows we assume a dust *p* = 0

2dΩ<sup>2</sup> 

, (48)

 *f L*2 *M* *T*

−<sup>1</sup> ×

and its trace is given by:

$$\frac{f\_R R}{L\_M^2} - \frac{2f}{L\_M^2} + 3\Delta \left(\frac{f\_R}{L\_M^2}\right) = \frac{8\pi G}{c^4} T - \left(\frac{f}{L\_M^2}\right)\_T \left[T + \Theta\right]\_{\text{'}}\tag{42}$$

where the subscripts *R* and *T* stand for the partial derivatives with respect to those quantities, i.e.

$$\left(\begin{array}{c} \\ \\ \end{array}\right)\_R := \frac{\partial}{\partial R'} \qquad \text{and} \qquad \left(\begin{array}{c} \\ \\ \end{array}\right)\_T := \frac{\partial}{\partial T}.\tag{43}$$

The tensor <sup>Θ</sup>*µν* is such that <sup>Θ</sup>*µνδgµν* :<sup>=</sup> *<sup>g</sup>αβδTαβ* and for the case of an ideal fluid it can be written as [14]:

$$
\Theta\_{\mu\nu} = -2T\_{\mu\nu} - pg\_{\mu\nu}.\tag{44}
$$

Note that equation (41) or (42) converge to the field (18) and trace (20) relations as discussed in section 3 when one considers a point mass generating the gravitational field, i.e. when *LM* = const. and so *<sup>∂</sup>*/*∂<sup>R</sup>* = *<sup>L</sup>*<sup>2</sup> *<sup>M</sup>∂*/*∂χ*.

In general terms, the *F*(*R*, *T*) theory described by Harko et al. [14] produces non-geodesic motion of test particles since:

$$\begin{aligned} \nabla^{\mu}T\_{\mu\nu} &= \left(\frac{f}{L\_{M}^{2}}\right)\_{T} \left\{ \frac{8\pi G}{c^{4}} - \left(\frac{f}{L\_{M}^{2}}\right)\_{T} \right\}^{-1} \times \\ & \left[ \left(T\_{\mu\nu} + \Theta\_{\mu\nu}\right) \nabla^{\mu} \ln\left(\frac{f}{L\_{M}^{2}}\right)\_{T} (R\_{\nu}T) + \nabla^{\mu}\Theta\_{\mu\nu} \right] \neq 0, \end{aligned} \tag{45}$$

and as such the geodesic equation has a force term:

$$\frac{d^2\mathbf{x}^\mu}{ds^2} + \Gamma^\mu\_{\nu\lambda}\boldsymbol{u}^\nu\boldsymbol{u}^\lambda = \lambda^\mu\,. \tag{46}$$

where the four-force

12 Open Questions in Cosmology

and its trace is given by:

i.e.

written as [14]:

*LM* = const. and so *<sup>∂</sup>*/*∂<sup>R</sup>* = *<sup>L</sup>*<sup>2</sup>

motion of test particles since:

∇*µTµν* =

   *f L*2 *M*

*<sup>T</sup>µν* + <sup>Θ</sup>*µν*

equation (40) is given by Harko et al. [14]:

 *fR L*2 *M*

*fR R L*2 *M*

− 2 *f L*2 *M*

*R*

:<sup>=</sup> *<sup>∂</sup>*

*<sup>M</sup>∂*/*∂χ*.

*T*

 ∇*<sup>µ</sup>* ln *f L*2 *M*

 8*πG <sup>c</sup>*<sup>4</sup> <sup>−</sup>

*<sup>R</sup>µν*<sup>−</sup> <sup>1</sup> 2*L*<sup>2</sup> *M*

> + 3∆ *fR L*2 *M*

8*πG <sup>c</sup>*<sup>4</sup> *<sup>T</sup>µν* <sup>−</sup>

it is possible to use all their results for our particular case expressed in equation (40). For example, the null variations of the complete action *<sup>S</sup>*<sup>f</sup> + *<sup>S</sup>*<sup>m</sup> for the particular case of

 *f L*2 *M*

*<sup>g</sup>µν*<sup>∆</sup> − ∇*µ*∇*<sup>ν</sup>*

 *f L*2 *M*

*T*

:<sup>=</sup> *<sup>∂</sup>*

<sup>Θ</sup>*µν* = −2*Tµν* − *pgµν*. (44)

*T*

 *T* + Θ 

*T*

 *fR L*2 *M*

> ,

*<sup>T</sup>µν* + <sup>Θ</sup>*µν*

 =

(41)

(45)

, (42)

*<sup>∂</sup><sup>T</sup>* . (43)

*f gµν* +

*∂R*, and

= 8*πG <sup>c</sup>*<sup>4</sup> *<sup>T</sup>* <sup>−</sup>

where the subscripts *R* and *T* stand for the partial derivatives with respect to those quantities,

The tensor <sup>Θ</sup>*µν* is such that <sup>Θ</sup>*µνδgµν* :<sup>=</sup> *<sup>g</sup>αβδTαβ* and for the case of an ideal fluid it can be

Note that equation (41) or (42) converge to the field (18) and trace (20) relations as discussed in section 3 when one considers a point mass generating the gravitational field, i.e. when

In general terms, the *F*(*R*, *T*) theory described by Harko et al. [14] produces non-geodesic

 *f L*2 *M*

*T*

*T*

−<sup>1</sup> ×

(*R*, *<sup>T</sup>*) + ∇*µ*Θ*µν*

 �= 0,

$$\begin{split} \lambda^{\mu} := & \frac{8\pi G}{c^4} \left( \rho c^2 + p \right)^{-1} \left[ \frac{8\pi G}{c^4} + \left( \frac{f}{L\_M^2} \right)\_T \right]^{-1} \times \\ \left( g^{\mu \nu} - \mu^{\mu} \boldsymbol{u}^{\nu} \right) \nabla\_{\boldsymbol{\nu}} p\_{\boldsymbol{\nu}} \end{split} \tag{47}$$

is perpendicular to the four velocity d*xα*/d*s*. As discussed by Harko et al. [14], the motion of test particles is geodesic, i.e. *<sup>λ</sup><sup>µ</sup>* <sup>=</sup> 0 and/or <sup>∇</sup>*αTαβ* <sup>=</sup> 0, (i) for the case of a pressureless *<sup>p</sup>* = 0 (dust) fluid and (ii) for the cases in which *FT*(*R*, *<sup>T</sup>*) = 0.

In what follows we will see how all the previous ideas can be applied to a Friedmann-Lemaître-Robertson-Walker dust universe and so, the divergence of the energy momentum tensor in equation (45) is null. It is worth noting that this condition on the energy-momentum tensor for many applications needs to be zero, including applications to the universe at any epoch.

## **5. Cosmological applications**

There are many good and interesting attempts to explain many cosmological observations using modified theories of gravity [see e.g. 32, and references therein], however these theories are not generally fully consistent with the gravitational anomalies shown at galactic and extragalactic scales discussed in sections 2 and 3. To see whether the gravitational *f*(*χ*) theory developed in the previous sections can deal with cosmological data, let us now apply the results obtained in those sections to an isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) universe following the procedures first explored by Carranza et al. [10]. In this case, the interval d*s* is given by [21]:

$$\mathbf{ds}^2 = c^2 \mathbf{d}t^2 - a^2(t) \left\{ \frac{\mathbf{d}r^2}{1 - \kappa r^2} + r^2 \mathbf{d}\Omega^2 \right\},\tag{48}$$

where *<sup>a</sup>*(*t*) is the scale factor of the universe normalised to unity, i.e. *<sup>a</sup>*<sup>0</sup> = 1, at the present epoch *<sup>t</sup>*0, and the angular displacement dΩ<sup>2</sup> := <sup>d</sup>*θ*<sup>2</sup> + sin<sup>2</sup> *<sup>θ</sup>* <sup>d</sup>*ϕ*<sup>2</sup> for the polar d*<sup>θ</sup>* and azimuthal d*ϕ* angular displacements with a comoving distance coordinate *r*. In what follows we assume a null space curvature *κ* = 0 at the present epoch in accordance with observations and deal with the expansion of the universe dictated by the field equations (41), avoiding any form of dark unknown component. Since we are interested on the compatibility of this cosmological model with SNIa observations, in what follows we assume a dust *p* = 0 model for which the covariant divergence of the energy-momentum tensor vanishes, and so as discussed in section 4 the trajectories of test particles are geodesic.

To begin with, let us rewrite the field equations (41) inspired by the approach first introduced by Capozziello & Fang [8] (see also Capozziello & Faraoni [9]) as follows:

$$G\_{\mu\nu} = \frac{8\pi G}{c^4} \left\{ \left( 1 + \frac{c^4}{8\pi G} F\_T \right) \frac{T\_{\mu\nu}}{F\_R} + T\_{\mu\nu}^{\text{curv}} \right\} \tag{49}$$

where the Einstein tensor is given by its usual form:

$$\mathcal{G}\_{\mu\nu} := \mathcal{R}\_{\mu\nu} - \frac{1}{2} \mathcal{R} \mathcal{g}\_{\mu\nu}. \tag{50}$$

10.5772/53878

147

. (55)

Extending Cosmology: The Metric Approach http://dx.doi.org/10.5772/53878

<sup>d</sup>*<sup>t</sup>* . (56)

*<sup>c</sup>*<sup>2</sup> , (57)

*<sup>H</sup>*<sup>3</sup> , (60)

*p*curv := *ωc*2*ρ*curv, (58)

<sup>2</sup> + <sup>3</sup>*H*d*FR*/d*<sup>t</sup> <sup>c</sup>*<sup>2</sup> (*RFR* <sup>−</sup> *<sup>F</sup>*) /2 <sup>−</sup> <sup>3</sup>*H*d*FR*/d*<sup>t</sup>* . (59)

between Ricci's scalar and the derivatives of the scale factor for a FLRW universe, then the

*c*4 <sup>8</sup>*π<sup>G</sup> FT*

The energy conservation equation is given by the null divergence of the energy-momentum

For completeness, we write down the correspondent generalisation of Raychadhuri's

<sup>+</sup> *<sup>H</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>8*πGp*curv

 *ρ FR*

(*ρ*˙ + 3*Hρ*) = −*ρ*

<sup>+</sup> *<sup>ρ</sup>*curv

d*FT*

dynamical Friedman's-like equation for a dust flat universe is:

*<sup>H</sup>*<sup>2</sup> <sup>=</sup> <sup>8</sup>*π<sup>G</sup>* 3

8*πG*

*<sup>c</sup>*<sup>4</sup> <sup>+</sup> *FT*

2 *a*¨ *a*

*<sup>w</sup>* <sup>=</sup> *<sup>c</sup>*<sup>2</sup> (*<sup>F</sup>* <sup>−</sup> *RFR*) /2 <sup>+</sup> <sup>d</sup><sup>2</sup>*FR*/d*<sup>t</sup>*

*M* = 4*π*

 *<sup>r</sup>*<sup>H</sup> 0 *ρ r*

On the other hand, note that the mass *M* that appears on the length *LM* must be the causally connected mass at a certain cosmic time *t*, since particles beyond Hubble's (or particle) horizon with respect to a given fundamental observer do not have any gravitational influence on him. At any particular cosmic epoch, this Hubble mass satisfies the spherically symmetric

> <sup>2</sup> <sup>d</sup>*<sup>r</sup>* <sup>=</sup> <sup>4</sup> 3 *πρ c*3

tensor:

and

where

equation for a dust flat universe:

where the "curvature-pressure"

condition implicit in equation (39) and so,

1 +

and

$$\begin{aligned} T\_{\mu\nu}^{\text{curr}} &:= \frac{c^4}{8\pi G F\_{\mathbb{R}}} \left[ \left( \frac{1}{2} \left( F - R F\_{\mathbb{R}} \right) - \Delta F\_{\mathbb{R}} \right) g\_{\mu\nu} + \\ & \quad \nabla\_{\mu} \nabla\_{\nu} F\_{\mathbb{R}} \right], \end{aligned} \tag{51}$$

represents the "*energy-momentum*" curvature tensor. Since *<sup>T</sup>*<sup>00</sup> = *<sup>ρ</sup>c*2, then it will be useful the identification *<sup>T</sup>*<sup>00</sup> := *<sup>ρ</sup>*curv*c*2. With this last definition and using the fact that the Laplace-Beltrami operator applied to a scalar field *ψ* is given by [see e.g. 19]:

$$
\Delta\psi = \frac{1}{\sqrt{-g}} \partial\_{\mu} \left( \sqrt{-g} \, g^{\mu\nu} \partial\_{\nu} \psi \right) , \tag{52}
$$

then

$$\rho\_{\rm curv} = \frac{c^2}{8\pi G F\_R} \left[ \frac{1}{2} \left( R F\_R - F \right) - \frac{3H}{c^2} \frac{\mathrm{d}F\_R}{\mathrm{d}t} \right],\tag{53}$$

where *H* := *a*˙(*t*)/*a*(*t*) represents Hubble's constant.

With the above definitions and using the 00 component of the field's equations (49) and the relation [cf. 11]:

$$R = -\frac{6}{c^2} \left[ \frac{\ddot{a}}{a} + \left( \frac{\dot{a}}{a} \right)^2 + \frac{\kappa c^2}{a^2} \right],\tag{54}$$

between Ricci's scalar and the derivatives of the scale factor for a FLRW universe, then the dynamical Friedman's-like equation for a dust flat universe is:

$$H^2 = \frac{8\pi G}{3} \left[ \left( 1 + \frac{c^4}{8\pi G} F\_T \right) \frac{\rho}{F\_R} + \rho\_{\text{curv}} \right]. \tag{55}$$

The energy conservation equation is given by the null divergence of the energy-momentum tensor:

$$\left(\frac{8\pi G}{c^4} + F\_T\right)(\dot{\rho} + 3H\rho) = -\rho \frac{\mathbf{d}F\_T}{\mathbf{d}t}.\tag{56}$$

For completeness, we write down the correspondent generalisation of Raychadhuri's equation for a dust flat universe:

2 *a*¨ *a* <sup>+</sup> *<sup>H</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>8*πGp*curv *<sup>c</sup>*<sup>2</sup> , (57)

where the "curvature-pressure"

$$p\_{\rm curv} := \omega c^2 \rho\_{\rm curv} \tag{58}$$

and

14 Open Questions in Cosmology

and

then

relation [cf. 11]:

model for which the covariant divergence of the energy-momentum tensor vanishes, and so

To begin with, let us rewrite the field equations (41) inspired by the approach first introduced

*c*4 <sup>8</sup>*π<sup>G</sup> FT*

*<sup>G</sup>µν* :<sup>=</sup> *<sup>R</sup>µν* <sup>−</sup> <sup>1</sup>

1

∇*µ*∇*νFR*

represents the "*energy-momentum*" curvature tensor. Since *<sup>T</sup>*<sup>00</sup> = *<sup>ρ</sup>c*2, then it will be useful the identification *<sup>T</sup>*<sup>00</sup> := *<sup>ρ</sup>*curv*c*2. With this last definition and using the fact that the

2

<sup>2</sup> (*<sup>F</sup>* <sup>−</sup> *RFR*) <sup>−</sup> <sup>∆</sup>*FR*

−*g gµν∂νψ*

<sup>2</sup> (*RFR* <sup>−</sup> *<sup>F</sup>*) <sup>−</sup> <sup>3</sup>*<sup>H</sup>*

*c*2

d*FR* d*t* 

 ,  *<sup>T</sup>µν FR*

+ *T*curv *µν*

> *<sup>g</sup>µν* +

*Rgµν*. (50)

, (52)

, (54)

, (53)

, (49)

(51)

as discussed in section 4 the trajectories of test particles are geodesic.

*<sup>G</sup>µν* <sup>=</sup> <sup>8</sup>*π<sup>G</sup> c*4

where the Einstein tensor is given by its usual form:

*T*curv

*µν* :<sup>=</sup> *<sup>c</sup>*<sup>4</sup>

8*πGFR*

Laplace-Beltrami operator applied to a scalar field *ψ* is given by [see e.g. 19]:

<sup>∆</sup>*<sup>ψ</sup>* <sup>=</sup> <sup>1</sup> √−*<sup>g</sup> ∂µ*

8*πGFR*

*<sup>R</sup>* <sup>=</sup> <sup>−</sup> <sup>6</sup> *c*2 *a*¨ *a* + *a*˙ *a* 2 + *κc*<sup>2</sup> *a*2 

 1

With the above definitions and using the 00 component of the field's equations (49) and the

*<sup>ρ</sup>*curv <sup>=</sup> *<sup>c</sup>*<sup>2</sup>

where *H* := *a*˙(*t*)/*a*(*t*) represents Hubble's constant.

by Capozziello & Fang [8] (see also Capozziello & Faraoni [9]) as follows:

 1 +

$$w = \frac{c^2 \left(F - RF\_R\right)/2 + \mathbf{d}^2 F\_R/\mathbf{d}t^2 + 3H \mathbf{d}F\_R/\mathbf{d}t}{c^2 \left(RF\_R - F\right)/2 - 3H \mathbf{d}F\_R/\mathbf{d}t}.\tag{59}$$

On the other hand, note that the mass *M* that appears on the length *LM* must be the causally connected mass at a certain cosmic time *t*, since particles beyond Hubble's (or particle) horizon with respect to a given fundamental observer do not have any gravitational influence on him. At any particular cosmic epoch, this Hubble mass satisfies the spherically symmetric condition implicit in equation (39) and so,

$$M = 4\pi \int\_0^{r\_\text{H}} \rho \, r^2 \, \text{d}r = \frac{4}{3} \pi \rho \frac{c^3}{H^3} \,\text{}\tag{60}$$

where

$$r\_{\rm H} := \frac{c}{H(t)},\tag{61}$$

149

. (69)

*c H*0, (70)

, (68)

Extending Cosmology: The Metric Approach http://dx.doi.org/10.5772/53878

*<sup>H</sup>*<sup>2</sup> <sup>=</sup> <sup>8</sup>*πG<sup>ρ</sup>* 3 *Z FR*

<sup>1</sup> <sup>−</sup> *<sup>q</sup>* <sup>−</sup> <sup>4</sup> (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*)

An important result can be obtained evaluating equation (68) at the present epoch, yielding:

2/(*b*−1) Ω(0)

matt :<sup>=</sup> <sup>3</sup>*H*2*<sup>ρ</sup>*

In other words, the value of Milgrom's acceleration constant a<sup>0</sup> at the current cosmic epoch

The numerical coincidence between the value of Milgrom's acceleration constant a<sup>0</sup> and the multiplication of the speed of light *c* by the current value of Hubble's constant *H*<sup>0</sup> has been noted since the early development of MOND [see e.g. 13, and references therein]. Note that equation (72) means that this coincidence relation occurs at approximately the present cosmic epoch in complete agreement with the results by Bernal, Capozziello, Cristofano & de Laurentis [4] where it is shown that a<sup>0</sup> shows no cosmological evolution and hence it can

For the power law (21) and the assumptions made above, it follows that the energy

<sup>+</sup> *B H*

*<sup>R</sup><sup>b</sup> <sup>L</sup>*2(*b*−1)

*<sup>M</sup>* <sup>=</sup> 0, (73)

*<sup>b</sup>* <sup>+</sup>

3 2 *β* + 3 *α*

matt(3*b*−5)/(*b*−1)

matt at the present epoch has been defined by it's usual

<sup>8</sup>*π<sup>G</sup>* . (71)

<sup>a</sup><sup>0</sup> ≈ *<sup>c</sup>* × *<sup>H</sup>*0. (72)

*Z* := 1 + (*b* − 1)

*<sup>ζ</sup>*<sup>4</sup> (<sup>1</sup> − *<sup>q</sup>*0)

be postulated as a fundamental constant of nature.

(*ρ*˙ + 3*Hρ*) +

*c*2 8*πG*

 *A ρ*˙ *ρ*

conservation equation (56) is given by:

*j* − *q* − 2

<sup>2</sup> (*bZ*0)

Ω(0)

where

relation:

is such that

is a dimensionless function.

<sup>a</sup><sup>0</sup> = 9 4

where the density parameter <sup>Ω</sup>(0)

is the Hubble radius or the distance of causal contact at a particular cosmic epoch [21]. In this respect the mass *M* is measured from the point of view of any given fundamental observer at a particular cosmic time *t* and so, it does not depend on which system of reference (or coordinates) is measured. As such, the mass *M* represents an invariant scalar quantity. From this last relation it follows that the length (31) is given by:

$$L\_M = \zeta \frac{\left(\frac{4}{3}\pi c^3 G\right)^{3/4}}{c \,\mathrm{a}\_0^{1/4}} \frac{\rho^{3/4}}{H^{9/4}}\,\mathrm{}\,\tag{62}$$

and so, by using relation (21) and the standard power-law assumptions:

$$a(t) = a(t\_0) \left(\frac{t}{t\_0}\right)^a, \qquad \rho(t) = \rho\_0 \left(\frac{a}{a(t\_0)}\right)^\beta. \tag{63}$$

for the unknown constant powers *α* and *β*, it follows that:

$$\frac{\mathbf{d}F\_R}{\mathbf{d}t} = b(b-1)R^{b-1}L\_M^{2(b-1)}H\left[\frac{j-q-2}{1-q} + \frac{3}{2}\left(\beta + \frac{3}{a}\right)\right],\tag{64}$$

$$\frac{\text{d}F\_T}{\text{d}t} = \frac{3}{2}(b-1)\frac{R^b L\_M^{2b-2}}{\rho c^2},\tag{65}$$

where

$$q(t) := -\frac{1}{a} \frac{\mathrm{d}^2 a}{\mathrm{d}t^2} H^{-2}, \qquad \text{and} \qquad j := \frac{1}{a} \frac{\mathrm{d}^3 a}{\mathrm{d}t^3} H^{-3}. \tag{66}$$

are the deceleration parameter and the jerk respectively. With these and the value of *LM* from equation (62), the curvature density (53) is given by:

$$\rho\_{\rm curv} = \frac{3H^2}{8\pi G}(b-1)\left[ (1-q) - \frac{j-q-2}{1-q} - \frac{3}{2}\left(\beta + \frac{3}{a}\right) \right].\tag{67}$$

Substitution of the previous relations on Friedmann's equation (55) gives:

$$H^2 = \frac{8\pi G\rho}{3\,\text{Z}\,\text{F}\_\text{R}},\tag{68}$$

where

16 Open Questions in Cosmology

*<sup>r</sup>*<sup>H</sup> :<sup>=</sup> *<sup>c</sup> H*(*t*)

is the Hubble radius or the distance of causal contact at a particular cosmic epoch [21]. In this respect the mass *M* is measured from the point of view of any given fundamental observer at a particular cosmic time *t* and so, it does not depend on which system of reference (or coordinates) is measured. As such, the mass *M* represents an invariant scalar quantity. From

3/4

, *<sup>ρ</sup>*(*t*) = *<sup>ρ</sup>*<sup>0</sup>

*ρ*3/4

 *a <sup>a</sup>*(*t*0)

+ 3 2 *β* + 3 *α* 

> *a* d3*a*

2 *β* + 3 *α* 

*β*

*c* a1/4 0

this last relation it follows that the length (31) is given by:

*<sup>a</sup>*(*t*) = *<sup>a</sup>*(*t*0)

for the unknown constant powers *α* and *β*, it follows that:

*<sup>q</sup>*(*t*) :<sup>=</sup> <sup>−</sup><sup>1</sup>

are the deceleration parameter and the jerk respectively.

*<sup>ρ</sup>*curv <sup>=</sup> <sup>3</sup>*H*<sup>2</sup>

*a* d2*a*

<sup>8</sup>*π<sup>G</sup>* (*<sup>b</sup>* <sup>−</sup> <sup>1</sup>)

Substitution of the previous relations on Friedmann's equation (55) gives:

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> *<sup>b</sup>*(*<sup>b</sup>* <sup>−</sup> <sup>1</sup>)*Rb*−1*L*2(*b*−1)

d*FT* <sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>3</sup> 2 (*b* − 1)

d*FR*

where

*LM* = *<sup>ζ</sup>*

and so, by using relation (21) and the standard power-law assumptions:

 *t t*0 *α*

4 <sup>3</sup>*πc*3*<sup>G</sup>*

*<sup>M</sup> H*

<sup>d</sup>*t*<sup>2</sup> *<sup>H</sup>*<sup>−</sup>2, and *<sup>j</sup>* :<sup>=</sup> <sup>1</sup>

(<sup>1</sup> <sup>−</sup> *<sup>q</sup>*) <sup>−</sup> *<sup>j</sup>* <sup>−</sup> *<sup>q</sup>* <sup>−</sup> <sup>2</sup>

<sup>1</sup> <sup>−</sup> *<sup>q</sup>* <sup>−</sup> <sup>3</sup>

With these and the value of *LM* from equation (62), the curvature density (53) is given by:

*j* − *q* − 2 1 − *q*

*<sup>R</sup>bL*<sup>2</sup>*b*−<sup>2</sup> *M*

, (61)

*<sup>H</sup>*9/4 , (62)

*<sup>ρ</sup>c*<sup>2</sup> , (65)

<sup>d</sup>*t*<sup>3</sup> *<sup>H</sup>*<sup>−</sup>3, (66)

. (63)

, (64)

. (67)

$$Z := 1 + (b - 1) \left[ \frac{j - q - 2}{1 - q} - \frac{4 \left( 1 - q \right)}{b} + \frac{3}{2} \left( \beta + \frac{3}{a} \right) \right]. \tag{69}$$

is a dimensionless function.

An important result can be obtained evaluating equation (68) at the present epoch, yielding:

$$\mathbf{a}\_{0} = \left[\frac{9}{4}\zeta^{4}\left(1-q\_{0}\right)^{2}\left(bZ\_{0}\right)^{2/(b-1)}\left(\Omega\_{\text{matt}}^{(0)}\right)^{(3b-5)/(b-1)}\right]c\,H\_{0},\tag{70}$$

where the density parameter <sup>Ω</sup>(0) matt at the present epoch has been defined by it's usual relation:

$$
\Omega\_{\rm matt}^{(0)} := \frac{3H^2 \rho}{8\pi G}. \tag{71}
$$

In other words, the value of Milgrom's acceleration constant a<sup>0</sup> at the current cosmic epoch is such that

$$
\mathfrak{a}\_0 \approx \mathfrak{c} \times H\_0. \tag{72}
$$

The numerical coincidence between the value of Milgrom's acceleration constant a<sup>0</sup> and the multiplication of the speed of light *c* by the current value of Hubble's constant *H*<sup>0</sup> has been noted since the early development of MOND [see e.g. 13, and references therein]. Note that equation (72) means that this coincidence relation occurs at approximately the present cosmic epoch in complete agreement with the results by Bernal, Capozziello, Cristofano & de Laurentis [4] where it is shown that a<sup>0</sup> shows no cosmological evolution and hence it can be postulated as a fundamental constant of nature.

For the power law (21) and the assumptions made above, it follows that the energy conservation equation (56) is given by:

$$\left(\dot{\rho} + 3H\rho\right) + \frac{c^2}{8\pi G} \left(A\frac{\dot{\rho}}{\rho} + \mathcal{B}H\right) \mathcal{R}^b L\_M^{2(b-1)} = 0,\tag{73}$$

where:

$$A := \frac{9}{4} \left( b - 1 \right)^2.$$

$$B := \frac{9}{2} \frac{b - 1}{b} + \frac{27}{4} \frac{(b - 1)^2}{a} + \frac{3}{2} \frac{b \left( b - 1 \right) \left( j - q - 2 \right)}{1 - q}.$$

Direct substitution of the density power law (63) into relation (73) gives a constraint equation between *α*, *β* and *b*:

$$
\beta = \frac{1}{\alpha} \left( \frac{9 - 5b}{3b - 5} \right). \tag{74}
$$

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z

**Figure 3.** Redshift magnitude plot for SNIa showing the distance modulus *µ* as a function of the redshift *z* for SNIa as presented by Riess et al. [34]. The dotted red line shows the best fit to the data with the *f*(*χ*) gravity theory applied to a flat dust FLRW universe (see text) with no dark components. The continuous blue line represents the best fit according to the

The parameter *β* can be found from conservation of mass arguments, since the total mass

the radius of the whole universe. Since *a*(*t*) and *ρ*(*t*) are time dependent functions, the only way the mass of the universe is conserved is by requiring *a*<sup>3</sup> *ρ* = const. and so, *β* = 3. This argument is exactly the one used in standard cosmology when dealing with a dust FLRW universe [see e.g. 21]. Using this value of *β* and the one already found for *α*, it follows that *b* = 1.57 ± 0.56, which is within the expected value of *b* = 3/2 discussed in section 3.

For completeness, we write down a few of the cosmographycal parameters obtained by this

*<sup>h</sup>* = 0.64 ± 0.009, *<sup>q</sup>*<sup>0</sup> = −0.2642 ± 0.075,

As explained by Carranza et al. [10], the obtained value *b* ≈ 3/2 is a completely expected result due to the following arguments. As explained in section 2, a gravitational system for which its characteristic size *r* is such that *x* := *lM*/*r* 1 is in the MONDian gravity regime. For the case of the universe, *x* ∼ a few and as such if not totally in the MONDian regime of gravity, then it is far away from the regime of Newtonian gravity. The relativistic version of this means that the universe is close to the regime for which *f*(*χ*) = *χ*3/2 and so *b* = 3/2. This is a very important result since seen in this way, the accelerated expansion of the universe is due to an extended gravity theory deviating from general relativity. It is quite interesting to note that the function *f*(*χ*) = *χ*3/2 which at its non-relativistic limit is

<sup>0</sup> *<sup>ρ</sup> <sup>r</sup>*<sup>2</sup> *<sup>a</sup>*<sup>3</sup> <sup>d</sup>*<sup>r</sup>* <sup>=</sup> const., where the upper limit of the integral is

*<sup>j</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.1246 <sup>±</sup> 0.004. (79)

0 0.5 1 1.5 2

µ(z)

standard concordance dust ΛCDM model.

of the universe *<sup>M</sup>*tot = <sup>4</sup>*<sup>π</sup> <sup>R</sup>*max

*f*(*χ*) gravity applied to the universe:

**6. Discussion**

46

44

42

40

38

36

Let us now proceed to fix the so far unknown parameters of the theory *α*, *β* and *b*. To do so, we need reliable observational data and as such, we use the redshift-magnitude SNIa data obtained by Riess et al. [34] and the following well known standard cosmological relations [see e.g. 21]:

$$1 + z = a(t\_0) / a(t),\tag{75}$$

$$
\mu\left(z\right) = 5\log\_{10}\left[H\_0 d\_L\left(z\right)\right] - 5\log\_{10}h + 42.38,\tag{76}
$$

$$d\_{\mathcal{L}}(z) = (1+z) \int\_0^z \frac{c}{H(z)} \,\,\mathrm{d}z,\tag{77}$$

for the cosmological redshift *z*, the distance modulus *µ*, the luminosity distance *dL* and where the normalised Hubble constant *h* at the present epoch is given by *h* := *H*0/ 100 km s<sup>−</sup>1/ Mpc . Also, from equation (63) it follows that

$$H(a) = H\_0 \left(\frac{a}{a(t\_0)}\right)^{-1/a} = H\_0 (1+z)^{1/a} \,\text{s}\tag{78}$$

and the substitution of this into equation (77) gives the distance modulus *dL* as a function of the redshift *z*. This means that the redshift magnitude relation (76) is a function that depends on the values of the current Hubble constant *H*<sup>0</sup> and the value of *α*. Figure 3 shows the best fit to the redshift magnitude relation of SNIa observed by Riess et al. [34], yielding *α* = 1.359 ± 0.139 and *h* = 0.64 ± 0.009. The best fit presented on the figure was obtained using the Marquardt-Levenberg fit provided by gnuplot (http://www.gnuplot.info) for non-linear functions. These values do not provide the whole description of the problem, since *β* and *b* are still unknown. However, according to the constraint equation (74) only one of them is needed in order to know the other once *α* is known.

**Figure 3.** Redshift magnitude plot for SNIa showing the distance modulus *µ* as a function of the redshift *z* for SNIa as presented by Riess et al. [34]. The dotted red line shows the best fit to the data with the *f*(*χ*) gravity theory applied to a flat dust FLRW universe (see text) with no dark components. The continuous blue line represents the best fit according to the standard concordance dust ΛCDM model.

The parameter *β* can be found from conservation of mass arguments, since the total mass of the universe *<sup>M</sup>*tot = <sup>4</sup>*<sup>π</sup> <sup>R</sup>*max <sup>0</sup> *<sup>ρ</sup> <sup>r</sup>*<sup>2</sup> *<sup>a</sup>*<sup>3</sup> <sup>d</sup>*<sup>r</sup>* <sup>=</sup> const., where the upper limit of the integral is the radius of the whole universe. Since *a*(*t*) and *ρ*(*t*) are time dependent functions, the only way the mass of the universe is conserved is by requiring *a*<sup>3</sup> *ρ* = const. and so, *β* = 3. This argument is exactly the one used in standard cosmology when dealing with a dust FLRW universe [see e.g. 21]. Using this value of *β* and the one already found for *α*, it follows that *b* = 1.57 ± 0.56, which is within the expected value of *b* = 3/2 discussed in section 3.

For completeness, we write down a few of the cosmographycal parameters obtained by this *f*(*χ*) gravity applied to the universe:

$$\begin{aligned} h &= 0.64 \pm 0.009, \quad q\_0 = -0.2642 \pm 0.075, \\ j\_0 &= -0.1246 \pm 0.004. \end{aligned} \tag{79}$$

## **6. Discussion**

18 Open Questions in Cosmology

between *α*, *β* and *b*:

relations [see e.g. 21]:

100 km s<sup>−</sup>1/ Mpc

*H*0/  *<sup>B</sup>* :<sup>=</sup> <sup>9</sup> 2 *b* − 1 *<sup>b</sup>* <sup>+</sup> *<sup>A</sup>* :<sup>=</sup> <sup>9</sup>

(*b* − 1) 2

*α*

*<sup>β</sup>* <sup>=</sup> <sup>1</sup> *α*

*<sup>d</sup>*<sup>L</sup> (*z*) = (<sup>1</sup> + *<sup>z</sup>*)

 *a <sup>a</sup>*(*t*0)

*<sup>H</sup>*(*a*) = *<sup>H</sup>*<sup>0</sup>

of them is needed in order to know the other once *α* is known.

27 4

<sup>4</sup> (*<sup>b</sup>* <sup>−</sup> <sup>1</sup>)

+ 3 2

Direct substitution of the density power law (63) into relation (73) gives a constraint equation

9 − 5*b* 3*b* − 5

Let us now proceed to fix the so far unknown parameters of the theory *α*, *β* and *b*. To do so, we need reliable observational data and as such, we use the redshift-magnitude SNIa data obtained by Riess et al. [34] and the following well known standard cosmological

> *<sup>z</sup>* 0

for the cosmological redshift *z*, the distance modulus *µ*, the luminosity distance *dL* and where the normalised Hubble constant *h* at the present epoch is given by *h* :=

<sup>−</sup>1/*<sup>α</sup>*

and the substitution of this into equation (77) gives the distance modulus *dL* as a function of the redshift *z*. This means that the redshift magnitude relation (76) is a function that depends on the values of the current Hubble constant *H*<sup>0</sup> and the value of *α*. Figure 3 shows the best fit to the redshift magnitude relation of SNIa observed by Riess et al. [34], yielding *α* = 1.359 ± 0.139 and *h* = 0.64 ± 0.009. The best fit presented on the figure was obtained using the Marquardt-Levenberg fit provided by gnuplot (http://www.gnuplot.info) for non-linear functions. These values do not provide the whole description of the problem, since *β* and *b* are still unknown. However, according to the constraint equation (74) only one

. Also, from equation (63) it follows that

*c*

2 ,

*b* (*b* − 1) (*j* − *q* − 2)

<sup>1</sup> <sup>−</sup> *<sup>q</sup>* .

<sup>1</sup> + *<sup>z</sup>* = *<sup>a</sup>*(*t*0)/*a*(*t*), (75)

*<sup>µ</sup>* (*z*) <sup>=</sup> 5 log10 [*H*<sup>0</sup> *dL* (*z*)] <sup>−</sup> 5 log10 *<sup>h</sup>* <sup>+</sup> 42.38, (76)

. (74)

*<sup>H</sup>* (*z*) <sup>d</sup>*z*, (77)

= *<sup>H</sup>*0(<sup>1</sup> + *<sup>z</sup>*)1/*α*, (78)

where:

As explained by Carranza et al. [10], the obtained value *b* ≈ 3/2 is a completely expected result due to the following arguments. As explained in section 2, a gravitational system for which its characteristic size *r* is such that *x* := *lM*/*r* 1 is in the MONDian gravity regime. For the case of the universe, *x* ∼ a few and as such if not totally in the MONDian regime of gravity, then it is far away from the regime of Newtonian gravity. The relativistic version of this means that the universe is close to the regime for which *f*(*χ*) = *χ*3/2 and so *b* = 3/2. This is a very important result since seen in this way, the accelerated expansion of the universe is due to an extended gravity theory deviating from general relativity. It is quite interesting to note that the function *f*(*χ*) = *χ*3/2 which at its non-relativistic limit is capable of predicting the correct dynamical behaviour of many astrophysical phenomena, is also able to explain the behaviour of the current accelerated expansion of the universe.

Seen in this way, the behaviour of gravity towards the past (for sufficiently large redshifts *z*) will differ from *f*(*χ*) = *χ*3/2 and eventually converge to *f*(*χ*) = *χ*, i.e. the gravitational regime of gravity is general relativity for sufficiently large redshifts. A very detailed investigation into this needs to be done at different levels in order to be coherent many different cosmological observations [see e.g. 20]. This in turn can serve to calibrate the index *p* of the transfer function *f*(*χ*) as presented in equation (37), which has a very soft transition when *p* = −1, i.e.,

$$f(\chi) = \frac{\chi^{3/2}}{1 + \chi^{3/2}},\tag{80}$$

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scales. There are many more astrophysical challenges that this theory needs to address, in particular with respect to lensing at different scales and the dynamics associated to galaxy

This work was supported by a DGAPA-UNAM grant (PAPIIT IN116210-3) and CONACyT 26344. The author acknowledges fruitful discussions at different stages with Tula Bernal, Diego Carranza, Salvatore Capozziello, Rituparno Goswami, Xavier Hernandez, Juan Carlos

Instituto de Astronomia, Universidad Nacional Autónoma de Mexico, Ciudad Universitaria,

[1] Bekenstein, J. [2006a]. The modified Newtonian dynamics - MOND and its implications

[2] Bekenstein, J. [2006b]. The modified Newtonian dynamics - MOND and its implications

[3] Bekenstein, J. D. [2004]. Relativistic gravitation theory for the modified Newtonian

[4] Bernal, T., Capozziello, S., Cristofano, G. & de Laurentis, M. [2011]. Mond's Acceleration

[5] Bernal, T., Capozziello, S., Hidalgo, J. C. & Mendoza, S. [2011]. Recovering MOND from

[6] Blanchet, L. & Marsat, S. [2012]. Relativistic MOND theory based on the Khronon scalar

[7] Capozziello, S., de Laurentis, M. & Faraoni, V. [2010]. A Bird's Eye View of f(R)-Gravity,

[8] Capozziello, S. & Fang, L. Z. [2002]. Curvature Quintessence, *International Journal of*

[9] Capozziello, S. & Faraoni, V. [2010]. *Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics*, Fundamental Theories of Physics, Springer.

[10] Carranza, D. A., Mendoza, S. & Torres, L. A. [2012]. A cosmological dust model with

Scale as a Fundamental Quantity, *Modern Physics Letters A* 26: 2677–2687.

extended metric theories of gravity, *European Physical Journal C* 71: 1794.

clusters. These will be addressed elsewhere.

**Acknowledgements**

Hidalgo and Luis Torres.

Distrito Federal CP 04510, Mexico

field, *ArXiv e-prints* .

*The Open Astronomy Journal* 3: 49–72.

extended *f*(*χ*) gravity, *ArXiv e-prints* .

*Modern Physics D* 11: 483–491.

for new physics, *Contemporary Physics* 47: 387–403.

for new physics, *Contemporary Physics* 47: 387–403.

dynamics paradigm, *Physical Review D* 70(8): 083509.

**Author details** Sergio Mendoza

**References**

and also has a very sharp transition when *p* → ∞, with the step function:

$$f(\chi) = \begin{cases} \chi^{3/2}, & \text{for } 0 \le \chi \le 1, \\ \chi, & \text{for } \chi \ge 1. \end{cases} \tag{81}$$

In this respect, perhaps something close to a sharp transition (81) will be observed since, as mentioned in section 2, at the non-relativistic level different astrophysical observations show a sharp transition from the Newtonian to the MONDian regimes. This sort of decision has to be taken with care and such a full description requires to analyse in full detail the whole Friedmann-like equations:

$$\begin{aligned} \left(\frac{8\pi G}{c^4} + F\_T\right) \left(\dot{\rho} + 3H\rho + \frac{3Hp}{c^2}\right) &= \\ -\rho \frac{\mathbf{d}F\_T}{\mathbf{d}t} + \frac{1}{c^2} \left(p\frac{\mathbf{d}F\_T}{\mathbf{d}t} + F\_T \frac{\mathbf{d}p}{\mathbf{d}t}\right) \end{aligned} \tag{82}$$

$$H^2 = \frac{8\pi G}{3} \left[ \left( 1 + \frac{c^4 F\_T}{8\pi G} \right) \frac{\rho}{F\_R} + \rho\_{\text{curv}} \right] - \frac{\kappa c^2}{a^2} \,\text{}\tag{83}$$

$$2\frac{\ddot{a}}{a} + H^2 + \frac{\kappa c^2}{a^2} = -\frac{8\pi Gp}{c^2 F\_R} - \frac{2pc^2 F\_T}{F\_R} - \frac{8\pi Gp\_{\text{curv}}}{c^2}.\tag{84}$$

These equations are directly obtained from taking the null covariant divergence of the energy momentum tensor, the 00 component of the field equations (49) and the density *ρ* contains all species of matter and/or radiation. The curvature density *ρ*curv and the curvature pressure *p*curv are related to one another by relation (58) with *ω* given by equation (59).

It is quite remarkable that a metric extended theory of gravity is able to reproduce phenomena from mass and length scales associated to the solar system up to cosmological

scales. There are many more astrophysical challenges that this theory needs to address, in particular with respect to lensing at different scales and the dynamics associated to galaxy clusters. These will be addressed elsewhere.

## **Acknowledgements**

20 Open Questions in Cosmology

when *p* = −1, i.e.,

Friedmann-like equations:

capable of predicting the correct dynamical behaviour of many astrophysical phenomena, is also able to explain the behaviour of the current accelerated expansion of the universe.

Seen in this way, the behaviour of gravity towards the past (for sufficiently large redshifts *z*) will differ from *f*(*χ*) = *χ*3/2 and eventually converge to *f*(*χ*) = *χ*, i.e. the gravitational regime of gravity is general relativity for sufficiently large redshifts. A very detailed investigation into this needs to be done at different levels in order to be coherent many different cosmological observations [see e.g. 20]. This in turn can serve to calibrate the index *p* of the transfer function *f*(*χ*) as presented in equation (37), which has a very soft transition

*<sup>f</sup>*(*χ*) = *<sup>χ</sup>*3/2

*χ*3/2, for 0 ≤ *χ* ≤ 1,

In this respect, perhaps something close to a sharp transition (81) will be observed since, as mentioned in section 2, at the non-relativistic level different astrophysical observations show a sharp transition from the Newtonian to the MONDian regimes. This sort of decision has to be taken with care and such a full description requires to analyse in full detail the whole

*ρ*˙ + 3*Hρ* +

*c*<sup>4</sup>*FT* 8*πG*

These equations are directly obtained from taking the null covariant divergence of the energy momentum tensor, the 00 component of the field equations (49) and the density *ρ* contains all species of matter and/or radiation. The curvature density *ρ*curv and the curvature pressure

It is quite remarkable that a metric extended theory of gravity is able to reproduce phenomena from mass and length scales associated to the solar system up to cosmological

3*H p c*2 =

<sup>−</sup> <sup>2</sup>*pc*<sup>2</sup>*FT FR*

+ *ρ*curv

 <sup>−</sup> *<sup>κ</sup>c*<sup>2</sup>

1 *c*2 *p* d*FT* <sup>d</sup>*<sup>t</sup>* <sup>+</sup> *FT*

 *ρ FR*

and also has a very sharp transition when *p* → ∞, with the step function:

*f*(*χ*) =

8*πG*

*<sup>c</sup>*<sup>4</sup> <sup>+</sup> *FT*

*<sup>H</sup>*<sup>2</sup> <sup>=</sup> <sup>8</sup>*π<sup>G</sup>* 3

+ *H*<sup>2</sup> +

2 *a*¨ *a*  1 +

*κc*<sup>2</sup>

− *ρ* d*FT* <sup>d</sup>*<sup>t</sup>* <sup>+</sup>

*<sup>a</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>8*πGp c*<sup>2</sup>*FR*

*p*curv are related to one another by relation (58) with *ω* given by equation (59).

<sup>1</sup> <sup>+</sup> *<sup>χ</sup>*3/2 , (80)

*<sup>χ</sup>*, for *<sup>χ</sup>* <sup>≥</sup> 1. (81)

d*p* d*t* ,

− 8*πGp*curv

(82)

*<sup>a</sup>*<sup>2</sup> , (83)

*<sup>c</sup>*<sup>2</sup> . (84)

This work was supported by a DGAPA-UNAM grant (PAPIIT IN116210-3) and CONACyT 26344. The author acknowledges fruitful discussions at different stages with Tula Bernal, Diego Carranza, Salvatore Capozziello, Rituparno Goswami, Xavier Hernandez, Juan Carlos Hidalgo and Luis Torres.

## **Author details**

## Sergio Mendoza

Instituto de Astronomia, Universidad Nacional Autónoma de Mexico, Ciudad Universitaria, Distrito Federal CP 04510, Mexico

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[22] Mendoza, S., Bernal, T., Hernandez, X., Hidalgo, J. C. & Torres, L. A. [2012]. Gravitational lensing with *f*(*χ*) = *χ*3/2 gravity in accordance with astrophysical

[23] Mendoza, S., Hernandez, X., Hidalgo, J. C. & Bernal, T. [2011]. A natural approach to extended Newtonian gravity: tests and predictions across astrophysical scales, *MNRAS*

[24] Mendoza, S. & Rosas-Guevara, Y. M. [2007]. Gravitational waves and lensing of the metric theory proposed by Sobouti, *Astronomy and Astrophysics* 472: 367–371.

[25] Meyer, H., Lohrmann, E., Schubert, S., Bartel, W., Glazov, A., Loehr, B., Niebuhr, C., Wuensch, E., Joensson, L. & Kempf, G. [2011]. Test of the Law of Gravitation at small

[26] Milgrom, M. [1983]. A modification of the Newtonian dynamics - Implications for

Observational Phenomenology and Relativistic Extensions, *ArXiv e-prints* .


**Chapter 7**

**Provisional chapter**

**Introduction to Palatini Theories of Gravity and**

**Introduction to Palatini Theories of Gravity and**

The impact of Einstein's fundamental idea of gravitation as a curved space-time phenomenon on our current understanding of the Universe has been enormously successful. A key aspect of his celebrated theory of General Relativity (GR) is that the spatial sections of four dimensional space-time need not be Euclidean. The Minkowskian description is just an approximation valid on (relatively) local portions of space-time. On larger scales, however, one must consider deformations induced by the matter on the geometry, which must be dictated by some set of field equations. In this respect, the predictions of GR are in agreement with experiments in scales that range from millimeters to astronomical units, scales in which weak and strong field phenomena can be observed [39]. The theory is so successful in those regimes and scales that it is generally accepted that it should work also at larger and shorter scales, and at weaker and stronger regimes. The validity of these assumptions, obviously, is not guaranteed *a priori* regardless of how beautiful and elegant the theory might appear. Therefore, not only must we keep confronting the predictions of the theory with experiments and/or observations at new scales, but also we have to demand theoretical consistency with

For the above reasons, we believe that scrutinizing the implicit assumptions and mathematical structures behind the classical formulation of GR could help better understand the starting point of some current approaches that go beyond our standard model of gravitational physics. At the same time, this could provide new insights useful to address from a different perspective some current open questions, such as the existence of black hole and big bang singularities or the cosmic speedup problem. In this sense, Einstein himself stated that "*the question whether the structure of [the spacetime] continuum is Euclidean, or in accordance with Riemann's general scheme, or otherwise, is . . . a physical question which must be answered by experience, and not a question of a mere convention to be selected on practical grounds*" [10]. From these words it follows that questioning the regime of applicability of the Riemannian nature of the geometry associated with the gravitational field and considering

> ©2012 Olmo, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Olmo; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

the other physical interactions and, in particular, in the quantum regime.

**Nonsingular Cosmologies**

**Nonsingular Cosmologies**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Gonzalo J. Olmo

**1. Introduction**

Gonzalo J. Olmo

http://dx.doi.org/10.5772/51807

**Provisional chapter**

## **Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies Nonsingular Cosmologies**

**Introduction to Palatini Theories of Gravity and**

Gonzalo J. Olmo Additional information is available at the end of the chapter

Gonzalo J. Olmo

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51807

## **1. Introduction**

The impact of Einstein's fundamental idea of gravitation as a curved space-time phenomenon on our current understanding of the Universe has been enormously successful. A key aspect of his celebrated theory of General Relativity (GR) is that the spatial sections of four dimensional space-time need not be Euclidean. The Minkowskian description is just an approximation valid on (relatively) local portions of space-time. On larger scales, however, one must consider deformations induced by the matter on the geometry, which must be dictated by some set of field equations. In this respect, the predictions of GR are in agreement with experiments in scales that range from millimeters to astronomical units, scales in which weak and strong field phenomena can be observed [39]. The theory is so successful in those regimes and scales that it is generally accepted that it should work also at larger and shorter scales, and at weaker and stronger regimes. The validity of these assumptions, obviously, is not guaranteed *a priori* regardless of how beautiful and elegant the theory might appear. Therefore, not only must we keep confronting the predictions of the theory with experiments and/or observations at new scales, but also we have to demand theoretical consistency with the other physical interactions and, in particular, in the quantum regime.

For the above reasons, we believe that scrutinizing the implicit assumptions and mathematical structures behind the classical formulation of GR could help better understand the starting point of some current approaches that go beyond our standard model of gravitational physics. At the same time, this could provide new insights useful to address from a different perspective some current open questions, such as the existence of black hole and big bang singularities or the cosmic speedup problem. In this sense, Einstein himself stated that "*the question whether the structure of [the spacetime] continuum is Euclidean, or in accordance with Riemann's general scheme, or otherwise, is . . . a physical question which must be answered by experience, and not a question of a mere convention to be selected on practical grounds*" [10]. From these words it follows that questioning the regime of applicability of the Riemannian nature of the geometry associated with the gravitational field and considering

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Olmo; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Olmo, licensee InTech. This is an open access chapter distributed under the terms of the Creative

more general frameworks are legitimate questions that should be explored by all available means (theoretical and experimental). These are some of the basic points to be addressed in this work.

*t* ∈ [0, 1] → M. In a chart (U, *ϕ*), the points of the curve have the following coordinate representation: *x* = *ϕ*(*pt*) = *ϕ* ◦ *γ*(*t*). If we consider now a function *f* on M, where *f* is a map that to every *p* ∈ M assigns a real number (*f* : M → **R**), the rate of change of *f* along

*dt* <sup>=</sup> *<sup>∂</sup> <sup>f</sup>*

where we have defined the components of the tangent vector to the curve in this chart as *<sup>X</sup><sup>µ</sup>* ≡ *dxµ*(*t*)/*dt*. Vectors can thus be seen as differential operators *<sup>X</sup>* = *<sup>X</sup>µ∂µ* whose action on functions is of the form *<sup>X</sup>*[ *<sup>f</sup>* ] = *<sup>X</sup>µ∂µ <sup>f</sup>* , thus providing a natural notion of directional derivative for functions. The set {*e<sup>µ</sup>* ≡ *∂µ*} defines a (coordinate) basis of the tangent space of vectors at the point p, which we denote *Tp*M. Obviously, vectors exist without specifying the coordinates. Under changes of coordinates, we have *<sup>V</sup>* = *<sup>V</sup>µe<sup>µ</sup>* = *<sup>V</sup>*˜ *<sup>α</sup>e*˜*<sup>α</sup>* = *<sup>V</sup>*˜ *<sup>α</sup> <sup>∂</sup>x<sup>µ</sup>*

When a vector is assigned smoothly to each point of M, it is called a vector field over M. Each component of a vector field is thus a smooth function from M → **R**. Given a vector field *X*, an integral curve of *X* is defined as the curve whose tangent vector coincides with *X*. For infinitesimal displacements of magnitude *ǫ* in the direction of *X*, a given point

correspondence between vectors of the tangent spaces *Tx*<sup>M</sup> and *<sup>T</sup>σǫ* (*x*)M. The effect of these transformations on a vector field *Y*(*x*) leads to the concept of Lie derivative, whose action on

This derivative operator is independent of the choice of coordinates and follows naturally from the differential structure of the manifold. It satisfies a number of useful properties such as bilinearity in its two arguments, L*X*(*<sup>Y</sup>* + *<sup>Z</sup>*) = L*XY* + L*XZ*, L*X*+*YZ* = L*XZ* + L*YZ*, and the chain rule L*<sup>X</sup> f Y* = (L*<sup>X</sup> <sup>f</sup>*)*<sup>Y</sup>* + *<sup>f</sup>*L*XY*, with L*<sup>X</sup> <sup>f</sup>* = *<sup>X</sup>*[ *<sup>f</sup>* ]. Though the Lie derivative provides a natural *directional derivative* for functions, it does not work in the same way for vectors and tensors of higher rank. In fact, since the partial derivatives of the vector *X* appear explicitly in L*XY*, two vectors whose components at a given point have the same values but whose partial derivatives at the point differ do not yield a vector that points in the same direction, i.e., they are not proportional. Therefore, in order to introduce a proper notion of *directional derivative* for vectors and tensors, we need to introduce a new structure called **connection** which specifies how vectors (and tensors in general) are transported along a

**Manifolds with a connection.** We are thus going to introduce a derivative operator, which we denote by ∇, such that given two vector fields *X* and *Y* we obtain a new vector field *Z* defined by *<sup>Z</sup>* ≡ ∇*XY*. This derivative operator must be bilinear in its two arguments, ∇*X*(*<sup>Y</sup>* + *<sup>Z</sup>*) = ∇*XY* + ∇*XZ*, ∇*X*+*YZ* = ∇*XZ* + ∇*YZ*, must satisfy the chain rule ∇*X*(*f Y*)=(∇*<sup>X</sup> <sup>f</sup>*)*<sup>Y</sup>* + *<sup>f</sup>* ∇*XY*, with ∇*<sup>X</sup> <sup>f</sup>* = *<sup>X</sup>*[ *<sup>f</sup>* ], and must also behave as a natural directional derivative in the sense that ∇*f XY* = *<sup>f</sup>* ∇*XY* to guarantee that any two proportional vectors yield a result that points in the same direction. In a given coordinate basis, we have <sup>∇</sup>*XY* <sup>=</sup> *<sup>X</sup>µ*∇*e<sup>µ</sup>* (*Yνeν*) =

*∂x<sup>µ</sup>*

*dxµ*(*t*)

*dt* <sup>≡</sup> *<sup>X</sup>µ*(*t*) *<sup>∂</sup> <sup>f</sup>*

Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

*<sup>ǫ</sup>* (*x*) = *<sup>x</sup><sup>µ</sup>* <sup>+</sup> *<sup>ǫ</sup>Xµ*(*x*). This transformation also induces a

L*XY* = [*Xν∂νYµ*(*x*) − *<sup>Y</sup>ν∂νXµ*(*x*)]*e<sup>µ</sup>* ≡ [*X*,*Y*] . (2)

. If our manifold is *m*−dimensional, defining *m*<sup>3</sup> functions called

*<sup>∂</sup>x<sup>µ</sup>* , (1)

http://dx.doi.org/10.5772/51807

*<sup>∂</sup>x*˜*<sup>α</sup>* for the vector components.

*<sup>∂</sup>x*˜*<sup>α</sup> eµ*,

159

the curve *γ*(*t*) using the coordinates of the chart (U, *ϕ*) is given by

*dt* <sup>=</sup> *d f*(*x*(*t*))

which implies the well-known transformation law *V<sup>µ</sup>* = *V*˜ *<sup>α</sup> <sup>∂</sup>x<sup>µ</sup>*

*d f*(*ϕ* ◦ *γ*(*t*))

*<sup>p</sup>* of coordinates *<sup>x</sup><sup>µ</sup>* becomes *<sup>σ</sup><sup>µ</sup>*

vector fields is defined as

curve.

*X<sup>µ</sup>* 

*<sup>e</sup>µ*[*Yν*]*e<sup>ν</sup>* <sup>+</sup> *<sup>Y</sup>ν*∇*e<sup>µ</sup> <sup>e</sup><sup>ν</sup>*

In this chapter we explore in some detail the implications of relaxing the Riemannian condition on the geometry by allowing the connection to be determined from first principles, not by choice or convention. This approach, known as metric-affine or Palatini formalism [24], assumes that metric and connection are equally fundamental and independent geometrical entities. In consequence, any geometrical theory of gravity formulated in this approach must provide enough equations to determine the form of the metric and the connection (within the unavoidable indeterminacy imposed by the underlying gauge freedom). We derive and discuss the field equations of a rather general family of Palatini theories and then focus on two particular subfamilies which have attracted special attention in recent years, namely, *f*(*R*) and *f*(*R*, *Q*) theories. The interest in studying these particular theories lies in their ability to avoid (or soften in some cases) big bang and black hole singularities and their relation with recent approaches to quantum gravity. Here we will focus on the early-time cosmology of such theories.

The content is organized as follows. We begin by briefly reviewing in section 2 the basics of differentiable manifolds with affine and metric structures, to emphasize that metric and connection are equally fundamental and independent geometrical objects. In section 3 a derivation of the field equations for a generic action depending on the metric and the Riemann tensor is presented taking into account also the presence of torsion. In section 4 we discuss a particular family of Lagrangians of the form *<sup>f</sup>*(*R*, *<sup>R</sup>µνRµν*) in combination with perfect fluid matter, and prepare the notation and field equations needed to study the dynamics of those theories. We then focus on the early-time characteristics of isotropic and anisotropic homogeneous cosmologies 5 and show that nonsingular bouncing solutions exist for *f*(*R*) and *f*(*R*, *Q*) models (subsections 5.5 and 5.6, respectively). We conclude with a discussion of the results presented and point out some open questions that should be addressed in the future.

## **2. Differentiable manifolds, affine connections, and the metric**

In this section we quickly review some of the mathematical structures needed to construct a geometric theory of the gravitational interactions. The goal is to put forward that metric and connection are equally fundamental and independent geometrical entities, an aspect usually overlooked in the construction of phenomenological extensions of GR. We will thus be more sketchy than mathematically accurate. For a more exhaustive and precise discussion of these topics see your favorite book on differentiable manifolds (or, for instance, [20]).

In the geometric description of gravitational theories, one begins by identifying physical events with points on an n-dimensional manifold M. The next natural step is to provide this manifold with a differentiable structure. One then labels the points *p* ∈ M with a set of charts (U*i*, *<sup>ϕ</sup>i*), where the U*<sup>i</sup>* are subsets of M and *<sup>ϕ</sup><sup>i</sup>* are maps from U*<sup>i</sup>* to **<sup>R</sup>***<sup>n</sup>* (or an open subset of **<sup>R</sup>***n*) such that every *<sup>p</sup>* ∈ M lies in at least one of the charts (U*i*, *<sup>ϕ</sup>i*). If for any two charts (U*i*, *<sup>ϕ</sup>i*) and (U*j*, *<sup>ϕ</sup>j*) that overlap at some nonzero subset of points the map *<sup>ϕ</sup><sup>i</sup>* ◦ *<sup>ϕ</sup><sup>j</sup>* −1 is not just continuous but differentiable, then we say that M is a differentiable manifold. Since the Euclidean view of vectors as arrows connecting two points of the manifold is not valid in general, to get a consistent definition we need to introduce first the concept of curve and tangent vector to a curve at a point. We thus say that a smooth curve *γ*(*t*) in M is a differentiable map that to each point of a segment associates a point in M, *γ*(*t*) :

*t* ∈ [0, 1] → M. In a chart (U, *ϕ*), the points of the curve have the following coordinate representation: *x* = *ϕ*(*pt*) = *ϕ* ◦ *γ*(*t*). If we consider now a function *f* on M, where *f* is a map that to every *p* ∈ M assigns a real number (*f* : M → **R**), the rate of change of *f* along the curve *γ*(*t*) using the coordinates of the chart (U, *ϕ*) is given by

2 Open Questions in Cosmology

addressed in the future.

focus on the early-time cosmology of such theories.

this work.

more general frameworks are legitimate questions that should be explored by all available means (theoretical and experimental). These are some of the basic points to be addressed in

In this chapter we explore in some detail the implications of relaxing the Riemannian condition on the geometry by allowing the connection to be determined from first principles, not by choice or convention. This approach, known as metric-affine or Palatini formalism [24], assumes that metric and connection are equally fundamental and independent geometrical entities. In consequence, any geometrical theory of gravity formulated in this approach must provide enough equations to determine the form of the metric and the connection (within the unavoidable indeterminacy imposed by the underlying gauge freedom). We derive and discuss the field equations of a rather general family of Palatini theories and then focus on two particular subfamilies which have attracted special attention in recent years, namely, *f*(*R*) and *f*(*R*, *Q*) theories. The interest in studying these particular theories lies in their ability to avoid (or soften in some cases) big bang and black hole singularities and their relation with recent approaches to quantum gravity. Here we will

The content is organized as follows. We begin by briefly reviewing in section 2 the basics of differentiable manifolds with affine and metric structures, to emphasize that metric and connection are equally fundamental and independent geometrical objects. In section 3 a derivation of the field equations for a generic action depending on the metric and the Riemann tensor is presented taking into account also the presence of torsion. In section 4 we discuss a particular family of Lagrangians of the form *<sup>f</sup>*(*R*, *<sup>R</sup>µνRµν*) in combination with perfect fluid matter, and prepare the notation and field equations needed to study the dynamics of those theories. We then focus on the early-time characteristics of isotropic and anisotropic homogeneous cosmologies 5 and show that nonsingular bouncing solutions exist for *f*(*R*) and *f*(*R*, *Q*) models (subsections 5.5 and 5.6, respectively). We conclude with a discussion of the results presented and point out some open questions that should be

In this section we quickly review some of the mathematical structures needed to construct a geometric theory of the gravitational interactions. The goal is to put forward that metric and connection are equally fundamental and independent geometrical entities, an aspect usually overlooked in the construction of phenomenological extensions of GR. We will thus be more sketchy than mathematically accurate. For a more exhaustive and precise discussion of these

In the geometric description of gravitational theories, one begins by identifying physical events with points on an n-dimensional manifold M. The next natural step is to provide this manifold with a differentiable structure. One then labels the points *p* ∈ M with a set of charts (U*i*, *<sup>ϕ</sup>i*), where the U*<sup>i</sup>* are subsets of M and *<sup>ϕ</sup><sup>i</sup>* are maps from U*<sup>i</sup>* to **<sup>R</sup>***<sup>n</sup>* (or an open subset of **<sup>R</sup>***n*) such that every *<sup>p</sup>* ∈ M lies in at least one of the charts (U*i*, *<sup>ϕ</sup>i*). If for any two charts (U*i*, *<sup>ϕ</sup>i*) and (U*j*, *<sup>ϕ</sup>j*) that overlap at some nonzero subset of points the map *<sup>ϕ</sup><sup>i</sup>* ◦ *<sup>ϕ</sup><sup>j</sup>*

is not just continuous but differentiable, then we say that M is a differentiable manifold. Since the Euclidean view of vectors as arrows connecting two points of the manifold is not valid in general, to get a consistent definition we need to introduce first the concept of curve and tangent vector to a curve at a point. We thus say that a smooth curve *γ*(*t*) in M is a differentiable map that to each point of a segment associates a point in M, *γ*(*t*) :

−1

**2. Differentiable manifolds, affine connections, and the metric**

topics see your favorite book on differentiable manifolds (or, for instance, [20]).

$$\frac{df(\varphi \circ \gamma(t))}{dt} = \frac{df(\mathbf{x}(t))}{dt} = \frac{\partial f}{\partial \mathbf{x}^{\mu}} \frac{d\mathbf{x}^{\mu}(t)}{dt} \equiv \mathbf{X}^{\mu}(t) \frac{\partial f}{\partial \mathbf{x}^{\mu}} \,, \tag{1}$$

where we have defined the components of the tangent vector to the curve in this chart as *<sup>X</sup><sup>µ</sup>* ≡ *dxµ*(*t*)/*dt*. Vectors can thus be seen as differential operators *<sup>X</sup>* = *<sup>X</sup>µ∂µ* whose action on functions is of the form *<sup>X</sup>*[ *<sup>f</sup>* ] = *<sup>X</sup>µ∂µ <sup>f</sup>* , thus providing a natural notion of directional derivative for functions. The set {*e<sup>µ</sup>* ≡ *∂µ*} defines a (coordinate) basis of the tangent space of vectors at the point p, which we denote *Tp*M. Obviously, vectors exist without specifying the coordinates. Under changes of coordinates, we have *<sup>V</sup>* = *<sup>V</sup>µe<sup>µ</sup>* = *<sup>V</sup>*˜ *<sup>α</sup>e*˜*<sup>α</sup>* = *<sup>V</sup>*˜ *<sup>α</sup> <sup>∂</sup>x<sup>µ</sup> <sup>∂</sup>x*˜*<sup>α</sup> eµ*, which implies the well-known transformation law *V<sup>µ</sup>* = *V*˜ *<sup>α</sup> <sup>∂</sup>x<sup>µ</sup> <sup>∂</sup>x*˜*<sup>α</sup>* for the vector components. When a vector is assigned smoothly to each point of M, it is called a vector field over M. Each component of a vector field is thus a smooth function from M → **R**. Given a vector field *X*, an integral curve of *X* is defined as the curve whose tangent vector coincides with *X*. For infinitesimal displacements of magnitude *ǫ* in the direction of *X*, a given point *<sup>p</sup>* of coordinates *<sup>x</sup><sup>µ</sup>* becomes *<sup>σ</sup><sup>µ</sup> <sup>ǫ</sup>* (*x*) = *<sup>x</sup><sup>µ</sup>* <sup>+</sup> *<sup>ǫ</sup>Xµ*(*x*). This transformation also induces a correspondence between vectors of the tangent spaces *Tx*<sup>M</sup> and *<sup>T</sup>σǫ* (*x*)M. The effect of these transformations on a vector field *Y*(*x*) leads to the concept of Lie derivative, whose action on vector fields is defined as

$$\mathcal{L}\_X Y = \left[ X^\nu \partial\_\nu Y^\mu(\mathbf{x}) - Y^\nu \partial\_\nu X^\mu(\mathbf{x}) \right] e\_\mu \equiv \left[ X, Y \right] \,. \tag{2}$$

This derivative operator is independent of the choice of coordinates and follows naturally from the differential structure of the manifold. It satisfies a number of useful properties such as bilinearity in its two arguments, L*X*(*<sup>Y</sup>* + *<sup>Z</sup>*) = L*XY* + L*XZ*, L*X*+*YZ* = L*XZ* + L*YZ*, and the chain rule L*<sup>X</sup> f Y* = (L*<sup>X</sup> <sup>f</sup>*)*<sup>Y</sup>* + *<sup>f</sup>*L*XY*, with L*<sup>X</sup> <sup>f</sup>* = *<sup>X</sup>*[ *<sup>f</sup>* ]. Though the Lie derivative provides a natural *directional derivative* for functions, it does not work in the same way for vectors and tensors of higher rank. In fact, since the partial derivatives of the vector *X* appear explicitly in L*XY*, two vectors whose components at a given point have the same values but whose partial derivatives at the point differ do not yield a vector that points in the same direction, i.e., they are not proportional. Therefore, in order to introduce a proper notion of *directional derivative* for vectors and tensors, we need to introduce a new structure called **connection** which specifies how vectors (and tensors in general) are transported along a curve.

**Manifolds with a connection.** We are thus going to introduce a derivative operator, which we denote by ∇, such that given two vector fields *X* and *Y* we obtain a new vector field *Z* defined by *<sup>Z</sup>* ≡ ∇*XY*. This derivative operator must be bilinear in its two arguments, ∇*X*(*<sup>Y</sup>* + *<sup>Z</sup>*) = ∇*XY* + ∇*XZ*, ∇*X*+*YZ* = ∇*XZ* + ∇*YZ*, must satisfy the chain rule ∇*X*(*f Y*)=(∇*<sup>X</sup> <sup>f</sup>*)*<sup>Y</sup>* + *<sup>f</sup>* ∇*XY*, with ∇*<sup>X</sup> <sup>f</sup>* = *<sup>X</sup>*[ *<sup>f</sup>* ], and must also behave as a natural directional derivative in the sense that ∇*f XY* = *<sup>f</sup>* ∇*XY* to guarantee that any two proportional vectors yield a result that points in the same direction. In a given coordinate basis, we have <sup>∇</sup>*XY* <sup>=</sup> *<sup>X</sup>µ*∇*e<sup>µ</sup>* (*Yνeν*) = *X<sup>µ</sup> <sup>e</sup>µ*[*Yν*]*e<sup>ν</sup>* <sup>+</sup> *<sup>Y</sup>ν*∇*e<sup>µ</sup> <sup>e</sup><sup>ν</sup>* . If our manifold is *m*−dimensional, defining *m*<sup>3</sup> functions called connection coefficients Γ*<sup>λ</sup> µν* by <sup>∇</sup>*e<sup>µ</sup> <sup>e</sup><sup>ν</sup>* <sup>≡</sup> <sup>Γ</sup>*<sup>λ</sup> µνe<sup>λ</sup>* we find that the last requirement, <sup>∇</sup>*f XY* <sup>=</sup> *<sup>f</sup>* ∇*XY*, is naturally satisfied. We thus find that

$$\nabla\_X Y = X^\mu \left[ \frac{\partial Y^\lambda}{\partial x^\mu} + \Gamma^\lambda\_{\mu\nu} Y^\nu \right] e\_\lambda \,. \tag{3}$$

tensor provides a notion of distance between nearby points and allows, among other things, to determine lengths, angles, areas, and volumes of objects which are locally defined in space-time. Formally, a (pseudo-Riemannian) metric tensor is a symmetric bilinear form that at each *p* ∈ M satisfies *gp*(*U*, *V*) = *gp*(*V*, *U*) for any two vectors *U*, *V* ∈ *Tp*M and *gp*(*U*, *V*) = 0 for any *U* ∈ *Tp*M iff *V* = 0. The metric tensor allows to define an inner product between vectors and also gives rise to an isomorphism between *Tp*M and the dual

impose a particular relation between the metric and the connection by demanding that the scalar product of any two vectors which are parallel transported along any curve remains covariantly constant. This condition can be translated into1 <sup>∇</sup>*µgαβ* <sup>=</sup> 0, which implies that

*gσλ* = *gλρ*

From this definition it follows that when the torsion vanishes, the connection is symmetric

the associated geometry is Riemannian. It should be noted that though connections are not tensors, the difference between any two connections is a tensor. This, in particular, allowed us to construct the torsion tensor. With more generality, when the manifold is provided with

*µν* <sup>+</sup> *<sup>A</sup><sup>λ</sup>*

*µν* is a tensor (which needs not be symmetric in its lower indices). Therefore, Palatini

theories of gravity, in which metric and connection are regarded as independent fields, can

From the above quick review of the properties of differentiable manifolds with metric and affine structures, it is clear that metric and connection are equally fundamental and independent geometrical entities. In the construction of theories of gravity based on geometry, we will thus assume this independence and will require those theories to yield equations that allow to determine both the metric and the connection and the possible relations between them. For simplicity, we will assume that the matter is only coupled to the

*<sup>p</sup>*M. In a coordinate basis, it can be represented by *<sup>g</sup>* <sup>=</sup> *<sup>g</sup>µνdx<sup>µ</sup> dxν*,

*<sup>p</sup>*M. In manifolds with a metric, one can

http://dx.doi.org/10.5772/51807

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Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

*∂µgρν* + *∂ν <sup>g</sup>ρµ* − *∂ρ <sup>g</sup>µν* . (10)

*µν* <sup>=</sup> *<sup>L</sup><sup>λ</sup>*

*µν* , (12)

*µν* has been added to the

*µν*, we say that

*∂µgρν* + *∂ν <sup>g</sup>ρµ* − *∂ρ <sup>g</sup>µν* . (11)

space of one-forms *<sup>T</sup>*<sup>∗</sup>

*βσ* <sup>≡</sup> <sup>Γ</sup>*<sup>ρ</sup>*

2Γ*<sup>λ</sup>* (*µν*) <sup>+</sup> *Tρ*

a metric, any connection Γ*<sup>λ</sup>*

gravitational Lagrangian.

**3. Dynamics of Palatini theories**

<sup>1</sup> From now on we use the more standard notation <sup>∇</sup>*<sup>µ</sup>* ≡ ∇*<sup>e</sup><sup>µ</sup>*

where *A<sup>λ</sup>*

(recall that *T<sup>ρ</sup>*

where the differentials *dx<sup>µ</sup>* form a basis of *<sup>T</sup>*<sup>∗</sup>

*βσ* <sup>−</sup> <sup>Γ</sup>*<sup>ρ</sup> σβ*)

*νσgρµ* <sup>+</sup> *<sup>T</sup><sup>ρ</sup>*

*Lλ µν* <sup>≡</sup> *<sup>g</sup>λρ* 2 

*µσgρν*

and coincides with the Levi-Civita connection. In that case, when Γ*<sup>λ</sup>*

*µν* can be expressed as

Γ*λ µν* <sup>=</sup> *<sup>L</sup><sup>λ</sup>*

be seen as theories in which an additional rank-three tensor field *A<sup>λ</sup>*

From the right-hand side of this equation, one defines the Levi-Civita connection as

The connection coefficients specify how the basis vectors change from point to point and, in principle, can be arbitrarily defined. Under changes of coordinates, these coefficients transform as follows:

$$\nabla\_{\varepsilon\mu}e\_{\nu} \equiv \Gamma^{\lambda}\_{\mu\nu}e\_{\lambda} = \frac{\partial \mathfrak{x}^{\mu}}{\partial \mathbf{x}^{\mu}} \nabla\_{\mathcal{E}\_{k}} \left(\frac{\partial \mathfrak{x}^{\oint}}{\partial \mathbf{x}^{\nu}} \tilde{e}\_{\beta}\right) = \frac{\partial \mathfrak{x}^{\mu}}{\partial \mathbf{x}^{\mu}} \left[\frac{\partial \mathbf{x}^{\lambda}}{\partial \mathbf{x}^{\nu}} \frac{\partial^{2} \mathfrak{x}^{\gamma}}{\partial \mathbf{x}^{\lambda} \partial \mathbf{x}^{\nu}} + \frac{\partial \mathfrak{x}^{\oint}}{\partial \mathbf{x}^{\gamma}} \tilde{\Gamma}^{\gamma}\_{\mathbf{a}\beta}\right] \tilde{e}\_{\gamma} = \Gamma^{\lambda}\_{\mu\nu} \frac{\partial \mathfrak{x}^{\gamma}}{\partial \mathbf{x}^{\lambda}} \tilde{e}\_{\gamma} \quad (4)$$

which implies

$$
\Gamma^{\lambda}\_{\mu\nu} = \frac{\partial \mathfrak{x}^{\lambda}}{\partial \mathfrak{x}^{\gamma}} \frac{\partial \mathfrak{x}^{\alpha}}{\partial \mathfrak{x}^{\mu}} \frac{\partial \mathfrak{x}^{\beta}}{\partial \mathfrak{x}^{\nu}} \Gamma^{\gamma}\_{\alpha\beta} + \frac{\partial \mathfrak{x}^{\lambda}}{\partial \mathfrak{x}^{\gamma}} \frac{\partial^{2} \mathfrak{x}^{\gamma}}{\partial \mathfrak{x}^{\mu} \partial \mathfrak{x}^{\nu}} \ . \tag{5}
$$

This transformation law indicates that the connection coefficients do not transform as tensorial quantities. Therefore, the connection cannot have an intrinsic geometrical meaning as a measure of how much a manifold is curved. As intrinsic geometric objects, we can define the torsion tensor

$$T(X,Y) = \nabla\_X Y - \nabla\_Y X - [X,Y] \; \tag{6}$$

and the Riemann curvature tensor

$$R(X,Y)Z = \nabla\_X \nabla\_Y Z - \nabla\_Y \nabla\_X Z - \nabla\_{[X,Y]} Z \,. \tag{7}$$

In a coordinate basis, these tensors have the following components:

$$T(\mathfrak{e}\_{\mu}, \mathfrak{e}\_{\nu}) = \left(\Gamma\_{\mu\nu}^{\lambda} - \Gamma\_{\nu\mu}^{\lambda}\right) \mathfrak{e}\_{\lambda} \,. \tag{8}$$

$$\mathcal{R}(e\_{\mu}, e\_{\nu})e\_{\lambda} = \left[\partial\_{\mu}\Gamma^{\beta}\_{\nu\lambda} - \partial\_{\nu}\Gamma^{\beta}\_{\mu\lambda} + \Gamma^{\kappa}\_{\nu\lambda}\Gamma^{\gamma}\_{\mu\kappa} - \Gamma^{\kappa}\_{\mu\lambda}\Gamma^{\gamma}\_{\nu\kappa}\right]e\_{\gamma} \,. \tag{9}$$

With the introduction of the connection, one can define the notion of parallel transport. Given a curve *γ*(*t*) such that its tangent vector in a given chart has coordinates *X<sup>µ</sup>* = *dxµ*(*t*)/*dt*, we say that a vector *<sup>Y</sup>* is parallel transported along *<sup>γ</sup>*(*t*) if ∇*XY* = 0. In components, this equation reads *dY<sup>µ</sup> dt* <sup>+</sup> <sup>Γ</sup>*<sup>µ</sup> αβ dx<sup>α</sup>*(*t*) *dt <sup>Y</sup><sup>β</sup>* <sup>=</sup> 0, where *<sup>d</sup>*/*dt* <sup>≡</sup> *<sup>X</sup>µ∂µ*. Geodesics are defined as those curves which are parallel transported along themselves, namely, ∇*XX* = 0 or *dX<sup>µ</sup> dt* <sup>+</sup> <sup>Γ</sup>*<sup>µ</sup> αβXαX<sup>β</sup>* <sup>=</sup> 0.

**Manifolds with a metric.** So far we have been able to construct a number of geometrical objects such as directional derivatives of vectors and tensors in general, the torsion and Riemann tensors, geodesic curves, . . . without the need to introduce a metric tensor. A metric tensor provides a notion of distance between nearby points and allows, among other things, to determine lengths, angles, areas, and volumes of objects which are locally defined in space-time. Formally, a (pseudo-Riemannian) metric tensor is a symmetric bilinear form that at each *p* ∈ M satisfies *gp*(*U*, *V*) = *gp*(*V*, *U*) for any two vectors *U*, *V* ∈ *Tp*M and *gp*(*U*, *V*) = 0 for any *U* ∈ *Tp*M iff *V* = 0. The metric tensor allows to define an inner product between vectors and also gives rise to an isomorphism between *Tp*M and the dual space of one-forms *<sup>T</sup>*<sup>∗</sup> *<sup>p</sup>*M. In a coordinate basis, it can be represented by *<sup>g</sup>* <sup>=</sup> *<sup>g</sup>µνdx<sup>µ</sup> dxν*, where the differentials *dx<sup>µ</sup>* form a basis of *<sup>T</sup>*<sup>∗</sup> *<sup>p</sup>*M. In manifolds with a metric, one can impose a particular relation between the metric and the connection by demanding that the scalar product of any two vectors which are parallel transported along any curve remains covariantly constant. This condition can be translated into1 <sup>∇</sup>*µgαβ* <sup>=</sup> 0, which implies that (recall that *T<sup>ρ</sup> βσ* <sup>≡</sup> <sup>Γ</sup>*<sup>ρ</sup> βσ* <sup>−</sup> <sup>Γ</sup>*<sup>ρ</sup> σβ*)

$$2\Gamma^{\lambda}\_{(\mu\nu)} + \left(T^{\rho}\_{\nu\sigma}\mathcal{g}\_{\rho\mu} + T^{\rho}\_{\mu\sigma}\mathcal{g}\_{\rho\nu}\right)\mathcal{g}^{\sigma\lambda} = \mathcal{g}^{\lambda\rho}\left[\partial\_{\mu}\mathcal{g}\_{\rho\nu} + \partial\_{\nu}\mathcal{g}\_{\rho\mu} - \partial\_{\rho}\mathcal{g}\_{\mu\nu}\right].\tag{10}$$

From the right-hand side of this equation, one defines the Levi-Civita connection as

$$L^{\lambda}\_{\mu\nu} \equiv \frac{\mathcal{G}^{\lambda\rho}}{2} \left[ \partial\_{\mu} \mathcal{g}\_{\rho\nu} + \partial\_{\nu} \mathcal{g}\_{\rho\mu} - \partial\_{\rho} \mathcal{g}\_{\mu\nu} \right] \,. \tag{11}$$

From this definition it follows that when the torsion vanishes, the connection is symmetric and coincides with the Levi-Civita connection. In that case, when Γ*<sup>λ</sup> µν* <sup>=</sup> *<sup>L</sup><sup>λ</sup> µν*, we say that the associated geometry is Riemannian. It should be noted that though connections are not tensors, the difference between any two connections is a tensor. This, in particular, allowed us to construct the torsion tensor. With more generality, when the manifold is provided with a metric, any connection Γ*<sup>λ</sup> µν* can be expressed as

$$
\Gamma^{\lambda}\_{\mu\nu} = L^{\lambda}\_{\mu\nu} + A^{\lambda}\_{\mu\nu} \,. \tag{12}
$$

where *A<sup>λ</sup> µν* is a tensor (which needs not be symmetric in its lower indices). Therefore, Palatini theories of gravity, in which metric and connection are regarded as independent fields, can be seen as theories in which an additional rank-three tensor field *A<sup>λ</sup> µν* has been added to the gravitational Lagrangian.

#### **3. Dynamics of Palatini theories**

4 Open Questions in Cosmology

transform as follows:

*µνe<sup>λ</sup>* <sup>=</sup> *<sup>∂</sup>x*˜*<sup>α</sup>*

and the Riemann curvature tensor

*<sup>∂</sup>x<sup>µ</sup>* <sup>∇</sup>*e*˜*<sup>α</sup>*

 *∂x*˜*<sup>β</sup> <sup>∂</sup>x<sup>ν</sup> <sup>e</sup>*˜*<sup>β</sup>*

Γ*λ µν* <sup>=</sup> *<sup>∂</sup>x<sup>λ</sup> ∂x*˜*<sup>γ</sup>*

In a coordinate basis, these tensors have the following components:

 Γ*λ µν* <sup>−</sup> <sup>Γ</sup>*<sup>λ</sup> νµ* 

 *∂µ*Γ*<sup>β</sup>*

*νλ* <sup>−</sup> *∂ν*Γ*<sup>β</sup>*

With the introduction of the connection, one can define the notion of parallel transport. Given a curve *γ*(*t*) such that its tangent vector in a given chart has coordinates *X<sup>µ</sup>* = *dxµ*(*t*)/*dt*, we say that a vector *<sup>Y</sup>* is parallel transported along *<sup>γ</sup>*(*t*) if ∇*XY* = 0. In components,

as those curves which are parallel transported along themselves, namely, ∇*XX* = 0 or

**Manifolds with a metric.** So far we have been able to construct a number of geometrical objects such as directional derivatives of vectors and tensors in general, the torsion and Riemann tensors, geodesic curves, . . . without the need to introduce a metric tensor. A metric

*µλ* <sup>+</sup> <sup>Γ</sup>*<sup>κ</sup>*

*νλ*Γ*<sup>γ</sup> µκ* <sup>−</sup> <sup>Γ</sup>*<sup>κ</sup>*

*<sup>T</sup>*(*eµ*,*eν*) =

*<sup>R</sup>*(*eµ*,*eν*)*e<sup>λ</sup>* <sup>=</sup>

*dt* <sup>+</sup> <sup>Γ</sup>*<sup>µ</sup> αβ dx<sup>α</sup>*(*t*)

<sup>∇</sup>*e<sup>µ</sup> <sup>e</sup><sup>ν</sup>* <sup>≡</sup> <sup>Γ</sup>*<sup>λ</sup>*

which implies

the torsion tensor

this equation reads *dY<sup>µ</sup>*

*αβXαX<sup>β</sup>* <sup>=</sup> 0.

*dX<sup>µ</sup> dt* <sup>+</sup> <sup>Γ</sup>*<sup>µ</sup>*

connection coefficients Γ*<sup>λ</sup>*

*<sup>f</sup>* ∇*XY*, is naturally satisfied. We thus find that

*µν* by <sup>∇</sup>*e<sup>µ</sup> <sup>e</sup><sup>ν</sup>* <sup>≡</sup> <sup>Γ</sup>*<sup>λ</sup>*

∇*XY* = *<sup>X</sup><sup>µ</sup>*

<sup>=</sup> *<sup>∂</sup>x*˜*<sup>α</sup> ∂x<sup>µ</sup>*

*∂x*˜*<sup>α</sup> ∂x<sup>µ</sup>* *∂x*˜*<sup>β</sup> <sup>∂</sup>x<sup>ν</sup>* <sup>Γ</sup>˜ *<sup>γ</sup> αβ* <sup>+</sup>

This transformation law indicates that the connection coefficients do not transform as tensorial quantities. Therefore, the connection cannot have an intrinsic geometrical meaning as a measure of how much a manifold is curved. As intrinsic geometric objects, we can define

 *∂Y<sup>λ</sup> <sup>∂</sup>x<sup>µ</sup>* <sup>+</sup> <sup>Γ</sup>*<sup>λ</sup>*

The connection coefficients specify how the basis vectors change from point to point and, in principle, can be arbitrarily defined. Under changes of coordinates, these coefficients

> *∂x<sup>λ</sup> ∂x*˜*<sup>ν</sup>*

*µνY<sup>ν</sup>* 

*∂*2*x*˜*<sup>γ</sup> <sup>∂</sup>xλ∂x<sup>ν</sup>* <sup>+</sup>

> *∂x<sup>λ</sup> ∂x*˜*<sup>γ</sup>*

*∂*2*x*˜*<sup>γ</sup>*

*<sup>T</sup>*(*X*,*Y*) = ∇*XY* − ∇*YX* − [*X*,*Y*] , (6)

*e<sup>λ</sup>* , (8)

*e<sup>γ</sup>* . (9)

*µλ*Γ*<sup>γ</sup> νκ* 

*dt <sup>Y</sup><sup>β</sup>* <sup>=</sup> 0, where *<sup>d</sup>*/*dt* <sup>≡</sup> *<sup>X</sup>µ∂µ*. Geodesics are defined

*<sup>R</sup>*(*X*,*Y*)*<sup>Z</sup>* <sup>=</sup> <sup>∇</sup>*X*∇*YZ* − ∇*Y*∇*XZ* − ∇[*X*,*Y*]*<sup>Z</sup>* . (7)

*∂x*˜*<sup>β</sup> <sup>∂</sup>x<sup>ν</sup>* <sup>Γ</sup>˜ *<sup>γ</sup> αβ* 

*µνe<sup>λ</sup>* we find that the last requirement, <sup>∇</sup>*f XY* <sup>=</sup>

*e<sup>λ</sup>* . (3)

*<sup>e</sup>*˜*<sup>γ</sup>* = <sup>Γ</sup>*<sup>λ</sup> µν ∂x*˜*<sup>γ</sup>*

*<sup>∂</sup>xµ∂x<sup>ν</sup>* . (5)

*<sup>∂</sup>x<sup>λ</sup> <sup>e</sup>*˜*<sup>γ</sup>* , (4)

From the above quick review of the properties of differentiable manifolds with metric and affine structures, it is clear that metric and connection are equally fundamental and independent geometrical entities. In the construction of theories of gravity based on geometry, we will thus assume this independence and will require those theories to yield equations that allow to determine both the metric and the connection and the possible relations between them. For simplicity, we will assume that the matter is only coupled to the

<sup>1</sup> From now on we use the more standard notation <sup>∇</sup>*<sup>µ</sup>* ≡ ∇*<sup>e</sup><sup>µ</sup>*

metric (which is consistent with the experimental tests of the equivalence principle [39]) but will allow an independent connection to appear in the gravitational sector of the theory. As pointed out above, this is equivalent to having, besides the metric, a rank-three gravitational tensor field. From a geometric perspective, this possibility seems much more natural and fundamental than considering, for instance, scalar fields in the gravitational sector, though scalar-tensor theories have traditionally received much more attention in the literature.

We begin by deriving the field equations of Palatini theories in a very general case and then consider some simplifications to make contact with the literature. For a generic Palatini theory in which the connection appears through the Riemann tensor or contractions of it, the action can be written as follows [25]

$$S = \frac{1}{2\kappa^2} \int d^4x \sqrt{-g} f(g\_{\mu\nu}, \mathcal{R}^a{}\_{\beta\mu\nu}) + S\_m[g\_{\mu\nu}, \psi] \,, \tag{13}$$

where *S<sup>λ</sup>*

*I*<sup>Γ</sup> = *d*4*x*

where *<sup>J</sup><sup>µ</sup>* ≡ *<sup>P</sup><sup>α</sup>*

*∂ f*

*µν* <sup>≡</sup> (Γ*<sup>λ</sup>*

*<sup>∂</sup>R<sup>α</sup> βµν* . In order to put the *<sup>δ</sup>R<sup>α</sup>*

*µν* <sup>−</sup> <sup>Γ</sup>*<sup>λ</sup>*

−*gP<sup>α</sup>*

*βµνδ*Γ*<sup>α</sup>*

that (17) can be written as

*I*<sup>Γ</sup> = *d*4*x ∂µ*(

*<sup>δ</sup><sup>S</sup>* <sup>=</sup> <sup>1</sup> 2*κ*<sup>2</sup> *d*4*x* −*g*

> + − 1 √−*<sup>g</sup>* ∇*µ*

where *P<sup>α</sup>*

and ∇*<sup>µ</sup>*

the matter, and *H<sup>α</sup>*

(torsion) parts, Γ*<sup>α</sup>*

Using this result, (15) becomes

*βµν*∇*µδ*Γ*<sup>α</sup>*

*νβ* <sup>=</sup> *d*4*x* ∇*µ*(

*νβ*. Since, in general, <sup>∇</sup>*µ*(

−*g Jµ*) − *δ*Γ*<sup>α</sup>*

 *∂ f <sup>∂</sup>gµν* <sup>−</sup> *<sup>f</sup>*

We thus find that the field equations can be written as follows

*<sup>∂</sup>g*(*µν*) <sup>−</sup> *<sup>f</sup>*

√−*<sup>g</sup>* ∇*µ*

> 1 −*g δSm δ*Γ*<sup>α</sup> νβ*

*β*[*µν*] + *S<sup>ν</sup> σρP<sup>α</sup>*

<sup>−</sup>*gP<sup>α</sup>*

*<sup>κ</sup>*2*Tµν* <sup>=</sup> *<sup>∂</sup> <sup>f</sup>*

*νβ* = − 1

*βµν* − *<sup>P</sup><sup>α</sup>*

*νβ* = − √

*µν* <sup>=</sup> *<sup>C</sup><sup>α</sup>*

*µ*

*κ*2*H<sup>α</sup>*

**3.1. Example: f(R,Q) theories**

For simplicity, from now on we will assume that *H<sup>α</sup>*

*µν* <sup>+</sup> *<sup>S</sup><sup>α</sup>*

√−*<sup>g</sup>* <sup>−</sup> *<sup>S</sup><sup>α</sup>*

*νβ* = − 1

for specific choices of the Lagrangian *<sup>f</sup>*(*gµν*, *<sup>R</sup><sup>α</sup>*

√−*<sup>g</sup>* ∇*C µ*

*κ*2*H<sup>α</sup>*

*<sup>β</sup>*[*µν*] = (*P<sup>α</sup>*

√−*<sup>g</sup>* <sup>=</sup> <sup>∇</sup>*<sup>C</sup>*

*νµ*)/2 now represents the torsion tensor (note the additional <sup>1</sup>

*βµν* term in (15) in suitable form, we need to note that

−*g Jµ*) − *δ*Γ*<sup>α</sup>*

*βµν*

*σµP<sup>α</sup> β*[*µν*] 2

√−*g Jµ*) = *∂µ*(

<sup>−</sup>*gP<sup>α</sup>*

*<sup>δ</sup>gµν* + *∂µ*

*βσρ* + 2*S<sup>σ</sup>*

*νβ*∇*<sup>µ</sup>*

Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

− 2*S<sup>σ</sup>*

<sup>2</sup> *<sup>g</sup>µν* (20)

*βσρ* + 2*S<sup>σ</sup>*

represents the coupling of matter to the connection.

*µνA<sup>α</sup>* <sup>−</sup> *<sup>S</sup><sup>α</sup>*

*<sup>β</sup>*[*µν*] <sup>−</sup> *<sup>S</sup><sup>β</sup>*

*µλP<sup>α</sup>*

*βµν*). To make contact with the literature [26],

*σµP<sup>α</sup>*

*<sup>δ</sup>gµν* is the energy-momentum tensor of

*νβ* = 0. Eq. (21) can be put in a

*µνA<sup>α</sup>* <sup>=</sup> <sup>∇</sup>*<sup>C</sup>*

√−*g Jµ*) + <sup>2</sup>*S<sup>σ</sup>*

*σµ*<sup>−</sup>*gP<sup>α</sup>*

−*g Jµ* (19)

*νβ*

−*gδ*Γ*<sup>α</sup>*

<sup>−</sup>*gP<sup>α</sup>*

compared to our initial definition in Eq.(8)) From now on we will use the notation *P<sup>α</sup>*

*νβ* ∇*µ*

<sup>2</sup> *<sup>g</sup>µν*

<sup>−</sup>*gP<sup>α</sup>*

*βνµ*)/2, *<sup>T</sup>µν* = − <sup>√</sup>

<sup>−</sup>*gP<sup>α</sup>*

*β*[*µν*] + *S<sup>ν</sup> σρP<sup>α</sup>*

more convenient form if the connection is decomposed into its symmetric and antisymmetric

*µα*√−*<sup>g</sup>*. By doing this, (21) turns into

*β*[*µν*] + *S<sup>λ</sup> µαP<sup>λ</sup>*

Eqs. (20) and (22) can be used to write the field equations for the metric and the connection

*µν*, such that <sup>∇</sup>*µA<sup>ν</sup>* <sup>=</sup> *∂µA<sup>ν</sup>* <sup>−</sup> *<sup>C</sup><sup>α</sup>*

2 −*g δSm* <sup>2</sup> factor as

http://dx.doi.org/10.5772/51807

*βµν* , (17)

*σµ*√−*g Jµ*, we find

*βµν* . (18)

+ *δSm* .

*<sup>β</sup>*[*µν*] , (21)

*<sup>µ</sup> <sup>A</sup><sup>ν</sup>* <sup>−</sup> *<sup>S</sup><sup>α</sup>*

*<sup>λ</sup>*[*µν*] . (22)

*µνA<sup>α</sup>*

*βµν* ≡

163

where *Sm* is the matter action, *ψ* represents collectively the matter fields, *κ*<sup>2</sup> is a constant with suitable dimensions (if *f* = *R*, then *κ*<sup>2</sup> = 8*πG*), and

$$R^{a}{}\_{\beta\mu\nu} = \partial\_{\mu} \Gamma^{a}\_{\nu\beta} - \partial\_{\nu} \Gamma^{a}\_{\mu\beta} + \Gamma^{a}\_{\mu\lambda} \Gamma^{\lambda}\_{\nu\beta} - \Gamma^{a}\_{\nu\lambda} \Gamma^{\lambda}\_{\mu\beta} \tag{14}$$

represents the components of the Riemann tensor, the field strength of the connection Γ*<sup>α</sup> µβ*. Note that since the connection is determined dynamically, i.e., we assume independence between the metric and affine structures of the theory, we cannot assume any *a priori* symmetry in its lower indices. This means that in the variation of the action to obtain the field equations we must bear in mind that Γ*<sup>α</sup> βγ* �<sup>=</sup> <sup>Γ</sup>*<sup>α</sup> γβ*, i.e., we admit the possibility of nonvanishing torsion. It should be noted that in GR energy and momentum are the sources of curvature, while torsion is sourced by the spin of particles [14]. The fact that torsion is usually not considered in introductory courses on gravitation may be rooted in the educational tradition of this subject and the fact that the spin of particles was discovered many years after the original formulation of GR by Einstein. Another reason may be that the effects of torsion are very weak in general, except at very high densities, where the role of torsion becomes dominant and may even avoid the formation of singularities (see [30] for a recent discussion and earlier literature on the topic). For these reasons, and to motivate and facilitate the exploration of the effects of torsion in extensions of GR, our derivation of the field equations will be as general as possible (within reasonable limits). We will assume a symmetric metric tensor *<sup>g</sup>µν* = *<sup>g</sup>νµ* and the usual definitions for the Ricci tensor *<sup>R</sup>µν* ≡ *<sup>R</sup><sup>ρ</sup> µρν* and the Ricci scalar *<sup>R</sup>* ≡ *<sup>g</sup>µνRµν*. The variation of the action (13) with respect to the metric and the connection can be expressed as

$$\delta \mathcal{S} = \frac{1}{2\kappa^2} \int d^4x \sqrt{-g} \left[ \left( \frac{\partial f}{\partial g^{\mu \nu}} - \frac{f}{2} g\_{\mu \nu} \right) \delta g^{\mu \nu} + \frac{\partial f}{\partial R^a} \delta R^a\_{\ \beta \mu \nu} \right] + \delta \mathcal{S}\_m \,. \tag{15}$$

Straightforward manipulations show that *δR<sup>α</sup> βµν* can be written as

$$
\delta R^{a}{}\_{\beta\mu\nu} = \nabla\_{\mu} \left( \delta \Gamma^{a}\_{\nu\beta} \right) - \nabla\_{\nu} \left( \delta \Gamma^{a}\_{\mu\beta} \right) + 2S^{\lambda}\_{\mu\nu} \delta \Gamma^{a}\_{\lambda\beta} \, \, \, \, \tag{16}
$$

where *S<sup>λ</sup> µν* <sup>≡</sup> (Γ*<sup>λ</sup> µν* <sup>−</sup> <sup>Γ</sup>*<sup>λ</sup> νµ*)/2 now represents the torsion tensor (note the additional <sup>1</sup> <sup>2</sup> factor as compared to our initial definition in Eq.(8)) From now on we will use the notation *P<sup>α</sup> βµν* ≡ *∂ f <sup>∂</sup>R<sup>α</sup> βµν* . In order to put the *<sup>δ</sup>R<sup>α</sup> βµν* term in (15) in suitable form, we need to note that

$$I\_{\Gamma} = \int d^4 \mathbf{x} \sqrt{-g} P\_{\mathbf{a}} \, {}^{\mathsf{f}\mu\nu} \nabla\_{\mu} \delta \Gamma\_{\nu\hat{\boldsymbol{\beta}}}^{\mathbf{a}} = \int d^4 \mathbf{x} \left[ \nabla\_{\mu} (\sqrt{-\mathbf{g}} \mathbf{J}^{\mu}) - \delta \Gamma\_{\nu\hat{\mathbf{g}}}^{\mathbf{a}} \nabla\_{\mu} \left( \sqrt{-\mathbf{g}} P\_{\mathbf{a}} {}^{\mathbf{f}} \mathbf{\hat{n}} \right) \right] \,, \tag{17}$$

where *<sup>J</sup><sup>µ</sup>* ≡ *<sup>P</sup><sup>α</sup> βµνδ*Γ*<sup>α</sup> νβ*. Since, in general, <sup>∇</sup>*µ*( √−*g Jµ*) = *∂µ*( √−*g Jµ*) + <sup>2</sup>*S<sup>σ</sup> σµ*√−*g Jµ*, we find that (17) can be written as

$$I\_{\Gamma} = \int d^4 \mathbf{x} \left[ \partial\_{\mu} (\sqrt{-g} I^{\mu}) - \delta \Gamma^{\mu}\_{\nu \oint} \left\{ \nabla\_{\mu} \left( \sqrt{-g} P\_{a}{}^{\beta \mu \nu} \right) - 2 S^{\sigma}\_{\sigma \mu} \sqrt{-g} P\_{a}{}^{\beta \mu \nu} \right\} \right] \,. \tag{18}$$

Using this result, (15) becomes

6 Open Questions in Cosmology

action can be written as follows [25]

*<sup>S</sup>* <sup>=</sup> <sup>1</sup> 2*κ*<sup>2</sup> *d*4*x*

suitable dimensions (if *f* = *R*, then *κ*<sup>2</sup> = 8*πG*), and

*Rα*

the field equations we must bear in mind that Γ*<sup>α</sup>*

and the connection can be expressed as

Straightforward manipulations show that *δR<sup>α</sup>*

*δR<sup>α</sup>*

*βµν* <sup>=</sup> <sup>∇</sup>*<sup>µ</sup>*

*<sup>δ</sup><sup>S</sup>* <sup>=</sup> <sup>1</sup> 2*κ*<sup>2</sup> *d*4*x* −*g*

*βµν* <sup>=</sup> *∂µ*Γ*<sup>α</sup>*

metric (which is consistent with the experimental tests of the equivalence principle [39]) but will allow an independent connection to appear in the gravitational sector of the theory. As pointed out above, this is equivalent to having, besides the metric, a rank-three gravitational tensor field. From a geometric perspective, this possibility seems much more natural and fundamental than considering, for instance, scalar fields in the gravitational sector, though scalar-tensor theories have traditionally received much more attention in the literature.

We begin by deriving the field equations of Palatini theories in a very general case and then consider some simplifications to make contact with the literature. For a generic Palatini theory in which the connection appears through the Riemann tensor or contractions of it, the

−*g f*(*gµν*, *<sup>R</sup><sup>α</sup>*

*νβ* <sup>−</sup> *∂ν*Γ*<sup>α</sup>*

where *Sm* is the matter action, *ψ* represents collectively the matter fields, *κ*<sup>2</sup> is a constant with

represents the components of the Riemann tensor, the field strength of the connection Γ*<sup>α</sup>*

Note that since the connection is determined dynamically, i.e., we assume independence between the metric and affine structures of the theory, we cannot assume any *a priori* symmetry in its lower indices. This means that in the variation of the action to obtain

of nonvanishing torsion. It should be noted that in GR energy and momentum are the sources of curvature, while torsion is sourced by the spin of particles [14]. The fact that torsion is usually not considered in introductory courses on gravitation may be rooted in the educational tradition of this subject and the fact that the spin of particles was discovered many years after the original formulation of GR by Einstein. Another reason may be that the effects of torsion are very weak in general, except at very high densities, where the role of torsion becomes dominant and may even avoid the formation of singularities (see [30] for a recent discussion and earlier literature on the topic). For these reasons, and to motivate and facilitate the exploration of the effects of torsion in extensions of GR, our derivation of the field equations will be as general as possible (within reasonable limits). We will assume a symmetric metric tensor *<sup>g</sup>µν* = *<sup>g</sup>νµ* and the usual definitions for the Ricci tensor *<sup>R</sup>µν* ≡ *<sup>R</sup><sup>ρ</sup>*

and the Ricci scalar *<sup>R</sup>* ≡ *<sup>g</sup>µνRµν*. The variation of the action (13) with respect to the metric

<sup>2</sup> *<sup>g</sup>µν* 

*δgµν* +

*βµν* can be written as

*∂ f ∂Rαβµν* *δR<sup>α</sup> βµν* 

*<sup>∂</sup> <sup>f</sup>*

 *δ*Γ*<sup>α</sup> νβ* − ∇*<sup>ν</sup> δ*Γ*<sup>α</sup> µβ* + 2*S<sup>λ</sup> µνδ*Γ*<sup>α</sup>*

*<sup>∂</sup>gµν* <sup>−</sup> *<sup>f</sup>*

*µβ* <sup>+</sup> <sup>Γ</sup>*<sup>α</sup>*

*µλ*Γ*<sup>λ</sup>*

*βγ* �<sup>=</sup> <sup>Γ</sup>*<sup>α</sup>*

*νβ* <sup>−</sup> <sup>Γ</sup>*<sup>α</sup>*

*νλ*Γ*<sup>λ</sup>*

*βµν*) + *Sm*[*gµν*, *<sup>ψ</sup>*] , (13)

*µβ* (14)

*γβ*, i.e., we admit the possibility

*µβ*.

*µρν*

+ *δSm* . (15)

*λβ* , (16)

$$\delta S = \frac{1}{2\kappa^2} \int d^4 \mathbf{x} \left[ \sqrt{-\mathbf{g}} \left( \frac{\partial f}{\partial \mathbf{g}^{\mu \nu}} - \frac{f}{2} \mathbf{g}\_{\mu \nu} \right) \delta \mathbf{g}^{\mu \nu} + \partial\_{\mu} \left( \sqrt{-\mathbf{g}} I^{\mu} \right) \tag{19}$$

$$+ \left\{ -\frac{1}{\sqrt{-\mathbf{g}}} \nabla\_{\mu} \left( \sqrt{-\mathbf{g}} P\_{\mathbf{a}} \delta^{[\mu \nu]} \right) + S\_{\sigma \rho}^{\nu} P\_{\mathbf{a}} \delta^{\sigma \rho} + 2 S\_{\upsilon \mu}^{\sigma} P\_{\mathbf{a}} \delta^{[\mu \nu]} \right\} 2 \sqrt{-\mathbf{g}} \delta \Gamma\_{\upsilon \beta}^{\mu} \right\} + \delta \mathbf{S}\_m \ .$$

We thus find that the field equations can be written as follows

$$
\kappa^2 T\_{\mu\nu} = \frac{\partial f}{\partial g^{(\mu\nu)}} - \frac{f}{2} g\_{\mu\nu} \tag{20}
$$

$$\kappa^2 H\_a^{\ v \beta} = -\frac{1}{\sqrt{-g}} \nabla\_\mu \left( \sqrt{-g} P\_a^{\beta[\mu\nu]} \right) + S\_{\sigma \rho}^{\nu} P\_a^{\beta \sigma \rho} + 2 S\_{\sigma \mu}^{\sigma} P\_a^{\beta[\mu\nu]} \, , \tag{21}$$

where *P<sup>α</sup> <sup>β</sup>*[*µν*] = (*P<sup>α</sup> βµν* − *<sup>P</sup><sup>α</sup> βνµ*)/2, *<sup>T</sup>µν* = − <sup>√</sup> 2 −*g δSm <sup>δ</sup>gµν* is the energy-momentum tensor of the matter, and *H<sup>α</sup> νβ* = − √ 1 −*g δSm δ*Γ*<sup>α</sup> νβ* represents the coupling of matter to the connection. For simplicity, from now on we will assume that *H<sup>α</sup> νβ* = 0. Eq. (21) can be put in a more convenient form if the connection is decomposed into its symmetric and antisymmetric (torsion) parts, Γ*<sup>α</sup> µν* <sup>=</sup> *<sup>C</sup><sup>α</sup> µν* <sup>+</sup> *<sup>S</sup><sup>α</sup> µν*, such that <sup>∇</sup>*µA<sup>ν</sup>* <sup>=</sup> *∂µA<sup>ν</sup>* <sup>−</sup> *<sup>C</sup><sup>α</sup> µνA<sup>α</sup>* <sup>−</sup> *<sup>S</sup><sup>α</sup> µνA<sup>α</sup>* <sup>=</sup> <sup>∇</sup>*<sup>C</sup> <sup>µ</sup> <sup>A</sup><sup>ν</sup>* <sup>−</sup> *<sup>S</sup><sup>α</sup> µνA<sup>α</sup>* and ∇*<sup>µ</sup>* √−*<sup>g</sup>* <sup>=</sup> <sup>∇</sup>*<sup>C</sup> µ* √−*<sup>g</sup>* <sup>−</sup> *<sup>S</sup><sup>α</sup> µα*√−*<sup>g</sup>*. By doing this, (21) turns into

$$\kappa^2 H\_\hbar{}^{\nu\beta} = -\frac{1}{\sqrt{-g}} \nabla\_\mu^\mathbb{C} \left( \sqrt{-g} P\_a{}^{\beta[\mu\nu]} \right) + \mathcal{S}\_{\mu a}^\lambda P\_\lambda{}^{\beta[\mu\nu]} - \mathcal{S}\_{\mu\lambda}^\emptyset P\_a{}^{\lambda[\mu\nu]} \,. \tag{22}$$

#### **3.1. Example: f(R,Q) theories**

Eqs. (20) and (22) can be used to write the field equations for the metric and the connection for specific choices of the Lagrangian *<sup>f</sup>*(*gµν*, *<sup>R</sup><sup>α</sup> βµν*). To make contact with the literature [26], [5] ,[23], we now focus on the case *<sup>f</sup>*(*R*, *<sup>Q</sup>*) = *<sup>f</sup>*(*gµνRµν*, *<sup>g</sup>µν <sup>g</sup>αβRµαRνβ*). For this family of Lagrangians, we obtain

$$P\_a{}^{\beta\mu\nu} = \delta\_a{}^{\mu}M^{\beta\nu} = \delta\_a{}^{\mu}\left(f\_R g^{\beta\nu} + 2f\_Q R^{\beta\nu}\right) \,, \tag{23}$$

In the recent literature on Palatini theories, only the torsionless case has been studied in detail. When torsion is considered in *f*(*R*) theories, Eqs. (30) and (31) recover the results presented in [24]. In general, those equations put forward that when the traceless torsion

of these theories, therefore, can be studied in different levels of complexity. The simplest case

When the torsion is set to zero, it can be shown [32], [12] that the vanishing of *R*[*µν*] guarantees the existence of a volume element that is covariantly conserved by Γ*<sup>α</sup>*

rank-two tensor that defines that volume element must be a solution of (30), which in this

Note that here *<sup>R</sup>βν*(Γ) is symmetric because we are taking *<sup>R</sup>*[*µν*](Γ) = 0. To obtain the solution

<sup>2</sup> *<sup>g</sup>µν* <sup>+</sup> <sup>2</sup> *fQRµαR<sup>α</sup>*

*<sup>ν</sup>* + <sup>2</sup> *fQB<sup>µ</sup>*

imply that *B*ˆ is an algebraic function of the components of the stress-energy tensor *T<sup>µ</sup>*

*B*ˆ = *B*ˆ(*T*ˆ). This relation is very important because it allows to express (32) in the form

*fRδ<sup>ν</sup>*

connection, therefore, can be obtained by elementary algebraic manipulations [23]. To do it,

*<sup>λ</sup>* <sup>+</sup> <sup>2</sup> *fQB<sup>λ</sup>*

*∂βhργ* <sup>+</sup> *∂γhρβ* <sup>−</sup> *∂ρhβγ*

*<sup>α</sup>B<sup>α</sup>*

*<sup>ν</sup>* = *<sup>κ</sup>*2*T<sup>µ</sup>*

*<sup>ν</sup>* ≡ *<sup>R</sup>µαgαν*. This equation can be seen as a second-order algebraic

*<sup>ν</sup>* <sup>≡</sup> *<sup>B</sup><sup>µ</sup>*

*ν*

√−*hhβν*

*<sup>ν</sup>* are functions of the stress-energy tensor of the matter. The

*fRδ<sup>ν</sup>*

*fRgβν* + <sup>2</sup> *fQRβν*(Γ)

*αλ* vanishes, the symmetric and antisymmetric parts of *<sup>M</sup>βν* decouple. The dynamics

*αλ* and *R*[*µν*] to zero. A more detailed discussion

Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

= 0 . (32)

*<sup>ν</sup>* = *<sup>κ</sup>*2*Tµν* , (33)

*αλ* and *R*[*µν*] set to zero),

http://dx.doi.org/10.5772/51807

*<sup>ν</sup>* , (34)

*<sup>ν</sup>*. The solutions to this equation

= 0 , (35)

= 0, and implies that Γ*<sup>α</sup>*

*ν*

. (36)

<sup>=</sup> √−*hhβν*,

*βν* is

*<sup>λ</sup>* <sup>+</sup> <sup>2</sup> *fQB<sup>λ</sup>*

*µν*. The

165

*<sup>ν</sup>*, i.e.,

tensor *S*˜*<sup>ν</sup>*

case takes the form

will be studied here and consists on setting *S<sup>ν</sup>*

**3.2. Volume-invariant and torsionless** *f*(*R*, *Q*)

∇<sup>Γ</sup> *α* −*g* 

of (32), we first consider (20) particularized to our theory (with *S<sup>ν</sup>*

*fRRµν* <sup>−</sup> *<sup>f</sup>*

*fRB<sup>µ</sup>*

equation for the matrix *<sup>B</sup>*ˆ, whose components are [*B*ˆ]*<sup>µ</sup>*

∇<sup>Γ</sup> *α*

which turns (35) into the well-known equation ∇*<sup>µ</sup>*

given by the Christoffel symbols of the tensor *hµν*, i.e.,

Γ*α βγ* <sup>=</sup> *<sup>h</sup>αρ* 2 

one defines a rank-two symmetric tensor *<sup>h</sup>µν* such that √−*ggβλ*

*<sup>ν</sup>* − *f* 2 *δµ*

−*ggβλ*

of the other cases can be found in [28].

and rewrite it in the following form

where we have defined *B<sup>µ</sup>*

where now *fR*, *fQ* and *B<sup>α</sup>*

where *fX* = *<sup>∂</sup><sup>X</sup> <sup>f</sup>* . Inserting this expression in (22) and tracing over *<sup>α</sup>* and *<sup>ν</sup>*, we find that ∇*C λ* [ √−*gMβλ*]=(<sup>2</sup> √−*<sup>g</sup>*/3)[*S<sup>σ</sup> λσMβλ* + (3/2)*S<sup>β</sup> λµMλµ*]. Using this result, the connection equation can be put as follows

$$\frac{1}{\sqrt{-g}}\nabla\_{\mathbf{a}}^{\mathbb{C}}\left[\sqrt{-g}M^{\mathbb{B}\nu}\right] = S\_{\mathbf{a}\lambda}^{\nu}M^{\mathbb{B}\lambda} - S\_{\beta\lambda}^{\nu}M^{\lambda\nu} - S\_{\mathbf{a}\lambda}^{\lambda}M^{\mathbb{B}\nu} + \frac{2}{3}\delta\_{\mathbf{a}}^{\nu}S\_{\lambda\sigma}^{\sigma}M^{\mathbb{B}\lambda} \tag{24}$$

The symmetric and antisymmetric combinations of this equation lead, respectively, to

$$\frac{1}{\sqrt{-\mathcal{g}}}\nabla\_{\mathfrak{a}}^{\mathbb{C}}\left[\sqrt{-\mathcal{g}}M^{\{\mathfrak{b}\}}\right] = S\_{\mathfrak{a}\lambda}^{\mathbb{V}}M^{\{\mathfrak{b}\lambda\}} - S\_{\mathfrak{a}\lambda}^{\mathcal{G}}M^{\{\mathfrak{v}\lambda\}} - S\_{\mathfrak{a}\lambda}^{\lambda}M^{\{\mathfrak{b}\}} + \frac{S\_{\lambda\sigma}^{\Gamma}}{3}\left(\delta\_{\mathfrak{a}}^{\nu}M^{\{\mathfrak{b}\lambda} + \delta\_{\mathfrak{a}}^{\mathcal{G}}M^{\{\lambda\}}\right) \text{(25)}$$

$$\frac{1}{\sqrt{-\mathcal{g}}}\nabla\_{\mathfrak{a}}^{\mathbb{C}}\left[\sqrt{-\mathcal{g}}M^{\{\mathfrak{b}\}}\right] = S\_{\mathfrak{a}\lambda}^{\nu}M^{\{\mathfrak{b}\lambda\}} - S\_{\mathfrak{a}\lambda}^{\mathcal{G}}M^{\{\nu\lambda\}} - S\_{\mathfrak{a}\lambda}^{\lambda}M^{\{\mathfrak{b}\nu\}} + \frac{S\_{\lambda\sigma}^{\sigma}}{3}\left(\delta\_{\mathfrak{a}}^{\nu}M^{\{\mathfrak{b}\lambda} - \delta\_{\mathfrak{a}}^{\mathcal{G}}M^{\{\lambda\}}\right) \text{ (26)}$$

Important simplifications can be achieved considering the new variables

Γ˜ *λ µν* <sup>=</sup> <sup>Γ</sup>*<sup>λ</sup> µν* <sup>+</sup> *αδ<sup>λ</sup> <sup>ν</sup> <sup>S</sup><sup>σ</sup> σµ* , (27)

and taking the parameter *α* = 2/3, which implies that *S*˜*<sup>λ</sup> µν* <sup>≡</sup> <sup>Γ</sup>˜ *<sup>λ</sup>* [*µν*] is such that *<sup>S</sup>*˜*<sup>σ</sup> σν* <sup>=</sup> 0. The symmetric and antisymmetric parts of Γ˜ *<sup>λ</sup> µν* are related to those of <sup>Γ</sup>*<sup>λ</sup> µν* by

$$\tilde{\mathcal{C}}\_{\mu\nu}^{\lambda} = \mathcal{C}\_{\mu\nu}^{\lambda} + \frac{1}{3} \left( \delta\_{\nu}^{\lambda} S\_{\sigma\mu}^{\sigma} + \delta\_{\mu}^{\lambda} S\_{\sigma\nu}^{\sigma} \right) \tag{28}$$

$$\tilde{S}^{\lambda}\_{\mu\nu} = S^{\lambda}\_{\mu\nu} + \frac{1}{3} \left( \delta^{\lambda}\_{\nu} S^{\sigma}\_{\sigma\mu} - \delta^{\lambda}\_{\mu} S^{\sigma}\_{\sigma\nu} \right) \tag{29}$$

Using these variables, Eqs. (25) and (26) take the following compact form

$$\frac{1}{\sqrt{-\mathcal{g}}} \nabla\_{\mathfrak{a}}^{\tilde{\mathsf{C}}} \left[ \sqrt{-\mathcal{g}} \mathcal{M}^{(\beta \nu)} \right] = \left[ \tilde{\mathcal{S}}\_{a\lambda}^{\nu} \mathcal{g}^{\beta \mathbf{x}} + \tilde{\mathcal{S}}\_{a\lambda}^{\beta} \mathcal{g}^{\nu \mathbf{x}} \right] \mathcal{g}^{\lambda \rho} \mathcal{M}\_{[\mathbf{x} \rho]} \tag{30}$$

$$\frac{1}{\sqrt{-\mathcal{g}}} \nabla\_{\mathfrak{a}}^{\mathbb{C}} \left[ \sqrt{-\mathcal{g}} \mathcal{M}^{[\mathcal{\theta}\nu]} \right] = \left[ \tilde{\mathcal{S}}\_{\mathfrak{a}\lambda}^{\nu} \mathcal{g}^{\mathbb{R}\mathfrak{x}} - \tilde{\mathcal{S}}\_{\mathfrak{a}\lambda}^{\mathbb{B}} \mathcal{g}^{\mathbb{V}\mathfrak{x}} \right] \mathcal{g}^{\lambda\rho} \mathcal{M}\_{(\mathsf{x}\rho)} \,. \tag{31}$$

In these equations, *<sup>M</sup>*(*βν*) <sup>=</sup> *fRgβν* <sup>+</sup> <sup>2</sup> *fQR*(*βν*)(Γ), and *<sup>M</sup>*[*βν*] <sup>=</sup> <sup>2</sup> *fQR*[*βν*] (Γ), where *<sup>R</sup>*(*βν*)(Γ) = *<sup>R</sup>*(*βν*)(Γ˜) and *<sup>R</sup>*[*βν*](Γ) = *<sup>R</sup>*[*βν*](Γ˜) + <sup>2</sup> 3 *∂βS<sup>σ</sup> σν* <sup>−</sup> *∂νS<sup>σ</sup> σβ* .

In the recent literature on Palatini theories, only the torsionless case has been studied in detail. When torsion is considered in *f*(*R*) theories, Eqs. (30) and (31) recover the results presented in [24]. In general, those equations put forward that when the traceless torsion tensor *S*˜*<sup>ν</sup> αλ* vanishes, the symmetric and antisymmetric parts of *<sup>M</sup>βν* decouple. The dynamics of these theories, therefore, can be studied in different levels of complexity. The simplest case will be studied here and consists on setting *S<sup>ν</sup> αλ* and *R*[*µν*] to zero. A more detailed discussion of the other cases can be found in [28].

## **3.2. Volume-invariant and torsionless** *f*(*R*, *Q*)

8 Open Questions in Cosmology

Lagrangians, we obtain

√−*gMβλ*]=(<sup>2</sup>

equation can be put as follows

1 √−*<sup>g</sup>* ∇*C α*

<sup>−</sup>*gM*(*βν*)

symmetric and antisymmetric parts of Γ˜ *<sup>λ</sup>*

1 √−*<sup>g</sup>* ∇*C*˜ *α*

1 √−*<sup>g</sup>* ∇*C*˜ *α*

*<sup>R</sup>*(*βν*)(Γ) = *<sup>R</sup>*(*βν*)(Γ˜) and *<sup>R</sup>*[*βν*](Γ) = *<sup>R</sup>*[*βν*](Γ˜) + <sup>2</sup>

<sup>−</sup>*gM*[*βν*]

∇*C λ* [

> 1 √−*<sup>g</sup>* ∇*C α*

1 √−*<sup>g</sup>* ∇*C α*

*Pα*

√−*<sup>g</sup>*/3)[*S<sup>σ</sup>*

−*gMβν*

 = *S<sup>ν</sup>*

and taking the parameter *α* = 2/3, which implies that *S*˜*<sup>λ</sup>*

*C*˜*λ µν* <sup>=</sup> *<sup>C</sup><sup>λ</sup>*

*S*˜*λ µν* <sup>=</sup> *<sup>S</sup><sup>λ</sup>*

 = *S<sup>ν</sup>*

*βµν* = *δα*

[5] ,[23], we now focus on the case *<sup>f</sup>*(*R*, *<sup>Q</sup>*) = *<sup>f</sup>*(*gµνRµν*, *<sup>g</sup>µν <sup>g</sup>αβRµαRνβ*). For this family of

*µ*

where *fX* = *<sup>∂</sup><sup>X</sup> <sup>f</sup>* . Inserting this expression in (22) and tracing over *<sup>α</sup>* and *<sup>ν</sup>*, we find that

*fRgβν* <sup>+</sup> <sup>2</sup> *fQRβν*

*βλMλν* <sup>−</sup> *<sup>S</sup><sup>λ</sup>*

, (23)

*λσMβλ* (24)

*β <sup>α</sup> <sup>M</sup>νλ*

*β <sup>α</sup> <sup>M</sup>νλ* (25)

.(26)

*σν* <sup>=</sup> 0. The

(Γ), where

(28)

(29)

*<sup>α</sup>Mβλ* <sup>+</sup> *<sup>δ</sup>*

*<sup>α</sup>Mβλ* <sup>−</sup> *<sup>δ</sup>*

*σµ* , (27)

[*µν*] is such that *<sup>S</sup>*˜*<sup>σ</sup>*

*<sup>g</sup>λρM*[*κρ*] (30)

*<sup>g</sup>λρM*(*κρ*) . (31)

*µν* by

*λµMλµ*]. Using this result, the connection

2 3 *δν αS<sup>σ</sup>*

*αλMβν* <sup>+</sup>

*Sσ λσ* 3 *δν*

*Sσ λσ* 3 *δν*

*αλM*(*βν*) <sup>+</sup>

*αλM*[*βν*] <sup>+</sup>

*µν* <sup>≡</sup> <sup>Γ</sup>˜ *<sup>λ</sup>*

*αλgνκ*

*αλgνκ*

*σν* <sup>−</sup> *∂νS<sup>σ</sup>*

*σβ* .

*<sup>µ</sup>Mβν* = *δα*

*λσMβλ* + (3/2)*S<sup>β</sup>*

*αλMβλ* <sup>−</sup> *<sup>S</sup><sup>ν</sup>*

The symmetric and antisymmetric combinations of this equation lead, respectively, to

*αλM*[*νλ*] <sup>−</sup> *<sup>S</sup><sup>λ</sup>*

*αλM*(*νλ*) <sup>−</sup> *<sup>S</sup><sup>λ</sup>*

*µν* <sup>+</sup> *αδ<sup>λ</sup>*

*<sup>ν</sup> <sup>S</sup><sup>σ</sup>*

*µν* are related to those of <sup>Γ</sup>*<sup>λ</sup>*

*αλgβκ* <sup>+</sup> *<sup>S</sup>*˜*<sup>β</sup>*

*αλgβκ* <sup>−</sup> *<sup>S</sup>*˜*<sup>β</sup>*

3 *∂βS<sup>σ</sup>*

= *S<sup>ν</sup>*

*αλM*[*βλ*] <sup>−</sup> *<sup>S</sup><sup>β</sup>*

*αλM*(*βλ*) <sup>−</sup> *<sup>S</sup><sup>β</sup>*

Important simplifications can be achieved considering the new variables

Γ˜ *λ µν* <sup>=</sup> <sup>Γ</sup>*<sup>λ</sup>*

> *µν* <sup>+</sup> 1 3 *δλ <sup>ν</sup> <sup>S</sup><sup>σ</sup> σµ* <sup>+</sup> *<sup>δ</sup><sup>λ</sup> <sup>µ</sup> <sup>S</sup><sup>σ</sup> σν*

> *µν* <sup>+</sup> 1 3 *δλ <sup>ν</sup> <sup>S</sup><sup>σ</sup> σµ* <sup>−</sup> *<sup>δ</sup><sup>λ</sup> <sup>µ</sup> <sup>S</sup><sup>σ</sup> σν*

 = *S*˜*ν*

 = *S*˜*ν*

In these equations, *<sup>M</sup>*(*βν*) <sup>=</sup> *fRgβν* <sup>+</sup> <sup>2</sup> *fQR*(*βν*)(Γ), and *<sup>M</sup>*[*βν*] <sup>=</sup> <sup>2</sup> *fQR*[*βν*]

Using these variables, Eqs. (25) and (26) take the following compact form

<sup>−</sup>*gM*(*βν*)

<sup>−</sup>*gM*[*βν*]

When the torsion is set to zero, it can be shown [32], [12] that the vanishing of *R*[*µν*] guarantees the existence of a volume element that is covariantly conserved by Γ*<sup>α</sup> µν*. The rank-two tensor that defines that volume element must be a solution of (30), which in this case takes the form

$$\nabla\_{\mathfrak{a}}^{\Gamma} \left[ \sqrt{-g} \left( f\_{\mathbb{R}} \mathbf{g}^{\mathbb{R}\nu} + 2 f\_{\mathbb{Q}} \mathbf{R}^{\mathbb{R}\nu} (\Gamma) \right) \right] = \mathbf{0} \,. \tag{32}$$

Note that here *<sup>R</sup>βν*(Γ) is symmetric because we are taking *<sup>R</sup>*[*µν*](Γ) = 0. To obtain the solution of (32), we first consider (20) particularized to our theory (with *S<sup>ν</sup> αλ* and *R*[*µν*] set to zero),

$$f\_{\mathcal{R}}R\_{\mu\nu} - \frac{f}{2}g\_{\mu\nu} + 2f\_{\mathcal{Q}}R\_{\mu\alpha}R^{\alpha}\_{\ \nu} = \kappa^2 T\_{\mu\nu} \tag{33}$$

and rewrite it in the following form

$$f\_{\mathbb{R}}B\_{\mu}{}^{\nu} - \frac{f}{2}\delta\_{\mu}{}^{\nu} + 2f\_{\mathbb{Q}}B\_{\mu}{}^{\kappa}B\_{\kappa}{}^{\nu} = \kappa^2 T\_{\mu}{}^{\nu} \,, \tag{34}$$

where we have defined *B<sup>µ</sup> <sup>ν</sup>* ≡ *<sup>R</sup>µαgαν*. This equation can be seen as a second-order algebraic equation for the matrix *<sup>B</sup>*ˆ, whose components are [*B*ˆ]*<sup>µ</sup> <sup>ν</sup>* <sup>≡</sup> *<sup>B</sup><sup>µ</sup> <sup>ν</sup>*. The solutions to this equation imply that *B*ˆ is an algebraic function of the components of the stress-energy tensor *T<sup>µ</sup> <sup>ν</sup>*, i.e., *B*ˆ = *B*ˆ(*T*ˆ). This relation is very important because it allows to express (32) in the form

$$\nabla\_{\mathfrak{a}}^{\Gamma} \left[ \sqrt{-\text{gg}}^{\mathcal{G}\lambda} \left( f\_{\mathcal{R}} \delta\_{\lambda}^{\vee} + 2f\_{\mathcal{Q}} \mathcal{B}\_{\lambda}{}^{\vee} \right) \right] = 0 \ , \tag{35}$$

where now *fR*, *fQ* and *B<sup>α</sup> <sup>ν</sup>* are functions of the stress-energy tensor of the matter. The connection, therefore, can be obtained by elementary algebraic manipulations [23]. To do it, one defines a rank-two symmetric tensor *<sup>h</sup>µν* such that √−*ggβλ fRδ<sup>ν</sup> <sup>λ</sup>* <sup>+</sup> <sup>2</sup> *fQB<sup>λ</sup> ν* <sup>=</sup> √−*hhβν*, which turns (35) into the well-known equation ∇*<sup>µ</sup>* √−*hhβν* = 0, and implies that Γ*<sup>α</sup> βν* is given by the Christoffel symbols of the tensor *hµν*, i.e.,

$$
\Gamma^{a}\_{\beta\gamma} = \frac{h^{a\rho}}{2} \left( \partial\_{\beta} h\_{\rho\gamma} + \partial\_{\gamma} h\_{\rho\beta} - \partial\_{\rho} h\_{\beta\gamma} \right) \,. \tag{36}
$$

From the defining expression of *hµν*, one finds that the relation between *hµν* and *gµν* can be expressed as follows

$$h\_{\mu\nu} = \sqrt{\det \hat{\Sigma}} [\Sigma^{-1}]\_{\mu}{}^{\mu} g\_{a\nu} \quad , \quad h^{\mu\nu} = \frac{g^{\mu a} \Sigma\_{a}{}^{\nu}}{\sqrt{\det \hat{\Sigma}}} \, , \tag{37}$$

Using (41) this equation can be rewritten as follows

<sup>2</sup> <sup>+</sup> *<sup>f</sup>* <sup>2</sup> *R* 8 *fQ* 

2 *fQ <sup>B</sup>*<sup>ˆ</sup> <sup>+</sup> *fR* 4 *fQ* ˆ*I* 2 = 

 2 *fQ <sup>B</sup>*<sup>ˆ</sup> <sup>+</sup> *fR* 4 *fQ* ˆ*I* = *s*1

*<sup>κ</sup>*2*<sup>P</sup>* <sup>+</sup> *<sup>f</sup>*

where ˆ*I*3*X*<sup>3</sup> denotes 3-dimensional identity matrix. Since the right-hand side of (44) is a

matrix with elements {*si* = ±1}. For consistency of the theory in the limit *fQ* → 0, we must

Note that we have kept the two signs ± in *<sup>σ</sup>*1. The reason for this will be understood later, when particular models are considered. The point is that in some cases of physical interest, at high densities one should take the negative sign in front of the square root to guarantee that *σ*<sup>1</sup> is continuous and differentiable accross the point where the square root vanishes.

*f*(*R*) + *αQ*

So far we have made progress without specifying the form of the Lagrangian *f*(*R*, *Q*).

*P* of the fluids, we must choose a Lagrangian explicitly. Restricting the function *f*(*R*, *Q*) to the

<sup>3</sup>*X*<sup>3</sup> = <sup>ˆ</sup>*I*3*X*3. This result allows to express <sup>Σ</sup><sup>ˆ</sup> as follows

Σˆ = *<sup>σ</sup>*<sup>1</sup> <sup>0</sup> <sup>0</sup> *<sup>σ</sup>*<sup>2</sup> <sup>ˆ</sup>*I*3*X*<sup>3</sup>

diagonal matrix, it is immediate to compute its square root, which leads to

where *s*<sup>1</sup> denotes a sign, which can be positive or negative, and *S*ˆ

*<sup>σ</sup>*<sup>1</sup> <sup>=</sup> *fR* 2 ± 2 *fQ* 

*<sup>σ</sup>*<sup>2</sup> <sup>=</sup> *fR* 2 + 

However, in order to find the explicit dependence of *<sup>R</sup>* = *<sup>B</sup><sup>µ</sup>*

<sup>2</sup> <sup>+</sup> *<sup>f</sup>* <sup>2</sup> *R* 8 *fQ* *<sup>λ</sup>*<sup>2</sup> <sup>−</sup> *<sup>κ</sup>*2(*<sup>ρ</sup>* <sup>+</sup> *<sup>P</sup>*) <sup>0</sup>

*λ*<sup>2</sup> <sup>−</sup> *<sup>κ</sup>*2(*<sup>ρ</sup>* <sup>+</sup> *<sup>P</sup>*) <sup>0</sup> <sup>0</sup> *<sup>λ</sup>S*<sup>ˆ</sup>

*λ*<sup>2</sup> − *κ*2(*ρ* + *P*)

*f*(*R*) + *αQ*, we will see that it is possible to find the generic dependence of

<sup>0</sup> *<sup>λ</sup>*<sup>2</sup> <sup>ˆ</sup>*I*3*X*<sup>3</sup>

and making explicit the matrix representation, (43) becomes

Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

3*X*3 

2 *fQλ* . (47)

*<sup>µ</sup>* and *<sup>Q</sup>* = *<sup>B</sup><sup>µ</sup>*

*<sup>α</sup>B<sup>α</sup>*

*<sup>µ</sup>* with the *ρ* and

<sup>ˆ</sup>*<sup>I</sup>* + *<sup>κ</sup>*2(*<sup>ρ</sup>* + *<sup>P</sup>*)*uµu<sup>µ</sup>* . (43)

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167

, (44)

, (45)

<sup>3</sup>*X*<sup>3</sup> denotes a 3*X*3 diagonal

, (46)

2 *fQ <sup>B</sup>*<sup>ˆ</sup> <sup>+</sup> *fR* 4 *fQ* ˆ*I* 2 = 

 *<sup>κ</sup>*2*<sup>P</sup>* + *<sup>f</sup>*

Denoting *λ*<sup>2</sup> ≡

have *<sup>s</sup>*<sup>1</sup> = 1 and *<sup>S</sup>*<sup>ˆ</sup>

where *σ*<sup>1</sup> and *σ*<sup>2</sup> take the form

This technical issue does not arise for *σ*2.

**4.1. Workable models:** *f*(*R*, *Q*) = ˜

family *f*(*R*, *Q*) = ˜

where we have defined the matrix <sup>Σ</sup>*<sup>α</sup> <sup>ν</sup>* ≡ *fRδ<sup>ν</sup> <sup>α</sup>* <sup>+</sup> <sup>2</sup> *fQB<sup>α</sup> ν* . With these relations and definitions, the field equations for the metric *hµν* can be written in compact form expressing (34) as *B<sup>µ</sup> <sup>α</sup>*Σ*<sup>α</sup> ν* = *f* 2 *δν <sup>µ</sup>* <sup>+</sup> *<sup>κ</sup>*2*T<sup>µ</sup> <sup>ν</sup>* and using the relation *B<sup>µ</sup> <sup>α</sup>*Σ*<sup>α</sup> <sup>ν</sup>* <sup>=</sup> <sup>√</sup> det <sup>Σ</sup><sup>ˆ</sup> *<sup>R</sup>µα*(*h*)*hαν* to obtain [27]

$$R\_{\mu}{}^{\nu}(h) = \frac{1}{\sqrt{\det \hat{\Sigma}}} \left( \frac{f}{2} \delta\_{\mu}^{\nu} + \kappa^2 T\_{\mu}{}^{\nu} \right) \,. \tag{38}$$

In general, it will be more convenient to work with the field equations for the auxiliary metric *hµν* because their form is more tractable. Nonetheless, if one insists on writing the field equations using the metric *gµν*, one must note that the connection (36) is related to the Levi-Civita connection of *gµν* by the tensor (recall Eq.(12))

$$A^{a}\_{\beta\gamma} \equiv \Gamma^{a}\_{\beta\gamma} - L^{a}\_{\beta\gamma} = \frac{h^{a\rho}}{2} \left[ \nabla^{L}\_{\mu} h\_{\beta\nu} + \nabla^{L}\_{\nu} h\_{\beta\mu} - \nabla^{L}\_{\rho} h\_{\mu\nu} \right] \tag{39}$$

The Riemann tensors of Γ*<sup>α</sup> βγ* and *<sup>L</sup><sup>α</sup> βγ* are thus related as follows

$$\mathcal{R}^{\mathfrak{a}}{}\_{\beta\mu\nu}(\Gamma) = \mathcal{R}^{\mathfrak{a}}{}\_{\beta\mu\nu}(L) + \nabla^{L}\_{\mu}A^{\mathfrak{a}}\_{\nu\beta} - \nabla^{L}\_{\nu}A^{\mathfrak{a}}\_{\mu\beta} + A^{\lambda}\_{\nu\beta}A^{\mathfrak{a}}\_{\mu\lambda} - A^{\lambda}\_{\mu\beta}A^{\mathfrak{a}}\_{\nu\lambda} \,. \tag{40}$$

which allows to express (38) in terms of the Ricci tensor of the metric *gµν*, the usual covariant derivatives of *L<sup>α</sup> βγ*, and the matter.

## **4.** *f*(*R*, *Q*) **theories with a perfect fluid**

The explicit form of the matrix <sup>Σ</sup><sup>ˆ</sup> that relates the metrics *<sup>h</sup>µν* and *<sup>g</sup>µν* can only be found once all the sources that make up *Tµν* have been specified. In our discussion we will just consider a perfect fluid or a sum of non-interacting perfect fluids such that

$$T\_{\mu\nu} = (\rho + P)\mu\_{\mu}\mu\_{\nu} + P\mathfrak{g}\_{\mu\nu} \tag{41}$$

with *<sup>ρ</sup>* <sup>=</sup> <sup>∑</sup>*<sup>i</sup> <sup>ρ</sup><sup>i</sup>* and *<sup>P</sup>* <sup>=</sup> <sup>∑</sup>*<sup>i</sup> Pi*. In order to find an expression for <sup>Σ</sup><sup>ˆ</sup> , we first rewrite (34) using matrix notation as

$$2f\_Q \hat{\mathcal{B}}^2 + f\_R \hat{\mathcal{B}} - \frac{f}{2} \hat{I} = \kappa^2 \hat{T} \,. \tag{42}$$

Using (41) this equation can be rewritten as follows

10 Open Questions in Cosmology

expressed as follows

*<sup>α</sup>*Σ*<sup>α</sup>*

*ν* = *f* 2 *δν*

(34) as *B<sup>µ</sup>*

[27]

*<sup>h</sup>µν* = 

*<sup>µ</sup>* <sup>+</sup> *<sup>κ</sup>*2*T<sup>µ</sup>*

*Rµ*

Levi-Civita connection of *gµν* by the tensor (recall Eq.(12))

*βγ* <sup>−</sup> *<sup>L</sup><sup>α</sup>*

*βγ* and *<sup>L</sup><sup>α</sup>*

*βµν*(*L*) + <sup>∇</sup>*<sup>L</sup>*

a perfect fluid or a sum of non-interacting perfect fluids such that

where we have defined the matrix <sup>Σ</sup>*<sup>α</sup>*

*Aα βγ* <sup>≡</sup> <sup>Γ</sup>*<sup>α</sup>*

*βµν*(Γ) = *<sup>R</sup><sup>α</sup>*

*βγ*, and the matter.

**4.** *f*(*R*, *Q*) **theories with a perfect fluid**

The Riemann tensors of Γ*<sup>α</sup>*

derivatives of *L<sup>α</sup>*

matrix notation as

*Rα*

From the defining expression of *hµν*, one finds that the relation between *hµν* and *gµν* can be

*α*

*<sup>ν</sup>* ≡

*<sup>ν</sup>* and using the relation *B<sup>µ</sup>*

definitions, the field equations for the metric *hµν* can be written in compact form expressing

 *f* 2 *δν <sup>µ</sup>* <sup>+</sup> *<sup>κ</sup>*2*T<sup>µ</sup>*

In general, it will be more convenient to work with the field equations for the auxiliary metric *hµν* because their form is more tractable. Nonetheless, if one insists on writing the field equations using the metric *gµν*, one must note that the connection (36) is related to the

*<sup>µ</sup>hρν* <sup>+</sup> <sup>∇</sup>*<sup>L</sup>*

*<sup>ν</sup> <sup>A</sup><sup>α</sup>*

*µβ* <sup>+</sup> *<sup>A</sup><sup>λ</sup>*

*βγ* are thus related as follows

*νβ* − ∇*<sup>L</sup>*

which allows to express (38) in terms of the Ricci tensor of the metric *gµν*, the usual covariant

The explicit form of the matrix <sup>Σ</sup><sup>ˆ</sup> that relates the metrics *<sup>h</sup>µν* and *<sup>g</sup>µν* can only be found once all the sources that make up *Tµν* have been specified. In our discussion we will just consider

with *<sup>ρ</sup>* <sup>=</sup> <sup>∑</sup>*<sup>i</sup> <sup>ρ</sup><sup>i</sup>* and *<sup>P</sup>* <sup>=</sup> <sup>∑</sup>*<sup>i</sup> Pi*. In order to find an expression for <sup>Σ</sup><sup>ˆ</sup> , we first rewrite (34) using

2

<sup>2</sup> *fQB*ˆ2 <sup>+</sup> *fRB*<sup>ˆ</sup> <sup>−</sup> *<sup>f</sup>*

*fRδ<sup>ν</sup>*

*<sup>g</sup>αν* , *<sup>h</sup>µν* <sup>=</sup> *<sup>g</sup>µα*Σ*<sup>α</sup>*

*<sup>α</sup>* <sup>+</sup> <sup>2</sup> *fQB<sup>α</sup>*

*<sup>α</sup>*Σ*<sup>α</sup>*

*ν* 

*<sup>ν</sup> <sup>h</sup>ρµ* − ∇*<sup>L</sup>*

*νβA<sup>α</sup>*

*<sup>T</sup>µν* = (*<sup>ρ</sup>* + *<sup>P</sup>*)*uµu<sup>ν</sup>* + *Pgµν* (41)

ˆ*I* = *κ*2*T*ˆ . (42)

*<sup>ρ</sup> hµν* 

*µλ* <sup>−</sup> *<sup>A</sup><sup>λ</sup>*

*µβA<sup>α</sup>*

√

*ν*

*<sup>ν</sup>* <sup>=</sup> <sup>√</sup>

*ν*

det <sup>Σ</sup><sup>ˆ</sup> , (37)

. With these relations and

. (38)

det <sup>Σ</sup><sup>ˆ</sup> *<sup>R</sup>µα*(*h*)*hαν* to obtain

. (39)

*νλ* , (40)

det <sup>Σ</sup><sup>ˆ</sup> [Σ−1]*<sup>µ</sup>*

*<sup>ν</sup>*(*h*) = <sup>1</sup> √ det Σˆ

> *βγ* <sup>=</sup> *<sup>h</sup>αρ* 2 ∇*L*

> > *µA<sup>α</sup>*

$$2f\_{\mathcal{Q}}\left(\hat{\mathcal{B}} + \frac{f\_{\mathcal{R}}}{4f\_{\mathcal{Q}}}\hat{I}\right)^2 = \left(\kappa^2 P + \frac{f}{2} + \frac{f\_{\mathcal{R}}^2}{8f\_{\mathcal{Q}}}\right)\hat{I} + \kappa^2 (\rho + P) u\_{\mu} u^{\mu} \,. \tag{43}$$

Denoting *λ*<sup>2</sup> ≡ *<sup>κ</sup>*2*<sup>P</sup>* + *<sup>f</sup>* <sup>2</sup> <sup>+</sup> *<sup>f</sup>* <sup>2</sup> *R* 8 *fQ* and making explicit the matrix representation, (43) becomes

$$2f\_Q \left(\hat{\mathcal{B}} + \frac{f\_R}{4f\_Q}\hat{I}\right)^2 = \begin{pmatrix} \lambda^2 - \kappa^2(\rho + P) & \vec{0} \\ \vec{0} & \lambda^2 \hat{I}\_{3X3} \end{pmatrix} \tag{44}$$

where ˆ*I*3*X*<sup>3</sup> denotes 3-dimensional identity matrix. Since the right-hand side of (44) is a diagonal matrix, it is immediate to compute its square root, which leads to

$$
\sqrt{2f\_Q} \left( \vec{\mathcal{B}} + \frac{f\_R}{4f\_Q} \hat{I} \right) = \begin{pmatrix} s\_1 \sqrt{\lambda^2 - \kappa^2(\rho + P)} & \vec{0} \\ \vec{0} & \lambda \hat{S}\_{3X3} \end{pmatrix} \tag{45}
$$

where *s*<sup>1</sup> denotes a sign, which can be positive or negative, and *S*ˆ <sup>3</sup>*X*<sup>3</sup> denotes a 3*X*3 diagonal matrix with elements {*si* = ±1}. For consistency of the theory in the limit *fQ* → 0, we must have *<sup>s</sup>*<sup>1</sup> = 1 and *<sup>S</sup>*<sup>ˆ</sup> <sup>3</sup>*X*<sup>3</sup> = <sup>ˆ</sup>*I*3*X*3. This result allows to express <sup>Σ</sup><sup>ˆ</sup> as follows

$$
\hat{\Sigma} = \begin{pmatrix} \sigma\_1 & \vec{0} \\ \vec{0} & \sigma\_2 \hat{I}\_{3X3} \end{pmatrix} \tag{46}
$$

where *σ*<sup>1</sup> and *σ*<sup>2</sup> take the form

$$
\sigma\_1 = \frac{f\_R}{2} \pm \sqrt{2f\_Q} \sqrt{\lambda^2 - \kappa^2(\rho + P)}
$$

$$
\sigma\_2 = \frac{f\_R}{2} + \sqrt{2f\_Q} \lambda \,. \tag{47}
$$

Note that we have kept the two signs ± in *<sup>σ</sup>*1. The reason for this will be understood later, when particular models are considered. The point is that in some cases of physical interest, at high densities one should take the negative sign in front of the square root to guarantee that *σ*<sup>1</sup> is continuous and differentiable accross the point where the square root vanishes. This technical issue does not arise for *σ*2.

#### **4.1. Workable models:** *f*(*R*, *Q*) = ˜ *f*(*R*) + *αQ*

So far we have made progress without specifying the form of the Lagrangian *f*(*R*, *Q*). However, in order to find the explicit dependence of *<sup>R</sup>* = *<sup>B</sup><sup>µ</sup> <sup>µ</sup>* and *<sup>Q</sup>* = *<sup>B</sup><sup>µ</sup> <sup>α</sup>B<sup>α</sup> <sup>µ</sup>* with the *ρ* and *P* of the fluids, we must choose a Lagrangian explicitly. Restricting the function *f*(*R*, *Q*) to the family *f*(*R*, *Q*) = ˜ *f*(*R*) + *αQ*, we will see that it is possible to find the generic dependence of *Q* with *ρ* and *P*, while *R* is found to depend only on the combination *T* = −*ρ* + 3*P* [23]. The reason for this follows from the trace of (33) with *gµν*, which for this family of Lagrangians gives the algebraic relation *R* ˜ *fR* − <sup>2</sup> ˜ *f* = *κ*2*T* and implies that *R* = *R*(*T*) (like in Palatini *f*(*R*) theories). For these theories, we have that *fQ* = *<sup>α</sup>*, which is a constant. Therefore, from the trace of (44) we find

$$
\sqrt{2f\_Q} \left( R + \frac{f\_R}{f\_Q} \right) = \sqrt{\lambda^2 - \kappa^2(\rho + P)} + 3\lambda \,\,\,\,\tag{48}
$$

that took place during the inflationary period may have washed out many of the relevant proper signatures needed to distinguish the predictions of different quantum theories of

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169

A conservative approach, therefore, consists on exploring the quantum properties and interactions of the matter fields in the very early universe using the well-established methods of quantum field theory in curved space-times [29]. The success of this approach has been confirmed in combination with models of inflation and sheds relevant light on the mechanisms that may have caused the primordial spectra of scalar and tensorial perturbations [38], [19], [9],[18], [15]. The applicability of this approach, however, becomes unreliable at increasing energies as the regime of the classical big bang singularity is approached and the quantum fluctuations of the gravitational field can no longer be neglected. At that stage, a complete quantum theory of gravity seems necessary to provide a consistent description of the ongoing physical processes. Obviously, different quantum theories could lead to completely different quantum gravitational scenarios and, therefore, a generic quantum origin for the universe cannot be guessed *a priori* by any logical means. In recent years, bouncing cosmological models have attracted much attention [21]. These are scenarios in which the big bang singularity is replaced by a quantum-induced bounce that connects an earlier phase of contraction with the subsequent expanding phase (in which we happen to exist). In such scenarios, aside from the quantum regime, the contracting and expanding phases are expected to asymptote an effective classical geometry whose dynamics, on consistency grounds, should match that of GR at low energies. In this context, and as an intermediate step between the quantum field theory approach in the (singular) curved background provided by GR and a (nonsingular) full theory of quantum gravity, one could consider the case of a smooth effective geometry free from big bang singularities on top of which quantum matter fields could still be treated perturbatively in a consistent way. This view would somehow disentangle the non-perturbative part of the quantum gravitational sector into an effective classical, nonsingular geometry, plus perturbative quantum corrections that propagate on top of the regular effective background. The absence of curvature singularities would make the treatment of quantum fields on the resulting geometry more reliable, and could help shed new light on the effects of the

In the literature there exist many interesting examples of (quantum and non-quantum) cosmological models that avoid the big bang singularity by means of a bounce. Roughly, those models can be classified in two large groups, depending on whether they contain a modified gravitational sector or a modified matter sector (see [21] for details and a very complete list of references). Generically, modified gravity theories imply the existence of new dynamical degrees of freedom, such as gravitational scalar fields (like in scalar-tensor theories), higher-derivatives of the metric, extra dimensions, . . . The consideration of exotic matter sources may be justified, in some cases, from an effective field theory approach, such as in the case of non-linear theories of electrodynamics, which naturally arise in low-energy limits of string theories. In the remainder of this chapter, we are going to study bouncing cosmological models from the modified gravity perspective provided by the Palatini theories discussed above. This approach is particularly interesting because, despite being a modified-gravity approach, the underlying mechanisms that modify the gravitational dynamics are not associated with new dynamical degrees of freedom or higher-derivative equations. In fact, it is the nontrivial role played by the matter in the determination of the

matter-gravity interaction in the very-early universe.

gravity.

which can be cast as

$$\left[\sqrt{2f\_Q}\left(R + \frac{f\_R}{f\_Q}\right) - 3\lambda\right]^2 = \lambda^2 - \kappa^2(\rho + P) \tag{49}$$

After a bit of algebra we find that

$$\lambda = \frac{\sqrt{2f\_Q}}{8} \left[ 3 \left( R + \frac{f\_R}{f\_Q} \right) \pm \sqrt{\left( R + \frac{f\_R}{f\_Q} \right)^2 - \frac{4\kappa^2(\rho + P)}{f\_Q}} \right] \tag{50}$$

From this expression and the definition of *λ*2, we find

$$\alpha Q = -\left(\tilde{f} + \frac{f\_R^2}{4f\_Q} + 2\kappa^2 P\right) + \frac{f\_Q}{16} \left[ 3\left(R + \frac{\tilde{f}\_R}{f\_Q}\right) \pm \sqrt{\left(R + \frac{\tilde{f}\_R}{f\_Q}\right)^2 - \frac{4\kappa^2(\rho + P)}{f\_Q}} \right]^2,\tag{51}$$

where *R*, ˜ *f* , and ˜ *fR* are functions of *<sup>T</sup>* = −*<sup>ρ</sup>* + <sup>3</sup>*P*.

## **5. Nonsingular cosmologies in** *f*(*R*, *Q*) **theories**

The difficulties faced by GR to provide a consistent description of singularities and quantum phenomena at high energies (microscopic or Planck scales) is generally seen as an indication that we should go beyond the standard geometric structures to successfully quantize the theory and avoid singularities. This idea has motivated a variety of approaches that range from the consideration of higher-dimensional superstrings and other extended objects [11], to non-commutative geometries or non-perturbative quantization methods [3], [31], [35] , to name just a few well-known cases. Unfortunately, the formidable task of building a satisfactory quantum theory of gravity is not yet complete. Moreover, even if we managed to get such a theory, we would still have to face the challenge of testing its predictions. In this sense, it should be noted that since the quantum gravitational regime is so far from our current and future experimental capabilities, our only hope might be to use the information available in the cosmic microwave background radiation to verify or rule out our theories [1]. How much of the quantum gravitational regime could be contrasted with these yet-to-come theories is not clear. This is due, in part, because the theorized rapid accelerated expansion that took place during the inflationary period may have washed out many of the relevant proper signatures needed to distinguish the predictions of different quantum theories of gravity.

12 Open Questions in Cosmology

trace of (44) we find

which can be cast as

*αQ* = −

where *R*, ˜

� ˜ *<sup>f</sup>* <sup>+</sup> ˜ *f* 2 *R* 4 *fQ*

*f* , and ˜

gives the algebraic relation *R* ˜

After a bit of algebra we find that

*λ* =

*Q* with *ρ* and *P*, while *R* is found to depend only on the combination *T* = −*ρ* + 3*P* [23]. The reason for this follows from the trace of (33) with *gµν*, which for this family of Lagrangians

theories). For these theories, we have that *fQ* = *<sup>α</sup>*, which is a constant. Therefore, from the

*<sup>R</sup>* <sup>+</sup> *fR fQ* �2

*f* = *κ*2*T* and implies that *R* = *R*(*T*) (like in Palatini *f*(*R*)

*λ*<sup>2</sup> − *κ*2(*ρ* + *P*) + 3*λ* , (48)

= *λ*<sup>2</sup> − *κ*2(*ρ* + *P*) (49)

<sup>−</sup> <sup>4</sup>*κ*2(*<sup>ρ</sup>* <sup>+</sup> *<sup>P</sup>*) *fQ*

(50)

 

2

, (51)

<sup>−</sup> <sup>4</sup>*κ*2(*<sup>ρ</sup>* <sup>+</sup> *<sup>P</sup>*) *fQ*

�2

*<sup>R</sup>* <sup>+</sup> ˜ *fR fQ*

*fR* − <sup>2</sup> ˜

� 2 *fQ* � *<sup>R</sup>* <sup>+</sup> *fR fQ* � = �

�� 2 *fQ* � *<sup>R</sup>* <sup>+</sup> *fR fQ* � − 3*λ* �2

�2 *fQ* 8

From this expression and the definition of *λ*2, we find

� <sup>+</sup> *fQ* 16

**5. Nonsingular cosmologies in** *f*(*R*, *Q*) **theories**

*fR* are functions of *<sup>T</sup>* = −*<sup>ρ</sup>* + <sup>3</sup>*P*.

 3 � *<sup>R</sup>* <sup>+</sup> ˜ *fR fQ* � ± �����

The difficulties faced by GR to provide a consistent description of singularities and quantum phenomena at high energies (microscopic or Planck scales) is generally seen as an indication that we should go beyond the standard geometric structures to successfully quantize the theory and avoid singularities. This idea has motivated a variety of approaches that range from the consideration of higher-dimensional superstrings and other extended objects [11], to non-commutative geometries or non-perturbative quantization methods [3], [31], [35] , to name just a few well-known cases. Unfortunately, the formidable task of building a satisfactory quantum theory of gravity is not yet complete. Moreover, even if we managed to get such a theory, we would still have to face the challenge of testing its predictions. In this sense, it should be noted that since the quantum gravitational regime is so far from our current and future experimental capabilities, our only hope might be to use the information available in the cosmic microwave background radiation to verify or rule out our theories [1]. How much of the quantum gravitational regime could be contrasted with these yet-to-come theories is not clear. This is due, in part, because the theorized rapid accelerated expansion

+ 2*κ*2*P*

 3 � *<sup>R</sup>* <sup>+</sup> *fR fQ* � ± �� A conservative approach, therefore, consists on exploring the quantum properties and interactions of the matter fields in the very early universe using the well-established methods of quantum field theory in curved space-times [29]. The success of this approach has been confirmed in combination with models of inflation and sheds relevant light on the mechanisms that may have caused the primordial spectra of scalar and tensorial perturbations [38], [19], [9],[18], [15]. The applicability of this approach, however, becomes unreliable at increasing energies as the regime of the classical big bang singularity is approached and the quantum fluctuations of the gravitational field can no longer be neglected. At that stage, a complete quantum theory of gravity seems necessary to provide a consistent description of the ongoing physical processes. Obviously, different quantum theories could lead to completely different quantum gravitational scenarios and, therefore, a generic quantum origin for the universe cannot be guessed *a priori* by any logical means.

In recent years, bouncing cosmological models have attracted much attention [21]. These are scenarios in which the big bang singularity is replaced by a quantum-induced bounce that connects an earlier phase of contraction with the subsequent expanding phase (in which we happen to exist). In such scenarios, aside from the quantum regime, the contracting and expanding phases are expected to asymptote an effective classical geometry whose dynamics, on consistency grounds, should match that of GR at low energies. In this context, and as an intermediate step between the quantum field theory approach in the (singular) curved background provided by GR and a (nonsingular) full theory of quantum gravity, one could consider the case of a smooth effective geometry free from big bang singularities on top of which quantum matter fields could still be treated perturbatively in a consistent way. This view would somehow disentangle the non-perturbative part of the quantum gravitational sector into an effective classical, nonsingular geometry, plus perturbative quantum corrections that propagate on top of the regular effective background. The absence of curvature singularities would make the treatment of quantum fields on the resulting geometry more reliable, and could help shed new light on the effects of the matter-gravity interaction in the very-early universe.

In the literature there exist many interesting examples of (quantum and non-quantum) cosmological models that avoid the big bang singularity by means of a bounce. Roughly, those models can be classified in two large groups, depending on whether they contain a modified gravitational sector or a modified matter sector (see [21] for details and a very complete list of references). Generically, modified gravity theories imply the existence of new dynamical degrees of freedom, such as gravitational scalar fields (like in scalar-tensor theories), higher-derivatives of the metric, extra dimensions, . . . The consideration of exotic matter sources may be justified, in some cases, from an effective field theory approach, such as in the case of non-linear theories of electrodynamics, which naturally arise in low-energy limits of string theories. In the remainder of this chapter, we are going to study bouncing cosmological models from the modified gravity perspective provided by the Palatini theories discussed above. This approach is particularly interesting because, despite being a modified-gravity approach, the underlying mechanisms that modify the gravitational dynamics are not associated with new dynamical degrees of freedom or higher-derivative equations. In fact, it is the nontrivial role played by the matter in the determination of the space-time connection that induces nonlinearities in the matter sector that end up changing the dynamics at very high matter-energy densities. In this sense, it should be noted that the gravitational field equations in vacuum exactly recover those of GR (with possibly a cosmological constant, depending on the particular Lagrangian chosen). For this reason, this type of theories can be regarded as a *minimal extension of the standard model of gravitational physics*, because they only appreciably depart from GR in regions that contain sources and when those sources reach the energy-density scales that characterize the correcting terms of the Lagrangian.

## **5.1. Homogeneous cosmologies in** *f*(*R*, *Q*) **theories**

In this section we introduce the basic definitions and formulas needed to derive the equations for the evolution of the expansion and shear [37] for an arbitrary Palatini *f*(*R*, *Q*) theory of the kind presented in Section 3.2 . These magnitudes will be very useful to extract information about the geometric properties of the space-time and to determine whether cosmic singularities are present or not. We focus on homogeneous cosmologies of the Bianchi I type (a different expansion factor for each spatial direction) because that will allow us to test the rebustness of our results against deviations from the idealized Friedmann-Robertson-Walker spacetimes (same expansion rate in all the spatial directions). We will also particularize our results to the case of *f*(*R*) theories, i.e., no dependence on *Q*.

We consider a Bianchi I spacetime with physical line element of the form

$$ds^2 = g\_{\mu\nu}dx^\mu dx^\nu = -dt^2 + \sum\_{i=1}^3 a\_i^2(t)(dx^i)^2 \tag{52}$$

The nonzero components of the corresponding Ricci tensor are

Ω¨ <sup>Ω</sup> <sup>−</sup>

> �3Ω˙ <sup>Ω</sup> <sup>−</sup> *<sup>S</sup>*˙ *S*

� Ω˙ Ω �2 + Ω˙ <sup>Ω</sup> ∑ *k*

where *Hk* ≡ *<sup>a</sup>*˙ *<sup>k</sup>*/*ak*. For completeness, we give an expression for the corresponding scalar

From the above formulas, one can readily find the corresponding ones in the isotropic, flat configuration by just replacing *Hi* → *<sup>H</sup>*. For the spatially nonflat case, the *Rtt*(*h*) component is the same as in the flat case. The *Rij*(*h*) component, however, picks up a new piece, 2*Kγij*,

From the previous formulas and the field equation (38), we find that the combination *Ri*

∑ *k*

*Hk* + 1 2

*<sup>i</sup>* and *Rj*

we see that it can be readily integrated regardless of the number and particular equations of

ln *Hij* <sup>+</sup> ln(*a*1*a*2*a*3) + ln <sup>Ω</sup>3/2 <sup>−</sup> ln *<sup>S</sup>*1/2�

�3Ω˙ <sup>Ω</sup> <sup>−</sup> *<sup>S</sup>*˙ *S*

*<sup>H</sup>*˙ *ij* <sup>+</sup> *Hij* �

where we have defined *Hij* ≡ *Hi* − *Hj*. Note that the final equality *Ri*

*Hk* + 1 2 Ω˙ Ω

� *S*˙ *<sup>S</sup>* <sup>−</sup> <sup>Ω</sup>˙ Ω �3Ω˙ <sup>Ω</sup> <sup>−</sup> *<sup>S</sup>*˙ *S* � +

Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

��� , (59)

��∑ *k*

*Hk* + <sup>3</sup>

*<sup>i</sup> γij*. The Ricci scalar then becomes

��� <sup>=</sup> 0 , (61)

*<sup>j</sup>* = 0, follows

= 0 , (62)

*<sup>i</sup>* − *Rj*

*<sup>j</sup>* as given by (38) are equal. Expressing

Ω¨ <sup>Ω</sup> <sup>−</sup> <sup>3</sup> 2 Ω˙ Ω *S*˙ *S*

*Hi* (58)

171

http://dx.doi.org/10.5772/51807

 

(60)

*i* −

*<sup>H</sup>*˙ *<sup>i</sup>* <sup>−</sup> ∑ *i H*2 *<sup>i</sup>* <sup>−</sup> <sup>3</sup> 2 Ω¨ <sup>Ω</sup> <sup>+</sup> 3 4 Ω˙ Ω � *S*˙ *S* + Ω˙ Ω � + 1 2 � *S*˙ *<sup>S</sup>* <sup>−</sup> <sup>2</sup>Ω˙ Ω � ∑ *i*

*Hk* + 1 2

where *<sup>γ</sup>ij* represents the nonflat spatial metric of *gij* = *<sup>a</sup>*<sup>2</sup>

*<sup>j</sup>* = 1 *S* �

*<sup>j</sup>* = *d dt* �

*<sup>a</sup>*2<sup>Ω</sup> .

*<sup>j</sup>* (no summation over indices) leads to

from the fact that the right hand sides of *Ri*

*Ri <sup>i</sup>* − *Rj*

*Ri <sup>i</sup>* − *Rj*

state of the fluids involved. The result is

*Rtt*(*h*) = <sup>−</sup> ∑

*Rij*(*h*) =

 2∑ *k H*˙ *<sup>k</sup>* <sup>+</sup> ∑ *k H*2 *k* + � ∑ *k Hk* �2 + � 3 Ω˙ <sup>Ω</sup> <sup>−</sup>

*<sup>R</sup>*(*h*) → *<sup>R</sup>K*=0(*h*) + <sup>6</sup>*<sup>K</sup>*

**5.2. Shear**

(61) in the form

*Rj*

curvature

*<sup>R</sup>*(*h*) = <sup>1</sup> *S* *i*

Ω *S* � <sup>2</sup>*H*˙ *<sup>i</sup>* +

� ∑ *k*

*δija*<sup>2</sup> *i* 2

+ <sup>2</sup>*Hi*

In terms of this line element, using the relation between metrics (37) and the expression (46) for the matrix Σˆ of a collection of perfect fluids, the nonzero components of the auxiliary metric *hµν* become

$$h\_{tt} = -\left(\frac{\sigma\_2^2}{\sqrt{\sigma\_1 \sigma\_2}}\right) \equiv -S \tag{53}$$

$$
\delta\_{\rm ij} = \sqrt{\sigma\_1 \sigma\_2} a\_i^2 \delta\_{\rm ij} \equiv \Omega a\_i^2 \delta\_{\rm ij} \tag{54}
$$

The relevant Christoffel symbols associated with *hµν* are the following:

$$
\Gamma^t\_{tt} = \frac{\dot{\mathcal{S}}}{2\mathcal{S}}\tag{55}
$$

$$
\Gamma^t\_{ij} = \frac{\Omega a\_i^2}{2S} \left[ \frac{\dot{\Omega}}{\Omega} + \frac{2\dot{a}\_i}{a\_i} \right] \delta\_{ij} \tag{56}
$$

$$
\Gamma^i\_{tj} = \frac{\delta^l\_j}{2} \left[ \frac{\dot{\Omega}}{\Omega} + \frac{2\dot{a}\_i}{a\_i} \right] \tag{57}
$$

The nonzero components of the corresponding Ricci tensor are

$$\begin{split} R\_{ll}(\boldsymbol{h}) &= -\sum\_{i} \boldsymbol{H}\_{i} - \sum\_{i} \boldsymbol{H}\_{i}^{2} - \frac{3}{2} \frac{\boldsymbol{\tilde{\Omega}}}{\boldsymbol{\Omega}} + \frac{3}{4} \frac{\boldsymbol{\Omega}}{\boldsymbol{\Omega}} \left( \frac{\boldsymbol{\tilde{\boldsymbol{S}}}}{\boldsymbol{\tilde{\boldsymbol{S}}}} + \frac{\boldsymbol{\Omega}}{\boldsymbol{\Omega}} \right) + \frac{1}{2} \left( \frac{\boldsymbol{\tilde{\boldsymbol{S}}}}{\boldsymbol{\tilde{\boldsymbol{S}}}} - \frac{2 \boldsymbol{\tilde{\Omega}}}{\boldsymbol{\Omega}} \right) \sum\_{i} \boldsymbol{H}\_{i} \\ R\_{lj}(\boldsymbol{h}) &= \frac{\delta\_{lj} a\_{i}^{2}}{2} \frac{\boldsymbol{\Omega}}{\boldsymbol{\mathcal{S}}} \left[ 2\boldsymbol{H}\_{i} + \frac{\boldsymbol{\tilde{\boldsymbol{\Omega}}}}{\boldsymbol{\Omega}} - \left( \frac{\boldsymbol{\Omega}}{\boldsymbol{\Omega}} \right)^{2} + \frac{\boldsymbol{\Omega}}{\boldsymbol{\Omega}} \sum\_{k} \boldsymbol{H}\_{k} + \frac{1}{2} \frac{\boldsymbol{\Omega}}{\boldsymbol{\Omega}} \left( \frac{3 \boldsymbol{\hat{\Omega}}}{\boldsymbol{\Omega}} - \frac{\boldsymbol{\hat{\boldsymbol{S}}}}{\boldsymbol{\mathcal{S}}} \right) + \\ &\quad + 2 \boldsymbol{H}\_{i} \left\{ \sum\_{k} \boldsymbol{H}\_{k} + \frac{1}{2} \left( \frac{3 \boldsymbol{\hat{\Omega}}}{\boldsymbol{\Omega}} - \frac{\boldsymbol{\hat{S}}}{\boldsymbol{\mathcal{S}}} \right) \right\} \right], \tag{59}$$

where *Hk* ≡ *<sup>a</sup>*˙ *<sup>k</sup>*/*ak*. For completeness, we give an expression for the corresponding scalar curvature

$$R(h) = \frac{1}{S} \left| 2\sum\_{k} H\_{k} + \sum\_{k} H\_{k}^{2} + \left(\sum\_{k} H\_{k}\right)^{2} + \left(3\frac{\dot{\Omega}}{\Omega} - \left\{\frac{\dot{S}}{S} - \frac{\dot{\Omega}}{\Omega}\right\}\right) \sum\_{k} H\_{k} + 3\frac{\ddot{\Omega}}{\Omega} - \frac{3}{2}\frac{\dot{\Omega}}{\Omega}\frac{\dot{S}}{S}\right| \tag{60}$$

From the above formulas, one can readily find the corresponding ones in the isotropic, flat configuration by just replacing *Hi* → *<sup>H</sup>*. For the spatially nonflat case, the *Rtt*(*h*) component is the same as in the flat case. The *Rij*(*h*) component, however, picks up a new piece, 2*Kγij*, where *<sup>γ</sup>ij* represents the nonflat spatial metric of *gij* = *<sup>a</sup>*<sup>2</sup> *<sup>i</sup> γij*. The Ricci scalar then becomes *<sup>R</sup>*(*h*) → *<sup>R</sup>K*=0(*h*) + <sup>6</sup>*<sup>K</sup> <sup>a</sup>*2<sup>Ω</sup> .

#### **5.2. Shear**

14 Open Questions in Cosmology

the Lagrangian.

metric *hµν* become

**5.1. Homogeneous cosmologies in** *f*(*R*, *Q*) **theories**

We consider a Bianchi I spacetime with physical line element of the form

*ds*<sup>2</sup> = *<sup>g</sup>µνdxµdx<sup>ν</sup>* = −*dt*<sup>2</sup> +

*htt* = −

The relevant Christoffel symbols associated with *hµν* are the following:

Γ*t tt* <sup>=</sup> *<sup>S</sup>*˙

Γ*t*

Γ*i tj* <sup>=</sup> *<sup>δ</sup><sup>i</sup> j* 2 Ω˙ <sup>Ω</sup> <sup>+</sup> 2*a*˙*<sup>i</sup> ai* 

*ij* <sup>=</sup> <sup>Ω</sup>*a*<sup>2</sup> *i* 2*S* Ω˙ <sup>Ω</sup> <sup>+</sup> 2*a*˙*<sup>i</sup> ai* 

*hij* <sup>=</sup> <sup>√</sup>*σ*1*σ*2*a*<sup>2</sup>

In terms of this line element, using the relation between metrics (37) and the expression (46) for the matrix Σˆ of a collection of perfect fluids, the nonzero components of the auxiliary

> *σ*<sup>2</sup> √ 2 *σ*1*σ*2

*<sup>i</sup> <sup>δ</sup>ij* <sup>≡</sup> <sup>Ω</sup>*a*<sup>2</sup>

space-time connection that induces nonlinearities in the matter sector that end up changing the dynamics at very high matter-energy densities. In this sense, it should be noted that the gravitational field equations in vacuum exactly recover those of GR (with possibly a cosmological constant, depending on the particular Lagrangian chosen). For this reason, this type of theories can be regarded as a *minimal extension of the standard model of gravitational physics*, because they only appreciably depart from GR in regions that contain sources and when those sources reach the energy-density scales that characterize the correcting terms of

In this section we introduce the basic definitions and formulas needed to derive the equations for the evolution of the expansion and shear [37] for an arbitrary Palatini *f*(*R*, *Q*) theory of the kind presented in Section 3.2 . These magnitudes will be very useful to extract information about the geometric properties of the space-time and to determine whether cosmic singularities are present or not. We focus on homogeneous cosmologies of the Bianchi I type (a different expansion factor for each spatial direction) because that will allow us to test the rebustness of our results against deviations from the idealized Friedmann-Robertson-Walker spacetimes (same expansion rate in all the spatial directions). We will also particularize our results to the case of *f*(*R*) theories, i.e., no dependence on *Q*.

> 3 ∑ *i*=1 *a*2 *<sup>i</sup>* (*t*)(*dx<sup>i</sup>*

)<sup>2</sup> (52)

≡ −*S* (53)

<sup>2</sup>*<sup>S</sup>* (55)

*<sup>i</sup> δij* (54)

*δij* (56)

(57)

From the previous formulas and the field equation (38), we find that the combination *Ri i* − *Rj <sup>j</sup>* (no summation over indices) leads to

$$R\_i{}^j - R\_j{}^j = \frac{1}{S} \left[ \dot{H}\_{ij} + H\_{ij} \left\{ \sum\_k H\_k + \frac{1}{2} \left( \frac{3\dot{\Omega}}{\Omega} - \frac{\dot{S}}{S} \right) \right\} \right] = 0 \,, \tag{61}$$

where we have defined *Hij* ≡ *Hi* − *Hj*. Note that the final equality *Ri <sup>i</sup>* − *Rj <sup>j</sup>* = 0, follows from the fact that the right hand sides of *Ri <sup>i</sup>* and *Rj <sup>j</sup>* as given by (38) are equal. Expressing (61) in the form

$$R\_l^{\ j} - R\_{\dot{\jmath}}^{\ j} = \frac{d}{dt} \left[ \ln H\_{l\dot{\jmath}} + \ln (a\_1 a\_2 a\_3) + \ln \Omega^{3/2} - \ln S^{1/2} \right] = 0 \,,\tag{62}$$

we see that it can be readily integrated regardless of the number and particular equations of state of the fluids involved. The result is

$$H\_{ij} = \mathbb{C}\_{i\bar{j}} \frac{\mathbb{S}^{\frac{1}{2}}}{\Omega^{\frac{3}{2}}} \frac{\mathbb{C}\_{ij}}{(a\_1 a\_2 a\_3)} = \frac{\mathbb{C}\_{i\bar{j}}}{\sigma\_1} \frac{V\_0}{V(t)}\,'\,\tag{63}$$

which in combination with (67) yields

this result, (69) can be written as

where we have defined

following form:

2( ˙

**5.4. Limit to** *f*(*R*)

*θ* + 3*θ*2) + *θ*

3 *θ* + Ω˙ 2Ω 2

3*θ*<sup>2</sup> 1 + 3 2 ∆1 2

of the universe according to *<sup>ρ</sup>i*(*t*) = *<sup>ρ</sup>i*(*t*0)

<sup>∆</sup><sup>1</sup> <sup>=</sup> <sup>−</sup> ∑

*i*

*V*(*t*) as well. This will be very useful later for our discussion of particular models.

H<sup>2</sup> = 1 6*σ*<sup>1</sup>

6Ω˙ <sup>Ω</sup> <sup>−</sup> *<sup>S</sup>*˙ *S* + Ω¨ <sup>Ω</sup> <sup>+</sup> 1 2 Ω˙ Ω Ω˙ <sup>Ω</sup> <sup>−</sup> *<sup>S</sup>*˙ *S*

<sup>=</sup> *<sup>f</sup>* <sup>+</sup> *<sup>κ</sup>*2(*<sup>ρ</sup>* <sup>+</sup> <sup>3</sup>*P*) 2*σ*<sup>1</sup>

<sup>=</sup> *<sup>f</sup>* <sup>+</sup> *<sup>κ</sup>*2(*<sup>ρ</sup>* <sup>+</sup> <sup>3</sup>*P*) 2*σ*<sup>1</sup>

*∂ρi*<sup>Ω</sup>

(<sup>1</sup> + *wi*)*ρ<sup>i</sup>*

Note that in this last equation *wi* = *wi*(*ρi*), i.e., they need not be constants. For fluids with constant *wi*, the conservation equation implies that their density depends on the volume

> *V*<sup>0</sup> *V*(*t*)

Lagrangian is specified, the equations of state *Pi* = *wiρ<sup>i</sup>* are given, and the anisotropy constants *Cij* are chosen, the right-hand side of Eqs. (65) and (70) can be parametrized in terms of *<sup>V</sup>*(*t*). This, in turn, allows us to parametrize the *Hi* functions of (64) in terms of

In the isotropic case (*σ*<sup>2</sup> = 0 , *θ* = *a*˙/*a* ≡ H) with nonzero spatial curvature, (70) takes the

 1 + <sup>3</sup> <sup>2</sup>∆<sup>1</sup>

The evolution equation for the expansion can be obtained by noting that the *Rij* equations, which are of the form *Rij* ≡ (Ω/2*S*)*gij* [...] = (*<sup>f</sup>* /2 + *<sup>κ</sup>*2*P*)*gij*/*σ*2, can be summed up to give

We now consider the limit *fQ* → 0, namely, the case in which the Lagrangian only depends on the Ricci scalar *R*. Doing this we will obtain the corresponding equations for shear and

<sup>1</sup>+*wi*

*f* + *κ*2(*ρ* + 3*P*) − <sup>6</sup>*Kσ*<sup>2</sup>

*a*2 

> =

For a set of non-interacting fluids with equations of state *wi* = *Pi*/*ρi*, we have that <sup>Ω</sup> = <sup>Ω</sup>(*ρi*, *wi*) and, therefore, <sup>Ω</sup>˙ <sup>=</sup> <sup>∑</sup>*<sup>i</sup>* <sup>Ω</sup>*ρ<sup>i</sup> <sup>ρ</sup>*˙*i*, where <sup>Ω</sup>*ρ<sup>i</sup>* <sup>≡</sup> *<sup>∂</sup>*Ω/*∂ρi*. Since for those fluids the conservation equation is *<sup>ρ</sup>*˙*<sup>i</sup>* <sup>=</sup> <sup>−</sup>3*θ*(<sup>1</sup> <sup>+</sup> *<sup>ω</sup>i*)*ρi*, we find that <sup>Ω</sup>˙ <sup>=</sup> <sup>−</sup>3*<sup>θ</sup>* <sup>∑</sup>*i*(<sup>1</sup> <sup>+</sup> *<sup>ω</sup>i*)*ρi*Ω*ρ<sup>i</sup>*

+ *σ*2

Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

+ *σ*2

<sup>2</sup> . (69)

http://dx.doi.org/10.5772/51807

<sup>2</sup> , (70)

Ω . (71)

. This implies that once a particular

<sup>2</sup> (72)

*f* + 2*κ*2*P*

*σ*1

. (73)

. With

173

where the constants *Cij* = −*Cji* satisfy the relation *<sup>C</sup>*<sup>12</sup> + *<sup>C</sup>*<sup>23</sup> + *<sup>C</sup>*<sup>31</sup> = 0, *<sup>V</sup>*<sup>0</sup> represents a reference volume, and *<sup>V</sup>*(*t*) = *<sup>V</sup>*0*a*1*a*2*a*<sup>3</sup> represents the volume of the universe. It is worth noting that writing explicitly the three equations (63) and combining them in pairs, one can write the individual Hubble rates as follows

$$\begin{aligned} H\_1 &= \theta + \frac{(\mathbf{C}\_{12} - \mathbf{C}\_{31})}{3\sigma\_1} \left(\frac{V\_0}{V(t)}\right) \\ H\_2 &= \theta + \frac{(\mathbf{C}\_{23} - \mathbf{C}\_{12})}{3\sigma\_1} \left(\frac{V\_0}{V(t)}\right) \\ H\_3 &= \theta + \frac{(\mathbf{C}\_{31} - \mathbf{C}\_{23})}{3\sigma\_1} \left(\frac{V\_0}{V(t)}\right) \end{aligned} \tag{64}$$

where *<sup>θ</sup>* is the expansion of a congruence of comoving observers and is defined as 3*<sup>θ</sup>* <sup>=</sup> <sup>∑</sup>*<sup>i</sup> Hi*. Using these relations, the shear *<sup>σ</sup>*<sup>2</sup> <sup>=</sup> <sup>∑</sup>*<sup>i</sup>* (*Hi* <sup>−</sup> *<sup>θ</sup>*) <sup>2</sup> of the congruence takes the form

$$
\sigma^2 = \frac{(\mathbb{C}\_{12}^2 + \mathbb{C}\_{23}^2 + \mathbb{C}\_{31}^2)}{9\sigma\_1^2} \left(\frac{V\_0}{V(t)}\right)^2,\tag{65}
$$

where we have used the relation (*C*<sup>12</sup> + *<sup>C</sup>*<sup>23</sup> + *<sup>C</sup>*31)<sup>2</sup> = 0.

### **5.3. Expansion**

We now derive an equation for the evolution of the expansion with time and a relation between expansion and shear. From previous results, one finds that

$$\mathcal{G}\_{tt}(h) \equiv -\frac{1}{2}\sum\_{k} H\_k^2 + \frac{1}{2}\left(\sum\_{k} H\_k\right)^2 + \frac{\dot{\Omega}}{\Omega} \sum\_{k} H\_k + \frac{3}{4}\left(\frac{\dot{\Omega}}{\Omega}\right)^2\tag{66}$$

In terms of the expansion and shear, this equation becomes

$$\mathbf{G}\_{ll} \equiv \Im \left( \theta + \frac{\dot{\Omega}}{2\Omega} \right)^2 - \frac{\sigma^2}{2} \,. \tag{67}$$

From the field equation (38), we find that

$$\mathbf{G}\_{tt} = \frac{f + \kappa^2(\rho + 3P)}{2\sigma\_1} \,\,\,\,\,\,\tag{68}$$

which in combination with (67) yields

16 Open Questions in Cosmology

**5.3. Expansion**

*Hij* = *Cij*

write the individual Hubble rates as follows

Using these relations, the shear *<sup>σ</sup>*<sup>2</sup> <sup>=</sup> <sup>∑</sup>*<sup>i</sup>* (*Hi* <sup>−</sup> *<sup>θ</sup>*)

*Gtt*(*h*) ≡ −<sup>1</sup>

From the field equation (38), we find that

*<sup>σ</sup>*<sup>2</sup> <sup>=</sup> (*C*<sup>2</sup>

between expansion and shear. From previous results, one finds that

*Gtt* ≡ 3

 *θ* + Ω˙ 2Ω 2 <sup>−</sup> *<sup>σ</sup>*<sup>2</sup>

*Gtt* <sup>=</sup> *<sup>f</sup>* <sup>+</sup> *<sup>κ</sup>*2(*<sup>ρ</sup>* <sup>+</sup> <sup>3</sup>*P*) 2*σ*<sup>1</sup>

2 ∑ *k H*2 *k* + 1 2 ∑ *k Hk* 2 + Ω˙ <sup>Ω</sup> ∑ *k*

In terms of the expansion and shear, this equation becomes

where we have used the relation (*C*<sup>12</sup> + *<sup>C</sup>*<sup>23</sup> + *<sup>C</sup>*31)<sup>2</sup> = 0.

*<sup>H</sup>*<sup>1</sup> <sup>=</sup> *<sup>θ</sup>* <sup>+</sup> (*C*<sup>12</sup> <sup>−</sup> *<sup>C</sup>*31)

*<sup>H</sup>*<sup>2</sup> <sup>=</sup> *<sup>θ</sup>* <sup>+</sup> (*C*<sup>23</sup> <sup>−</sup> *<sup>C</sup>*12)

*<sup>H</sup>*<sup>3</sup> <sup>=</sup> *<sup>θ</sup>* <sup>+</sup> (*C*<sup>31</sup> <sup>−</sup> *<sup>C</sup>*23)

<sup>12</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup>

9*σ*<sup>2</sup> 1

*Cij* (*a*1*a*2*a*3) <sup>=</sup> *Cij*

where the constants *Cij* = −*Cji* satisfy the relation *<sup>C</sup>*<sup>12</sup> + *<sup>C</sup>*<sup>23</sup> + *<sup>C</sup>*<sup>31</sup> = 0, *<sup>V</sup>*<sup>0</sup> represents a reference volume, and *<sup>V</sup>*(*t*) = *<sup>V</sup>*0*a*1*a*2*a*<sup>3</sup> represents the volume of the universe. It is worth noting that writing explicitly the three equations (63) and combining them in pairs, one can

3*σ*<sup>1</sup>

3*σ*<sup>1</sup>

3*σ*<sup>1</sup>

where *<sup>θ</sup>* is the expansion of a congruence of comoving observers and is defined as 3*<sup>θ</sup>* <sup>=</sup> <sup>∑</sup>*<sup>i</sup> Hi*.

<sup>23</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> 31)

We now derive an equation for the evolution of the expansion with time and a relation

*σ*1

 *V*<sup>0</sup> *V*(*t*) 

 *V*<sup>0</sup> *V*(*t*) 

 *V*<sup>0</sup> *V*(*t*) 

 *V*<sup>0</sup> *V*(*t*) 2

*Hk* + 3 4  Ω˙ Ω 2

<sup>2</sup> . (67)

, (68)

<sup>2</sup> of the congruence takes the form

, (65)

*V*0

*<sup>V</sup>*(*t*) , (63)

(64)

(66)

$$3\left(\theta + \frac{\dot{\Omega}}{2\Omega}\right)^2 = \frac{f + \kappa^2(\rho + 3P)}{2\sigma\_1} + \frac{\sigma^2}{2} \ . \tag{69}$$

For a set of non-interacting fluids with equations of state *wi* = *Pi*/*ρi*, we have that <sup>Ω</sup> = <sup>Ω</sup>(*ρi*, *wi*) and, therefore, <sup>Ω</sup>˙ <sup>=</sup> <sup>∑</sup>*<sup>i</sup>* <sup>Ω</sup>*ρ<sup>i</sup> <sup>ρ</sup>*˙*i*, where <sup>Ω</sup>*ρ<sup>i</sup>* <sup>≡</sup> *<sup>∂</sup>*Ω/*∂ρi*. Since for those fluids the conservation equation is *<sup>ρ</sup>*˙*<sup>i</sup>* <sup>=</sup> <sup>−</sup>3*θ*(<sup>1</sup> <sup>+</sup> *<sup>ω</sup>i*)*ρi*, we find that <sup>Ω</sup>˙ <sup>=</sup> <sup>−</sup>3*<sup>θ</sup>* <sup>∑</sup>*i*(<sup>1</sup> <sup>+</sup> *<sup>ω</sup>i*)*ρi*Ω*ρ<sup>i</sup>* . With this result, (69) can be written as

$$3\theta^2 \left(1 + \frac{3}{2}\Delta\_1\right)^2 = \frac{f + \kappa^2(\rho + 3P)}{2\sigma\_1} + \frac{\sigma^2}{2} \,' \,. \tag{70}$$

where we have defined

$$
\Delta\_1 = -\sum\_i (1 + w\_i) \rho\_i \frac{\partial\_{\rho\_l} \Omega}{\Omega} \,. \tag{71}
$$

Note that in this last equation *wi* = *wi*(*ρi*), i.e., they need not be constants. For fluids with constant *wi*, the conservation equation implies that their density depends on the volume of the universe according to *<sup>ρ</sup>i*(*t*) = *<sup>ρ</sup>i*(*t*0) *V*<sup>0</sup> *V*(*t*) <sup>1</sup>+*wi* . This implies that once a particular Lagrangian is specified, the equations of state *Pi* = *wiρ<sup>i</sup>* are given, and the anisotropy constants *Cij* are chosen, the right-hand side of Eqs. (65) and (70) can be parametrized in terms of *<sup>V</sup>*(*t*). This, in turn, allows us to parametrize the *Hi* functions of (64) in terms of *V*(*t*) as well. This will be very useful later for our discussion of particular models.

In the isotropic case (*σ*<sup>2</sup> = 0 , *θ* = *a*˙/*a* ≡ H) with nonzero spatial curvature, (70) takes the following form:

$$\mathcal{H}^2 = \frac{1}{6\sigma\_1} \frac{\left[f + \kappa^2(\rho + 3P) - \frac{6K\sigma\_1}{a^2}\right]}{\left[1 + \frac{3}{2}\Delta\_1\right]^2} \tag{72}$$

The evolution equation for the expansion can be obtained by noting that the *Rij* equations, which are of the form *Rij* ≡ (Ω/2*S*)*gij* [...] = (*<sup>f</sup>* /2 + *<sup>κ</sup>*2*P*)*gij*/*σ*2, can be summed up to give

$$2(\dot{\theta} + 3\theta^2) + \theta \left(\frac{6\dot{\Omega}}{\Omega} - \frac{\dot{S}}{S}\right) + \left\{\frac{\ddot{\Omega}}{\Omega} + \frac{1}{2}\frac{\dot{\Omega}}{\Omega} \left(\frac{\dot{\Omega}}{\Omega} - \frac{\dot{S}}{S}\right)\right\} = \frac{\left[f + 2\kappa^2 P\right]}{\sigma\_1} \,. \tag{73}$$

## **5.4. Limit to** *f*(*R*)

We now consider the limit *fQ* → 0, namely, the case in which the Lagrangian only depends on the Ricci scalar *R*. Doing this we will obtain the corresponding equations for shear and expansion in the *f*(*R*) case without the need for extra work. From the definitions of *λ*<sup>2</sup> (see below eq.(43)), and *<sup>σ</sup>*<sup>1</sup> and *<sup>σ</sup>*<sup>2</sup> in (47), it is easy to see that in the limit *fQ* → 0 we get

$$
\sigma\_1 \to \sigma\_2 \to f\_R \tag{74}
$$

*<sup>θ</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> 9 *V*˙ *V* 2 = 

where *<sup>ρ</sup><sup>d</sup>* <sup>=</sup> *<sup>ρ</sup>d*,0 *<sup>V</sup>*<sup>0</sup>

on the isotropic case, *C*<sup>2</sup>

*ρ<sup>d</sup>* reaches the value *ρ<sup>B</sup>*

density *<sup>ρ</sup><sup>B</sup>* = *<sup>ρ</sup><sup>P</sup>*

equation of state given by

introductory work).

*wrad eff* <sup>=</sup> <sup>1</sup>

<sup>3</sup> <sup>−</sup> <sup>5</sup>*α*<sup>2</sup> 18*π*<sup>2</sup>  *Nc* + <sup>5</sup> <sup>4</sup> *Nf*

*V*(*t*)  *<sup>ρ</sup><sup>d</sup>* <sup>+</sup> *<sup>ρ</sup><sup>r</sup>* <sup>+</sup> *<sup>a</sup>ρ<sup>d</sup>*

3 1 − *<sup>a</sup>ρ<sup>d</sup> ρP*

and *<sup>ρ</sup><sup>r</sup>* <sup>=</sup> *<sup>ρ</sup>r*,0 *<sup>V</sup>*<sup>0</sup>

<sup>23</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup>

<sup>12</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup>

2*ρ<sup>P</sup>*

*V*(*t*) 4/3 . In general, an homogeneous cosmological model experiences a bounce when the expansion *θ* vanishes, which implies an extremum (a maximum or a minimum) of the volume of the Universe. If *V*(*t*) vanishes at some finite time, then a big bang or big crunch singularity is found, depending on whether *V*˙ > 0 or *V*˙ < 0 at that time. Focusing for the moment

 <sup>1</sup> + <sup>2</sup>*aρ<sup>d</sup> ρP* 

*fR* = <sup>1</sup> − <sup>2</sup>*aρd*/*ρ<sup>P</sup>* = 0. This condition, *fR* = 0, characterizes the location of the bounce in Palatini *f*(*R*) theories with a single fluid with constant equation of state [5]. For our quadratic model, in particular, bouncing solutions exist if the dynamics allows to reach the

initial singularity, whereas for *a* < 0 it takes *w* < 1/3. The case *a* = 0 naturally recovers the equations of GR. It is worth noting that a cosmic bounce may arise even for presureless matter, *w* = 0, if *a* < 0, which implies that exotic sources of matter-energy that violate the energy conditions are not necessary to avoid the big bang singularity in this framework. The reason for this is that at high energies gravitation may become repulsive for matter sources with *w* > −1, whereas it is attractive at low energy-densities for those same sources. Note also that the pure radiation universe, *w* = 1/3, is a peculiar case because it does not produce any modified dynamics in Palatini *f*(*R*) theories. On physical grounds, however, it should be noted that due to quantum effects related with the trace anomaly of the electromagnetic field, a gas of photons in a *SU*(*N*) gauge theory with *Nf* fermion flavors has an effective

<sup>2</sup> <sup>+</sup> (*C*<sup>2</sup>

<sup>12</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup>

<sup>54</sup>

*<sup>d</sup>* <sup>=</sup> *<sup>ρ</sup>P*/(2|*a*|) [see Fig.1)] . This value of the density implies that

<sup>2</sup>*a*(3*w*−1) <sup>&</sup>gt; 0. This means that for *<sup>a</sup>* <sup>&</sup>gt; 0 fluids with *<sup>w</sup>* <sup>&</sup>gt; 1/3 avoid the

11

2 + <sup>7</sup> 2 *NcNf N*<sup>2</sup> *<sup>c</sup>* <sup>−</sup><sup>1</sup>

where *Nc* is the color number of the gauge theory (which has *Nc*(*Nc* − 1) generators) [13], [17]. Therefore, a universe filled with photons should be able to avoid the singularity if *a* > 0. In physically realistic scenarios, one should consider the co-existence of several fluids and take into account the time dependence of the number of effective degrees of freedom and the transfer of energy among different species [15], which leads to the possibility of having different effective fluids at different stages of the cosmic expansion. In this sense, the "dust plus radiation" model represented by (80) needs not be accurate at all times because dust particles may become relativistic at high energies and contribute to *ρ<sup>r</sup>* rather than to *ρd*. This suggests that the choice/determination of the sign of the parameter *a* is not a trivial issue and would require a very careful and elaborate analysis (which goes beyond the scope of this

When anisotropies are taken into account, one finds that bouncing solutions are still possible as long as the amount of anisotropy is not too large. In Fig.2, we see that increasing the

<sup>3</sup> *Nc* <sup>−</sup> <sup>2</sup>

<sup>3</sup> *Nf* 

, (81)

<sup>23</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> 31)

 *V*<sup>0</sup> *V*(*t*) 2

http://dx.doi.org/10.5772/51807

, (80)

175

1 − *<sup>a</sup>ρ<sup>d</sup> ρP* 2

Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

<sup>31</sup> <sup>=</sup> 0, we find that a bounce occurs if *<sup>a</sup>* <sup>&</sup>lt; 0 when

$$S \to \Omega \to f\_{\mathbb{R}} \,. \tag{75}$$

With these rules it is easy to see that *<sup>h</sup>µν* = *fRgµν*, which makes (38) boil down to the expected field equations for Palatini *<sup>f</sup>*(*R*) theories, namely, *fRRµν*(*h*) <sup>−</sup> *<sup>f</sup>* <sup>2</sup> *<sup>g</sup>µν* <sup>=</sup> *<sup>κ</sup>*2*Tµν*. Equation (63) turns into

$$H\_{ij} = \frac{C\_{ij}}{f\_{\mathbb{R}}} \frac{V\_0}{V(t)},\tag{76}$$

from which one can easily obtain expressions for *H*1, *H*<sup>2</sup> and *H*<sup>3</sup> as in (64). The shear becomes

$$
\sigma^2 = \frac{(\mathcal{C}\_{12}^2 + \mathcal{C}\_{23}^2 + \mathcal{C}\_{31}^2)}{9f\_R^2} \left(\frac{V\_0}{V(t)}\right)^2,\tag{77}
$$

where *<sup>C</sup>*<sup>12</sup> + *<sup>C</sup>*<sup>23</sup> + *<sup>C</sup>*<sup>31</sup> = 0. The relation between expansion and shear for a collection of non-interacting perfect fluids now becomes

$$3\theta^2 \left(1 + \frac{3}{2}\tilde{\Lambda}\_1\right)^2 = \frac{f + \kappa^2(\rho + 3P)}{2f\_R} + \frac{\sigma^2}{2} \tag{78}$$

where ∆˜ <sup>1</sup> is given by (71) but with Ω replaced by *fR*. In the isotropic case with nonzero *K* we find

$$\mathcal{H}^2 = \frac{1}{6f\_R} \frac{\left[f + \kappa^2(\rho + 3P) - \frac{6Kf\_R}{a^2}\right]}{\left[1 + \frac{3}{2}\tilde{\Delta}\_1\right]^2} \,. \tag{79}$$

## **5.5. Bouncing** *f*(*R*) **cosmologies**

We now present the cosmological dynamics of simple *f*(*R*) models to illustrate how this family of theories modifies the standard Big Bang picture of the early universe. Consider, for instance, the model2 *<sup>f</sup>*(*R*) = *<sup>R</sup>* + *aR*2/*RP*, where *RP* = *<sup>l</sup>* −2 *<sup>P</sup>* <sup>=</sup> *<sup>c</sup>*3/¯*hG* is the Planck curvature. From the trace equation *R fR* − <sup>2</sup> *<sup>f</sup>* = *<sup>κ</sup>*2*<sup>T</sup>* (see Sec.4.1), we find that this model leads to the same relation between the matter and the scalar curvature as in GR, namely, *R* = −*κ*2*T*. This implies that the theory behaves as GR whenever the energy density is much smaller than the Planck density scale *<sup>ρ</sup><sup>P</sup>* ≡ *RP*/*κ*2. Since by definition *<sup>θ</sup>* = <sup>1</sup> <sup>3</sup> <sup>∑</sup>*<sup>i</sup> a*˙*i ai* <sup>=</sup> <sup>1</sup> 3 *d dt* ln *<sup>a</sup>*1*a*2*a*<sup>3</sup> <sup>=</sup> <sup>1</sup> 3 *V*˙ *V* , where *<sup>V</sup>* = *<sup>V</sup>*0*a*1*a*2*a*<sup>3</sup> represents the volume of the universe (with *<sup>V</sup>*<sup>0</sup> = *<sup>V</sup>*(*t*0)), Eq. (78) for this quadratic model with dust and radiation leads to

<sup>2</sup> Note that the constant *a* could be absorbed into a redefinition of *RP* and, therefore, only its sign is relevant.

$$\theta^2 = \frac{1}{9} \left(\frac{\mathcal{V}}{\mathcal{V}}\right)^2 = \frac{\left(\rho\_d + \rho\_r + \frac{a\rho\_d}{2\rho\_p}\right)\left(1 + \frac{2a\rho\_d}{\rho\_p}\right)}{3\left(1 - \frac{a\rho\_d}{\rho\_p}\right)^2} + \frac{\left(\mathcal{C}\_{12}^2 + \mathcal{C}\_{23}^2 + \mathcal{C}\_{31}^2\right)}{54\left(1 - \frac{a\rho\_d}{\rho\_p}\right)^2} \left(\frac{V\_0}{V(t)}\right)^2,\tag{80}$$

where *<sup>ρ</sup><sup>d</sup>* <sup>=</sup> *<sup>ρ</sup>d*,0 *<sup>V</sup>*<sup>0</sup> *V*(*t*) and *<sup>ρ</sup><sup>r</sup>* <sup>=</sup> *<sup>ρ</sup>r*,0 *<sup>V</sup>*<sup>0</sup> *V*(*t*) 4/3 . In general, an homogeneous cosmological model experiences a bounce when the expansion *θ* vanishes, which implies an extremum (a maximum or a minimum) of the volume of the Universe. If *V*(*t*) vanishes at some finite time, then a big bang or big crunch singularity is found, depending on whether *V*˙ > 0 or *V*˙ < 0 at that time. Focusing for the moment on the isotropic case, *C*<sup>2</sup> <sup>12</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> <sup>23</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> <sup>31</sup> <sup>=</sup> 0, we find that a bounce occurs if *<sup>a</sup>* <sup>&</sup>lt; 0 when *ρ<sup>d</sup>* reaches the value *ρ<sup>B</sup> <sup>d</sup>* <sup>=</sup> *<sup>ρ</sup>P*/(2|*a*|) [see Fig.1)] . This value of the density implies that *fR* = <sup>1</sup> − <sup>2</sup>*aρd*/*ρ<sup>P</sup>* = 0. This condition, *fR* = 0, characterizes the location of the bounce in Palatini *f*(*R*) theories with a single fluid with constant equation of state [5]. For our quadratic model, in particular, bouncing solutions exist if the dynamics allows to reach the density *<sup>ρ</sup><sup>B</sup>* = *<sup>ρ</sup><sup>P</sup>* <sup>2</sup>*a*(3*w*−1) <sup>&</sup>gt; 0. This means that for *<sup>a</sup>* <sup>&</sup>gt; 0 fluids with *<sup>w</sup>* <sup>&</sup>gt; 1/3 avoid the initial singularity, whereas for *a* < 0 it takes *w* < 1/3. The case *a* = 0 naturally recovers the equations of GR. It is worth noting that a cosmic bounce may arise even for presureless matter, *w* = 0, if *a* < 0, which implies that exotic sources of matter-energy that violate the energy conditions are not necessary to avoid the big bang singularity in this framework. The reason for this is that at high energies gravitation may become repulsive for matter sources with *w* > −1, whereas it is attractive at low energy-densities for those same sources. Note also that the pure radiation universe, *w* = 1/3, is a peculiar case because it does not produce any modified dynamics in Palatini *f*(*R*) theories. On physical grounds, however, it should be noted that due to quantum effects related with the trace anomaly of the electromagnetic field, a gas of photons in a *SU*(*N*) gauge theory with *Nf* fermion flavors has an effective equation of state given by

18 Open Questions in Cosmology

turns into

we find

expansion in the *f*(*R*) case without the need for extra work. From the definitions of *λ*<sup>2</sup> (see

With these rules it is easy to see that *<sup>h</sup>µν* = *fRgµν*, which makes (38) boil down to the expected

from which one can easily obtain expressions for *H*1, *H*<sup>2</sup> and *H*<sup>3</sup> as in (64). The shear becomes

<sup>23</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> 31)

where *<sup>C</sup>*<sup>12</sup> + *<sup>C</sup>*<sup>23</sup> + *<sup>C</sup>*<sup>31</sup> = 0. The relation between expansion and shear for a collection of

where ∆˜ <sup>1</sup> is given by (71) but with Ω replaced by *fR*. In the isotropic case with nonzero *K*

 1 + <sup>3</sup> <sup>2</sup>∆˜ <sup>1</sup>

We now present the cosmological dynamics of simple *f*(*R*) models to illustrate how this family of theories modifies the standard Big Bang picture of the early universe. Consider, for

From the trace equation *R fR* − <sup>2</sup> *<sup>f</sup>* = *<sup>κ</sup>*2*<sup>T</sup>* (see Sec.4.1), we find that this model leads to the same relation between the matter and the scalar curvature as in GR, namely, *R* = −*κ*2*T*. This implies that the theory behaves as GR whenever the energy density is much smaller than

where *<sup>V</sup>* = *<sup>V</sup>*0*a*1*a*2*a*<sup>3</sup> represents the volume of the universe (with *<sup>V</sup>*<sup>0</sup> = *<sup>V</sup>*(*t*0)), Eq. (78) for

<sup>2</sup> Note that the constant *a* could be absorbed into a redefinition of *RP* and, therefore, only its sign is relevant.

<sup>=</sup> *<sup>f</sup>* <sup>+</sup> *<sup>κ</sup>*2(*<sup>ρ</sup>* <sup>+</sup> <sup>3</sup>*P*) 2 *fR*

*<sup>f</sup>* + *<sup>κ</sup>*2(*<sup>ρ</sup>* + <sup>3</sup>*P*) − <sup>6</sup>*K fR*

*V*0 *V*(*t*)

> *V*<sup>0</sup> *V*(*t*)

2

+ *σ*2

*a*2 

−2

<sup>3</sup> <sup>∑</sup>*<sup>i</sup> a*˙*i ai* <sup>=</sup> <sup>1</sup> 3 *d*

*Hij* <sup>=</sup> *Cij fR*

<sup>12</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup>

9 *f* <sup>2</sup> *R*

*<sup>σ</sup>*<sup>1</sup> → *<sup>σ</sup>*<sup>2</sup> → *fR* (74) *<sup>S</sup>* → <sup>Ω</sup> → *fR* . (75)

<sup>2</sup> *<sup>g</sup>µν* <sup>=</sup> *<sup>κ</sup>*2*Tµν*. Equation (63)

, (77)

<sup>2</sup> (78)

<sup>2</sup> . (79)

*<sup>P</sup>* <sup>=</sup> *<sup>c</sup>*3/¯*hG* is the Planck curvature.

*dt* ln *<sup>a</sup>*1*a*2*a*<sup>3</sup> <sup>=</sup> <sup>1</sup>

3 *V*˙ *V* ,

, (76)

below eq.(43)), and *<sup>σ</sup>*<sup>1</sup> and *<sup>σ</sup>*<sup>2</sup> in (47), it is easy to see that in the limit *fQ* → 0 we get

field equations for Palatini *<sup>f</sup>*(*R*) theories, namely, *fRRµν*(*h*) <sup>−</sup> *<sup>f</sup>*

*<sup>σ</sup>*<sup>2</sup> <sup>=</sup> (*C*<sup>2</sup>

non-interacting perfect fluids now becomes

**5.5. Bouncing** *f*(*R*) **cosmologies**

3*θ*<sup>2</sup> 1 + 3 2 ∆˜ 1 2

instance, the model2 *<sup>f</sup>*(*R*) = *<sup>R</sup>* + *aR*2/*RP*, where *RP* = *<sup>l</sup>*

this quadratic model with dust and radiation leads to

the Planck density scale *<sup>ρ</sup><sup>P</sup>* ≡ *RP*/*κ*2. Since by definition *<sup>θ</sup>* = <sup>1</sup>

H<sup>2</sup> = 1 6 *fR* 
$$w\_{eff}^{rad} = \frac{1}{3} - \frac{5a^2}{18\pi^2} \frac{\left(N\_c + \frac{5}{4}N\_f\right)\left(\frac{11}{3}N\_c - \frac{2}{3}N\_f\right)}{2 + \frac{7}{2}\frac{N\_c N\_f}{N\_c^2 - 1}}\,,\tag{81}$$

where *Nc* is the color number of the gauge theory (which has *Nc*(*Nc* − 1) generators) [13], [17]. Therefore, a universe filled with photons should be able to avoid the singularity if *a* > 0. In physically realistic scenarios, one should consider the co-existence of several fluids and take into account the time dependence of the number of effective degrees of freedom and the transfer of energy among different species [15], which leads to the possibility of having different effective fluids at different stages of the cosmic expansion. In this sense, the "dust plus radiation" model represented by (80) needs not be accurate at all times because dust particles may become relativistic at high energies and contribute to *ρ<sup>r</sup>* rather than to *ρd*. This suggests that the choice/determination of the sign of the parameter *a* is not a trivial issue and would require a very careful and elaborate analysis (which goes beyond the scope of this introductory work).

When anisotropies are taken into account, one finds that bouncing solutions are still possible as long as the amount of anisotropy is not too large. In Fig.2, we see that increasing the

the crossing through *fR* = 0 implies a divergence in some curvature scalars of the theory. Whether this divergence is a true (or strong) physical singularity in the sense defined in [36], [8], [16] is an open question that will be explored elsewhere. In any case, we remark that the existence of that divergence does not have any effect on the time evolution of the expansion

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177

In the previous section we have seen that Palatini *f*(*R*) models are able to avoid the big bang singularity in idealized homogeneous and isotropic scenarios but run into trouble when anisotropies are present. The divergence of the shear is a generic problem for those *<sup>f</sup>*(*R*) theories in which the function *fR* vanishes at some point, regardless of the number and equation of state of the fluids involved. Though the nature of this divergence has not been identified yet with that of a strong singularity, which besides the divergence of some components of the Riemann, Ricci, and Weyl tensors also requires the divergence of some of their integrals, its very presence is a disturbing aspect that one would like to overcome within the framework of Palatini theories. In this sense, a natural step is to study the behavior in anisotropic scenarios of some simple generalization of the *f*(*R*) family to see if the situation improves. Using Lagrangians of the form presented in (4.1), we will show next that completely regular bouncing solutions exist for both isotropic and anisotropic

**5.6. Nonsingular universes in** *f*(*R*, *Q*) **Palatini theories**

Consider Eq.(72) particularized to the following *f*(*R*, *Q*) Lagrangian

*f*(*R*, *Q*) = *R* + *a*

Eq.(51) establishes that there may exist a maximum for the combination *ρ* + *P*.

limit and GR at low curvatures, we must take the positive sign, i.e., *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> *<sup>σ</sup>*<sup>+</sup>

some high density. Beyond that point, we may need to switch from *<sup>σ</sup>*<sup>+</sup>

*R*2 *RP*

For this theory, we find that *<sup>R</sup>* = *<sup>κ</sup>*2(*<sup>ρ</sup>* − <sup>3</sup>*P*) and *<sup>Q</sup>* = *<sup>Q</sup>*(*ρ*, *<sup>P</sup>*) is given by (51) with *<sup>α</sup>* ≡ *<sup>b</sup>*/*RP*. From now on we assume that the parameter *b* of the Lagrangian is positive and has been absorbed into a redefinition of *RP*, which is assumed positive. This restriction is necessary (though not sufficient) if one wants the scalar *Q* to be bounded from above when fluids with *<sup>w</sup>* > −1 are considered. Stated differently, when *<sup>b</sup>*/*RP* > 0, positivity of the square root of

In order to have (72) well defined, one must make sure that the choice of sign in front of the square root of *<sup>σ</sup>*<sup>1</sup> in (47) is the correct one. In this sense, we find that to recover the *<sup>f</sup>*(*R*)

when considering particular models, which are characterized by the constant *a* and, for instance, a constant equation of state *w*, one realizes that the square root may reach a zero at

that *σ*<sup>1</sup> is a continuous and differentiable function (see Fig.3 for an illustration of this point). Bearing in mind this technical subtlety, one can then proceed to represent the Hubble function for different choices of parameters and fluid combinations to determine whether bouncing

<sup>+</sup> *<sup>b</sup> <sup>Q</sup> RP*

(82)

<sup>1</sup> . However,

<sup>1</sup> to guarantee

<sup>1</sup> to *<sup>σ</sup>*<sup>−</sup>

*θ*, as can be seen in Fig.2.

homogeneous cosmologies.

*5.6.1. Isotropic universe*

solutions exist or not.

**Figure 1.** Representation of the Hubble function (left) and volume of the Universe (right) as a function of time for the model *f*(*R*) = *R* − *R*2/2*RP* in a universe filled with dust and radiation (for the numerical integration *ρd*,0 = 103*ρr*,0, and *V* = 105*V*0). The GR solutions corresponding to a contracting branch, which ends in a big crunch, and an expanding branch, which begins with a big bang, are represented together with the bouncing solution of the Palatini model that interpolates between those singular solutions.

**Figure 2.** Representation of the expansion (left) and volume of the Universe (right) as a function of time for the model *f*(*R*) = *R* − *R*2/2*RP* in a universe filled with dust and radiation with anisotropies (for the numerical integration *ρd*,0 = 103*ρr*,0, and *V* = 105*V*0). From right to left, we have plotted the bouncing cases *C*<sup>2</sup> = 0, 40, 40.60211073, 40.60211073942454489657, and the collapsing case with *C*<sup>2</sup> = 40.60211073942454489658. Fine tunning the value of *C*<sup>2</sup> even more should allow to keep the universe in its minimum for longer periods of time in the past, which eventually should lead to an asymptotically static solution.

value of *C*<sup>2</sup> ≡ *C*<sup>2</sup> <sup>12</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> <sup>23</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> <sup>31</sup> from zero, the volume of the universe presents a minimum as long as *C*<sup>2</sup> < *C*<sup>2</sup> *<sup>c</sup>* . If *<sup>C</sup>*<sup>2</sup> > *<sup>C</sup>*<sup>2</sup> *<sup>c</sup>* , the collapse is unavoidable and *<sup>V</sup>* <sup>→</sup> 0 in a finite time. The critical case *C*<sup>2</sup> → *C*<sup>2</sup> *<sup>c</sup>* represents a configuration that is neither a bouncing universe nor a big bang. It corresponds to a state in which the volume of the universe remains constant in the past and expands in the future. Though this solution is clearly unstable and fine-tuned, its existence puts forward the possibility of obtaining static regular solutions corresponding to ultracompact objects, which could shed new light on the internal structure of black holes and/or topological deffects when Planck scale corrections to the gravitational action are taken into account. It should be noted, however, that in order to obtain this asymptotically static solution one must cross from the domain where *fR* > 0 to the region where *fR* < 0. Since the shear, as defined in (77) for *f*(*R*) theories with perfect fluids, is proportional to 1/ *f* <sup>2</sup> *R*, the crossing through *fR* = 0 implies a divergence in some curvature scalars of the theory. Whether this divergence is a true (or strong) physical singularity in the sense defined in [36], [8], [16] is an open question that will be explored elsewhere. In any case, we remark that the existence of that divergence does not have any effect on the time evolution of the expansion *θ*, as can be seen in Fig.2.

## **5.6. Nonsingular universes in** *f*(*R*, *Q*) **Palatini theories**

In the previous section we have seen that Palatini *f*(*R*) models are able to avoid the big bang singularity in idealized homogeneous and isotropic scenarios but run into trouble when anisotropies are present. The divergence of the shear is a generic problem for those *<sup>f</sup>*(*R*) theories in which the function *fR* vanishes at some point, regardless of the number and equation of state of the fluids involved. Though the nature of this divergence has not been identified yet with that of a strong singularity, which besides the divergence of some components of the Riemann, Ricci, and Weyl tensors also requires the divergence of some of their integrals, its very presence is a disturbing aspect that one would like to overcome within the framework of Palatini theories. In this sense, a natural step is to study the behavior in anisotropic scenarios of some simple generalization of the *f*(*R*) family to see if the situation improves. Using Lagrangians of the form presented in (4.1), we will show next that completely regular bouncing solutions exist for both isotropic and anisotropic homogeneous cosmologies.

#### *5.6.1. Isotropic universe*

20 Open Questions in Cosmology

GR

0.4

singular solutions.

0.3 0.2 0.1

solution.

value of *C*<sup>2</sup> ≡ *C*<sup>2</sup>

long as *C*<sup>2</sup> < *C*<sup>2</sup>

critical case *C*<sup>2</sup> → *C*<sup>2</sup>

<sup>12</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup>

<sup>23</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup>

*<sup>c</sup>* . If *<sup>C</sup>*<sup>2</sup> > *<sup>C</sup>*<sup>2</sup>

0.1 0.2 0.3 Ht

0.2

0.2

0.4 Ht

GR

<sup>375</sup> <sup>370</sup> <sup>365</sup> <sup>360</sup> <sup>355</sup> 350t

<sup>400</sup> <sup>390</sup> <sup>380</sup> <sup>370</sup> <sup>360</sup> 350t

2 *RP*

Anisotropic Bounce in fRR *<sup>R</sup>*<sup>2</sup>

GR

Bounce

<sup>410</sup> <sup>400</sup> <sup>390</sup> <sup>380</sup> <sup>370</sup> 360t

<sup>31</sup> from zero, the volume of the universe presents a minimum as

*<sup>c</sup>* , the collapse is unavoidable and *<sup>V</sup>* <sup>→</sup> 0 in a finite time. The

*<sup>c</sup>* represents a configuration that is neither a bouncing universe nor a

Anisotropic Bounce in fRR *<sup>R</sup>*<sup>2</sup>

1

*R*,

2

3

4

5 Vt

2 *RP*

5

**Figure 1.** Representation of the Hubble function (left) and volume of the Universe (right) as a function of time for the model *f*(*R*) = *R* − *R*2/2*RP* in a universe filled with dust and radiation (for the numerical integration *ρd*,0 = 103*ρr*,0, and *V* = 105*V*0). The GR solutions corresponding to a contracting branch, which ends in a big crunch, and an expanding branch, which begins with a big bang, are represented together with the bouncing solution of the Palatini model that interpolates between those

**Figure 2.** Representation of the expansion (left) and volume of the Universe (right) as a function of time for the model *f*(*R*) = *R* − *R*2/2*RP* in a universe filled with dust and radiation with anisotropies (for the numerical integration *ρd*,0 = 103*ρr*,0, and *V* = 105*V*0). From right to left, we have plotted the bouncing cases *C*<sup>2</sup> = 0, 40, 40.60211073, 40.60211073942454489657, and the collapsing case with *C*<sup>2</sup> = 40.60211073942454489658. Fine tunning the value of *C*<sup>2</sup> even more should allow to keep the universe in its minimum for longer periods of time in the past, which eventually should lead to an asymptotically static

big bang. It corresponds to a state in which the volume of the universe remains constant in the past and expands in the future. Though this solution is clearly unstable and fine-tuned, its existence puts forward the possibility of obtaining static regular solutions corresponding to ultracompact objects, which could shed new light on the internal structure of black holes and/or topological deffects when Planck scale corrections to the gravitational action are taken into account. It should be noted, however, that in order to obtain this asymptotically static solution one must cross from the domain where *fR* > 0 to the region where *fR* < 0. Since the shear, as defined in (77) for *f*(*R*) theories with perfect fluids, is proportional to 1/ *f* <sup>2</sup>

10

15

20

Vt

<sup>370</sup> <sup>368</sup> <sup>366</sup> <sup>364</sup> <sup>362</sup> 360t

Big Crunch Big Bang

<sup>2</sup> *RP* compared to GR

fRR *<sup>R</sup>*<sup>2</sup>

Bounce

<sup>2</sup> *RP* compared to GR

fRR *<sup>R</sup>*<sup>2</sup>

Consider Eq.(72) particularized to the following *f*(*R*, *Q*) Lagrangian

$$f(R, \mathbb{Q}) = R + a \frac{R^2}{R\_P} + b \frac{\mathbb{Q}}{R\_P} \tag{82}$$

For this theory, we find that *<sup>R</sup>* = *<sup>κ</sup>*2(*<sup>ρ</sup>* − <sup>3</sup>*P*) and *<sup>Q</sup>* = *<sup>Q</sup>*(*ρ*, *<sup>P</sup>*) is given by (51) with *<sup>α</sup>* ≡ *<sup>b</sup>*/*RP*. From now on we assume that the parameter *b* of the Lagrangian is positive and has been absorbed into a redefinition of *RP*, which is assumed positive. This restriction is necessary (though not sufficient) if one wants the scalar *Q* to be bounded from above when fluids with *<sup>w</sup>* > −1 are considered. Stated differently, when *<sup>b</sup>*/*RP* > 0, positivity of the square root of Eq.(51) establishes that there may exist a maximum for the combination *ρ* + *P*.

In order to have (72) well defined, one must make sure that the choice of sign in front of the square root of *<sup>σ</sup>*<sup>1</sup> in (47) is the correct one. In this sense, we find that to recover the *<sup>f</sup>*(*R*) limit and GR at low curvatures, we must take the positive sign, i.e., *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> *<sup>σ</sup>*<sup>+</sup> <sup>1</sup> . However, when considering particular models, which are characterized by the constant *a* and, for instance, a constant equation of state *w*, one realizes that the square root may reach a zero at some high density. Beyond that point, we may need to switch from *<sup>σ</sup>*<sup>+</sup> <sup>1</sup> to *<sup>σ</sup>*<sup>−</sup> <sup>1</sup> to guarantee that *σ*<sup>1</sup> is a continuous and differentiable function (see Fig.3 for an illustration of this point). Bearing in mind this technical subtlety, one can then proceed to represent the Hubble function for different choices of parameters and fluid combinations to determine whether bouncing solutions exist or not.

**Figure 3.** Illustration of the need to combine the two branches of *σ*<sup>1</sup> to obtain a continuous and differentiable curve. The branch that starts at *σ*<sup>1</sup> = 1 has the plus sign in front of the square root (continuous green line). When the square root vanishes (at the blue dot), the function must be continued through the dashed red branch, which corresponds to the negative sign in front of the square root.

The classification of the bouncing solutions of the model (82) with a fluid with constant *w* was carried out in [5]. It was found that for every value of the parameter *a* there exist an infinite number of bouncing solutions, which depend on the particular equation of state *w*. The bouncing solutions can be divided into two large classes:

• **Class I:** *a* ≥ 0**.** The bounce occurs when the scalar *Q* reaches its maximum value and happens for all equations of state satisfying the condition

$$
\hbar w > w\_{\rm min} = \frac{a}{2 + 3a} \,\, . \tag{83}
$$

• **If** −1 ≤ *a* ≤ −1/3**.** In this case, one finds numerically that the bouncing solutions are

• **If** *a* ≤ −1**.** Similarly as the family *a* ≥ 0, this set of models also allows for a simple characterization of the bouncing solutions, which correspond to the interval −1 < *w* < *a*/(2 + 3*a*). In the limiting case *a* = −1 we obtain the condition −1 < *w* < 1 (compare

this with the numerical fit above, which gives −1 < *w* < 1.12).

*θ*<sup>2</sup> = *H*<sup>2</sup> +

1 6

where *H* represents the Hubble function in the *K* = 0 isotropic case. To better understand the behavior of *θ*2, let us consider when and why *H*<sup>2</sup> vanishes. Using the results of [5] summarized above, one finds that *H*<sup>2</sup> vanishes either when the density reaches the value *ρQmax* or when the function *σ*<sup>1</sup> vanishes. These two conditions imply a divergence in the

vanishing of *H*<sup>2</sup> (isotropic bounce). Technically, these two types of divergences can be easily characterized. From the definition of <sup>∆</sup><sup>1</sup> in (71), one can see that <sup>∆</sup><sup>1</sup> ∼ *∂ρ*Ω/Ω. Since <sup>Ω</sup> <sup>≡</sup> <sup>√</sup>*σ*1*σ*2, it is clear that <sup>∆</sup><sup>1</sup> diverges when *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> 0. The divergence due to reaching *<sup>ρ</sup>Qmax* is a bit more elaborate. One must note that *∂ρ*<sup>Ω</sup> contain terms that are finite plus a term of the form *∂ρλ*, with *λ* defined below Eq. (43). In this *λ* there is a *Q* term hidden in the function *<sup>f</sup>*(*R*, *<sup>Q</sup>*), which implies that *∂ρλ* ∼ *∂ρQ*/*RP* plus other finite terms. From the definition of *<sup>Q</sup>* it follows that *∂ρ<sup>Q</sup>* has finite contributions plus the term *∂ρ*Φ/

where <sup>Φ</sup> ≡ (<sup>1</sup> + (<sup>1</sup> + <sup>2</sup>*a*)*R*/*RP*)<sup>2</sup> − <sup>4</sup>*κ*2(*<sup>ρ</sup>* + *<sup>P</sup>*)/*RP*, which diverges when <sup>Φ</sup> vanishes. This divergence of *∂ρQ* indicates that *Q* cannot be extended beyond the maximum value *Qmax*. Now, since the shear goes like *<sup>σ</sup>*<sup>2</sup> ∼ 1/(*σ*1)<sup>2</sup> [see Eq.(65)], we see that the condition *<sup>σ</sup>*<sup>1</sup> = <sup>0</sup> implies a divergence on *σ*<sup>2</sup> (though *θ*<sup>2</sup> remains finite). This is exactly the same type of divergence that we already found in the *<sup>f</sup>*(*R*) models, where *<sup>σ</sup>*<sup>1</sup> → *fR*. Since in the *<sup>f</sup>*(*R*) models the bounce can only occur when *fR* = 0, there is no way to avoid the divergence of the shear in the anisotropic case within the *f*(*R*) setting. On the contrary, since the quadratic *<sup>f</sup>*(*R*, *<sup>Q</sup>*) model (82) allows for a second mechanism for the bounce, which takes place at *<sup>ρ</sup>Qmax* , there is a natural way out of the problem with the shear. Summarizing, we conclude that for universes governed by the Lagrangian (82) and containing a single stiff fluid there exist

*<sup>w</sup>*<sup>0</sup> < *<sup>w</sup>* < <sup>∞</sup> if −1/3 ≤ *<sup>a</sup>* ≤ 0, for *<sup>w</sup>*<sup>0</sup> < *<sup>w</sup>* < (*<sup>α</sup>* + *<sup>β</sup>a*)/(<sup>1</sup> + <sup>3</sup>*a*)<sup>2</sup> if −<sup>1</sup> ≤ *<sup>a</sup>* ≤ −1/3, and for −1/3 < *<sup>w</sup>* < *<sup>a</sup>*/(<sup>2</sup> + <sup>3</sup>*a*) if *<sup>a</sup>* ≤ −1, where *<sup>w</sup>*<sup>0</sup> < 0 is defined as the equation of state for which the (isotropic) bounce occurs when *<sup>Q</sup>* = *Qmax* and *<sup>σ</sup>*<sup>1</sup> = 0 simultaneously (see [5] for details). These results imply that for *a* < 0 the interval 0 ≤ *w* ≤ 1/3 is always included in the family of completely regular isotropic and anisotropic bouncing solutions, which contain the dust and radiation cases. For *a* ≥ 0, the radiation case is always nonsingular too.

completely regular bouncing solutions in the anisotropic case for *w* > *<sup>a</sup>*

*σ*2 (1 + <sup>3</sup>

<sup>2</sup>∆1)2, which appears in the denominator of *<sup>H</sup>*<sup>2</sup> and, therefore, force the

Using Eqs. (70) and (72), the expansion can be written as follows:

(1+3*a*)<sup>2</sup> <sup>&</sup>gt; 1, where *<sup>α</sup>* <sup>=</sup> 1.1335 and *<sup>β</sup>* <sup>=</sup> <sup>−</sup>3.3608.

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Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

2∆1)<sup>2</sup> , (85)

√Φ,

<sup>2</sup>+3*<sup>a</sup>* if *<sup>a</sup>* <sup>≥</sup> 0, for

restricted to the interval <sup>−</sup><sup>1</sup> <sup>&</sup>lt; *<sup>w</sup>* <sup>&</sup>lt; *<sup>α</sup>*+*β<sup>a</sup>*

**5.7. Anisotropic universe**

quantity (1 + <sup>3</sup>

From this equation it follows that a radiation dominated universe, with *w* = 1/3, always bounces for any *a* > 0.

	- **If** −1/4 < *a* ≤ 0**.** The bounce occurs if

$$-\frac{1}{3} + \frac{1}{3}\sqrt{\frac{1+4a}{1+a}} < w < \infty \tag{84}$$

We see that when *a* = 0 we find agreement with the discussion of case I. As *a* approaches the limiting value −1/4, the bouncing solutions extend up to *w* → −1/3.

• **If** −1/3 ≤ *a* ≤ −1/4**.** Numerically one finds that the bouncing solutions cannot be extended below *w* < −1 and occur if −1 < *w* < ∞, where *w* = −1 is excluded.


#### **5.7. Anisotropic universe**

22 Open Questions in Cosmology

0.2

front of the square root.

bounces for any *a* > 0.

0.4

0.6

0.8

1.0 Σ1

> <sup>a</sup>��<sup>1</sup> 2

> > <sup>w</sup>�<sup>1</sup> 3

<sup>2</sup> <sup>+</sup> <sup>3</sup>*<sup>a</sup>* . (83)

< *w* < ∞ (84)

<sup>1</sup> . To proceed with the

w�0

� and Σ<sup>1</sup> �

<sup>w</sup>��<sup>1</sup> 3

Discontinuity of Σ<sup>1</sup>

0.0 0.1 0.2 0.3 0.4 <sup>Ρ</sup>

**Figure 3.** Illustration of the need to combine the two branches of *σ*<sup>1</sup> to obtain a continuous and differentiable curve. The branch that starts at *σ*<sup>1</sup> = 1 has the plus sign in front of the square root (continuous green line). When the square root vanishes (at the blue dot), the function must be continued through the dashed red branch, which corresponds to the negative sign in

The classification of the bouncing solutions of the model (82) with a fluid with constant *w* was carried out in [5]. It was found that for every value of the parameter *a* there exist an infinite number of bouncing solutions, which depend on the particular equation of state *w*.

• **Class I:** *a* ≥ 0**.** The bounce occurs when the scalar *Q* reaches its maximum value and

*<sup>w</sup>* <sup>&</sup>gt; *wmin* <sup>=</sup> *<sup>a</sup>*

From this equation it follows that a radiation dominated universe, with *w* = 1/3, always

1 + 4*a* 1 + *a*

We see that when *a* = 0 we find agreement with the discussion of case I. As *a* approaches the limiting value −1/4, the bouncing solutions extend up to *w* → −1/3. • **If** −1/3 ≤ *a* ≤ −1/4**.** Numerically one finds that the bouncing solutions cannot be extended below *w* < −1 and occur if −1 < *w* < ∞, where *w* = −1 is excluded.

• **Class II:** *a* ≤ 0**.** This case is more involved because the bounce can occur either at the point where *Q* reaches its maximum or when *σ*<sup>1</sup> vanishes. This last case can only happen

The bouncing solutions can be divided into two large classes:

happens for all equations of state satisfying the condition

at high curvatures when we are in the branch defined by *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> *<sup>σ</sup>*<sup>−</sup>

− 1 3 + 1 3

classification, we divide this sector into several intervals:

• **If** −1/4 < *a* ≤ 0**.** The bounce occurs if

Using Eqs. (70) and (72), the expansion can be written as follows:

$$
\theta^2 = H^2 + \frac{1}{6} \frac{\sigma^2}{(1 + \frac{3}{2}\Delta\_1)^2} \,'\,\tag{85}
$$

where *H* represents the Hubble function in the *K* = 0 isotropic case. To better understand the behavior of *θ*2, let us consider when and why *H*<sup>2</sup> vanishes. Using the results of [5] summarized above, one finds that *H*<sup>2</sup> vanishes either when the density reaches the value *ρQmax* or when the function *σ*<sup>1</sup> vanishes. These two conditions imply a divergence in the quantity (1 + <sup>3</sup> <sup>2</sup>∆1)2, which appears in the denominator of *<sup>H</sup>*<sup>2</sup> and, therefore, force the vanishing of *H*<sup>2</sup> (isotropic bounce). Technically, these two types of divergences can be easily characterized. From the definition of <sup>∆</sup><sup>1</sup> in (71), one can see that <sup>∆</sup><sup>1</sup> ∼ *∂ρ*Ω/Ω. Since <sup>Ω</sup> <sup>≡</sup> <sup>√</sup>*σ*1*σ*2, it is clear that <sup>∆</sup><sup>1</sup> diverges when *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> 0. The divergence due to reaching *<sup>ρ</sup>Qmax* is a bit more elaborate. One must note that *∂ρ*<sup>Ω</sup> contain terms that are finite plus a term of the form *∂ρλ*, with *λ* defined below Eq. (43). In this *λ* there is a *Q* term hidden in the function *<sup>f</sup>*(*R*, *<sup>Q</sup>*), which implies that *∂ρλ* ∼ *∂ρQ*/*RP* plus other finite terms. From the definition of *<sup>Q</sup>* it follows that *∂ρ<sup>Q</sup>* has finite contributions plus the term *∂ρ*Φ/ √Φ, where <sup>Φ</sup> ≡ (<sup>1</sup> + (<sup>1</sup> + <sup>2</sup>*a*)*R*/*RP*)<sup>2</sup> − <sup>4</sup>*κ*2(*<sup>ρ</sup>* + *<sup>P</sup>*)/*RP*, which diverges when <sup>Φ</sup> vanishes. This divergence of *∂ρQ* indicates that *Q* cannot be extended beyond the maximum value *Qmax*. Now, since the shear goes like *<sup>σ</sup>*<sup>2</sup> ∼ 1/(*σ*1)<sup>2</sup> [see Eq.(65)], we see that the condition *<sup>σ</sup>*<sup>1</sup> = <sup>0</sup> implies a divergence on *σ*<sup>2</sup> (though *θ*<sup>2</sup> remains finite). This is exactly the same type of divergence that we already found in the *<sup>f</sup>*(*R*) models, where *<sup>σ</sup>*<sup>1</sup> → *fR*. Since in the *<sup>f</sup>*(*R*) models the bounce can only occur when *fR* = 0, there is no way to avoid the divergence of the shear in the anisotropic case within the *f*(*R*) setting. On the contrary, since the quadratic *<sup>f</sup>*(*R*, *<sup>Q</sup>*) model (82) allows for a second mechanism for the bounce, which takes place at *<sup>ρ</sup>Qmax* , there is a natural way out of the problem with the shear. Summarizing, we conclude that for universes governed by the Lagrangian (82) and containing a single stiff fluid there exist completely regular bouncing solutions in the anisotropic case for *w* > *<sup>a</sup>* <sup>2</sup>+3*<sup>a</sup>* if *<sup>a</sup>* <sup>≥</sup> 0, for *<sup>w</sup>*<sup>0</sup> < *<sup>w</sup>* < <sup>∞</sup> if −1/3 ≤ *<sup>a</sup>* ≤ 0, for *<sup>w</sup>*<sup>0</sup> < *<sup>w</sup>* < (*<sup>α</sup>* + *<sup>β</sup>a*)/(<sup>1</sup> + <sup>3</sup>*a*)<sup>2</sup> if −<sup>1</sup> ≤ *<sup>a</sup>* ≤ −1/3, and for −1/3 < *<sup>w</sup>* < *<sup>a</sup>*/(<sup>2</sup> + <sup>3</sup>*a*) if *<sup>a</sup>* ≤ −1, where *<sup>w</sup>*<sup>0</sup> < 0 is defined as the equation of state for which the (isotropic) bounce occurs when *<sup>Q</sup>* = *Qmax* and *<sup>σ</sup>*<sup>1</sup> = 0 simultaneously (see [5] for details). These results imply that for *a* < 0 the interval 0 ≤ *w* ≤ 1/3 is always included in the family of completely regular isotropic and anisotropic bouncing solutions, which contain the dust and radiation cases. For *a* ≥ 0, the radiation case is always nonsingular too.

**Figure 4.** Representation of the expansion squared (left) and volume of the Universe (right) as a function of time for the model *f*(*R*, *Q*) = *R* − *l* 2 *<sup>P</sup>R*2/2 <sup>+</sup> *<sup>l</sup>* 2 *<sup>P</sup><sup>Q</sup>* in a radiation universe with anisotropies. We have plotted the bouncing cases *<sup>C</sup>*<sup>2</sup> <sup>=</sup> 0, 50, 102, 103. Note that the bounce always occurs at the same maximum density (minimum volume). Note also that the time spent in the bouncing region decreases as the anisotropy grows. The starting point of the time integration is chosen such that at *t* = 0 the two branches of *σ*<sup>1</sup> coincide.

#### **5.8. Example: Radiation universe**

As an illustrative example, we consider here the particular case of a universe filled with radiation. Besides its obvious physical interest, this case leads to a number of algebraic simplifications that make more transparent the form of some basic definitions

$$Q = \frac{3R\_P^2}{8} \left[ 1 - \frac{8\kappa^2 \rho}{3R\_P} - \sqrt{1 - \frac{16\kappa^2 \rho}{3R\_P}} \right] \tag{86}$$

*C*2 �10<sup>3</sup>

Dust�Radiation �Vs� Pure Radiation

*C*2 �10<sup>2</sup>

Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

*C*2 �0

> 1 *V*

http://dx.doi.org/10.5772/51807

181

0.05 0.10 0.15 0.20 0.25

**Figure 5.** Comparison of the expansion in a universe filled with dust and radiation (*ρ*0,*rad* <sup>=</sup> <sup>10</sup>−3*ρ*0,*dust*) and a radiation

gravity based on a geometry in which the connection has been forced to be given by the Christoffel symbols of the metric admits an alternative formulation in which the form of the connection is dictated by the theory itself, i.e., it is not given by convention or selected on

In our exploration of Palatini theories, we have seen that assuming that metric and connection are independent geometrical objects has non-trivial effects on the resulting field equations as compared with the usual *metric* formulation of the same theories. For the particular family of *f*(*R*, *Q*) models studied here, we have seen that the metric is governed by second-order equations that boil down to GR in vacuum. This is in sharp contrast with the usual *metric* formulation of those same theories, where one finds fourth-order derivatives of the metric (see, for instance, [2] for a detailed analysis of the cosmology of the quadratic model (82) in metric formalism). The absence of higher-order derivatives in the Palatini formulation is a remarkable point that seems not to have been sufficiently appreciated in the literature. In fact, having second-order field equations is very important because it automatically implies the absence of ghosts and other dynamical instabilities. In this sense, it should be noted that Lovelock<sup>3</sup> theories [40], which are generally regarded as the natural extension of the Einstein-Hilbert Lagrangian to higher dimensions, have received a lot of attention in the literature because they are seen as the most general actions for gravity that give at most second-order field equations for the metric. As we have seen here, this property is shared (at least) by all Palatini theories of the *f*(*R*, *Q*) type (with or without torsion). This puts forward

that Palatini theories, are natural candidates to explore new dynamics beyond GR.

exactly the same as one finds in their usual *metric* formulation [7].

Before concluding, we would like to stress the fact that the quadratic Palatini model (82) is able to avoid the big bang singularity in very natural situations, such as in pure radiation, pure dust, or dust plus radiation universes with or without anisotropies (see Fig.5). This observation has been possible thanks to the formulas presented in section 5, where we have

<sup>3</sup> We would like to mention that when Lovelock theories are formulated à la Palatini, the resulting field equations are

0.02

practical grounds.

dominated universe (dashed lines) for several values of the anisotropy.

0.04

0.06

0.08

0.10

0.12 Θ2

$$
\sigma\_1^{\pm} = \frac{1}{2} \pm \frac{1}{2\sqrt{2}} \sqrt{5 - 3\sqrt{1 - \frac{16\kappa^2 \rho}{3R\_P}}} - \frac{24\kappa^2 \rho}{R\_P} \tag{87}
$$

$$
\sigma\_2 = \frac{1}{2} + \frac{1}{2\sqrt{2}}\sqrt{5 - 3\sqrt{1 - \frac{16\kappa^2 \rho}{3R\_P}} - \frac{8\kappa^2 \rho}{3R\_P}}\tag{88}
$$

Note that the coincidence of the two branches of *<sup>σ</sup>*<sup>1</sup> occurs at *<sup>κ</sup>*2*<sup>ρ</sup>* <sup>=</sup> *RP*/6, where *<sup>σ</sup>*<sup>±</sup> <sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> . It is easy to see that at low densities (86) leads to *<sup>Q</sup>* <sup>≈</sup> <sup>4</sup>(*κ*2*ρ*)2/3 <sup>+</sup> <sup>32</sup>(*κ*2*ρ*)3/9*RP* <sup>+</sup> 320(*κ*2*ρ*)4/27*R*<sup>2</sup> *<sup>P</sup>* <sup>+</sup> . . ., which recovers the expected result for GR, namely, *<sup>Q</sup>* <sup>=</sup> <sup>3</sup>*P*<sup>2</sup> <sup>+</sup> *<sup>ρ</sup>*2. From this formula we also see that the maximum value of *<sup>Q</sup>* occurs at *<sup>κ</sup>*2*ρmax* = <sup>3</sup>*RP*/16 and leads to *Qmax* = 3*R*<sup>2</sup> *<sup>P</sup>*/16. At this point the shear also takes its maximum allowed value, namely, *σ*<sup>2</sup> *max* <sup>=</sup> <sup>√</sup>3/16*R*3/2 *<sup>P</sup>* (*C*<sup>2</sup> <sup>12</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> <sup>23</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> <sup>31</sup>), which is always finite. At *<sup>ρ</sup>max* the expansion vanishes producing a cosmic bounce regardless of the amount of anisotropy [see Fig.4].

#### **6. Conclusions and open questions**

In this chapter we have tried to convey the idea that in the construction of extended theories of gravity, one should bear in mind the fact that metric and connection are equally fundamental and independent objects. This observation allows to broaden the spectrum of available possibilities to go beyond the *standard model of gravitation*. In fact, any theory of

24 Open Questions in Cosmology

Expansion in fR,Q�R�*lP*

0.02 0.04 0.06 0.08 0.10 0.12 Θ2

model *f*(*R*, *Q*) = *R* − *l*

320(*κ*2*ρ*)4/27*R*<sup>2</sup>

namely, *σ*<sup>2</sup>

and leads to *Qmax* = 3*R*<sup>2</sup>

*max* <sup>=</sup> <sup>√</sup>3/16*R*3/2

**6. Conclusions and open questions**

*C*2 �10<sup>3</sup>

2 *<sup>P</sup>R*2/2 <sup>+</sup> *<sup>l</sup>* 2

**5.8. Example: Radiation universe**

at *t* = 0 the two branches of *σ*<sup>1</sup> coincide.

*C*2 �102

*C*2 10<sup>3</sup> *C*<sup>2</sup> 10<sup>2</sup>

Volume in fR,Q<sup>R</sup>*lP*

*<sup>P</sup><sup>Q</sup>* in a radiation universe with anisotropies. We have plotted the bouncing cases *<sup>C</sup>*<sup>2</sup> <sup>=</sup>

<sup>1</sup> <sup>−</sup> <sup>16</sup>*κ*2*<sup>ρ</sup>* 3*RP*

<sup>1</sup> <sup>−</sup> <sup>16</sup>*κ*2*<sup>ρ</sup>* 3*RP*

<sup>1</sup> <sup>−</sup> <sup>16</sup>*κ*2*<sup>ρ</sup>* 3*RP*

*<sup>P</sup>* <sup>+</sup> . . ., which recovers the expected result for GR, namely, *<sup>Q</sup>* <sup>=</sup> <sup>3</sup>*P*<sup>2</sup> <sup>+</sup> *<sup>ρ</sup>*2.

*<sup>P</sup>*/16. At this point the shear also takes its maximum allowed value,

<sup>−</sup> <sup>24</sup>*κ*2*<sup>ρ</sup> RP*

<sup>−</sup> <sup>8</sup>*κ*2*<sup>ρ</sup>* 3*RP*

<sup>31</sup>), which is always finite. At *<sup>ρ</sup>max* the expansion

1.0 0.5 <sup>t</sup>

3.6 3.8

(86)

(87)

(88)

4.2 4.4 4.6 Vt

<sup>2</sup> aR2bQ

*C*2 50

> *C*2 0

**Figure 4.** Representation of the expansion squared (left) and volume of the Universe (right) as a function of time for the

0, 50, 102, 103. Note that the bounce always occurs at the same maximum density (minimum volume). Note also that the time spent in the bouncing region decreases as the anisotropy grows. The starting point of the time integration is chosen such that

As an illustrative example, we consider here the particular case of a universe filled with radiation. Besides its obvious physical interest, this case leads to a number of algebraic

simplifications that make more transparent the form of some basic definitions

<sup>1</sup> <sup>−</sup> <sup>8</sup>*κ*2*<sup>ρ</sup>* 3*RP* − �

����<sup>5</sup> <sup>−</sup> <sup>3</sup>

����<sup>5</sup> <sup>−</sup> <sup>3</sup>

Note that the coincidence of the two branches of *<sup>σ</sup>*<sup>1</sup> occurs at *<sup>κ</sup>*2*<sup>ρ</sup>* <sup>=</sup> *RP*/6, where *<sup>σ</sup>*<sup>±</sup> <sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> . It is easy to see that at low densities (86) leads to *<sup>Q</sup>* <sup>≈</sup> <sup>4</sup>(*κ*2*ρ*)2/3 <sup>+</sup> <sup>32</sup>(*κ*2*ρ*)3/9*RP* <sup>+</sup>

From this formula we also see that the maximum value of *<sup>Q</sup>* occurs at *<sup>κ</sup>*2*ρmax* = <sup>3</sup>*RP*/16

In this chapter we have tried to convey the idea that in the construction of extended theories of gravity, one should bear in mind the fact that metric and connection are equally fundamental and independent objects. This observation allows to broaden the spectrum of available possibilities to go beyond the *standard model of gravitation*. In fact, any theory of

vanishes producing a cosmic bounce regardless of the amount of anisotropy [see Fig.4].

�

�

<sup>2</sup> <sup>±</sup> <sup>1</sup> 2 √2

*C*2 �0

*<sup>Q</sup>* <sup>=</sup> <sup>3</sup>*R*<sup>2</sup> *P* 8

*σ*± <sup>1</sup> <sup>=</sup> <sup>1</sup>

*<sup>σ</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> 2 +

*<sup>P</sup>* (*C*<sup>2</sup>

<sup>12</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup>

<sup>23</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup>

<sup>2</sup> aR2�bQ with radiation

0.05 0.10 0.15 0.20 0.25 <sup>Ρ</sup>rad

**Figure 5.** Comparison of the expansion in a universe filled with dust and radiation (*ρ*0,*rad* <sup>=</sup> <sup>10</sup>−3*ρ*0,*dust*) and a radiation dominated universe (dashed lines) for several values of the anisotropy.

gravity based on a geometry in which the connection has been forced to be given by the Christoffel symbols of the metric admits an alternative formulation in which the form of the connection is dictated by the theory itself, i.e., it is not given by convention or selected on practical grounds.

In our exploration of Palatini theories, we have seen that assuming that metric and connection are independent geometrical objects has non-trivial effects on the resulting field equations as compared with the usual *metric* formulation of the same theories. For the particular family of *f*(*R*, *Q*) models studied here, we have seen that the metric is governed by second-order equations that boil down to GR in vacuum. This is in sharp contrast with the usual *metric* formulation of those same theories, where one finds fourth-order derivatives of the metric (see, for instance, [2] for a detailed analysis of the cosmology of the quadratic model (82) in metric formalism). The absence of higher-order derivatives in the Palatini formulation is a remarkable point that seems not to have been sufficiently appreciated in the literature. In fact, having second-order field equations is very important because it automatically implies the absence of ghosts and other dynamical instabilities. In this sense, it should be noted that Lovelock<sup>3</sup> theories [40], which are generally regarded as the natural extension of the Einstein-Hilbert Lagrangian to higher dimensions, have received a lot of attention in the literature because they are seen as the most general actions for gravity that give at most second-order field equations for the metric. As we have seen here, this property is shared (at least) by all Palatini theories of the *f*(*R*, *Q*) type (with or without torsion). This puts forward that Palatini theories, are natural candidates to explore new dynamics beyond GR.

Before concluding, we would like to stress the fact that the quadratic Palatini model (82) is able to avoid the big bang singularity in very natural situations, such as in pure radiation, pure dust, or dust plus radiation universes with or without anisotropies (see Fig.5). This observation has been possible thanks to the formulas presented in section 5, where we have

<sup>3</sup> We would like to mention that when Lovelock theories are formulated à la Palatini, the resulting field equations are exactly the same as one finds in their usual *metric* formulation [7].

extended the analysis carried out in [5] for a single perfect fluid with constant equation of state to include several perfect fluids with arbitrary equation of state *w*(*ρ*). This allows to explore the dynamics of realistic cosmological models with several fluids and is a necessary step prior to the consideration of the growth and evolution of inhomogeneities in these nonsingular backgrounds. Though the model (82) has been proposed on grounds of mathematical simplicity and motivated by the form of the effective action provided by perturbative quantization schemes in curved backgrounds, its ability to successfully deal with cosmological [5] and black hole singularities [27] as well as other aspects of quantum gravity phenomenology [26] demands further theoretical work to provide a more solid ground to it. In this sense, we note that the effective dynamics of loop quantum cosmology [4] in a Friedmann-Robertson-Walker background filled with a massless scalar can be exactly reproduced by a Palatini *f*(*R*) theory [22]. The extension of that result to more general spacetimes and matter sources could shed new light on the potential relation of (82) with a more fundamental theory of quantum gravity. All these open questions will be considered in detail elsewhere.

[10] Feigl, H. and Brodbeck, M. (eds.), *Readings in the phylosophy of science*,

Introduction to Palatini Theories of Gravity and Nonsingular Cosmologies

http://dx.doi.org/10.5772/51807

183

[11] Green, M. , Schwarz, J. , and Witten, E. , "Superstring Theory", Cambridge University

[12] Hehl, F. W., McCrea, J. D., Mielke, E. W., and Ne'eman, Y., Phys. Rept. 258, 1 (1995)

[13] Kajantie, K., Laine, M. , Rummukainen, K. and Schroder, Y. , Phys. Rev. D 67, 105008

[18] Liddle A. R. and Lyth D.H., *Cosmological inflation and large-scale structure*, Cambridge

[19] Lyth D.H. and Liddle A.R. The primordial density perturbation, Cambridge University

[23] Olmo, G. J. , Sanchis-Alepuz, H., and Tripathi, S., Phys. Rev. D80, 024013 (2009).

[27] Olmo, G. J. , and Rubiera-Garcia,D. , Phys.Rev. D86 (2012) 044014; Int.J.Mod.Phys. D21

[29] Parker L. and Toms D.J., *Quantum field theory in curved spacetime: quantized fields and*

[30] Poplawski, N. J., Phys. Lett. B 694, 181 (2010) [Erratum-ibid. B 701, 672 (2011)]

*gravity*, (Cambridge University Press, Cambridge, England, 2009).

[20] Nakahara, M. , *Geometry, Topology and Physics*, Adam Hilger, Bristol (1990).

[22] Olmo, G. J. , and Singh, P., JCAP 0901, 030 (2009) [arXiv:0806.2783 [gr-qc]].

[21] Novello, M. and Perez Bergliaffa, S.E. , *Phys.Rep.* 463, 127-213 (2008).

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[15] Kolb, E.W. and Turner, M.S. *The early universe*, Westview Press (1990).

Press, Cambridge, England (1987).

[14] Kibble, T. W. B., J. Math. Phys. 2, 212 (1961).

[16] Krolak, A. , Class. Quant. Grav. 3, 267 (1998).

University Press, (2000).

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(2012) 1250067; Eur.Phys.J. C72 (2012) 2098.

[arXiv:1007.0587 [astro-ph.CO]].

[28] Olmo, G. J. , and Rubiera-Garcia,D. , to appear (2012).

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[gr-qc/9402012].

(2003).

This work has been supported by the Spanish grants FIS2008-06078-C03-02, FIS2011-29813-C02-02, the Consolider Program CPAN (CSD2007-00042), and the JAE-doc program of the Spanish Research Council (CSIC).

## **Author details**

#### Gonzalo J. Olmo

Departamento de Física Teórica & IFIC, Centro Mixto Universidad de Valencia & CSIC, Facultad de Física, Universidad de Valencia, Burjassot, Valencia, Spain

## **References**


26 Open Questions in Cosmology

in detail elsewhere.

**Author details**

Gonzalo J. Olmo

**References**

[gr-qc]].

[hep-th]].

program of the Spanish Research Council (CSIC).

(2011)] [arXiv:1106.4240 [astro-ph.CO]].

[2] Anderson, P., Phys. Rev. D 28, 271 (1983).

Press, Cambridge, England, 1982).

extended the analysis carried out in [5] for a single perfect fluid with constant equation of state to include several perfect fluids with arbitrary equation of state *w*(*ρ*). This allows to explore the dynamics of realistic cosmological models with several fluids and is a necessary step prior to the consideration of the growth and evolution of inhomogeneities in these nonsingular backgrounds. Though the model (82) has been proposed on grounds of mathematical simplicity and motivated by the form of the effective action provided by perturbative quantization schemes in curved backgrounds, its ability to successfully deal with cosmological [5] and black hole singularities [27] as well as other aspects of quantum gravity phenomenology [26] demands further theoretical work to provide a more solid ground to it. In this sense, we note that the effective dynamics of loop quantum cosmology [4] in a Friedmann-Robertson-Walker background filled with a massless scalar can be exactly reproduced by a Palatini *f*(*R*) theory [22]. The extension of that result to more general spacetimes and matter sources could shed new light on the potential relation of (82) with a more fundamental theory of quantum gravity. All these open questions will be considered

This work has been supported by the Spanish grants FIS2008-06078-C03-02, FIS2011-29813-C02-02, the Consolider Program CPAN (CSD2007-00042), and the JAE-doc

Departamento de Física Teórica & IFIC, Centro Mixto Universidad de Valencia & CSIC,

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[4] Ashtekar, A. and Singh, P., Class. Quant. Grav. 28, 213001 (2011) [arXiv:1108.0893

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[7] Borunda, M., Janssen, B. and Bastero-Gil, M., JCAP 0811, 008 (2008) [arXiv:0804.4440

Facultad de Física, Universidad de Valencia, Burjassot, Valencia, Spain

[3] Ashtekar, A. and Lewandowski, J. , *Class. Quant. Grav.* 21 (2004) R53.

[8] Clarke, C. J. S. and Krolak, A. , Journ. Geom. Phys. 2, 17 (1985).

[9] Dodelson S., *Modern Cosmology*, Academic Press, (2003).


**Chapter 8**

**Provisional chapter**

**Inter-Universal Entanglement**

Additional information is available at the end of the chapter

**Inter-Universal Entanglement**

Additional information is available at the end of the chapter

The concept of the multiverse changes many of the preconceptions made in the physics and cosmology of the last century, providing us with a new paradigm that has inevitably influence on major philosophical ideas. The creation of the universe stops being a singled out event to become part of a more general and mediocre process, what can be thought of as a new "Copernican turn" in the natural philosophy of the XXI century. The multiverse also opens the door to new approaches for traditional questions in quantum cosmology. The origin of the universe, the problem of the cosmological constant and the arrow of time, which would eventually depend on the boundary conditions that are imposed on the state of the whole multiverse, challenge us to adopt new and open-minded attitudes for facing up these

It would mean a crucial step for the multiverse proposals if a particular theory could make observable and distinguishable predictions about the current properties of our universe. That would bring the multiverse into the category of a physical theory at the same footing as any other. Then, once the concept of the multiverse has reached a wider acceptance in theoretical cosmology, it is now imperiously needed to develop a precise characterization of the concept of a physical multiverse: one for which the theory could be not only falseable but also indirectly tested, at least in principle. Some claims have been made to that respect [16, 17, 31], although we are far from being able to state the observability of any kind of

In order to see the effects of other universes in the properties of our own universe, it seems to be essential considering any kind of interaction or correlation among the universes of the multiverse. Classical correlations in the state of the multiverse would be induced by the existence of wormholes that would crop up and connect different regions of two or more universes [15, 28, 44, 84]. Quantum correlations in the form of entanglement among the universal states provide us with a another interaction paradigm in the context of the quantum multiverse and it opens the door to a completely new and wider vision of the multiverse.

> ©2012 Pérez, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Pérez; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Salvador J. Robles Pérez

Salvador J. Robles Pérez

http://dx.doi.org/10.5772/52012

**1. Introduction**

problems.

multiverse nowadays.


## **Chapter 8**

**Provisional chapter**

## **Inter-Universal Entanglement**

**Inter-Universal Entanglement**

Salvador J. Robles Pérez Additional information is available at the end of the chapter

Salvador J. Robles Pérez

28 Open Questions in Cosmology

184 Open Questions in Cosmology

[31] Rovelli, C. , *Quantum Gravity*, (Cambridge U. Press, 2004).

[38] Weinberg S., Cosmology, Oxford University Press, (2008).

Press, Cambridge; *Living Rev.Rel.* 9,3,(2005), gr-qc/0510072 .

[34] Sciama, D. W., Rev. Mod. Phys. 36, 463 (1964).

[36] Tipler, F. J. , Phys. Lett. A 64, 8 ( 1977).

[40] Zanelli, J. (2005). arXiv:hep-th/0502193v4 .

[32] Schouten, J.A. , *Tensor analysis for physicists*, Oxford University Press, London (1951).

[33] Sciama, D. W. , in: Recent Developments in General Relativity, p. 415 (Pergamon, 1962).

[35] Thiemann, T. ,*Modern canonical quantum general relativity*, (Cambridge U. Press, 2007).

[39] Will, C.M. (1993). *Theory and Experiment in Gravitational Physics*, Cambridge University

[37] Wald, R.M., *General Relativity*, (The University of Chicago Press, Chicago, 1984).

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52012

## **1. Introduction**

The concept of the multiverse changes many of the preconceptions made in the physics and cosmology of the last century, providing us with a new paradigm that has inevitably influence on major philosophical ideas. The creation of the universe stops being a singled out event to become part of a more general and mediocre process, what can be thought of as a new "Copernican turn" in the natural philosophy of the XXI century. The multiverse also opens the door to new approaches for traditional questions in quantum cosmology. The origin of the universe, the problem of the cosmological constant and the arrow of time, which would eventually depend on the boundary conditions that are imposed on the state of the whole multiverse, challenge us to adopt new and open-minded attitudes for facing up these problems.

It would mean a crucial step for the multiverse proposals if a particular theory could make observable and distinguishable predictions about the current properties of our universe. That would bring the multiverse into the category of a physical theory at the same footing as any other. Then, once the concept of the multiverse has reached a wider acceptance in theoretical cosmology, it is now imperiously needed to develop a precise characterization of the concept of a physical multiverse: one for which the theory could be not only falseable but also indirectly tested, at least in principle. Some claims have been made to that respect [16, 17, 31], although we are far from being able to state the observability of any kind of multiverse nowadays.

In order to see the effects of other universes in the properties of our own universe, it seems to be essential considering any kind of interaction or correlation among the universes of the multiverse. Classical correlations in the state of the multiverse would be induced by the existence of wormholes that would crop up and connect different regions of two or more universes [15, 28, 44, 84]. Quantum correlations in the form of entanglement among the universal states provide us with a another interaction paradigm in the context of the quantum multiverse and it opens the door to a completely new and wider vision of the multiverse.

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Pérez; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Pérez, licensee InTech. This is an open access chapter distributed under the terms of the Creative

On the one hand, together with the classical laws of thermodynamics, we can also consider the novel laws of entanglement thermodynamics. This adds a new tool for studying the properties of both the universe and the multiverse. Furthermore, we would expect that the classical and the quantum thermodynamical laws were complementary provided that the quantum theory is a more general framework from which the classical one is recovered as a particular limiting case. Then, local entropic processes of a single universe could be related to the thermodynamical properties of entanglement among universes [60].

perhaps because the mediocre perception that it entails for our world and for the human

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 187

As it happened historically, the controversy disappears when it is properly defined what it is meant by the word 'world'. If Bruno meant by the word 'world' what is now known as a solar system, von Humbolt meant by 'island universes' what we currently know as galaxies. We now uncontroversially know that there exist many solar systems in billions of different galaxies. Maybe, the controversy of the current multiverse proposals could partially be unravelled by first defining precisely what we mean by the word 'universe', in the physics

Since the advent of the theory of relativity, in the early XX century, we can understand by the word 'universe' a particular geometrical configuration of the space-time as a whole that, following Einstein's equations, is determined by a given distribution of energy-matter in the universe. Furthermore, the geometrical description of the space-time encapsulates the causal relation between material points and, thus, the universe entails everything that may have a causal connection with a particular observer. In other words, the universe is everything we

Being this true, it does not close the door for the observation of the quantum effects that other universes might have in the properties of our own universe and, thus, it does not prevent us to consider a multiverse scenario. For instance, let us consider a spatially flat space-time endorsed with a cosmological constant. It is well-known that, for a given observer, there is an event horizon beyond which no classical information can be transmitted or received. Thus, two far distant observers are surrounded by their respective event horizons becoming then causally disconnected from each other. These causal enclosures may be interpreted as different universes within the whole space-time manifold2. However, cosmic fields are defined upon the whole space-time and, then, some quantum correlations might be present in the state of the field for two distant regions of the space-time, in the same way as non-local correlations appear in an EPR state of light in quantum optics. Therefore, being two observers classically disconnected, they may share common cosmological quantum fields allowing us, in principle, to study the quantum influence that other regions of the space-time may have

This is an example of a more general kind of multiverse proposals for which it can be defined a common space-time to the universes. It includes the multiverse that comes out in the scenario of eternal inflation [45, 46]. There are other proposals3 in which there is no common space-time among the universes, being the most notable example the landscape of the string theories [9, 71]. In such multidimensional theories, the dimensional reduction that gives rise to our four dimensional universe may contain up to 10<sup>500</sup> different vacua that can be populated with inflationary universes [29]. Two universes belonging to different vacua share no common space-time. However, it might well be that relic quantum correlations may appear between their quantum states, and even some kind of interaction has been proposed

Therefore, even if we have not been exhaustive in the justification of a multiverse scenario, it can easily be envisaged that the multiverse is a plausible cosmological scenario within the framework of the quantum theory provided that this has to be applied to the space-time as

<sup>3</sup> A more exhaustive classification of multiverses and their properties can be found in Refs. [49, 72, 73].

being itself.

of the XXI century.

can observe.

in the properties of their isolated patches.

to be observable [31, 48], in principle.

<sup>2</sup> This is the so-called Level I multiverse in Refs. [72, 73].

a whole. That is the basic assumption of the present chapter.

On the other hand, the quantum effects of the space-time are customary restricted to the obscure region of the Planck scale or to the neighbourhood of space-time singularities (both local and cosmological). However, cosmic entanglement among different universes of the multiverse could avoid such restriction and still be present along the whole history of a large parent universe [31, 62]. Thus, the effects of inter-universal entanglement on a single universe, and even the boundary conditions of the whole multiverse from which such entanglement would be consequence of, could in principle be tested in a large parent universe like ours. This adds a completely novel feature to the quantum theory of the universe.

The chapter is outlined as follows. In Sec. 2, we shall describe the customary picture in which the universes are spontaneously created from the gravitational vacuum or *space-time foam*. The universes are quantum mechanically described by a wave function that can represent, in the semiclassical regime, either an expanding or a contracting universe. Then, it will be introduced the so-called 'third quantization formalism', where creation and annihilation operators of universes can be defined and it can be given a wave function that represent the quantum state of the multiverse. Afterwards, it will be shown that an appropriate boundary condition of the multiverse allows us to interpret it as made up of entangled pairs of universes.

In Sec. 3, we shall briefly summarize the main features of quantum entanglement in quantum optics, making special emphasis in the characteristics that completely departure from the classical description of light. In Sec. 4, we shall address the question of whether quantum entanglement in the multiverse may induce observable effects in the properties of a single universe. We shall pose a pair of entangled universes and compute the thermodynamical properties of entanglement for each single universe of the entangled pair. It will be shown that the entropy of entanglement can be considered as an arrow of time for single universes and that the vacuum energy of entanglement might allow us to test the whole multiverse proposal. Finally, in Sec. 4, we shall draw some tentative conclusions.

## **2. Quantum multiverse**

#### **2.1. Introduction**

A many-world interpretation of Nature can be dated back to the very ancient Greek philosophy1 or, in a more recent epoch, to the many-world interpretation that Giordano Bruno derived from the heliocentric theory of Copernicus [68], in the XV century, and to the Kant's idea of 'island-universes', term coined by the Prussian naturalist Alenxander von Humboldt in the XIX century [33]. In any case, it was always a very controversial proposal

<sup>1</sup> The interpretation was posed, of course, in a radically different cultural context. However, it is curious reading some of the pieces that have survived from Greek philosophers like Anaximander, Heraclitus or Democritus, in relation to a 'many-world' interpretation on Nature.

perhaps because the mediocre perception that it entails for our world and for the human being itself.

2 Open Questions in Cosmology

pairs of universes.

**2. Quantum multiverse**

a 'many-world' interpretation on Nature.

**2.1. Introduction**

On the one hand, together with the classical laws of thermodynamics, we can also consider the novel laws of entanglement thermodynamics. This adds a new tool for studying the properties of both the universe and the multiverse. Furthermore, we would expect that the classical and the quantum thermodynamical laws were complementary provided that the quantum theory is a more general framework from which the classical one is recovered as a particular limiting case. Then, local entropic processes of a single universe could be related

On the other hand, the quantum effects of the space-time are customary restricted to the obscure region of the Planck scale or to the neighbourhood of space-time singularities (both local and cosmological). However, cosmic entanglement among different universes of the multiverse could avoid such restriction and still be present along the whole history of a large parent universe [31, 62]. Thus, the effects of inter-universal entanglement on a single universe, and even the boundary conditions of the whole multiverse from which such entanglement would be consequence of, could in principle be tested in a large parent universe like ours. This adds a completely novel feature to the quantum theory of the universe.

The chapter is outlined as follows. In Sec. 2, we shall describe the customary picture in which the universes are spontaneously created from the gravitational vacuum or *space-time foam*. The universes are quantum mechanically described by a wave function that can represent, in the semiclassical regime, either an expanding or a contracting universe. Then, it will be introduced the so-called 'third quantization formalism', where creation and annihilation operators of universes can be defined and it can be given a wave function that represent the quantum state of the multiverse. Afterwards, it will be shown that an appropriate boundary condition of the multiverse allows us to interpret it as made up of entangled

In Sec. 3, we shall briefly summarize the main features of quantum entanglement in quantum optics, making special emphasis in the characteristics that completely departure from the classical description of light. In Sec. 4, we shall address the question of whether quantum entanglement in the multiverse may induce observable effects in the properties of a single universe. We shall pose a pair of entangled universes and compute the thermodynamical properties of entanglement for each single universe of the entangled pair. It will be shown that the entropy of entanglement can be considered as an arrow of time for single universes and that the vacuum energy of entanglement might allow us to test the whole multiverse

A many-world interpretation of Nature can be dated back to the very ancient Greek philosophy1 or, in a more recent epoch, to the many-world interpretation that Giordano Bruno derived from the heliocentric theory of Copernicus [68], in the XV century, and to the Kant's idea of 'island-universes', term coined by the Prussian naturalist Alenxander von Humboldt in the XIX century [33]. In any case, it was always a very controversial proposal

<sup>1</sup> The interpretation was posed, of course, in a radically different cultural context. However, it is curious reading some of the pieces that have survived from Greek philosophers like Anaximander, Heraclitus or Democritus, in relation to

proposal. Finally, in Sec. 4, we shall draw some tentative conclusions.

to the thermodynamical properties of entanglement among universes [60].

As it happened historically, the controversy disappears when it is properly defined what it is meant by the word 'world'. If Bruno meant by the word 'world' what is now known as a solar system, von Humbolt meant by 'island universes' what we currently know as galaxies. We now uncontroversially know that there exist many solar systems in billions of different galaxies. Maybe, the controversy of the current multiverse proposals could partially be unravelled by first defining precisely what we mean by the word 'universe', in the physics of the XXI century.

Since the advent of the theory of relativity, in the early XX century, we can understand by the word 'universe' a particular geometrical configuration of the space-time as a whole that, following Einstein's equations, is determined by a given distribution of energy-matter in the universe. Furthermore, the geometrical description of the space-time encapsulates the causal relation between material points and, thus, the universe entails everything that may have a causal connection with a particular observer. In other words, the universe is everything we can observe.

Being this true, it does not close the door for the observation of the quantum effects that other universes might have in the properties of our own universe and, thus, it does not prevent us to consider a multiverse scenario. For instance, let us consider a spatially flat space-time endorsed with a cosmological constant. It is well-known that, for a given observer, there is an event horizon beyond which no classical information can be transmitted or received. Thus, two far distant observers are surrounded by their respective event horizons becoming then causally disconnected from each other. These causal enclosures may be interpreted as different universes within the whole space-time manifold2. However, cosmic fields are defined upon the whole space-time and, then, some quantum correlations might be present in the state of the field for two distant regions of the space-time, in the same way as non-local correlations appear in an EPR state of light in quantum optics. Therefore, being two observers classically disconnected, they may share common cosmological quantum fields allowing us, in principle, to study the quantum influence that other regions of the space-time may have in the properties of their isolated patches.

This is an example of a more general kind of multiverse proposals for which it can be defined a common space-time to the universes. It includes the multiverse that comes out in the scenario of eternal inflation [45, 46]. There are other proposals3 in which there is no common space-time among the universes, being the most notable example the landscape of the string theories [9, 71]. In such multidimensional theories, the dimensional reduction that gives rise to our four dimensional universe may contain up to 10<sup>500</sup> different vacua that can be populated with inflationary universes [29]. Two universes belonging to different vacua share no common space-time. However, it might well be that relic quantum correlations may appear between their quantum states, and even some kind of interaction has been proposed to be observable [31, 48], in principle.

Therefore, even if we have not been exhaustive in the justification of a multiverse scenario, it can easily be envisaged that the multiverse is a plausible cosmological scenario within the framework of the quantum theory provided that this has to be applied to the space-time as a whole. That is the basic assumption of the present chapter.

<sup>2</sup> This is the so-called Level I multiverse in Refs. [72, 73].

<sup>3</sup> A more exhaustive classification of multiverses and their properties can be found in Refs. [49, 72, 73].

#### **2.2. Classical universes**

In next sections, we shall describe the quantum state of a multiverse made up of homogeneous and isotropic universes. Then, it is worth first noting that homogeneity and isotropy are assumable conditions as far as we deal with large parent universes, where by *large* we mean universes with a length scale which is much greater than the Planck scale even though it can be rather small compared to macroscopic scales. At the Planck length the quantum fluctuations of the metric become of the same order of the metric and the assumptions of homogeneity and isotropy are meaningless. However, except for its very early phase the universe can properly be modeled by a homogeneous and isotropic metric, at least as a first approximation.

We will also consider homogeneous and isotropic scalar fields. This can be more objectionable. It can be considered a good approximation after the inflationary expansion of the universe has rapidly smoothed out the large inhomogeneities of the distribution of matter in the universe, and it clearly is an appropriate assumption for the large scale of the current universe. However, we should keep in mind that the study of inhomogeneities is a keystone for the observational tests of the inflationary scenario. Similarly, they might encode valuable information for testing the properties of inter-universal entanglement. However, as a first approach to the problem, we shall mainly be concerned with a multiverse made up of fully homogeneous and isotropic universes and matter fields.

Therefore, let us consider a space-time described by a closed Friedmann-Robertson-Walker (FRW) metric,

$$ds^2 = -\mathcal{N}^2 dt^2 + a^2(t)d\Omega\_{3\prime}^2 \tag{1}$$

scalar field, *ϕ*. Variation of the action with respect to the lapse function, fixing afterwards

Let us focus on a slow-varying scalar field, which constitutes a particularly interesting case

A limiting case is that of a constant value of the field, *<sup>ϕ</sup>*˙ = 0 and *ρϕ* = *<sup>V</sup>*(*ϕ*0) ≡ <sup>Λ</sup>. It effectively describes a de-Sitter space-time with a value Λ of the cosmological constant. Then,

*dt* )<sup>2</sup> <sup>≪</sup> *<sup>V</sup>*(*ϕ*), and *<sup>V</sup>*(*ϕ*) <sup>≈</sup> *<sup>V</sup>*(*ϕ*0) represents the nearly constant energy density of the

+ *V*(*ϕ*)

*∂V*(*ϕ*)

≡ −<sup>1</sup> + *<sup>a</sup>*2*σ*2*ρϕ*, (4)

, with *MP* ∼ 1019GeV

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 189

*dt*<sup>2</sup> <sup>≪</sup> <sup>3</sup> *a da dt dϕ dt*

*<sup>H</sup>* , the real solution

<sup>2</sup>*<sup>H</sup>* ),

3*M*<sup>2</sup> *P*

*∂ϕ* <sup>=</sup> 0. (5)

*a*2*H*<sup>2</sup> − 1, (6)

*<sup>H</sup>* cosh *Ht*, (7)

*<sup>H</sup>* at *<sup>t</sup>* <sup>=</sup> 0, and eventually follows

*<sup>E</sup>H*2, (8)

*<sup>H</sup>* cos *<sup>H</sup>τ*, (9)

*<sup>H</sup>* . However, we can perform a

the value N = 1, gives the Friedmann equation

= −1 + *a*2*σ*<sup>2</sup>

where *ρϕ* is the energy density of the scalar field, and [45] *<sup>σ</sup>*<sup>2</sup> = <sup>8</sup>*<sup>π</sup>*

*d*2*ϕ dt*<sup>2</sup> <sup>+</sup> 3 *a da dt dϕ dt* <sup>+</sup>

 1 2 *dϕ dt* 2

being the Planck mass. Variation of Eq. (2) with respect to the scalar field yields

that can model the inflationary stage of the universe. In that case [45, 47], *<sup>d</sup>*2*<sup>ϕ</sup>*

*da dt* <sup>=</sup>

*<sup>a</sup>*(*t*) = <sup>1</sup>

Wick rotation to Euclidean time, *τ* = *it*, by mean of which Eq. (6) transforms into

*aE*(*τ*) = <sup>1</sup>

*daE <sup>d</sup><sup>τ</sup>* <sup>=</sup>

an exponential expansion. It corresponds to the Lorentzian regime of the universe. On the

 1 − *a*<sup>2</sup>

is the analytic continuation to Euclidean time of the Lorentzian solution (7). The solution given by Eq. (9) represents an Euclidean space-time that originates at *aE* = 0 (for *<sup>τ</sup>* = − *<sup>π</sup>*

where, *H*<sup>2</sup> ≡ *σ*2Λ. It can be distinguished two regimes. For values, *a* ≥ <sup>1</sup>

represents a universe that starts out from a value *<sup>a</sup>*<sup>0</sup> = <sup>1</sup>

other hand, there is no real solution of Eq. (6) for values *a* < <sup>1</sup>

 *da dt* 2

and ( *<sup>d</sup><sup>ϕ</sup>*

whose solution,

scalar field, i.e. *ρϕ* ≈ *<sup>V</sup>*(*ϕ*0).

the Friedmann equation (4) can be written as

where N is the lapse function that parameterizes the different foliations of the space-time into space and time, *a*(*t*) is the scale factor, and *d*Ω<sup>2</sup> <sup>3</sup> is the usual line element on *<sup>S</sup>*<sup>3</sup> [39, 50, 80]. The degrees of freedom of the minisuperspace being considered are then the lapse function, N , the scale factor, *<sup>a</sup>*, and *<sup>n</sup>* scalar fields, *<sup>ϕ</sup>* = (*ϕ*1,..., *<sup>ϕ</sup>n*), that represent the matter content of the universe. The total action of the space-time minimally coupled to the scalar fields can conveniently be written as [39]

$$S = \int dt \mathcal{L} = \int dt \mathcal{N} \left( \frac{1}{2} \frac{G\_{AB}}{\mathcal{N}^2} \frac{dq^A}{dt} \frac{dq^B}{dt} - \mathcal{V}(q^I) \right) \tag{2}$$

for *<sup>I</sup>*, *<sup>A</sup>*, *<sup>B</sup>* = 0, . . . , *<sup>n</sup>*, where *GAB* ≡ *GAB*(*qI*), is the minisupermetric of the *<sup>n</sup>* + 1 dimensional minisuperspace, with {*qI*}≡{*a*, *ϕ*}, and the summation over repeated indices is implicitly understood in Eq. (2). The minisupermetric *GAB* is given by [39], *GAB* = diag(−*a*, *<sup>a</sup>*3,..., *<sup>a</sup>*3), and the potential V(*qI*) by

$$\mathcal{V}(q^I) \equiv \mathcal{V}(a, \vec{\varphi}) = a^\Im \left( V\_1(\varphi\_1), \dots, V\_n(\varphi\_n) \right) - a,\tag{3}$$

where *Vi*(*ϕi*) is the potential that corresponds to the field *<sup>ϕ</sup>i*. The classical equations of motion are obtained by variation of the action (2). Let us for simplicity consider only one scalar field, *ϕ*. Variation of the action with respect to the lapse function, fixing afterwards the value N = 1, gives the Friedmann equation

$$\left(\frac{da}{dt}\right)^2 = -1 + a^2 \sigma^2 \left(\frac{1}{2} \left(\frac{d\varphi}{dt}\right)^2 + V(\varphi)\right) \equiv -1 + a^2 \sigma^2 \rho\_{\varphi\nu} \tag{4}$$

where *ρϕ* is the energy density of the scalar field, and [45] *<sup>σ</sup>*<sup>2</sup> = <sup>8</sup>*<sup>π</sup>* 3*M*<sup>2</sup> *P* , with *MP* ∼ 1019GeV being the Planck mass. Variation of Eq. (2) with respect to the scalar field yields

$$\frac{d^2\varrho}{dt^2} + \frac{3}{a}\frac{da}{dt}\frac{d\varrho}{dt} + \frac{\partial V(\varrho)}{\partial \varrho} = 0.\tag{5}$$

Let us focus on a slow-varying scalar field, which constitutes a particularly interesting case that can model the inflationary stage of the universe. In that case [45, 47], *<sup>d</sup>*2*<sup>ϕ</sup> dt*<sup>2</sup> <sup>≪</sup> <sup>3</sup> *a da dt dϕ dt* and ( *<sup>d</sup><sup>ϕ</sup> dt* )<sup>2</sup> <sup>≪</sup> *<sup>V</sup>*(*ϕ*), and *<sup>V</sup>*(*ϕ*) <sup>≈</sup> *<sup>V</sup>*(*ϕ*0) represents the nearly constant energy density of the scalar field, i.e. *ρϕ* ≈ *<sup>V</sup>*(*ϕ*0).

A limiting case is that of a constant value of the field, *<sup>ϕ</sup>*˙ = 0 and *ρϕ* = *<sup>V</sup>*(*ϕ*0) ≡ <sup>Λ</sup>. It effectively describes a de-Sitter space-time with a value Λ of the cosmological constant. Then, the Friedmann equation (4) can be written as

$$\frac{da}{dt} = \sqrt{a^2 H^2 - 1},\tag{6}$$

where, *H*<sup>2</sup> ≡ *σ*2Λ. It can be distinguished two regimes. For values, *a* ≥ <sup>1</sup> *<sup>H</sup>* , the real solution

$$a(t) = \frac{1}{H} \cosh Ht,\tag{7}$$

represents a universe that starts out from a value *<sup>a</sup>*<sup>0</sup> = <sup>1</sup> *<sup>H</sup>* at *<sup>t</sup>* <sup>=</sup> 0, and eventually follows an exponential expansion. It corresponds to the Lorentzian regime of the universe. On the other hand, there is no real solution of Eq. (6) for values *a* < <sup>1</sup> *<sup>H</sup>* . However, we can perform a Wick rotation to Euclidean time, *τ* = *it*, by mean of which Eq. (6) transforms into

*daE <sup>d</sup><sup>τ</sup>* <sup>=</sup> 1 − *a*<sup>2</sup> *<sup>E</sup>H*2, (8)

whose solution,

4 Open Questions in Cosmology

**2.2. Classical universes**

at least as a first approximation.

(FRW) metric,

In next sections, we shall describe the quantum state of a multiverse made up of homogeneous and isotropic universes. Then, it is worth first noting that homogeneity and isotropy are assumable conditions as far as we deal with large parent universes, where by *large* we mean universes with a length scale which is much greater than the Planck scale even though it can be rather small compared to macroscopic scales. At the Planck length the quantum fluctuations of the metric become of the same order of the metric and the assumptions of homogeneity and isotropy are meaningless. However, except for its very early phase the universe can properly be modeled by a homogeneous and isotropic metric,

We will also consider homogeneous and isotropic scalar fields. This can be more objectionable. It can be considered a good approximation after the inflationary expansion of the universe has rapidly smoothed out the large inhomogeneities of the distribution of matter in the universe, and it clearly is an appropriate assumption for the large scale of the current universe. However, we should keep in mind that the study of inhomogeneities is a keystone for the observational tests of the inflationary scenario. Similarly, they might encode valuable information for testing the properties of inter-universal entanglement. However, as a first approach to the problem, we shall mainly be concerned with a multiverse made up of

Therefore, let us consider a space-time described by a closed Friedmann-Robertson-Walker

*ds*<sup>2</sup> = −N <sup>2</sup>*dt*<sup>2</sup> + *a*2(*t*)*d*Ω<sup>2</sup>

where N is the lapse function that parameterizes the different foliations of the space-time

The degrees of freedom of the minisuperspace being considered are then the lapse function, N , the scale factor, *<sup>a</sup>*, and *<sup>n</sup>* scalar fields, *<sup>ϕ</sup>* = (*ϕ*1,..., *<sup>ϕ</sup>n*), that represent the matter content of the universe. The total action of the space-time minimally coupled to the scalar fields can

for *<sup>I</sup>*, *<sup>A</sup>*, *<sup>B</sup>* = 0, . . . , *<sup>n</sup>*, where *GAB* ≡ *GAB*(*qI*), is the minisupermetric of the *<sup>n</sup>* + 1 dimensional minisuperspace, with {*qI*}≡{*a*, *ϕ*}, and the summation over repeated indices is implicitly understood in Eq. (2). The minisupermetric *GAB* is given by [39], *GAB* = diag(−*a*, *<sup>a</sup>*3,..., *<sup>a</sup>*3),

where *Vi*(*ϕi*) is the potential that corresponds to the field *<sup>ϕ</sup>i*. The classical equations of motion are obtained by variation of the action (2). Let us for simplicity consider only one

*dq<sup>A</sup> dt*

*dq<sup>B</sup>*

*dt* − V(*q<sup>I</sup>*

) ≡ V(*a*, *<sup>ϕ</sup>*) = *<sup>a</sup>*<sup>3</sup> (*V*1(*ϕ*1),..., *Vn*(*ϕn*)) − *<sup>a</sup>*, (3)

) 

3, (1)

, (2)

<sup>3</sup> is the usual line element on *<sup>S</sup>*<sup>3</sup> [39, 50, 80].

fully homogeneous and isotropic universes and matter fields.

into space and time, *a*(*t*) is the scale factor, and *d*Ω<sup>2</sup>

*S* = 

V(*q<sup>I</sup>*

*dtL* = *dt*N 1 2 *GAB* N 2

conveniently be written as [39]

and the potential V(*qI*) by

$$a\_E(\tau) = \frac{1}{H} \cos H\tau,\tag{9}$$

is the analytic continuation to Euclidean time of the Lorentzian solution (7). The solution given by Eq. (9) represents an Euclidean space-time that originates at *aE* = 0 (for *<sup>τ</sup>* = − *<sup>π</sup>* <sup>2</sup>*<sup>H</sup>* ), and expands to the value *aE* = <sup>1</sup> *<sup>H</sup>* at *<sup>τ</sup>* <sup>=</sup> 0. The transition from the Euclidean region to the Lorentzian region occurs at the boundary hypersurface <sup>Σ</sup><sup>0</sup> ≡ <sup>Σ</sup>(*a*0), at *<sup>t</sup>* = <sup>0</sup> = *<sup>τ</sup>*. This transition should not be seen as a process happening *in time* because the Euclidean time is not actual time (it is *imaginary time*). On the contrary, it precisely corresponds to the appearance of time [39] and to the appearance of the (real) universe, actually.

Eq. (2). The Hamiltonian then reads

*<sup>H</sup>* <sup>≡</sup> *dq<sup>I</sup>*

quantization of the momenta, *pI* → *<sup>p</sup>*ˆ*<sup>I</sup>* ≡ −*ih*¯ *<sup>∂</sup>*

the Wheeler-De Witt equation (11) explicitly yields

*∂ ∂q<sup>A</sup>*

*<sup>h</sup>*¯ <sup>2</sup> *<sup>∂</sup>*2*<sup>φ</sup> <sup>∂</sup>a*<sup>2</sup> <sup>+</sup>

*φ*±

*<sup>S</sup>*(*a*, *<sup>ϕ</sup>*) =

of factor ordering, it can be written as [39]

 <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup> √−*<sup>G</sup>*

*<sup>∂</sup><sup>a</sup>* and M ≡˙ *<sup>∂</sup>*<sup>M</sup>

where *N*(*ϕ*) is a normalization factor, and

The positive and negative signs of *<sup>φ</sup>*<sup>±</sup>

(12) as

where, *φ*˙ ≡ *∂φ*

region, as

*dt pI* <sup>−</sup> *<sup>L</sup>* <sup>=</sup> NH≡N

The invariance of general relativity under time reparametrizations implies that the variation of the Hamiltonian (10) with respect to the lapse function vanishes. We obtain thus the classical Hamiltonian constraint, H = 0, which gives rise to the Friedmann equation (4). The wave function of the universe, *φ*, can then be obtained by performing a canonical

the Hamiltonian constraint to the wave function *φ*, i.e. Hˆ *φ* = 0. With an appropriate choice

*∂q<sup>B</sup>* 

determinant of *GAB*. For a homogeneous and isotropic universe with a slow-varying field

where, *φ* ≡ *φ*(*a*, *ϕ*). Let us note that if we replace *V*(*ϕ*) by Λ, the wave function *φ* ≡ *φ*Λ(*a*) represents the quantum state of a de-Sitter universe. For later convenience, let us write Eq.

will be useful later on to recall the formal resemblance of Eq. (13) to the equation of motion of a harmonic oscillator. The WKB solutions of Eq. (13) can be written, in the Lorentzian

*da <sup>ω</sup>*(*a*, *<sup>ϕ</sup>*) = <sup>1</sup>

branches of the universe, respectively. This can be seen by noticing that, for sufficiently

M(*a*)*ω*(*a*, *ϕ*)

*h*¯

*e* ±*iS*(*a*,*ϕ*)

(*a*2*V*(*ϕ*) − 1)

*<sup>∂</sup><sup>a</sup>* , with M≡M(*a*) = *<sup>a</sup>*, and, *<sup>ω</sup>* <sup>≡</sup> *<sup>ω</sup>*(*a*, *<sup>ϕ</sup>*) = *<sup>a</sup>*

+ V(*q<sup>I</sup>* ) *φ*(*q<sup>I</sup>*

√−*G GAB <sup>∂</sup>*

where *<sup>G</sup>AB* is the inverse of the minisupermetric *GAB*, with *<sup>G</sup>ABGBC* = *<sup>δ</sup><sup>A</sup>*

*h*¯ 2 *a ∂φ*

*φ*¨ + M˙

*WKB*(*a*, *<sup>ϕ</sup>*) = *<sup>N</sup>*(*ϕ*)

*<sup>G</sup>AB pA pB* + V(*q<sup>I</sup>*

) 

*<sup>∂</sup>qI* , and applying the quantum version of

*<sup>∂</sup><sup>a</sup>* + (*a*4*V*(*ϕ*) <sup>−</sup> *<sup>a</sup>*2)*<sup>φ</sup>* <sup>=</sup> 0, (12)

<sup>M</sup>*φ*˙ <sup>+</sup> *<sup>ω</sup>*2*<sup>φ</sup>* <sup>=</sup> 0, (13)

3 2

*WKB* correspond to the contracting and expanding

*h*¯

<sup>3</sup>*V*(*ϕ*) . (15)

, (14)

. (10)

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 191

) = 0, (11)

*<sup>C</sup>* , and *G* is the

*a*2*V*(*ϕ*) − 1. It

This is, briefly sketched, the classical picture for the nucleation of a universe from *nothing* [26, 39, 77], depicted in Fig. 1, where by *nothing* we should understand a state of the universe where it does not exist space, time and matter, in the customary sense4. Within that picture, the quantum fluctuations of the gravitational vacuum provide it with a *foam structure* [13, 19, 25, 85] where tiny black holes, wormholes and baby universes [70] are virtually created and annihilated (see, Fig. 2). Some of the baby universes may branch off from the parent space-time and become isolated universes that, subsequently, may undergo an inflationary stage and develop into a large parent universe like ours.

**Figure 1.** The creation of a De–Sitter universe from a De–Sitter instanton.

**Figure 2.** Space-time foam: some of the baby universes may branch off from the parent space-time.

#### **2.3. Quantum state of the multiverse**

Following the canonical quantization formalism, the momenta conjugated to the configuration variables *<sup>q</sup><sup>I</sup>* are given by *pI* ≡ *<sup>δ</sup><sup>L</sup> <sup>δ</sup>*( *dqI dt* ) , where *L* ≡ *L*(*q<sup>I</sup>* , *dq<sup>I</sup> dt* ) is the Lagrangian of

<sup>4</sup> However, it does not correspond to the absolute meaning of 'nothing', in a similar way as the vacuum of a quantum field theory is not 'empty' (see, Ref. [77]).

Eq. (2). The Hamiltonian then reads

6 Open Questions in Cosmology

and expands to the value *aE* = <sup>1</sup>

*<sup>H</sup>* at *<sup>τ</sup>* <sup>=</sup> 0. The transition from the Euclidean region to

the Lorentzian region occurs at the boundary hypersurface <sup>Σ</sup><sup>0</sup> ≡ <sup>Σ</sup>(*a*0), at *<sup>t</sup>* = <sup>0</sup> = *<sup>τ</sup>*. This transition should not be seen as a process happening *in time* because the Euclidean time is not actual time (it is *imaginary time*). On the contrary, it precisely corresponds to the appearance

This is, briefly sketched, the classical picture for the nucleation of a universe from *nothing* [26, 39, 77], depicted in Fig. 1, where by *nothing* we should understand a state of the universe where it does not exist space, time and matter, in the customary sense4. Within that picture, the quantum fluctuations of the gravitational vacuum provide it with a *foam structure* [13, 19, 25, 85] where tiny black holes, wormholes and baby universes [70] are virtually created and annihilated (see, Fig. 2). Some of the baby universes may branch off from the parent space-time and become isolated universes that, subsequently, may undergo an inflationary

of time [39] and to the appearance of the (real) universe, actually.

stage and develop into a large parent universe like ours.

**Figure 1.** The creation of a De–Sitter universe from a De–Sitter instanton.

**2.3. Quantum state of the multiverse**

field theory is not 'empty' (see, Ref. [77]).

configuration variables *<sup>q</sup><sup>I</sup>* are given by *pI* ≡ *<sup>δ</sup><sup>L</sup>*

**Figure 2.** Space-time foam: some of the baby universes may branch off from the parent space-time.

Following the canonical quantization formalism, the momenta conjugated to the

*<sup>δ</sup>*( *dqI dt* )

<sup>4</sup> However, it does not correspond to the absolute meaning of 'nothing', in a similar way as the vacuum of a quantum

, where *L* ≡ *L*(*q<sup>I</sup>*

, *dq<sup>I</sup>*

*dt* ) is the Lagrangian of

$$H \equiv \frac{dq^I}{dt} p\_I - L = \mathcal{N} \mathcal{H} \equiv \mathcal{N} \left( \mathbb{G}^{AB} p\_A p\_B + \mathcal{V}(q^I) \right). \tag{10}$$

The invariance of general relativity under time reparametrizations implies that the variation of the Hamiltonian (10) with respect to the lapse function vanishes. We obtain thus the classical Hamiltonian constraint, H = 0, which gives rise to the Friedmann equation (4). The wave function of the universe, *φ*, can then be obtained by performing a canonical quantization of the momenta, *pI* → *<sup>p</sup>*ˆ*<sup>I</sup>* ≡ −*ih*¯ *<sup>∂</sup> <sup>∂</sup>qI* , and applying the quantum version of the Hamiltonian constraint to the wave function *φ*, i.e. Hˆ *φ* = 0. With an appropriate choice of factor ordering, it can be written as [39]

$$\left\{-\frac{\hbar^2}{\sqrt{-G}}\frac{\partial}{\partial q^A}\left(\sqrt{-G}\,G^{AB}\frac{\partial}{\partial q^B}\right) + \mathcal{V}(q^I)\right\}\phi(q^I) = 0,\tag{11}$$

where *<sup>G</sup>AB* is the inverse of the minisupermetric *GAB*, with *<sup>G</sup>ABGBC* = *<sup>δ</sup><sup>A</sup> <sup>C</sup>* , and *G* is the determinant of *GAB*. For a homogeneous and isotropic universe with a slow-varying field the Wheeler-De Witt equation (11) explicitly yields

$$
\hbar^2 \frac{\partial^2 \phi}{\partial a^2} + \frac{\hbar^2}{a} \frac{\partial \phi}{\partial a} + (a^4 V(\varphi) - a^2) \phi = 0,\tag{12}
$$

where, *φ* ≡ *φ*(*a*, *ϕ*). Let us note that if we replace *V*(*ϕ*) by Λ, the wave function *φ* ≡ *φ*Λ(*a*) represents the quantum state of a de-Sitter universe. For later convenience, let us write Eq. (12) as

$$
\ddot{\phi} + \frac{\dot{\mathcal{M}}}{\mathcal{M}} \dot{\phi} + \omega^2 \phi = 0,\tag{13}
$$

where, *φ*˙ ≡ *∂φ <sup>∂</sup><sup>a</sup>* and M ≡˙ *<sup>∂</sup>*<sup>M</sup> *<sup>∂</sup><sup>a</sup>* , with M≡M(*a*) = *<sup>a</sup>*, and, *<sup>ω</sup>* <sup>≡</sup> *<sup>ω</sup>*(*a*, *<sup>ϕ</sup>*) = *<sup>a</sup> h*¯ *a*2*V*(*ϕ*) − 1. It will be useful later on to recall the formal resemblance of Eq. (13) to the equation of motion of a harmonic oscillator. The WKB solutions of Eq. (13) can be written, in the Lorentzian region, as

$$\phi\_{WKB}^{\pm}(a,\varphi) = \frac{N(\varphi)}{\sqrt{\mathcal{M}(a)\omega(a,\varphi)}} e^{\pm iS(a,\varphi)}\,,\tag{14}$$

where *N*(*ϕ*) is a normalization factor, and

$$S(a,\varphi) = \int da \,\omega(a,\varphi) = \frac{1}{\hbar} \frac{(a^2 V(\varphi) - 1)^{\frac{3}{2}}}{3V(\varphi)}.\tag{15}$$

The positive and negative signs of *<sup>φ</sup>*<sup>±</sup> *WKB* correspond to the contracting and expanding branches of the universe, respectively. This can be seen by noticing that, for sufficiently

on the Euclidean sector of the wave function. In the case being considered, it is equivalent to

M(*a*)*ω*(*a*, *ϕ*)

given by a linear combination of expanding and contracting branches of the universe [27],

Both expanding and contracting branches suffer subsequently a very effective decoherence process [21, 37] becoming quantum mechanically independent. Thus, observers inhabiting a

Let us now introduce the so-called 'third quantization' formalism [70], where the creation and the annihilation of universes is naturally incorporated in a parallel way as the creation and annihilation of particles is naturally formulated in a quantum field theory. The third quantization formalism consists of considering the wave function of the universe, *φ*(*a*, *ϕ*), as a field defined upon the minisuperspace of variables (*a*, *ϕ*). The minisupermetric of the minisuperspace, *GAB* = diag(−*a*, *<sup>a</sup>*3,..., *<sup>a</sup>*3), where *Gaa* = −*a*, has a Lorentzian signature and it allows us to formally interpret the scale factor as an intrinsic time variable of the minisuperspace. This has not to be confused with a time variable in terms of 'clocks and rods' measured by any observer. The consideration of the scale factor as a time variable within a single universe is a tricky task (see Refs. [23, 30, 34, 38–40, 78]) that will partially be

We already noticed the formal analogy between the Wheeler-de Witt equation (13) and the equation of motion of a harmonic oscillator. Taking further the analogy, we can find a (third

quantized) action for which the variational principle gives rise to Eq. (13), given by

2 *da*

2M*P*<sup>2</sup> *φ* +

where M≡M(*a*) and *ω* ≡ *ω*(*a*, *ϕ*) are defined after Eq. (13). The configuration variable of the third quantization formalism is the wave function of the universe, *φ*, and the quantum

M*ω*<sup>2</sup>

*da* (3)*<sup>L</sup>* <sup>=</sup> <sup>1</sup>

Lagrangian of the action (20), and the third quantized Hamiltonian then reads

(3)*<sup>H</sup>* <sup>=</sup> <sup>1</sup>

The third quantized momentum is defined as, (3)*P<sup>φ</sup>* <sup>≡</sup> *<sup>δ</sup>* (3)*<sup>L</sup>*

cos *<sup>S</sup>* <sup>∝</sup> *<sup>e</sup>*

branch of the universe cannot perceive any effect of the quantum superposition.

*e* −*I*(*a*,*ϕ*)

> *e*

M*φ*˙ <sup>2</sup> − M*ω*2*φ*<sup>2</sup>

. (20)

*δφ*˙ <sup>=</sup> <sup>M</sup>*φ*˙, where (3)*<sup>L</sup>* is the

<sup>2</sup> *<sup>φ</sup>*2, (21)

<sup>+</sup>*iS*(*a*,*ϕ*) <sup>+</sup> *<sup>e</sup>*

<sup>3</sup>*V*(*ϕ*) [22, 39]. In the Lorentzian sector, the wave function turns out to be

+ <sup>1</sup> 3*V*(*ϕ*) M(*a*)*ω*(*a*, *ϕ*) , (18)

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 193

−*iS*(*a*,*ϕ*) 

. (19)

*<sup>E</sup>* (*a*, *<sup>ϕ</sup>*) <sup>≈</sup> *<sup>N</sup>*(*ϕ*)

impose regularity conditions [22], and thus

+ <sup>1</sup>

*<sup>φ</sup>T*(*a*, *<sup>ϕ</sup>*) <sup>≈</sup> *<sup>e</sup>*

**2.4. Third quantization formalism**

addressed on subsequent sections.

(3) *<sup>S</sup>* <sup>=</sup> <sup>1</sup> 2 

with *N*(*ϕ*) = *e*

i.e.

*φNB*

+ <sup>1</sup> 3*V*(*ϕ*) M(*a*)*ω*(*a*, *ϕ*)

**Figure 3.** Boundary conditions of the universe.

large values of the scale factor, the Fourier transform of *<sup>φ</sup>*<sup>±</sup> *WKB*(*a*, *<sup>ϕ</sup>*) is highly peaked around the value of the classical momentum *p<sup>c</sup> <sup>a</sup>* [20], i.e. *<sup>φ</sup>*˜<sup>±</sup> *WKB*(*pa*, *<sup>ϕ</sup>*) <sup>≈</sup> *<sup>δ</sup>*(*pa* <sup>−</sup> *<sup>p</sup><sup>c</sup> <sup>a</sup>*). The classical momentum reads, *p<sup>c</sup> <sup>a</sup>* <sup>=</sup> <sup>−</sup>*<sup>a</sup> <sup>∂</sup><sup>a</sup> <sup>∂</sup><sup>t</sup>* , and quantum mechanically, for large values of the scale factor, *<sup>p</sup>*ˆ*a<sup>φ</sup>* <sup>=</sup> <sup>−</sup>*ih*¯ *<sup>φ</sup>*˙ ≈ ±*ωφ*, where the positive and negative signs correspond to the signs of *<sup>φ</sup>*<sup>±</sup> *WKB*. Then, *<sup>∂</sup><sup>a</sup> <sup>∂</sup><sup>t</sup>* ≈ ∓*<sup>ω</sup> <sup>a</sup>* , where the negative sign describes a contracting universe and the positive sign an expanding universe. Thus, the solutions *<sup>φ</sup>*<sup>±</sup> *WKB* of the Wheeler-de Witt equation (13) describe the contracting and expanding branches of the universe, respectively.

In order to fix the state of the universe, a boundary condition has to be imposed on the wave function *φWKB*. The tunneling boundary condition [78, 79] states that the only modes that survive the Euclidean barrier are the outgoing modes of the minisuperspace that correspond, in the Lorentzian region, to the expanding branches of the universe (see, Fig. 3). Then, the wave function of the universe reads

$$\phi^T(a,\varphi) \approx \frac{N(\varphi)}{\sqrt{\mathcal{M}(a)\omega(a,\varphi)}} e^{-iS(a,\varphi)}\text{,}\tag{16}$$

with [39, 79], *N*(*ϕ*) = *e* − <sup>1</sup> <sup>3</sup>*V*(*ϕ*) . By using the matching conditions, the wave function (16) turns out to be given in the Euclidean region by

$$\phi\_E^T(a,\varphi) \approx \frac{e^{-\frac{1}{3V(\varphi)}}}{\sqrt{\mathcal{M}(a)\omega(a,\varphi)}} \left( e^{+I(a,\varphi)} + e^{-I(a,\varphi)} \right),\tag{17}$$

where, *I* = *iS*, is the Euclidean action. The first term in Eq. (17) may diverge as the scale factor degenerates. However, this is not a problem in terms of Vilenkin's reasoning [78] because the tunneling boundary condition is mainly intended for fixing the state of the wave function on the Lorentzian region where the current probability is defined, cf. [78]. The philosophy of the 'no-boundary' proposal of Hartle and Hawking [24] is quite the contrary. For these authors, the actual quantum description of the universe is given by a path integral performed over all compact Euclidean metrics. The no-boundary condition is then imposed on the Euclidean sector of the wave function. In the case being considered, it is equivalent to impose regularity conditions [22], and thus

$$\phi\_E^{NB}(a,\varphi) \approx \frac{N(\varphi)}{\sqrt{\mathcal{M}(a)\omega(a,\varphi)}} e^{-I(a,\varphi)}\,,\tag{18}$$

with *N*(*ϕ*) = *e* + <sup>1</sup> <sup>3</sup>*V*(*ϕ*) [22, 39]. In the Lorentzian sector, the wave function turns out to be given by a linear combination of expanding and contracting branches of the universe [27], i.e.

$$\phi^T(a,\varphi) \approx \frac{e^{+\frac{1}{\mathcal{N}(\varphi)}}}{\sqrt{\mathcal{M}(a)\omega(a,\varphi)}} \cos S \propto \frac{e^{+\frac{1}{\mathcal{N}(\varphi)}}}{\sqrt{\mathcal{M}(a)\omega(a,\varphi)}} \left(e^{+iS(a,\varphi)} + e^{-iS(a,\varphi)}\right). \tag{19}$$

Both expanding and contracting branches suffer subsequently a very effective decoherence process [21, 37] becoming quantum mechanically independent. Thus, observers inhabiting a branch of the universe cannot perceive any effect of the quantum superposition.

#### **2.4. Third quantization formalism**

8 Open Questions in Cosmology

**Figure 3.** Boundary conditions of the universe.

wave function of the universe reads

momentum reads, *p<sup>c</sup>*

*<sup>∂</sup><sup>t</sup>* ≈ ∓*<sup>ω</sup>*

with [39, 79], *N*(*ϕ*) = *e*

Then, *<sup>∂</sup><sup>a</sup>*

the value of the classical momentum *p<sup>c</sup>*

*<sup>a</sup>* <sup>=</sup> <sup>−</sup>*<sup>a</sup> <sup>∂</sup><sup>a</sup>*

sign an expanding universe. Thus, the solutions *<sup>φ</sup>*<sup>±</sup>

− <sup>1</sup>

out to be given in the Euclidean region by

*φT*

*<sup>E</sup>* (*a*, *<sup>ϕ</sup>*) <sup>≈</sup> *<sup>e</sup>*

large values of the scale factor, the Fourier transform of *<sup>φ</sup>*<sup>±</sup>

*<sup>a</sup>* [20], i.e. *<sup>φ</sup>*˜<sup>±</sup>

*<sup>p</sup>*ˆ*a<sup>φ</sup>* <sup>=</sup> <sup>−</sup>*ih*¯ *<sup>φ</sup>*˙ ≈ ±*ωφ*, where the positive and negative signs correspond to the signs of *<sup>φ</sup>*<sup>±</sup>

In order to fix the state of the universe, a boundary condition has to be imposed on the wave function *φWKB*. The tunneling boundary condition [78, 79] states that the only modes that survive the Euclidean barrier are the outgoing modes of the minisuperspace that correspond, in the Lorentzian region, to the expanding branches of the universe (see, Fig. 3). Then, the

M(*a*)*ω*(*a*, *ϕ*)

 *e*

where, *I* = *iS*, is the Euclidean action. The first term in Eq. (17) may diverge as the scale factor degenerates. However, this is not a problem in terms of Vilenkin's reasoning [78] because the tunneling boundary condition is mainly intended for fixing the state of the wave function on the Lorentzian region where the current probability is defined, cf. [78]. The philosophy of the 'no-boundary' proposal of Hartle and Hawking [24] is quite the contrary. For these authors, the actual quantum description of the universe is given by a path integral performed over all compact Euclidean metrics. The no-boundary condition is then imposed

describe the contracting and expanding branches of the universe, respectively.

*<sup>φ</sup>T*(*a*, *<sup>ϕ</sup>*) <sup>≈</sup> *<sup>N</sup>*(*ϕ*)

− <sup>1</sup> 3*V*(*ϕ*) M(*a*)*ω*(*a*, *ϕ*) *WKB*(*a*, *<sup>ϕ</sup>*) is highly peaked around

, (16)

, (17)

*WKB* of the Wheeler-de Witt equation (13)

*<sup>a</sup>*). The classical

*WKB*.

*WKB*(*pa*, *<sup>ϕ</sup>*) <sup>≈</sup> *<sup>δ</sup>*(*pa* <sup>−</sup> *<sup>p</sup><sup>c</sup>*

*<sup>∂</sup><sup>t</sup>* , and quantum mechanically, for large values of the scale factor,

*<sup>a</sup>* , where the negative sign describes a contracting universe and the positive

*e* −*iS*(*a*,*ϕ*)

<sup>3</sup>*V*(*ϕ*) . By using the matching conditions, the wave function (16) turns

<sup>+</sup>*I*(*a*,*ϕ*) <sup>+</sup> *<sup>e</sup>*

−*I*(*a*,*ϕ*)  Let us now introduce the so-called 'third quantization' formalism [70], where the creation and the annihilation of universes is naturally incorporated in a parallel way as the creation and annihilation of particles is naturally formulated in a quantum field theory. The third quantization formalism consists of considering the wave function of the universe, *φ*(*a*, *ϕ*), as a field defined upon the minisuperspace of variables (*a*, *ϕ*). The minisupermetric of the minisuperspace, *GAB* = diag(−*a*, *<sup>a</sup>*3,..., *<sup>a</sup>*3), where *Gaa* = −*a*, has a Lorentzian signature and it allows us to formally interpret the scale factor as an intrinsic time variable of the minisuperspace. This has not to be confused with a time variable in terms of 'clocks and rods' measured by any observer. The consideration of the scale factor as a time variable within a single universe is a tricky task (see Refs. [23, 30, 34, 38–40, 78]) that will partially be addressed on subsequent sections.

We already noticed the formal analogy between the Wheeler-de Witt equation (13) and the equation of motion of a harmonic oscillator. Taking further the analogy, we can find a (third quantized) action for which the variational principle gives rise to Eq. (13), given by

$$\mathcal{S}^{(3)}\mathcal{S} = \frac{1}{2}\int da \,\,^{(3)}L = \frac{1}{2}\int da \,\left(\mathcal{M}\dot{\phi}^2 - \mathcal{M}\omega^2\phi^2\right). \tag{20}$$

The third quantized momentum is defined as, (3)*P<sup>φ</sup>* <sup>≡</sup> *<sup>δ</sup>* (3)*<sup>L</sup> δφ*˙ <sup>=</sup> <sup>M</sup>*φ*˙, where (3)*<sup>L</sup>* is the Lagrangian of the action (20), and the third quantized Hamiltonian then reads

$$P^{(3)}H = \frac{1}{2\mathcal{M}}P^2\_{\phi} + \frac{\mathcal{M}\omega^2}{2}\phi^2. \tag{21}$$

where M≡M(*a*) and *ω* ≡ *ω*(*a*, *ϕ*) are defined after Eq. (13). The configuration variable of the third quantization formalism is the wave function of the universe, *φ*, and the quantum state of the multiverse is thus given by another wave function, Ψ ≡ Ψ(*φ*, *a*), which is the solution of the (third quantized) Schrödinger equation [61, 70]

$$\hat{H}^{(3)}\hat{H}(\phi,-i\hbar\frac{\partial}{\partial\phi'},a)\Psi(\phi,a)=i\hbar\frac{\partial\Psi(\phi,a)}{\partial a}.\tag{22}$$

The customary interpretation of the wave function Ψ is the following [70]: let us expand the quantum state of the multiverse, |Ψ�, in an orthonormal basis of number states, |*N*�, i.e.

$$|\Psi\rangle = \sum\_{N} \Psi\_N(\phi, a) |N\rangle,\tag{23}$$

field. The kind of universes created and annihilated by ˆ

the universe. If otherwise the 'no-boundary' proposal is chosen, ˆ

linear combinations of expanding and contracting branches.

boundary condition is imposed, then, ˆ

**2.5. Boundary conditions of the multiverse**

However, in terms of the constant operators ˆ

*dN*ˆ 0,*<sup>i</sup> da* <sup>≡</sup> *<sup>i</sup> h*¯ [

ˆ *<sup>b</sup>ω*,*<sup>i</sup>* <sup>≡</sup>

ˆ *b*† *<sup>ω</sup>*,*<sup>i</sup>* <sup>≡</sup>

the number operator, *<sup>N</sup>*<sup>ˆ</sup> *<sup>ω</sup>*,*<sup>i</sup>* <sup>≡</sup> <sup>ˆ</sup>

*b*† *<sup>i</sup>* and <sup>ˆ</sup>

of universes of the multiverse is not conserved because *<sup>N</sup>*<sup>ˆ</sup> 0,*<sup>i</sup>* ≡ <sup>ˆ</sup>

oscillator (21) with the proper frequency *ω* of the Hamiltonian, i.e.

M(*a*)*ωi*(*a*, *<sup>ϕ</sup>*) *h*¯

M(*a*)*ωi*(*a*, *<sup>ϕ</sup>*) *h*¯

> *b*† *ω*,*i* ˆ

*dN*<sup>ˆ</sup> *<sup>ω</sup>*,*<sup>i</sup> da* <sup>≡</sup> *<sup>i</sup> h*¯ [

For a given representation, ˆ

multiverse.

operator, i.e.

*b*† 0,*<sup>i</sup>* and <sup>ˆ</sup>

on the boundary conditions imposed on the probability amplitudes *Ai*(*a*, *<sup>ϕ</sup>*) and *<sup>A</sup>*<sup>∗</sup>

*b*† 0,*<sup>i</sup>* (ˆ

Recalling the previous discussion on the boundary conditions of the universe, if the tunneling

Therefore, at least for universes with high order of symmetry, the third quantization formalism parallels that of a quantum field theory in a curved space-time, i.e. it can formally be seen as a quantum field theory defined on the curved minisuperspace described by the minisupermetric *GAB*. The scale factor formally plays the role of the time variable and the matter fields *ϕ* the role of the spatial coordinates. Creation and annihilation operators of universes can properly be defined in the curved minisuperspace. However, as it happens in a quantum field theory, different representations can be chosen to describe the quantum state of the universes. The meaning of such representations needs of a further analysis in terms of the boundary condition that has to be imposed on the quantum state of the whole

might be interpreted in the third quantization formalism as the number of *i*-universes in the multiverse, where the index *i* labels the different kinds of universes considered in the model.

For a large parent universe, i.e. for values *a* ≫ 1, the creation and annihilation operators can asymptotically be taken to be the usual creation and annihilation operators of the harmonic

> *<sup>φ</sup>*ˆ*<sup>i</sup>* +

for a given type of *i*-universes. However, in terms of the asymptotic representation (29-30)

(3)*H*ˆ*i*, *<sup>N</sup>*<sup>ˆ</sup> *<sup>ω</sup>*,*i*] + *<sup>∂</sup>N*<sup>ˆ</sup> *<sup>ω</sup>*,*<sup>i</sup>*

It would be expected that the number of universes in the multiverse would be a property of the multiverse independent of any internal property of a particular single universe.

(3)*H*ˆ*i*, *<sup>N</sup>*<sup>ˆ</sup> 0,*i*] + *<sup>∂</sup>N*<sup>ˆ</sup> 0,*<sup>i</sup>*

*b*0,*<sup>i</sup>* and ˆ *b*†

> *<sup>∂</sup><sup>a</sup>* <sup>=</sup> *<sup>i</sup> h*¯ [

*<sup>φ</sup>*ˆ*<sup>i</sup>* <sup>−</sup> *<sup>i</sup>*

*i* M(*a*)*ωi*(*a*, *<sup>ϕ</sup>*)

M(*a*)*ωi*(*a*, *<sup>ϕ</sup>*)

*bω*,*i*, is neither an invariant operator because

*<sup>∂</sup><sup>a</sup>* <sup>=</sup> *<sup>∂</sup>N*<sup>ˆ</sup> *<sup>ω</sup>*,*<sup>i</sup>*

*b*0,*i*, respectively, also depend

*b*0,*i*) creates (annihilates)

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012

*b*0,*i*) creates (annihilates) expanding branches of

*b*† 0,*<sup>i</sup>* (ˆ

*bi*, the eigenvalues of the number operator *<sup>N</sup>*ˆ*<sup>i</sup>* ≡ <sup>ˆ</sup>

*b*† 0,*i* ˆ

*P*ˆ *φi* 

*P*ˆ *φi* 

0,*<sup>i</sup>* defined in Eqs. (26-27), the number

(3)*H*ˆ*i*, *<sup>N</sup>*<sup>ˆ</sup> 0,*i*] �<sup>=</sup> 0. (28)

*b*0,*<sup>i</sup>* is not an invariant

, (29)

, (30)

*<sup>∂</sup><sup>a</sup>* �<sup>=</sup> 0. (31)

*<sup>i</sup>* (*a*, *<sup>ϕ</sup>*).

195

*b*† *i* ˆ *bi*

then, |Ψ*N*(*φ*, *<sup>a</sup>*0)| <sup>2</sup> gives the probability to find in the multiverse *N* universes with a value *a*<sup>0</sup> of the scale factor. We can consider different types of universes having different energy-matter contents represented by the fields *<sup>ϕ</sup>*(*i*) of the *<sup>i</sup>*-universe. The wave function of the whole multiverse is given then by a linear superposition of wave functions of the form [61, 62]

$$\Psi\_{\vec{N}}(\vec{\phi}, a) = \Psi\_{N\_1}(\phi\_1, a)\Psi\_{N\_2}(\phi\_2, a)\cdots\Psi\_{N\_n}(\phi\_n, a),\tag{24}$$

where, *<sup>φ</sup>* <sup>≡</sup> (*φ*1, *<sup>φ</sup>*2,..., *<sup>φ</sup>n*) and *<sup>N</sup>* <sup>≡</sup> (*N*1, *<sup>N</sup>*2,..., *Nn*), with *Ni* being the number of universes of type *<sup>i</sup>*, represented by the wave function *<sup>φ</sup><sup>i</sup>* <sup>≡</sup> *<sup>φ</sup>*(*<sup>ϕ</sup>*(*i*), *<sup>a</sup>*). Following the canonical interpretation of the wave function in quantum mechanics, <sup>|</sup>Ψ*<sup>N</sup>* ( *<sup>φ</sup>*, *<sup>a</sup>*0)| <sup>2</sup> gives the probability to find *<sup>N</sup>* universes in the multiverse with a value of the scale factor and the scalar fields given by, *<sup>a</sup>* <sup>=</sup> *<sup>a</sup>*<sup>0</sup> and *<sup>ϕ</sup>*(*i*) <sup>=</sup> *<sup>ϕ</sup>*(*i*) <sup>0</sup> , for the *i*-universe.

Let us just consider one type, *i*, of universes. The wave function *φ<sup>i</sup>* can be promoted to an operator *φ*ˆ*<sup>i</sup>* that can be written as

$$
\hat{\phi}\_i(a,\varphi) = A\_i(a,\varphi)\hat{b}\_{0,i}^\dagger + A\_i^\*(a,\varphi)\hat{b}\_{0,i} \tag{25}
$$

where the probability amplitudes *Ai*(*a*, *<sup>ϕ</sup>*) and *<sup>A</sup>*<sup>∗</sup> *<sup>i</sup>* (*a*, *<sup>ϕ</sup>*) satisfy the Wheeler-de Witt equation (13), and

$$
\hat{\mathfrak{h}}\_{0,i} \equiv \sqrt{\frac{\mathcal{M}\_0 \omega\_0}{\hbar}} \left( \hat{\mathfrak{p}}\_i + \frac{i}{\mathcal{M}\_0 \omega\_0} \hat{\mathfrak{p}}\_{\Phi\_i} \right) \,' \tag{26}
$$

$$
\hat{b}\_{0,i}^{\dagger} \equiv \sqrt{\frac{\mathcal{M}\_0 \omega\_0}{\hbar}} \left( \hat{\phi}\_i - \frac{i}{\mathcal{M}\_0 \omega\_0} \hat{\mathcal{P}}\_{\Phi\_i} \right) \,, \tag{27}
$$

are the customary creation and annihilation operators of the harmonic oscillator, with M<sup>0</sup> and *<sup>ω</sup>*<sup>0</sup> being the mass and frequency terms, M(*a*) and *<sup>ω</sup>*(*a*, *<sup>ϕ</sup>*), respectively, evaluated on the boundary hypersurface <sup>Σ</sup><sup>0</sup> for which, *<sup>a</sup>* = *<sup>a</sup>*<sup>0</sup> and *<sup>ϕ</sup>* = *<sup>ϕ</sup>*0. The operators <sup>ˆ</sup> *b*0,*<sup>i</sup>* and ˆ *b*† 0,*<sup>i</sup>* can then be interpreted as the annihilation and creation operators of universes with a value of the scale factor *<sup>a</sup>*<sup>0</sup> and an energy density given by *ρϕ* ≈ *<sup>V</sup>*(*ϕ*0), for the case of a slow-varying field. The kind of universes created and annihilated by ˆ *b*† 0,*<sup>i</sup>* and <sup>ˆ</sup> *b*0,*i*, respectively, also depend on the boundary conditions imposed on the probability amplitudes *Ai*(*a*, *<sup>ϕ</sup>*) and *<sup>A</sup>*<sup>∗</sup> *<sup>i</sup>* (*a*, *<sup>ϕ</sup>*). Recalling the previous discussion on the boundary conditions of the universe, if the tunneling boundary condition is imposed, then, ˆ *b*† 0,*<sup>i</sup>* (ˆ *b*0,*i*) creates (annihilates) expanding branches of the universe. If otherwise the 'no-boundary' proposal is chosen, ˆ *b*† 0,*<sup>i</sup>* (ˆ *b*0,*i*) creates (annihilates) linear combinations of expanding and contracting branches.

Therefore, at least for universes with high order of symmetry, the third quantization formalism parallels that of a quantum field theory in a curved space-time, i.e. it can formally be seen as a quantum field theory defined on the curved minisuperspace described by the minisupermetric *GAB*. The scale factor formally plays the role of the time variable and the matter fields *ϕ* the role of the spatial coordinates. Creation and annihilation operators of universes can properly be defined in the curved minisuperspace. However, as it happens in a quantum field theory, different representations can be chosen to describe the quantum state of the universes. The meaning of such representations needs of a further analysis in terms of the boundary condition that has to be imposed on the quantum state of the whole multiverse.

## **2.5. Boundary conditions of the multiverse**

10 Open Questions in Cosmology

then, |Ψ*N*(*φ*, *<sup>a</sup>*0)|

[61, 62]

where,

(13), and

state of the multiverse is thus given by another wave function, Ψ ≡ Ψ(*φ*, *a*), which is the

*∂φ*, *<sup>a</sup>*)Ψ(*φ*, *<sup>a</sup>*) = *ih*¯

The customary interpretation of the wave function Ψ is the following [70]: let us expand the quantum state of the multiverse, |Ψ�, in an orthonormal basis of number states, |*N*�, i.e.

*a*<sup>0</sup> of the scale factor. We can consider different types of universes having different energy-matter contents represented by the fields *<sup>ϕ</sup>*(*i*) of the *<sup>i</sup>*-universe. The wave function of the whole multiverse is given then by a linear superposition of wave functions of the form

of type *<sup>i</sup>*, represented by the wave function *<sup>φ</sup><sup>i</sup>* <sup>≡</sup> *<sup>φ</sup>*(*<sup>ϕ</sup>*(*i*), *<sup>a</sup>*). Following the canonical

to find *<sup>N</sup>* universes in the multiverse with a value of the scale factor and the scalar fields

Let us just consider one type, *i*, of universes. The wave function *φ<sup>i</sup>* can be promoted to an

*b*† 0,*<sup>i</sup>* <sup>+</sup> *<sup>A</sup>*<sup>∗</sup>

<sup>0</sup> , for the *i*-universe.

*<sup>φ</sup>*ˆ*i*(*a*, *<sup>ϕ</sup>*) = *Ai*(*a*, *<sup>ϕ</sup>*)<sup>ˆ</sup>

M0*ω*<sup>0</sup> *h*¯

M0*ω*<sup>0</sup> *h*¯

the boundary hypersurface <sup>Σ</sup><sup>0</sup> for which, *<sup>a</sup>* = *<sup>a</sup>*<sup>0</sup> and *<sup>ϕ</sup>* = *<sup>ϕ</sup>*0. The operators <sup>ˆ</sup>

 *<sup>φ</sup>*ˆ*<sup>i</sup>* +

are the customary creation and annihilation operators of the harmonic oscillator, with M<sup>0</sup> and *<sup>ω</sup>*<sup>0</sup> being the mass and frequency terms, M(*a*) and *<sup>ω</sup>*(*a*, *<sup>ϕ</sup>*), respectively, evaluated on

then be interpreted as the annihilation and creation operators of universes with a value of the scale factor *<sup>a</sup>*<sup>0</sup> and an energy density given by *ρϕ* ≈ *<sup>V</sup>*(*ϕ*0), for the case of a slow-varying

*<sup>φ</sup>* <sup>≡</sup> (*φ*1, *<sup>φ</sup>*2,..., *<sup>φ</sup>n*) and *<sup>N</sup>* <sup>≡</sup> (*N*1, *<sup>N</sup>*2,..., *Nn*), with *Ni* being the number of universes


*∂*Ψ(*φ*, *a*)

<sup>2</sup> gives the probability to find in the multiverse *N* universes with a value

*<sup>φ</sup>*, *<sup>a</sup>*) = <sup>Ψ</sup>*N*<sup>1</sup> (*φ*1, *<sup>a</sup>*)Ψ*N*<sup>2</sup> (*φ*2, *<sup>a</sup>*)··· <sup>Ψ</sup>*Nn* (*φn*, *<sup>a</sup>*), (24)

*<sup>i</sup>* (*a*, *<sup>ϕ</sup>*)<sup>ˆ</sup>

*i* M0*ω*<sup>0</sup> *P*ˆ *φi* 

*<sup>φ</sup>*ˆ*<sup>i</sup>* <sup>−</sup> *<sup>i</sup>* M0*ω*<sup>0</sup> *P*ˆ *φi* 

*<sup>∂</sup><sup>a</sup>* . (22)

<sup>Ψ</sup>*N*(*φ*, *<sup>a</sup>*)|*N*�, (23)

*<sup>φ</sup>*, *<sup>a</sup>*0)|

<sup>2</sup> gives the probability

*b*0,*i*, (25)

, (26)

, (27)

*b*0,*<sup>i</sup>* and ˆ *b*† 0,*<sup>i</sup>* can

*<sup>i</sup>* (*a*, *<sup>ϕ</sup>*) satisfy the Wheeler-de Witt equation

solution of the (third quantized) Schrödinger equation [61, 70]

<sup>Ψ</sup>*<sup>N</sup>* (

where the probability amplitudes *Ai*(*a*, *<sup>ϕ</sup>*) and *<sup>A</sup>*<sup>∗</sup>

ˆ *<sup>b</sup>*0,*<sup>i</sup>* ≡

ˆ *b*† 0,*<sup>i</sup>* <sup>≡</sup>

given by, *<sup>a</sup>* <sup>=</sup> *<sup>a</sup>*<sup>0</sup> and *<sup>ϕ</sup>*(*i*) <sup>=</sup> *<sup>ϕ</sup>*(*i*)

operator *φ*ˆ*<sup>i</sup>* that can be written as

interpretation of the wave function in quantum mechanics, <sup>|</sup>Ψ*<sup>N</sup>* (

(3)*H*<sup>ˆ</sup> (*φ*, <sup>−</sup>*ih*¯ *<sup>∂</sup>*

For a given representation, ˆ *b*† *<sup>i</sup>* and <sup>ˆ</sup> *bi*, the eigenvalues of the number operator *<sup>N</sup>*ˆ*<sup>i</sup>* ≡ <sup>ˆ</sup> *b*† *i* ˆ *bi* might be interpreted in the third quantization formalism as the number of *i*-universes in the multiverse, where the index *i* labels the different kinds of universes considered in the model. However, in terms of the constant operators ˆ *b*0,*<sup>i</sup>* and ˆ *b*† 0,*<sup>i</sup>* defined in Eqs. (26-27), the number of universes of the multiverse is not conserved because *<sup>N</sup>*<sup>ˆ</sup> 0,*<sup>i</sup>* ≡ <sup>ˆ</sup> *b*† 0,*i* ˆ *b*0,*<sup>i</sup>* is not an invariant operator, i.e.

$$\frac{d\hat{N}\_{0,i}}{da} \equiv \frac{i}{\hbar} [^{(3)}\hat{H}\_{\text{i}} \, \hat{N}\_{0,i}] + \frac{\partial \hat{N}\_{0,i}}{\partial a} = \frac{i}{\hbar} [^{(3)}\hat{H}\_{\text{i}} \, \hat{N}\_{0,i}] \neq 0. \tag{28}$$

For a large parent universe, i.e. for values *a* ≫ 1, the creation and annihilation operators can asymptotically be taken to be the usual creation and annihilation operators of the harmonic oscillator (21) with the proper frequency *ω* of the Hamiltonian, i.e.

$$\hat{\boldsymbol{\theta}}\_{\omega,i} \equiv \sqrt{\frac{\mathcal{M}(\boldsymbol{a})\omega\_{i}(\boldsymbol{a},\boldsymbol{\varphi})}{\hbar}} \left(\hat{\boldsymbol{\phi}}\_{i} + \frac{\mathrm{i}}{\mathcal{M}(\boldsymbol{a})\omega\_{i}(\boldsymbol{a},\boldsymbol{\varphi})}\hat{\boldsymbol{P}}\_{\boldsymbol{\Phi}\_{i}}\right) ,\tag{29}$$

$$\delta^{\dagger}\_{\omega,i} \equiv \sqrt{\frac{\mathcal{M}(a)\omega\_{i}(a,\varphi)}{\hbar}} \left(\hat{\phi}\_{i} - \frac{i}{\mathcal{M}(a)\omega\_{i}(a,\varphi)}\hat{P}\_{\Phi\_{i}}\right),\tag{30}$$

for a given type of *i*-universes. However, in terms of the asymptotic representation (29-30) the number operator, *<sup>N</sup>*<sup>ˆ</sup> *<sup>ω</sup>*,*<sup>i</sup>* <sup>≡</sup> <sup>ˆ</sup> *b*† *ω*,*i* ˆ *bω*,*i*, is neither an invariant operator because

$$\frac{d\hat{\mathcal{N}}\_{\omega,i}}{da} \equiv \frac{i}{\hbar} [^{(3)}\hat{H}\_{i\prime}\hat{\mathcal{N}}\_{\omega,i}] + \frac{\partial \hat{\mathcal{N}}\_{\omega,i}}{\partial a} = \frac{\partial \hat{\mathcal{N}}\_{\omega,i}}{\partial a} \neq 0. \tag{31}$$

It would be expected that the number of universes in the multiverse would be a property of the multiverse independent of any internal property of a particular single universe. Therefore, it seems appropriate to impose the following boundary condition on the multiverse:

The number of universes of the multiverse does not depend on the value of the scale factor of a particular single universe.

This boundary condition imposes the restriction that the number operator *N*ˆ*<sup>i</sup>* for a particular type of *i*-universes has to be an invariant operator5. We can then follow the theory of invariants developed by Lewis [43] and others [11, 41, 55, 57, 67, 69, 76], and find a Hermitian invariant operator, <sup>ˆ</sup>*Ii* = *<sup>h</sup>*¯(<sup>ˆ</sup> *b*† *i* ˆ *bi* + <sup>1</sup> <sup>2</sup> ), where [43]

$$\delta\_i(a) \equiv \sqrt{\frac{1}{2\hbar}} \left( \frac{1}{R\_i} \hat{\phi}\_i + i(\mathcal{R}\_i \hat{\mathcal{P}}\_{\Phi\_i} - \dot{\mathcal{R}}\_i \hat{\phi}\_i) \right), \tag{32}$$

The Hamiltonian (35) is formally the same Hamiltonian of a degenerated parametric amplifier used in quantum optics [66, 82] (see also, Sec. 3). The quadratic terms are interpreted therein as the creation and annihilation operators of pairs of entangled photons.

and annihilate, respectively, pairs of entangled universes. In the case that the universes were distinguishable, the Hamiltonian (35) would take the form of a non-degenerated parametric

where the indices 1 and 2 label the two universes of the entangled pair. The distinguishability of universes is certainly a tricky task. However, observers may exist in the two universes of an entangled pair because the universes share similar properties and, then, the plausible (classical and quantum) communications between these observers would make the universes be distinguishable. Classical communications between the observers of different universes can be conceivable by the presence of wormholes connecting the universes and quantum communications could then be implemented by using quantum correlated fields shared by the two observers. Therefore, it is at least plausible to pose a model of the multiverse made

The general quantum state of a multiverse formed by entangled pairs of de-Sitter universes

*<sup>φ</sup>* <sup>≡</sup> (*φ*1, *<sup>φ</sup>*2,..., *<sup>φ</sup>n*), and *<sup>N</sup>* <sup>≡</sup> (2*N*1, 2*N*2,...,2*Nn*), with *Ni* being the number of pairs

(*φi*, *<sup>a</sup>*) = *<sup>H</sup>*ˆ*i*(*φ*, *<sup>p</sup>φ*, *<sup>a</sup>*)ΨΛ*<sup>i</sup>*

1 2 *β*(*i*) 0 (ˆ *b* (*i*) <sup>1</sup> )† <sup>ˆ</sup> *b* (*i*) <sup>1</sup> + (<sup>ˆ</sup> *b* (*i*) <sup>2</sup> )† <sup>ˆ</sup> *b* (*i*) <sup>2</sup> <sup>+</sup> <sup>1</sup>

Back to the early years of the quantum development, in 1935, Schrödinger [64, 65] coined the word 'entanglement' to describe a puzzling feature of the quantum theory that was formerly

(*a*, *<sup>φ</sup>*2)··· <sup>Ψ</sup>Λ*<sup>n</sup>*

*Ni*

*Nn*

*Ni*

(*a*, *<sup>φ</sup>*1)ΨΛ<sup>2</sup> *N*<sup>2</sup>

would be given by linear combinations of terms like [61, 62] (see Eq. (24))

*N*<sup>1</sup>

*<sup>φ</sup>*, *<sup>a</sup>*) = <sup>Ψ</sup>Λ<sup>1</sup>

of universes of type *<sup>i</sup>*, represented by the wave function *<sup>φ</sup><sup>i</sup>* <sup>≡</sup> *<sup>φ</sup>*Λ*<sup>i</sup>*

value <sup>Λ</sup>*<sup>i</sup>* of the cosmological constant. The wave functions, <sup>Ψ</sup>Λ*<sup>i</sup>*

*b*† and ˆ

*b* of Eq. (35) as operators that create

, (38)

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 197

(*a*, *φn*), (39)

(*a*) that corresponds to the

(*φi*, *<sup>a</sup>*), in Eq. (39) are the

, (41)

(*φi*, *<sup>a</sup>*), (40)

Similarly, we can interpret the quadratic terms in ˆ

(3)*H*<sup>ˆ</sup> <sup>=</sup> *<sup>h</sup>*¯

up of entangled pairs of distinguishable universes.

<sup>Ψ</sup>*<sup>N</sup>* (

solutions of the third quantized Schrödinger equation

for each kind of *i*-universes in the multiverse [62].

*ih*¯ *<sup>∂</sup> ∂a* ΨΛ*<sup>i</sup> Ni*

 *β*<sup>+</sup> ˆ *b*† 1 ˆ *b*† <sup>2</sup> <sup>+</sup> *<sup>β</sup>*<sup>−</sup> <sup>ˆ</sup> *b*1 ˆ *<sup>b</sup>*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*<sup>0</sup> 2 (ˆ *b*† 1 ˆ *<sup>b</sup>*<sup>1</sup> + <sup>ˆ</sup> *b*† 2 ˆ *<sup>b</sup>*<sup>2</sup> + <sup>1</sup>) 

amplifier [82]

where,

with

*<sup>H</sup>*ˆ*<sup>i</sup>* = *<sup>h</sup>*¯

**3.1. Introduction**

 *β*(*i*) <sup>−</sup> ˆ *b* (*i*) 1 ˆ *b* (*i*) <sup>2</sup> <sup>+</sup> *<sup>β</sup>*(*i*) <sup>+</sup> (ˆ *b* (*i*) <sup>1</sup> )†(<sup>ˆ</sup> *b* (*i*) <sup>2</sup> )† <sup>+</sup>

**3. Quantum entanglement**

$$\hat{b}\_i^\dagger(a) \equiv \sqrt{\frac{1}{2\hbar}} \left( \frac{1}{R\_i} \hat{\phi}\_i - i(R\_i \hat{P}\_{\Phi\_i} - \dot{R}\_i \hat{\phi}\_i) \right), \tag{33}$$

with, *Ri* ≡ *Ri*(*a*, *<sup>ϕ</sup>*), that can be written as *<sup>R</sup>* = *φ*2 1,*<sup>i</sup>* <sup>+</sup> *<sup>φ</sup>*<sup>2</sup> 2,*i* , being *φ*1,*<sup>i</sup>* and *φ*2,*<sup>i</sup>* two independent solutions of the Wheeler-de Witt equation (13). In the semiclassical regime, we can use independent combinations of the solutions *φWKB* <sup>+</sup> and *φWKB* − so that

$$R\_i(a, \varphi) \approx \frac{e^{\pm \frac{1}{\mathcal{W}\_l(\varphi)}}}{\sqrt{\mathcal{M}(a)\omega\_i(a, \varphi)}},\tag{34}$$

where the positive sign corresponding to the choice of the no-boundary proposal and the negative sign to the tunneling boundary condition. The number operator for a particular kind of *<sup>i</sup>*-universes in the representation given by Eqs. (32-33), *<sup>N</sup>*ˆ*<sup>i</sup>* ≡ <sup>ˆ</sup> *b*† *i* ˆ *bi*, is then an invariant operator fulfilling the boundary condition of the multiverse and, thus, the eigenvalues *Ni*, with *<sup>N</sup>*ˆ*i*|*Ni*, *<sup>a</sup>*� = *Ni*|*Ni*, *<sup>a</sup>*� and *Ni* �= *Ni*(*a*), can properly be interpreted as the number of *i*-universes of the multiverse.

In terms of the invariant representation, the Hamiltonian (21) takes the form

$$
\hat{H}^{(3)}\hat{H} = \hbar \left( \beta\_+ \left( \hat{b}^\dagger \right)^2 + \beta\_- \hat{b}^2 + \beta\_0 \left( \hat{b}^\dagger \hat{b} + \frac{1}{2} \right) \right),
\tag{35}
$$

where,

$$\mathcal{J}\_{+}^{\*} = \mathcal{J}\_{-} = \frac{1}{4} \left\{ \left( \dot{\mathcal{R}} - \frac{i}{\mathcal{R}} \right)^{2} + \omega^{2} \mathcal{R}^{2} \right\}\_{\prime} \tag{36}$$

$$\beta\_0 = \frac{1}{2} \left( \dot{\mathcal{R}}^2 + \frac{1}{\overline{\mathcal{R}}^2} + \omega^2 \mathcal{R}^2 \right). \tag{37}$$

<sup>5</sup> We are not considering transitions from one kind of universes to another.

The Hamiltonian (35) is formally the same Hamiltonian of a degenerated parametric amplifier used in quantum optics [66, 82] (see also, Sec. 3). The quadratic terms are interpreted therein as the creation and annihilation operators of pairs of entangled photons. Similarly, we can interpret the quadratic terms in ˆ *b*† and ˆ *b* of Eq. (35) as operators that create and annihilate, respectively, pairs of entangled universes. In the case that the universes were distinguishable, the Hamiltonian (35) would take the form of a non-degenerated parametric amplifier [82]

$$\hat{H}^{(3)}\hat{H} = \hbar \left( \beta\_+ \hat{b}\_1^\dagger \hat{b}\_2^\dagger + \beta\_- \hat{b}\_1 \hat{b}\_2 + \frac{\beta\_0}{2} \left( \hat{b}\_1^\dagger \hat{b}\_1 + \hat{b}\_2^\dagger \hat{b}\_2 + 1 \right) \right), \tag{38}$$

where the indices 1 and 2 label the two universes of the entangled pair. The distinguishability of universes is certainly a tricky task. However, observers may exist in the two universes of an entangled pair because the universes share similar properties and, then, the plausible (classical and quantum) communications between these observers would make the universes be distinguishable. Classical communications between the observers of different universes can be conceivable by the presence of wormholes connecting the universes and quantum communications could then be implemented by using quantum correlated fields shared by the two observers. Therefore, it is at least plausible to pose a model of the multiverse made up of entangled pairs of distinguishable universes.

The general quantum state of a multiverse formed by entangled pairs of de-Sitter universes would be given by linear combinations of terms like [61, 62] (see Eq. (24))

$$\Psi\_{\vec{N}}(\vec{\phi},a) = \Psi\_{\mathcal{N}\_1}^{\Lambda\_1}(a,\phi\_1)\Psi\_{\mathcal{N}\_2}^{\Lambda\_2}(a,\phi\_2)\cdots \cdot \Psi\_{\mathcal{N}\_n}^{\Lambda\_n}(a,\phi\_n),\tag{39}$$

where, *<sup>φ</sup>* <sup>≡</sup> (*φ*1, *<sup>φ</sup>*2,..., *<sup>φ</sup>n*), and *<sup>N</sup>* <sup>≡</sup> (2*N*1, 2*N*2,...,2*Nn*), with *Ni* being the number of pairs of universes of type *<sup>i</sup>*, represented by the wave function *<sup>φ</sup><sup>i</sup>* <sup>≡</sup> *<sup>φ</sup>*Λ*<sup>i</sup>* (*a*) that corresponds to the value <sup>Λ</sup>*<sup>i</sup>* of the cosmological constant. The wave functions, <sup>Ψ</sup>Λ*<sup>i</sup> Ni* (*φi*, *<sup>a</sup>*), in Eq. (39) are the solutions of the third quantized Schrödinger equation

$$i\hbar \frac{\partial}{\partial a} \Psi\_{N\_l}^{\Lambda\_l}(\phi\_{i\prime}, a) = \hat{H}\_l(\phi\_{\prime} \, p\_{\phi\prime}, a) \Psi\_{N\_l}^{\Lambda\_l}(\phi\_{i\prime}, a), \tag{40}$$

with

12 Open Questions in Cosmology

invariant operator, <sup>ˆ</sup>*Ii* = *<sup>h</sup>*¯(<sup>ˆ</sup>

*i*-universes of the multiverse.

where,

factor of a particular single universe.

*b*† *i* ˆ *bi* + <sup>1</sup>

ˆ *bi*(*a*) ≡

ˆ *b*† *<sup>i</sup>* (*a*) <sup>≡</sup>

with, *Ri* ≡ *Ri*(*a*, *<sup>ϕ</sup>*), that can be written as *<sup>R</sup>* =

we can use independent combinations of the solutions *φWKB*

kind of *<sup>i</sup>*-universes in the representation given by Eqs. (32-33), *<sup>N</sup>*ˆ*<sup>i</sup>* ≡ <sup>ˆ</sup>

 *β*<sup>+</sup> (ˆ

<sup>+</sup> <sup>=</sup> *<sup>β</sup>*<sup>−</sup> <sup>=</sup> <sup>1</sup>

4 

*<sup>β</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> 2 *R*˙ <sup>2</sup> + 1 *<sup>R</sup>*<sup>2</sup> <sup>+</sup> *<sup>ω</sup>*2*R*<sup>2</sup>

(3)*H*<sup>ˆ</sup> <sup>=</sup> *<sup>h</sup>*¯

*β*∗

<sup>5</sup> We are not considering transitions from one kind of universes to another.

In terms of the invariant representation, the Hamiltonian (21) takes the form

multiverse:

Therefore, it seems appropriate to impose the following boundary condition on the

The number of universes of the multiverse does not depend on the value of the scale

This boundary condition imposes the restriction that the number operator *N*ˆ*<sup>i</sup>* for a particular type of *i*-universes has to be an invariant operator5. We can then follow the theory of invariants developed by Lewis [43] and others [11, 41, 55, 57, 67, 69, 76], and find a Hermitian

*<sup>φ</sup>*ˆ*<sup>i</sup>* + *<sup>i</sup>*(*RiP*<sup>ˆ</sup>

*<sup>φ</sup>*ˆ*<sup>i</sup>* − *<sup>i</sup>*(*RiP*<sup>ˆ</sup>

± <sup>1</sup> <sup>3</sup>*Vi*(*ϕ*) M(*a*)*ωi*(*a*, *<sup>ϕ</sup>*)

independent solutions of the Wheeler-de Witt equation (13). In the semiclassical regime,

where the positive sign corresponding to the choice of the no-boundary proposal and the negative sign to the tunneling boundary condition. The number operator for a particular

operator fulfilling the boundary condition of the multiverse and, thus, the eigenvalues *Ni*, with *<sup>N</sup>*ˆ*i*|*Ni*, *<sup>a</sup>*� = *Ni*|*Ni*, *<sup>a</sup>*� and *Ni* �= *Ni*(*a*), can properly be interpreted as the number of

*b*†)<sup>2</sup> + *β*<sup>−</sup> ˆ

*<sup>R</sup>*˙ <sup>−</sup> *<sup>i</sup> R* 2

*<sup>b</sup>*<sup>2</sup> + *<sup>β</sup>*<sup>0</sup> (<sup>ˆ</sup>

*b*† ˆ *b* + 1 2 ) 

+ *ω*2*R*<sup>2</sup>

*<sup>φ</sup><sup>i</sup>* <sup>−</sup> *<sup>R</sup>*˙ *<sup>i</sup>φ*ˆ*i*)

*<sup>φ</sup><sup>i</sup>* <sup>−</sup> *<sup>R</sup>*˙ *<sup>i</sup>φ*ˆ*i*)

 *φ*2 1,*<sup>i</sup>* <sup>+</sup> *<sup>φ</sup>*<sup>2</sup> 2,*i*

<sup>+</sup> and *φWKB*

, (32)

, (33)

, being *φ*1,*<sup>i</sup>* and *φ*2,*<sup>i</sup>* two

*bi*, is then an invariant

, (35)

, (36)

. (37)

− so that

*b*† *i* ˆ

, (34)

<sup>2</sup> ), where [43]

 1 *Ri*

 1 *Ri*

*Ri*(*a*, *<sup>ϕ</sup>*) <sup>≈</sup> *<sup>e</sup>*

 1 2¯*h*

 1 2¯*h*

$$\hat{H}\_{i} = \hbar \left\{ \beta\_{-}^{(i)} \delta\_{1}^{(i)} \delta\_{2}^{(i)} + \beta\_{+}^{(i)} (\delta\_{1}^{(i)})^{\\\\\dagger} (\delta\_{2}^{(i)})^{\\\dagger} + \frac{1}{2} \beta\_{0}^{(i)} \left( (\delta\_{1}^{(i)})^{\\\dagger} \delta\_{1}^{(i)} + (\delta\_{2}^{(i)})^{\\\dagger} \delta\_{2}^{(i)} + 1 \right) \right\},\tag{41}$$

for each kind of *i*-universes in the multiverse [62].

#### **3. Quantum entanglement**

#### **3.1. Introduction**

Back to the early years of the quantum development, in 1935, Schrödinger [64, 65] coined the word 'entanglement' to describe a puzzling feature of the quantum theory that was formerly posed by Einstein, Podolski and Rosen in a famous gedanken experiment [12]. Schrödinger also realized that entanglement is precisely *the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought* [64]. Let us briefly show it by following the example given in Ref. [66] (see also Ref. [82]). Let us consider the photo-disintegration of a Hg2 molecule formed by two atoms of Hg with spin <sup>1</sup> <sup>2</sup> . Before the disintegration, the molecule is taken to be in a state of zero angular momentum so that the composite state is given by

$$|\mathbf{H}\mathbf{g}\_2\rangle = \frac{1}{\sqrt{2}}\left(|\uparrow\_1\downarrow\_2\rangle - |\downarrow\_1\uparrow\_2\rangle\right),\tag{42}$$

where *a*ˆ† and *a*ˆ are the usual creation and annihilation operators of the harmonic oscillator. The operators *X*ˆ <sup>1</sup> and *X*ˆ <sup>2</sup> are essentially dimensionless position and momentum operators.

where, for a coherent state, <sup>∆</sup>*X*<sup>1</sup> = <sup>∆</sup>*X*<sup>2</sup> = 1. A squeezed state is defined as the quantum

Therefore, for a squeezed state the uncertainty of one of the quadratures is reduced below the limit of the Heisenberg principle at the expense of the increased fluctuations of the other

Unlike the generation of coherent states, which is associated with linear terms of the creation and annihilation operators in the Hamiltonian, the generation of squeezed states is associated with quadratic terms of such operators. For instance, let us consider the Hamiltonian that

> *χ* <sup>2</sup> ((*a*ˆ† )2−*a*ˆ<sup>2</sup>

*<sup>S</sup>*ˆ†(*χ*) = *<sup>S</sup>*ˆ−1(*χ*) = *<sup>S</sup>*ˆ(−*χ*). It is therefore a unitary operator. The Heisenberg equations of

*<sup>X</sup>*<sup>ˆ</sup> <sup>1</sup>(0) , *<sup>X</sup>*<sup>ˆ</sup> <sup>2</sup>(*t*) = *<sup>e</sup>*

Then, for an initial vacuum state, for which <sup>∆</sup>*Xi*(0) = 1, the variances of the quadratures

<sup>2</sup>*χ<sup>t</sup>* , <sup>∆</sup>*X*2(*t*) = *<sup>e</sup>*

*dt* <sup>=</sup> *<sup>χ</sup>X*<sup>ˆ</sup> <sup>1</sup> , *dX*<sup>ˆ</sup> <sup>2</sup>

)*t*

−*χt*

−2*χt*

represents in quantum optics a degenerated parametric amplifier [66, 82]

*H*ˆ = *ih*¯ *χ* 2 (*a*ˆ †)<sup>2</sup> − *a*ˆ 2 

where *χ* is a coupling constant. Then, the time evolution of the vacuum state,


<sup>∆</sup>*X*1∆*X*<sup>2</sup> ≥ 1, (44)

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 199

(∆*Xi*)<sup>2</sup> < <sup>1</sup> (*<sup>i</sup>* = 1 or 2). (45)

, (46)


*<sup>X</sup>*<sup>ˆ</sup> <sup>2</sup>(0). (49)

. (50)

*<sup>χ</sup>* being the squeezing operator which satisfies,

*dt* <sup>=</sup> <sup>−</sup>*χX*<sup>ˆ</sup> 2, (48)

The uncertainty relation for ∆*X*<sup>1</sup> and ∆*X*<sup>2</sup> reads

state for which one of the quadratures satisfies7

yields a squeezed (vacuum) state, |*s*(*t*)�, with *S*ˆ

<sup>7</sup> An ideal squeezed state also satisfies ∆*X*1∆*X*<sup>2</sup> = 1

with solutions given by

read

motion for the quadrature amplitudes turn out to be then

*dX*ˆ <sup>1</sup>

*<sup>X</sup>*<sup>ˆ</sup> <sup>1</sup>(*t*) = *<sup>e</sup>*

*χt*

<sup>∆</sup>*X*1(*t*) = *<sup>e</sup>*

quadrature.

where 1 and 2 refer to the atoms of Hg and | ↑ (↓)� refers to the value +<sup>1</sup> <sup>2</sup> (−<sup>1</sup> <sup>2</sup> ) of the projection of their spin along the *z*-axis. After the photo-disintegration, performed with no disturbance of the angular momentum, the two atoms separate each other in opposite directions so we can make independent measurements on them. Before doing any measurement we do not know the particular value of the spin of each atom. However, we do anticipatedly know that if a measurement of the spin projection is performed on the atom 1 yielding a value +<sup>1</sup> <sup>2</sup> (−<sup>1</sup> <sup>2</sup> ), then, the spin projection of the atom 2 is to be <sup>−</sup><sup>1</sup> <sup>2</sup> (+<sup>1</sup> 2 ). Furthermore, if it is performed a different measurement of the projection of the spin of the particle 1 along, say, the *x*-axis, we are determining the value of the spin projection of the particle 2 along the same axis, too. This non-local feature of the quantum theory is known as *entanglement* and the state (42) is called an *entangled state*.

In 1964, Bell derived certain inequalities [7, 8] that should be satisfied by any reasonable realistic6 theory of local variables. The experiments of Aspect [3] and others [2, 63, 74, 83] have shown that the entangled states of the quantum theory violate such inequalities. Furthermore, these states have not only provided us with an experimental test of the quantum postulates but they have also given rise to the development of a completely new branch of physics, the so-called quantum information theory [32, 35, 75], which includes interesting subjects like quantum computation, quantum cryptography, and quantum teleportation, which are currently under a promising state of development.

It is finally worth noticing that the kinematical non-locality of the quantum theory is also the feature that forces us to consider a wave function of the universe. As it is pointed out in Ref. [39], *if gravity is quantized, the kinematical non-separability of quantum theory demands that the whole universe must be described in quantum terms* (cf. p. 4). Every space-time region is entangled to its environment, which is entangled to another environment and so forth, ending up in a quantum description of the whole universe.

#### **3.2. Squeezed and entangled states of light**

Squeezed states of light [81] can be seen as a generalization of the coherent states. Let us define the quadrature operators

$$
\hat{X}\_1 \equiv \mathfrak{A} + \mathfrak{A}^\dagger \quad , \quad \hat{X}\_2 = i(\mathfrak{A}^\dagger - \mathfrak{A}) , \tag{43}
$$

<sup>6</sup> By a realistic theory we mean a theory that presupposes that the elements of the theory represent elements of physical reality (see Ref. [12]).

where *a*ˆ† and *a*ˆ are the usual creation and annihilation operators of the harmonic oscillator. The operators *X*ˆ <sup>1</sup> and *X*ˆ <sup>2</sup> are essentially dimensionless position and momentum operators. The uncertainty relation for ∆*X*<sup>1</sup> and ∆*X*<sup>2</sup> reads

$$
\Delta X\_1 \Delta X\_2 \ge 1,\tag{44}
$$

where, for a coherent state, <sup>∆</sup>*X*<sup>1</sup> = <sup>∆</sup>*X*<sup>2</sup> = 1. A squeezed state is defined as the quantum state for which one of the quadratures satisfies7

$$\left(\left(\Delta X\_{i}\right)^{2} < 1 \quad (i = 1 \text{ or } 2). \tag{45}$$

Therefore, for a squeezed state the uncertainty of one of the quadratures is reduced below the limit of the Heisenberg principle at the expense of the increased fluctuations of the other quadrature.

Unlike the generation of coherent states, which is associated with linear terms of the creation and annihilation operators in the Hamiltonian, the generation of squeezed states is associated with quadratic terms of such operators. For instance, let us consider the Hamiltonian that represents in quantum optics a degenerated parametric amplifier [66, 82]

$$\hat{H} = i\hbar \frac{\chi}{2} \left( (\mathfrak{d}^\dagger)^2 - \mathfrak{d}^2 \right) ,\tag{46}$$

where *χ* is a coupling constant. Then, the time evolution of the vacuum state,

$$|s(t)\rangle = \hat{S}(\chi)|0\rangle = e^{\frac{i}{2}\left((\mathbb{A}^\dagger)^2 - \mathbb{A}^2\right)t}|0\rangle,\tag{47}$$

yields a squeezed (vacuum) state, |*s*(*t*)�, with *S*ˆ *<sup>χ</sup>* being the squeezing operator which satisfies, *<sup>S</sup>*ˆ†(*χ*) = *<sup>S</sup>*ˆ−1(*χ*) = *<sup>S</sup>*ˆ(−*χ*). It is therefore a unitary operator. The Heisenberg equations of motion for the quadrature amplitudes turn out to be then

$$\frac{d\hat{X}\_1}{dt} = \chi \hat{X}\_1 \; \; \; \; \; \; \frac{d\hat{X}\_2}{dt} = -\chi \hat{X}\_2 \; \; \tag{48}$$

with solutions given by

14 Open Questions in Cosmology

composite state is given by

atom 1 yielding a value +<sup>1</sup>

posed by Einstein, Podolski and Rosen in a famous gedanken experiment [12]. Schrödinger also realized that entanglement is precisely *the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought* [64]. Let us briefly show it by following the example given in Ref. [66] (see also Ref. [82]). Let us consider the

disintegration, the molecule is taken to be in a state of zero angular momentum so that the

the projection of their spin along the *z*-axis. After the photo-disintegration, performed with no disturbance of the angular momentum, the two atoms separate each other in opposite directions so we can make independent measurements on them. Before doing any measurement we do not know the particular value of the spin of each atom. However, we do anticipatedly know that if a measurement of the spin projection is performed on the

Furthermore, if it is performed a different measurement of the projection of the spin of the particle 1 along, say, the *x*-axis, we are determining the value of the spin projection of the particle 2 along the same axis, too. This non-local feature of the quantum theory is known as

In 1964, Bell derived certain inequalities [7, 8] that should be satisfied by any reasonable realistic6 theory of local variables. The experiments of Aspect [3] and others [2, 63, 74, 83] have shown that the entangled states of the quantum theory violate such inequalities. Furthermore, these states have not only provided us with an experimental test of the quantum postulates but they have also given rise to the development of a completely new branch of physics, the so-called quantum information theory [32, 35, 75], which includes interesting subjects like quantum computation, quantum cryptography, and quantum

It is finally worth noticing that the kinematical non-locality of the quantum theory is also the feature that forces us to consider a wave function of the universe. As it is pointed out in Ref. [39], *if gravity is quantized, the kinematical non-separability of quantum theory demands that the whole universe must be described in quantum terms* (cf. p. 4). Every space-time region is entangled to its environment, which is entangled to another environment and so forth,

Squeezed states of light [81] can be seen as a generalization of the coherent states. Let us

† , *<sup>X</sup>*<sup>ˆ</sup> <sup>2</sup> = *<sup>i</sup>*(*a*<sup>ˆ</sup>

<sup>6</sup> By a realistic theory we mean a theory that presupposes that the elements of the theory represent elements of physical

† − *a*ˆ), (43)

teleportation, which are currently under a promising state of development.

*<sup>X</sup>*<sup>ˆ</sup> <sup>1</sup> ≡ *<sup>a</sup>*<sup>ˆ</sup> + *<sup>a</sup>*<sup>ˆ</sup>

<sup>2</sup> . Before the

<sup>2</sup> (−<sup>1</sup> <sup>2</sup> ) of

> <sup>2</sup> (+<sup>1</sup> 2 ).

(| ↑1↓2�−|↓1↑2�), (42)

<sup>2</sup> ), then, the spin projection of the atom 2 is to be <sup>−</sup><sup>1</sup>

photo-disintegration of a Hg2 molecule formed by two atoms of Hg with spin <sup>1</sup>

where 1 and 2 refer to the atoms of Hg and | ↑ (↓)� refers to the value +<sup>1</sup>

<sup>|</sup>Hg2� <sup>=</sup> <sup>1</sup> √2

<sup>2</sup> (−<sup>1</sup>

*entanglement* and the state (42) is called an *entangled state*.

ending up in a quantum description of the whole universe.

**3.2. Squeezed and entangled states of light**

define the quadrature operators

reality (see Ref. [12]).

$$
\hat{X}\_1(t) = e^{\chi t} \hat{X}\_1(0) \quad , \quad \hat{X}\_2(t) = e^{-\chi t} \hat{X}\_2(0). \tag{49}
$$

Then, for an initial vacuum state, for which <sup>∆</sup>*Xi*(0) = 1, the variances of the quadratures read

$$
\Delta X\_1(t) = e^{2\chi t} \quad \text{,} \quad \Delta X\_2(t) = e^{-2\chi t}. \tag{50}
$$

<sup>7</sup> An ideal squeezed state also satisfies ∆*X*1∆*X*<sup>2</sup> = 1

It can clearly be seen that one of the variances (∆*X*2) decreases in time at the expense of the increase of the other (∆*X*1), with <sup>∆</sup>*X*1(*t*)∆*X*2(*t*) = 1. The squeezed vacuum state is therefore an ideal squeezed state (see footnote 7).

The Hamiltonian given by Eq. (46) is associated with the generation of entangled pairs of photons of equal frequency. For that reason, squeezed states are usually dubbed two photon coherent states [87, 88]. The non-degenerate amplifier is a generalization of the Hamiltonian (46) which generates entangled pairs of distinguishable photons of frequency *ω*<sup>1</sup> and *ω*2, respectively. In that case, the Hamiltonian reads

$$
\hat{H} = i\hbar \chi (\mathfrak{a}\_1^\dagger \mathfrak{a}\_2^\dagger - \mathfrak{a}\_1 \mathfrak{a}\_2),
\tag{51}
$$

*<sup>g</sup>*(2)

and light can be described classically.

and, C, correlator.

quantum mechanics, actually.

that cannot be described classically [59].

(0) = �(*a*†)2*a*2�

�*a*†*a*�<sup>2</sup> <sup>≥</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>

There is then room for a quantum violation of the classical inequality *<sup>g</sup>*(2)(0) ≥ 1. For a large number of photons the quantum inequality (57) becomes the classical constraint *<sup>g</sup>*(2)(0) ≥ 1,

**Figure 4.** Experimental setup for testing photon antibunching [59]: S, source of light; BS, beam splitter; PD, photodetector;

It is worth noticing that what it is violated in an experimental setup involving squeezed and entangled states are some classical assumptions. For instance, in the experimental setup depicted in Fig. 4, the photon is not split into two photons by the beam splitter but it takes either the path that reaches the photo-counter 1 or the path that reaches the photo-counter 2. The fact that the photon is not divided into two photons, as it would happen to an electromagnetic wave, supports the consideration of the photon as a real and individual entity. Moreover, the corpuscular nature of the photon is the postulate that Einstein assumed in order to properly describe the photoelectric effect and it can be considered the germ of

However, such a conclusion does not imply that we can interpret the photon as a classical particle. The double-slit experiment clearly shows that the concept of photon as a localized particle is generally meaningless. The quantum concept of particle has rather to be understood as a global property of the field. Their localization and the space-time independence of different particles depend on the *separability* of their states. Furthermore, the violation of the classical inequalities is associated with negative values of the probability distributions. This can clearly be seen from Eq. (56), where a negative value of *P*(*I*) is needed to obtain a value *<sup>g</sup>*(2)(0) < 1. It plainly shows that there are quantum states of light

Another test for the non-classicality of some quantum states is given by the violation of the Bell's inequalities. This is achieved, for a two mode state of light, whenever it is satisfied [59]

> <sup>1</sup> *<sup>a</sup>*1*a*† <sup>2</sup> *<sup>a</sup>*2�

<sup>2</sup> *<sup>a</sup>*2� <sup>+</sup> �(*a*†

1)2*a*<sup>2</sup> 1� ≥ √2

<sup>2</sup> . (58)

*<sup>C</sup>* <sup>≡</sup> �*a*†

�*a*† <sup>1</sup> *<sup>a</sup>*1*a*† �*a*†*a*�

. (57)

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 201

where *a*ˆ† <sup>1</sup>, *<sup>a</sup>*ˆ1 and *<sup>a</sup>*ˆ† <sup>2</sup>, *a*ˆ2 are the creation and annihilation operators of modes with frequency *ω*<sup>1</sup> and *ω*2, respectively. The solutions of the Heisenberg equations read [82]

$$\mathfrak{a}\_1(t) = \mathfrak{a}\_1(0)\cosh\chi t + \mathfrak{a}\_2^\dagger(0)\sinh\chi t,\tag{52}$$

$$
\hat{a}\_2(t) = \hat{a}\_2(0)\cosh\chi t + \hat{a}\_1^\dagger(0)\sinh\chi t,\tag{53}
$$

and the evolution of the two-mode vacuum state is now given by

$$|s\_2\rangle = \hat{S}\_2(\chi)|0\_10\_2\rangle = e^{\left(\hbar\_1^t\hbar\_2^t - \hbar\_1\hbar\_2\right)\chi t}|0\_10\_2\rangle. \tag{54}$$

where *S*ˆ <sup>2</sup>(*χ*) is the two mode squeeze operator.

Squeezed and entangled states are usually dubbed non-classical states [59] because they may violate some inequalities that should be satisfied in the classical description of light. For instance, in Fig. 4 it is depicted the typical experimental setup to test the violation of the classical inequality *<sup>g</sup>*(2)(0) <sup>≥</sup> 1 (photon bunching [59, 82]), where *<sup>g</sup>*(2)(*τ*) is the second order correlation function that measures the correlation between the state of the field at two different times *<sup>t</sup>* and *<sup>t</sup>* + *<sup>τ</sup>*. Classically, a beam of light with an initial intensity *IA* is split into two beams of equal intensities, *IA*<sup>1</sup> = *IA*<sup>2</sup> ≡ *<sup>I</sup>*. If the averaged intensity is defined by

$$
\langle I \rangle = \int P(I) I \, dI,\tag{55}
$$

for a given positive distribution *<sup>P</sup>*(*I*), then, *<sup>g</sup>*(2)(0) can be written as

$$g^{(2)}(0) = \frac{\langle I\_{A1}I\_{A2}\rangle}{\langle I\_{A1}\rangle \langle I\_{A2}\rangle} = \frac{\langle I^2\rangle}{\langle I\rangle^2} = 1 + \frac{1}{\langle I\rangle^2} \int dI \, P(I)(I - \langle I\rangle)^2 \ge 1. \tag{56}$$

Quantum mechanically, however, the second order correlation function is defined, for a single mode, as [59, 82]

$$g^{(2)}(0) = \frac{\langle (a^\dagger)^2 a^2 \rangle}{\langle a^\dagger a \rangle^2} \ge 1 - \frac{1}{\langle a^\dagger a \rangle}.\tag{57}$$

There is then room for a quantum violation of the classical inequality *<sup>g</sup>*(2)(0) ≥ 1. For a large number of photons the quantum inequality (57) becomes the classical constraint *<sup>g</sup>*(2)(0) ≥ 1, and light can be described classically.

16 Open Questions in Cosmology

where *a*ˆ†

where *S*ˆ

<sup>1</sup>, *<sup>a</sup>*ˆ1 and *<sup>a</sup>*ˆ†

*<sup>g</sup>*(2)

mode, as [59, 82]

an ideal squeezed state (see footnote 7).

respectively. In that case, the Hamiltonian reads

It can clearly be seen that one of the variances (∆*X*2) decreases in time at the expense of the increase of the other (∆*X*1), with <sup>∆</sup>*X*1(*t*)∆*X*2(*t*) = 1. The squeezed vacuum state is therefore

The Hamiltonian given by Eq. (46) is associated with the generation of entangled pairs of photons of equal frequency. For that reason, squeezed states are usually dubbed two photon coherent states [87, 88]. The non-degenerate amplifier is a generalization of the Hamiltonian (46) which generates entangled pairs of distinguishable photons of frequency *ω*<sup>1</sup> and *ω*2,

> † 1 *a*ˆ †

<sup>2</sup>, *a*ˆ2 are the creation and annihilation operators of modes with frequency

†

†

1 *a*ˆ† <sup>2</sup>−*a*ˆ1 *<sup>a</sup>*ˆ2)*χ<sup>t</sup>*

<sup>2</sup> <sup>−</sup> *<sup>a</sup>*ˆ1*a*ˆ2), (51)

<sup>2</sup>(0) sinh *<sup>χ</sup>t*, (52)

<sup>1</sup>(0) sinh *<sup>χ</sup>t*, (53)

*P*(*I*)*I dI*, (55)

*dI P*(*I*)(*I* − �*I*�)<sup>2</sup> ≥ 1. (56)


*H*ˆ = *ih*¯ *χ*(*a*ˆ

*ω*<sup>1</sup> and *ω*2, respectively. The solutions of the Heisenberg equations read [82]

*<sup>a</sup>*ˆ1(*t*) = *<sup>a</sup>*ˆ1(0) cosh *<sup>χ</sup><sup>t</sup>* + *<sup>a</sup>*<sup>ˆ</sup>

*<sup>a</sup>*ˆ2(*t*) = *<sup>a</sup>*ˆ2(0) cosh *<sup>χ</sup><sup>t</sup>* + *<sup>a</sup>*<sup>ˆ</sup>

<sup>2</sup>(*χ*)|0102� <sup>=</sup> *<sup>e</sup>*(*a*ˆ†

two beams of equal intensities, *IA*<sup>1</sup> = *IA*<sup>2</sup> ≡ *<sup>I</sup>*. If the averaged intensity is defined by

�*I*�<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup>

Quantum mechanically, however, the second order correlation function is defined, for a single

1 �*I*�<sup>2</sup> 

�*I*� = 

for a given positive distribution *<sup>P</sup>*(*I*), then, *<sup>g</sup>*(2)(0) can be written as

�*IA*1��*IA*2� <sup>=</sup> �*I*2�

(0) = �*IA*<sup>1</sup> *IA*2�

Squeezed and entangled states are usually dubbed non-classical states [59] because they may violate some inequalities that should be satisfied in the classical description of light. For instance, in Fig. 4 it is depicted the typical experimental setup to test the violation of the classical inequality *<sup>g</sup>*(2)(0) <sup>≥</sup> 1 (photon bunching [59, 82]), where *<sup>g</sup>*(2)(*τ*) is the second order correlation function that measures the correlation between the state of the field at two different times *<sup>t</sup>* and *<sup>t</sup>* + *<sup>τ</sup>*. Classically, a beam of light with an initial intensity *IA* is split into

and the evolution of the two-mode vacuum state is now given by


<sup>2</sup>(*χ*) is the two mode squeeze operator.

**Figure 4.** Experimental setup for testing photon antibunching [59]: S, source of light; BS, beam splitter; PD, photodetector; and, C, correlator.

It is worth noticing that what it is violated in an experimental setup involving squeezed and entangled states are some classical assumptions. For instance, in the experimental setup depicted in Fig. 4, the photon is not split into two photons by the beam splitter but it takes either the path that reaches the photo-counter 1 or the path that reaches the photo-counter 2. The fact that the photon is not divided into two photons, as it would happen to an electromagnetic wave, supports the consideration of the photon as a real and individual entity. Moreover, the corpuscular nature of the photon is the postulate that Einstein assumed in order to properly describe the photoelectric effect and it can be considered the germ of quantum mechanics, actually.

However, such a conclusion does not imply that we can interpret the photon as a classical particle. The double-slit experiment clearly shows that the concept of photon as a localized particle is generally meaningless. The quantum concept of particle has rather to be understood as a global property of the field. Their localization and the space-time independence of different particles depend on the *separability* of their states. Furthermore, the violation of the classical inequalities is associated with negative values of the probability distributions. This can clearly be seen from Eq. (56), where a negative value of *P*(*I*) is needed to obtain a value *<sup>g</sup>*(2)(0) < 1. It plainly shows that there are quantum states of light that cannot be described classically [59].

Another test for the non-classicality of some quantum states is given by the violation of the Bell's inequalities. This is achieved, for a two mode state of light, whenever it is satisfied [59]

$$C \equiv \frac{\langle a\_1^\dagger a\_1 a\_2^\dagger a\_2 \rangle}{\langle a\_1^\dagger a\_1 a\_2^\dagger a\_2 \rangle + \langle (a\_1^\dagger)^2 a\_1^2 \rangle} \ge \frac{\sqrt{2}}{2}. \tag{58}$$

For the two mode squeezed operators (52-53), it can be checked that

$$
\langle a\_1^\dagger a\_1 a\_2^\dagger a\_2 \rangle = N^2 (6\mathbf{x}^4 + 6\mathbf{x}^2 + 1) + N(6\mathbf{x}^4 + 4\mathbf{x}^2) + \mathbf{x}^2 (2\mathbf{x}^2 + 1),
\tag{59}
$$

$$
\langle (a\_1^\dagger)^2 a\_1^2 \rangle = N^2 (6x^4 + 6x^2 + 1) + N(6x^4 + 2x^2 - 1) + 2x^4,\tag{60}
$$

where, *x* ≡ *x*(*t*) = sinh *χt*, and the mean value has been computed for initial number states, with *<sup>N</sup>*<sup>1</sup> = *<sup>N</sup>*<sup>2</sup> ≡ *<sup>N</sup>*. For an initial vacuum state, *<sup>x</sup>*(0) = 0 and *<sup>N</sup>* = 0, then *<sup>C</sup>* = <sup>1</sup> > 0.7, which implies a maximum violation of Bell's inequalities8. This result is expected because the quantum vacuum state is a highly non-local state. For a pair of entangled photons (*N* = 1), it is obtained

$$\mathcal{C} = \frac{14\mathbf{x}^4 + 11\mathbf{x}^2 + 1}{28\mathbf{x}^4 + 19\mathbf{x}^2 + 1},\tag{61}$$

*dE* = *δW* + *δQ*, (65)

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 203

*<sup>T</sup>* <sup>+</sup> *<sup>σ</sup>*. (67)

<sup>2</sup>(*r*), (68)

<sup>2</sup>−*a*ˆ1 *<sup>a</sup>*ˆ2)*r*(*t*), with *<sup>r</sup>*(*t*) = *<sup>χ</sup>t*, and <sup>|</sup>0102� ≡

�*N*2|*ρ*<sup>ˆ</sup> *<sup>N</sup>*2�, (69)

<sup>−</sup>Γ(*t*)*a*ˆ1 *<sup>a</sup>*ˆ2 , (70)

*<sup>S</sup>*(*ρ*) = −Tr(*ρ*ˆ(*t*)ln *<sup>ρ</sup>*ˆ(*t*)) = −Σ*iλi*(*t*)ln *<sup>λ</sup>i*(*t*), (66)

is then directly satisfied. The quantum informational analogue to the entropy is defined

where *<sup>λ</sup>i*(*t*) are the eigenvalues of the density matrix, and 0 ln 0 ≡ 0. For a pure state, *<sup>ρ</sup>*ˆ*<sup>n</sup>* = *<sup>ρ</sup>*<sup>ˆ</sup> and *<sup>λ</sup><sup>i</sup>* = *<sup>δ</sup>ij* for some value *<sup>j</sup>*. Then, the entropy vanishes. For a mixed state, *<sup>S</sup>* > 0. It can be

*dS* <sup>=</sup> *<sup>δ</sup><sup>Q</sup>*

the change of entropy has to be non-negative for any adiabatic process, i.e. *σ* ≥ 0.

*<sup>ρ</sup>*<sup>ˆ</sup> = |*s*2��*s*2| = *<sup>S</sup>*<sup>ˆ</sup>

obtained by tracing out the degrees of freedom of the partner mode, i.e.

*<sup>ρ</sup>*ˆ1 ≡ Tr2*ρ*<sup>ˆ</sup> =

Γ(*t*)*a*ˆ† 1 *a*ˆ† 2 *e* −*g*(*t*)(*a*ˆ†

with, *r*(*t*) = *χt*. We can thus compute the reduced density matrix (69), yielding

The first term corresponds to the variation of the entropy due to the change of heat. The second term in Eq. (67) is called [1] *entropy production*, and it accounts for the variation of entropy due to any adiabatic process. The second principle of thermodynamics states that

Let us now analyze the thermodynamical properties of a two mode squeezed state, Eq. (54),

<sup>2</sup>(*r*) <sup>≡</sup> *<sup>e</sup>*(*a*ˆ†


> ∞ ∑ *N*2=0

and similarly for *ρ*ˆ2 by replacing the indices 2 and 1. By making use of the disentangling

2(*r*)|0102��0102|*S*ˆ†

1 *a*ˆ†

<sup>2</sup>(*r*) can be written as

<sup>2</sup> *<sup>a</sup>*ˆ2+1) *e*

Γ(*t*) ≡ tanh *r*(*t*) , *g*(*t*) ≡ ln cosh *r*(*t*), (71)

<sup>1</sup> *<sup>a</sup>*ˆ1+*a*ˆ†

through the von Neumann formulae [14, 35, 54, 75]

distinguished two terms [1] in the variation of entropy,

represented by the density matrix

where the squeezing operator is given by, *S*ˆ

theorem [10, 86], the squeezing operator *S*ˆ

where

*S*ˆ <sup>2</sup>(*r*) = *<sup>e</sup>*

which implies a violation of Bell's inequalities for a value, 0.31 > sinh *χt* > 0. At later times, the effective number of photons, �*Neff*� = sinh2 *<sup>χ</sup>t*, produced by the parametric amplifier grows and the quantum correlations are destroyed. The radiation effectively becomes classical, then. However, at shorter times, the two mode squeezed states violate the Bell's inequalities showing their non-classical behaviour.

Therefore, entangled and squeezed states can essentially be seen as non-classical states, which is fundamentally related to the complementary principle of quantum mechanics. Generally speaking, the classical description of light in terms of waves and particles, separately, does not hold: i) the photon has to be considered as an individual entity (particle description), and ii) we have to complementary consider interference as well as non-local effects between the states of two distant photons (wave description).

#### **3.3. Thermodynamics of entanglement**

For a physical system whose quantum state is represented by a density matrix9, *ρ*ˆ(*t*), whose evolution is determined by a Hamiltonian, *H*ˆ ≡ *H*ˆ (*t*), we can define the following thermodynamical quantities [1, 14]

$$E(t) = \text{Tr}(\not p(t)\hat{H}(t)),\tag{62}$$

$$Q(t) = \int^t \text{Tr}\left(\frac{d\hat{\rho}(t')}{dt'}\hat{H}(t')\right)dt',\tag{63}$$

$$\mathcal{W}(t) = \int^{t} \text{Tr}\left(\not p(t') \frac{d\hat{H}(t')}{dt'}\right) dt',\tag{64}$$

where Tr(*O*ˆ) denotes the trace of the operator *O*ˆ. The quantities *E*(*t*), *Q*(*t*), and *W*(*t*), are the quantum informational analogue to the energy, heat and work, respectively. The first principle of thermodynamics,

<sup>8</sup> Let us notice that for a pure entangled state like |*ψ*� = <sup>√</sup><sup>1</sup> <sup>2</sup> (|00� <sup>+</sup> <sup>|</sup>11�), �*ψ*|(*a*† <sup>1</sup> )<sup>2</sup> *<sup>a</sup>*<sup>2</sup> <sup>1</sup>|*ψ*� <sup>=</sup> 0 and thus *<sup>C</sup>* <sup>=</sup> 1, too.

<sup>9</sup> In this section, it turns out to be convenient to use the density matrix formalism. This can generally be found in the bibliography (see, for instance, Refs. [14, 32, 35, 54, 75]). Let us just briefly note that for a pure state |*ψ*�, the density matrix is given by *<sup>ρ</sup>*<sup>ˆ</sup> = |*ψ*��*ψ*|, and for a mixed state, *<sup>ρ</sup>*<sup>ˆ</sup> = <sup>∑</sup>*<sup>i</sup> <sup>λ</sup>i*|*i*��*i*|, where *<sup>λ</sup><sup>i</sup>* < 1 are the eigenvalues of the density matrix, with <sup>∑</sup>*<sup>i</sup> <sup>λ</sup><sup>i</sup>* = 1, and the vectors {|*i*�} form an orthonormal basis.

$$dE = \delta W + \delta Q\_{\prime} \tag{65}$$

is then directly satisfied. The quantum informational analogue to the entropy is defined through the von Neumann formulae [14, 35, 54, 75]

$$S(\rho) = -\text{Tr}\left(\hat{\rho}(t)\ln\hat{\rho}(t)\right) = -\Sigma\_i\lambda\_i(t)\ln\lambda\_i(t),\tag{66}$$

where *<sup>λ</sup>i*(*t*) are the eigenvalues of the density matrix, and 0 ln 0 ≡ 0. For a pure state, *<sup>ρ</sup>*ˆ*<sup>n</sup>* = *<sup>ρ</sup>*<sup>ˆ</sup> and *<sup>λ</sup><sup>i</sup>* = *<sup>δ</sup>ij* for some value *<sup>j</sup>*. Then, the entropy vanishes. For a mixed state, *<sup>S</sup>* > 0. It can be distinguished two terms [1] in the variation of entropy,

$$d\mathcal{S} = \frac{\delta \mathcal{Q}}{T} + \sigma.\tag{67}$$

The first term corresponds to the variation of the entropy due to the change of heat. The second term in Eq. (67) is called [1] *entropy production*, and it accounts for the variation of entropy due to any adiabatic process. The second principle of thermodynamics states that the change of entropy has to be non-negative for any adiabatic process, i.e. *σ* ≥ 0.

Let us now analyze the thermodynamical properties of a two mode squeezed state, Eq. (54), represented by the density matrix

$$\mathfrak{f} = |\mathfrak{s}\_2\rangle\langle\mathfrak{s}\_2| = \hat{\mathfrak{S}}\_2(r)|0\_10\_2\rangle\langle 0\_10\_2|\hat{\mathfrak{S}}\_2^\dagger(r),\tag{68}$$

where the squeezing operator is given by, *S*ˆ <sup>2</sup>(*r*) <sup>≡</sup> *<sup>e</sup>*(*a*ˆ† 1 *a*ˆ† <sup>2</sup>−*a*ˆ1 *<sup>a</sup>*ˆ2)*r*(*t*), with *<sup>r</sup>*(*t*) = *<sup>χ</sup>t*, and <sup>|</sup>0102� ≡ |01�|02�, with |01� and |02� being the initial ground states of each single mode, respectively. The reduced density matrix that represents the quantum state of each single mode can be obtained by tracing out the degrees of freedom of the partner mode, i.e.

$$\not{p}\_1 \equiv \text{Tr}\_2 \not{p} = \sum\_{N\_2=0}^{\infty} \langle N\_2 | \not{p} | N\_2 \rangle \,\tag{69}$$

and similarly for *ρ*ˆ2 by replacing the indices 2 and 1. By making use of the disentangling theorem [10, 86], the squeezing operator *S*ˆ <sup>2</sup>(*r*) can be written as

$$\hat{S}\_2(r) = e^{\Gamma(t)\hbar\_1^\dagger \hbar\_2^\dagger} e^{-\mathcal{G}(t)(\hbar\_1^\dagger \hbar\_1 + \hbar\_2^\dagger \hbar\_2 + 1)} e^{-\Gamma(t)\hbar\_1 \hbar\_2} \,. \tag{70}$$

where

18 Open Questions in Cosmology

it is obtained

�*a*† <sup>1</sup> *<sup>a</sup>*1*a*†

> �(*a*† 1)2*a*<sup>2</sup>

For the two mode squeezed operators (52-53), it can be checked that

the Bell's inequalities showing their non-classical behaviour.

**3.3. Thermodynamics of entanglement**

thermodynamical quantities [1, 14]

principle of thermodynamics,

<sup>8</sup> Let us notice that for a pure entangled state like |*ψ*� = <sup>√</sup><sup>1</sup>

matrix, with <sup>∑</sup>*<sup>i</sup> <sup>λ</sup><sup>i</sup>* = 1, and the vectors {|*i*�} form an orthonormal basis.

effects between the states of two distant photons (wave description).

*Q*(*t*) =

*W*(*t*) =

 *<sup>t</sup>* Tr

 *<sup>t</sup>* Tr *ρ*ˆ(*t* ′ ) *dH*ˆ (*t* ′ ) *dt*′

<sup>2</sup> *<sup>a</sup>*2� <sup>=</sup> *<sup>N</sup>*2(6*x*<sup>4</sup> <sup>+</sup> <sup>6</sup>*x*<sup>2</sup> <sup>+</sup> <sup>1</sup>) + *<sup>N</sup>*(6*x*<sup>4</sup> <sup>+</sup> <sup>4</sup>*x*2) + *<sup>x</sup>*2(2*x*<sup>2</sup> <sup>+</sup> <sup>1</sup>), (59)

where, *x* ≡ *x*(*t*) = sinh *χt*, and the mean value has been computed for initial number states, with *<sup>N</sup>*<sup>1</sup> = *<sup>N</sup>*<sup>2</sup> ≡ *<sup>N</sup>*. For an initial vacuum state, *<sup>x</sup>*(0) = 0 and *<sup>N</sup>* = 0, then *<sup>C</sup>* = <sup>1</sup> > 0.7, which implies a maximum violation of Bell's inequalities8. This result is expected because the quantum vacuum state is a highly non-local state. For a pair of entangled photons (*N* = 1),

> *<sup>C</sup>* <sup>=</sup> <sup>14</sup>*x*<sup>4</sup> <sup>+</sup> <sup>11</sup>*x*<sup>2</sup> <sup>+</sup> <sup>1</sup> 28*x*<sup>4</sup> + 19*x*<sup>2</sup> + 1

which implies a violation of Bell's inequalities for a value, 0.31 > sinh *χt* > 0. At later times, the effective number of photons, �*Neff*� = sinh2 *<sup>χ</sup>t*, produced by the parametric amplifier grows and the quantum correlations are destroyed. The radiation effectively becomes classical, then. However, at shorter times, the two mode squeezed states violate

Therefore, entangled and squeezed states can essentially be seen as non-classical states, which is fundamentally related to the complementary principle of quantum mechanics. Generally speaking, the classical description of light in terms of waves and particles, separately, does not hold: i) the photon has to be considered as an individual entity (particle description), and ii) we have to complementary consider interference as well as non-local

For a physical system whose quantum state is represented by a density matrix9, *ρ*ˆ(*t*), whose evolution is determined by a Hamiltonian, *H*ˆ ≡ *H*ˆ (*t*), we can define the following

> *dρ*ˆ(*t* ′ ) *dt*′ *<sup>H</sup>*<sup>ˆ</sup> (*<sup>t</sup>*

where Tr(*O*ˆ) denotes the trace of the operator *O*ˆ. The quantities *E*(*t*), *Q*(*t*), and *W*(*t*), are the quantum informational analogue to the energy, heat and work, respectively. The first

<sup>9</sup> In this section, it turns out to be convenient to use the density matrix formalism. This can generally be found in the bibliography (see, for instance, Refs. [14, 32, 35, 54, 75]). Let us just briefly note that for a pure state |*ψ*�, the density matrix is given by *<sup>ρ</sup>*<sup>ˆ</sup> = |*ψ*��*ψ*|, and for a mixed state, *<sup>ρ</sup>*<sup>ˆ</sup> = <sup>∑</sup>*<sup>i</sup> <sup>λ</sup>i*|*i*��*i*|, where *<sup>λ</sup><sup>i</sup>* < 1 are the eigenvalues of the density

*E*(*t*) = Tr(*ρ*ˆ(*t*)*H*ˆ (*t*)), (62)

<sup>1</sup> )<sup>2</sup> *<sup>a</sup>*<sup>2</sup>

′ ) *dt*′

<sup>2</sup> (|00� <sup>+</sup> <sup>|</sup>11�), �*ψ*|(*a*†

 *dt*′

<sup>1</sup>� <sup>=</sup> *<sup>N</sup>*2(6*x*<sup>4</sup> <sup>+</sup> <sup>6</sup>*x*<sup>2</sup> <sup>+</sup> <sup>1</sup>) + *<sup>N</sup>*(6*x*<sup>4</sup> <sup>+</sup> <sup>2</sup>*x*<sup>2</sup> <sup>−</sup> <sup>1</sup>) + <sup>2</sup>*x*4, (60)

, (61)

, (63)

, (64)

<sup>1</sup>|*ψ*� <sup>=</sup> 0 and thus *<sup>C</sup>* <sup>=</sup> 1, too.

$$
\Gamma(t) \equiv \tanh r(t) \quad , \ \lg(t) \equiv \ln \cosh r(t) \,\tag{71}
$$

with, *r*(*t*) = *χt*. We can thus compute the reduced density matrix (69), yielding

$$\hat{\rho}\_1(t) = e^{-2g(t)} \sum\_{\text{Nl}=0}^{\infty} e^{2\text{N}\_1 \ln \Gamma(t)} |\text{N}\_1\rangle\langle\text{N}\_1| = \frac{1}{\cosh^2 r(t)} \sum\_{\text{Nl}=0}^{\infty} \left(\tanh^2 r(t)\right)^{\text{N}\_1} |\text{N}\_1\rangle\langle\text{N}\_1|. \tag{72}$$

It turns out to be that *ρ*ˆ1 describes a thermal state

$$\left|\rho\_1(t) = \frac{1}{Z(t)}\sum\_{N\_1=0}^{\infty} e^{-\frac{\omega\_1}{T(t)}\left(N\_1 + \frac{1}{2}\right)}|N\_1\rangle\langle N\_1|\right.\tag{73}$$

where, *<sup>Z</sup>*−<sup>1</sup> = 2 sinh *<sup>ω</sup>*<sup>1</sup> <sup>2</sup>*T*(*t*), with a time dependent temperature of entanglement given by

$$T(t) = \frac{\omega\_1}{2\ln\frac{1}{\Gamma(t)}},\tag{74}$$

It can also be checked that

*arrow of time* [36, 56].

where, ˆ

*bk* ≡ <sup>ˆ</sup>

*bk*(*a*0) and <sup>ˆ</sup>

with the mode-dependent frequency,

damped harmonic oscillator,

*<sup>σ</sup>* <sup>≡</sup> *dSent*

**4. Quantum entanglement in the multiverse**

*h*¯ <sup>2</sup>*φ*¨ +

considered, can be decomposed in normal modes as

*b*† *<sup>k</sup>* <sup>≡</sup> <sup>ˆ</sup> *b*†

*<sup>φ</sup>*ˆ(*a*, *<sup>ϕ</sup>*) =

*h*¯ 2 *<sup>a</sup> <sup>φ</sup>*˙ <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup>

where, *φ* ≡ *φ*(*a*, *ϕ*) is the wave function of the universe with, *φ*˙ ≡ *∂φ*

*dk e*

*<sup>ω</sup>k*(*a*) = <sup>1</sup>

*<sup>k</sup>*(*a*) + <sup>M</sup>˙ <sup>M</sup>*A*˙

evaluated at *<sup>a</sup>*0. The probability amplitudes *Ak*(*a*) and *<sup>A</sup>*<sup>∗</sup>

*A*¨

*h*¯

**4.1. Creation of entangled pairs of universes**

*dt* <sup>−</sup> <sup>1</sup> *T*(*t*) *δQ*

Therefore, the second principle of thermodynamics provides us with no arrow of time because the entropy production *σ* identically vanishes at any time. In a non-reversible process, however, the constraint *σ* > 0 would give rise to the *entanglement thermodynamical*

First, we shall present a plausible scenario for the nucleation of a pair of entangled universes. The Wheeler-de Witt equation (12) for a de-Sitter universe with a massless scalar field reads

the cosmological constant. As it was already pointed out in Sec. 2, in the third quantization formalism the wave function *φ* is promoted to an operator *φ*ˆ that, in the case now being

*ik<sup>ϕ</sup>Ak*(*a*)<sup>ˆ</sup>

*b*† *<sup>k</sup>* <sup>+</sup> *<sup>e</sup>*

Λ*a*<sup>4</sup> − *a*<sup>2</sup> +

*<sup>k</sup>*(*a*) + *<sup>ω</sup>*<sup>2</sup>

with, M≡M(*a*) = *<sup>a</sup>*, and *<sup>ω</sup><sup>k</sup>* ≡ *<sup>ω</sup>k*(*a*). Let us recall that the real values of the frequency (82) define the oscillatory regime of the wave function of the universe in the Lorentzian region, and the complex values define the exponential regime of the Euclidean region. Let us first consider the zero mode of the wave function, i.e. *k* = 0. Then, the wave function

<sup>−</sup>*ikϕA*<sup>∗</sup>

*h*¯ <sup>2</sup>*k*<sup>2</sup>

*<sup>k</sup>* (*a*)<sup>ˆ</sup> *bk* 

*<sup>k</sup>* (*a*0), are the constant operators defined in Eqs. (26-27), now

*<sup>δ</sup><sup>t</sup>* <sup>=</sup> 0 , <sup>∀</sup>*t*. (79)

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 205

*<sup>a</sup>*<sup>2</sup> *<sup>φ</sup>*′′ + (Λ*a*<sup>4</sup> <sup>−</sup> *<sup>a</sup>*2)*<sup>φ</sup>* <sup>=</sup> 0, (80)

*<sup>∂</sup><sup>a</sup>* and *<sup>φ</sup>*′ <sup>≡</sup> *∂φ*

*<sup>a</sup>*<sup>2</sup> , (82)

*<sup>k</sup> Ak*(*a*) = 0, (83)

*<sup>k</sup>* (*a*) satisfy the equation of the

*∂ϕ* , and <sup>Λ</sup> is

, (81)

with *ω*<sup>1</sup> being the frequency of the mode. It is worth mentioning that the thermal state (73) is indistinguishable from a classical mixture [36, 56]. In that sense, it can be seen as a classical state. However, it has been obtained from the partial trace of a composite entangled state which is, as it has previously been shown, a quantum state having no classical analogue.

We can now compute the thermodynamical quantities given by Eqs. (62-64) and Eq. (66) for the thermal state (73). The entropy of entanglement, i.e. the quantum entropy that corresponds to the reduced density matrix *ρ*ˆ1, reads

$$S\_{\rm eff}(t) = -\text{Tr}(\not p\_1 \ln \not p\_1) = \cosh^2 r(t) \ln \cosh^2 r(t) - \sinh^2 r(t) \ln \sinh^2 r(t). \tag{75}$$

The total energy *<sup>E</sup>*<sup>1</sup> ≡ *<sup>E</sup>*(*ρ*1) yields

$$E\_1(t) = \omega\_1(\sinh^2 r(t) + \frac{1}{2}) \equiv \omega\_1(\langle N(t) \rangle + \frac{1}{2}),\tag{76}$$

where �*N*(*t*)� is an effective mean number of photons due to the squeezing effect. For a mode of constant frequency *ω*1, the variation of work vanishes because

$$
\delta W\_1 = \frac{d\omega\_1}{dt} (\sinh^2 r(t) + \frac{1}{2}) = 0. \tag{77}
$$

The variation of heat is however different from zero. It reads

$$
\delta Q = \omega\_1 \sinh 2r(t) \frac{\partial r(t)}{\partial t} dt. \tag{78}
$$

It can also be checked that

20 Open Questions in Cosmology

*<sup>ρ</sup>*ˆ1(*t*) = *<sup>e</sup>*

where, *<sup>Z</sup>*−<sup>1</sup> = 2 sinh *<sup>ω</sup>*<sup>1</sup>

<sup>−</sup>2*g*(*t*) <sup>∞</sup> ∑ *N*1=0 *e* 2*N*<sup>1</sup> ln Γ(*t*)

It turns out to be that *ρ*ˆ1 describes a thermal state

corresponds to the reduced density matrix *ρ*ˆ1, reads

The total energy *<sup>E</sup>*<sup>1</sup> ≡ *<sup>E</sup>*(*ρ*1) yields

*<sup>ρ</sup>*ˆ1(*t*) = <sup>1</sup>

*Z*(*t*)

*<sup>E</sup>*1(*t*) = *<sup>ω</sup>*1(sinh<sup>2</sup> *<sup>r</sup>*(*t*) + <sup>1</sup>

mode of constant frequency *ω*1, the variation of work vanishes because

*<sup>δ</sup>W*<sup>1</sup> <sup>=</sup> *<sup>d</sup>ω*<sup>1</sup>

The variation of heat is however different from zero. It reads

<sup>|</sup>*N*1��*N*1<sup>|</sup> <sup>=</sup> <sup>1</sup>

∞ ∑ *N*1=0 *e*

*<sup>T</sup>*(*t*) = *<sup>ω</sup>*<sup>1</sup>

with *ω*<sup>1</sup> being the frequency of the mode. It is worth mentioning that the thermal state (73) is indistinguishable from a classical mixture [36, 56]. In that sense, it can be seen as a classical state. However, it has been obtained from the partial trace of a composite entangled state which is, as it has previously been shown, a quantum state having no classical analogue.

We can now compute the thermodynamical quantities given by Eqs. (62-64) and Eq. (66) for the thermal state (73). The entropy of entanglement, i.e. the quantum entropy that

*Sent*(*t*) = −Tr(*ρ*ˆ1 ln *<sup>ρ</sup>*ˆ1) = cosh<sup>2</sup> *<sup>r</sup>*(*t*)ln cosh2 *<sup>r</sup>*(*t*) − sinh2 *<sup>r</sup>*(*t*)ln sinh2 *<sup>r</sup>*(*t*). (75)

) ≡ *<sup>ω</sup>*1(�*N*(*t*)� +

2

*∂r*(*t*) *∂t*

1 2

), (76)

) = 0. (77)

*dt*. (78)

2

where �*N*(*t*)� is an effective mean number of photons due to the squeezing effect. For a

*dt* (sinh2 *<sup>r</sup>*(*t*) + <sup>1</sup>

*<sup>δ</sup><sup>Q</sup>* = *<sup>ω</sup>*<sup>1</sup> sinh 2*r*(*t*)

2 ln <sup>1</sup> Γ(*t*)

cosh2 *r*(*t*)

<sup>−</sup> *<sup>ω</sup>*<sup>1</sup> *<sup>T</sup>*(*t*) (*N*1<sup>+</sup> <sup>1</sup>

2 )

<sup>2</sup>*T*(*t*), with a time dependent temperature of entanglement given by

∞ ∑ *N*1=0 

tanh<sup>2</sup> *r*(*t*)

*<sup>N</sup>*<sup>1</sup>


, (74)


$$
\sigma \equiv \frac{d\mathcal{S}\_{ent}}{dt} - \frac{1}{T(t)} \frac{\delta \mathcal{Q}}{\delta t} = 0 \quad \text{, } \forall t. \tag{79}
$$

Therefore, the second principle of thermodynamics provides us with no arrow of time because the entropy production *σ* identically vanishes at any time. In a non-reversible process, however, the constraint *σ* > 0 would give rise to the *entanglement thermodynamical arrow of time* [36, 56].

## **4. Quantum entanglement in the multiverse**

#### **4.1. Creation of entangled pairs of universes**

First, we shall present a plausible scenario for the nucleation of a pair of entangled universes. The Wheeler-de Witt equation (12) for a de-Sitter universe with a massless scalar field reads

$$
\hbar^2 \ddot{\phi} + \frac{\hbar^2}{a} \phi - \frac{\hbar^2}{a^2} \phi^{\prime\prime} + (\Lambda a^4 - a^2) \phi = 0,\tag{80}
$$

where, *φ* ≡ *φ*(*a*, *ϕ*) is the wave function of the universe with, *φ*˙ ≡ *∂φ <sup>∂</sup><sup>a</sup>* and *<sup>φ</sup>*′ <sup>≡</sup> *∂φ ∂ϕ* , and <sup>Λ</sup> is the cosmological constant. As it was already pointed out in Sec. 2, in the third quantization formalism the wave function *φ* is promoted to an operator *φ*ˆ that, in the case now being considered, can be decomposed in normal modes as

$$
\hat{\Phi}(\mathfrak{a}, \mathfrak{q}) = \int dk \left( e^{ik\mathfrak{q}} A\_k(a) \mathfrak{f}\_k^\dagger + e^{-ik\mathfrak{q}} A\_k^\*(a) \mathfrak{f}\_k \right), \tag{81}
$$

where, ˆ *bk* ≡ <sup>ˆ</sup> *bk*(*a*0) and <sup>ˆ</sup> *b*† *<sup>k</sup>* <sup>≡</sup> <sup>ˆ</sup> *b*† *<sup>k</sup>* (*a*0), are the constant operators defined in Eqs. (26-27), now with the mode-dependent frequency,

$$
\omega\_k(a) = \frac{1}{\hbar} \sqrt{\Lambda a^4 - a^2 + \frac{\hbar^2 k^2}{a^2}},\tag{82}
$$

evaluated at *<sup>a</sup>*0. The probability amplitudes *Ak*(*a*) and *<sup>A</sup>*<sup>∗</sup> *<sup>k</sup>* (*a*) satisfy the equation of the damped harmonic oscillator,

$$
\ddot{A}\_k(a) + \frac{\dot{\mathcal{M}}}{\mathcal{M}} \dot{A}\_k(a) + \omega\_k^2 A\_k(a) = 0,\tag{83}
$$

with, M≡M(*a*) = *<sup>a</sup>*, and *<sup>ω</sup><sup>k</sup>* ≡ *<sup>ω</sup>k*(*a*). Let us recall that the real values of the frequency (82) define the oscillatory regime of the wave function of the universe in the Lorentzian region, and the complex values define the exponential regime of the Euclidean region. Let us first consider the zero mode of the wave function, i.e. *k* = 0. Then, the wave function

proposed by Barvinsky and Kamenshchik in Refs. [4–6], two instantons can be matched by identifying their hypersurfaces <sup>Σ</sup>′′ (see Fig. 6). The instantons can thus be created in pairs which would eventually give rise to an entangled pair of universes. Let us notice that this is a quantum effect having no classical analog because the quantum correction term in Eq. (82)

The matching hypersurface <sup>Σ</sup>′′ <sup>≡</sup> <sup>Σ</sup>′′(*a*−), where *<sup>a</sup>*<sup>−</sup> <sup>≡</sup> *<sup>a</sup>*−(*θk*) is given by Eq. (85) with Eq. (86), depends on the value *k* of the mode. Therefore, the matched instantons can only be joined for an equal value of the mode of their respective scalar fields. The universes created from such a double instanton are then entangled, with a composite quantum state given by

where *ϕI*,*I I* are the values of the scalar field of each single universe, labelled by *I* and *I I*,

universes because the orthonormality relations between the modes [62]. Then, the composite

It is also worth mentioning that, in the model being considered, there is no Euclidean regime for values *k* ≥ *km* and, therefore, no universes are created from the space-time foam with such values of the mode. Then, *km* can be considered the natural cut-off of the model. Let us also note that a similar behavior of the modes of the universe would be obtained for a non-massless scalar field provided that the potential of the scalar field, *V*(*ϕ*), satisfies the

Entangled states, like those found in the preceding section or those appearing in the phantom multiverse [18, 62], can generally be posed in the quantum multiverse. Furthermore, the canonical representations of the harmonic oscillator that represent the quantum state of the multiverse, in the model described in Sec. 2, are related by squeezed transformations [41].

As we saw in Sec. 3, entangled and squeezed states are usually dubbed 'non-classical' states because they are related to the violation of classical inequalities. Such violation is fundamentally associated to the complementary principle of quantum mechanics. In the multiverse, squeezed and entangled states may also violate the classical inequalities [62]. However, the conceptual meaning of such violation can be quite different from that given in quantum optics. For instance, if the existence of entangled and squeezed states would imply a violation of Bell's inequalities, then, it could not be interpreted in terms of locality or non-locality because these concepts are only well-defined inside a universe, where space and time are meaningful. In the quantum multiverse, there is generally no common space-time among the universes and, therefore, the violation of Bell's inequalities would be rather related to the interdependence of the quantum states that represent different universes of

Like in quantum optics [75], the violation of the classical inequalities in the multiverse depends on the representation which is chosen to describe the quantum states of the

<sup>−</sup>*ik*(*ϕI*+*ϕI I*)*A*<sup>∗</sup>

*I*,*k*(*a*)*A*<sup>∗</sup>

*I I*,*<sup>k</sup>* cannot be present in the state of the pair of

*I I*,*k*(*a*) <sup>ˆ</sup>

*bI*,*<sup>k</sup>* <sup>ˆ</sup> *bI I*,*<sup>k</sup>* , (87)

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 207

*b*† *I*,*k* ˆ *b*† *I I*,*<sup>k</sup>* <sup>+</sup> *<sup>e</sup>*

quantum state must necessarily be the entangled state represented by Eq. (87).

Thus, squeezed states may generally be considered in the quantum multiverse.

does not appear in the classical theory.

respectively. The cross terms like *AI*,*kA*<sup>∗</sup>

boundary condition [38, 39], *V*(*ϕ*) → 0 for *a* → 0.

**4.2. Entangled and squeezed states in the multiverse**

*ik*(*ϕI*+*ϕI I*)*AI*,*k*(*a*)*AI I*,*k*(*a*) <sup>ˆ</sup>

*<sup>φ</sup>I*,*I I* =

the multiverse.

 *dk e*

**Figure 5.** Before reaching the collapse, the instanton finds the transition hypersurface Σ′′.

**Figure 6.** Creation of a pair of entangled universes from a pair of instantons.

*φ*Λ(*a*) quantum mechanically describes the nucleation of a de-Sitter universe from a de-Sitter instanton [26, 39, 77, 78] depicted in Sec. 2, with a transition hypersurface <sup>Σ</sup><sup>0</sup> ≡ <sup>Σ</sup>(*a*0) located at *<sup>a</sup>*<sup>0</sup> = <sup>√</sup> 1 <sup>Λ</sup> (see, Fig 1).

For values of *k* different from zero, the quantum correction term given in Eq. (82) introduces a novelty. For the value, *km* > *k* > 0, where *k*<sup>2</sup> *<sup>m</sup>* <sup>≡</sup> <sup>4</sup> 27¯*h*<sup>2</sup> <sup>Λ</sup><sup>2</sup> , there are two transition hypersurfaces from the Euclidean to the Lorentzian region, <sup>Σ</sup>′ ≡ <sup>Σ</sup>(*a*+) and <sup>Σ</sup>′′ ≡ <sup>Σ</sup>(*a*−), respectively, located at [62]

$$a\_+ \equiv \frac{1}{\sqrt{3\Lambda}} \sqrt{1 + 2\cos\left(\frac{\theta\_k}{3}\right)}\,\tag{84}$$

$$a\_- \equiv \frac{1}{\sqrt{3\Lambda}} \sqrt{1 - 2\cos\left(\frac{\theta\_k + \pi}{3}\right)},\tag{85}$$

where, in units for which ¯*h* = 1,

$$\theta\_k \equiv \arctan \frac{2k\sqrt{k\_m^2 - k^2}}{k\_m^2 - 2k^2}. \tag{86}$$

The picture is then rather different from the one depicted in Fig. 1. First, at the transition hypersurface <sup>Σ</sup>′ the universe finds the Euclidean region (let us notice that for *<sup>k</sup>* <sup>→</sup> 0, *<sup>a</sup>*<sup>+</sup> <sup>→</sup> *<sup>a</sup>*<sup>0</sup> and *a*<sup>−</sup> → 0). However, before reaching the collapse, the Euclidean instanton finds a new transition hypersurface <sup>Σ</sup>′′ (see Fig 5). Then, following a mechanism that parallels that proposed by Barvinsky and Kamenshchik in Refs. [4–6], two instantons can be matched by identifying their hypersurfaces <sup>Σ</sup>′′ (see Fig. 6). The instantons can thus be created in pairs which would eventually give rise to an entangled pair of universes. Let us notice that this is a quantum effect having no classical analog because the quantum correction term in Eq. (82) does not appear in the classical theory.

The matching hypersurface <sup>Σ</sup>′′ <sup>≡</sup> <sup>Σ</sup>′′(*a*−), where *<sup>a</sup>*<sup>−</sup> <sup>≡</sup> *<sup>a</sup>*−(*θk*) is given by Eq. (85) with Eq. (86), depends on the value *k* of the mode. Therefore, the matched instantons can only be joined for an equal value of the mode of their respective scalar fields. The universes created from such a double instanton are then entangled, with a composite quantum state given by

$$\phi\_{\rm I,II} = \int dk \left( e^{i\mathbf{k}(\varphi\_{\rm I} + \varphi\_{\rm II})} A\_{\rm I,k}(a) A\_{\rm II,k}(a) \hat{b}\_{\rm I,k}^{\dagger} \hat{b}\_{\rm II,k}^{\dagger} + e^{-i\mathbf{k}(\varphi\_{\rm I} + \varphi\_{\rm II})} A\_{\rm I,k}^{\ast}(a) A\_{\rm II,k}^{\ast}(a) \hat{b}\_{\rm I,k} \hat{b}\_{\rm II,k} \right), \tag{87}$$

where *ϕI*,*I I* are the values of the scalar field of each single universe, labelled by *I* and *I I*, respectively. The cross terms like *AI*,*kA*<sup>∗</sup> *I I*,*<sup>k</sup>* cannot be present in the state of the pair of universes because the orthonormality relations between the modes [62]. Then, the composite quantum state must necessarily be the entangled state represented by Eq. (87).

It is also worth mentioning that, in the model being considered, there is no Euclidean regime for values *k* ≥ *km* and, therefore, no universes are created from the space-time foam with such values of the mode. Then, *km* can be considered the natural cut-off of the model. Let us also note that a similar behavior of the modes of the universe would be obtained for a non-massless scalar field provided that the potential of the scalar field, *V*(*ϕ*), satisfies the boundary condition [38, 39], *V*(*ϕ*) → 0 for *a* → 0.

## **4.2. Entangled and squeezed states in the multiverse**

22 Open Questions in Cosmology

at *<sup>a</sup>*<sup>0</sup> = <sup>√</sup>

located at [62]

1

<sup>Λ</sup> (see, Fig 1).

where, in units for which ¯*h* = 1,

novelty. For the value, *km* > *k* > 0, where *k*<sup>2</sup>

**Figure 5.** Before reaching the collapse, the instanton finds the transition hypersurface Σ′′.

**Figure 6.** Creation of a pair of entangled universes from a pair of instantons.

*<sup>a</sup>*<sup>+</sup> <sup>≡</sup> <sup>1</sup> √3Λ

*<sup>a</sup>*<sup>−</sup> <sup>≡</sup> <sup>1</sup> √3Λ

*φ*Λ(*a*) quantum mechanically describes the nucleation of a de-Sitter universe from a de-Sitter instanton [26, 39, 77, 78] depicted in Sec. 2, with a transition hypersurface <sup>Σ</sup><sup>0</sup> ≡ <sup>Σ</sup>(*a*0) located

For values of *k* different from zero, the quantum correction term given in Eq. (82) introduces a

from the Euclidean to the Lorentzian region, <sup>Σ</sup>′ ≡ <sup>Σ</sup>(*a*+) and <sup>Σ</sup>′′ ≡ <sup>Σ</sup>(*a*−), respectively,

*<sup>θ</sup><sup>k</sup>* ≡ arctan

*<sup>m</sup>* <sup>≡</sup> <sup>4</sup> 27¯*h*<sup>2</sup>

1 + 2 cos

1 − 2 cos

2*k k*<sup>2</sup> *<sup>m</sup>* − *<sup>k</sup>*<sup>2</sup>

The picture is then rather different from the one depicted in Fig. 1. First, at the transition hypersurface <sup>Σ</sup>′ the universe finds the Euclidean region (let us notice that for *<sup>k</sup>* <sup>→</sup> 0, *<sup>a</sup>*<sup>+</sup> <sup>→</sup> *<sup>a</sup>*<sup>0</sup> and *a*<sup>−</sup> → 0). However, before reaching the collapse, the Euclidean instanton finds a new transition hypersurface <sup>Σ</sup>′′ (see Fig 5). Then, following a mechanism that parallels that

*k*2

 *θ<sup>k</sup>* 3 

 *<sup>θ</sup><sup>k</sup>* + *<sup>π</sup>* 3

<sup>Λ</sup><sup>2</sup> , there are two transition hypersurfaces

*<sup>m</sup>* <sup>−</sup> <sup>2</sup>*k*<sup>2</sup> . (86)

, (84)

, (85)

Entangled states, like those found in the preceding section or those appearing in the phantom multiverse [18, 62], can generally be posed in the quantum multiverse. Furthermore, the canonical representations of the harmonic oscillator that represent the quantum state of the multiverse, in the model described in Sec. 2, are related by squeezed transformations [41]. Thus, squeezed states may generally be considered in the quantum multiverse.

As we saw in Sec. 3, entangled and squeezed states are usually dubbed 'non-classical' states because they are related to the violation of classical inequalities. Such violation is fundamentally associated to the complementary principle of quantum mechanics. In the multiverse, squeezed and entangled states may also violate the classical inequalities [62]. However, the conceptual meaning of such violation can be quite different from that given in quantum optics. For instance, if the existence of entangled and squeezed states would imply a violation of Bell's inequalities, then, it could not be interpreted in terms of locality or non-locality because these concepts are only well-defined inside a universe, where space and time are meaningful. In the quantum multiverse, there is generally no common space-time among the universes and, therefore, the violation of Bell's inequalities would be rather related to the interdependence of the quantum states that represent different universes of the multiverse.

Like in quantum optics [75], the violation of the classical inequalities in the multiverse depends on the representation which is chosen to describe the quantum states of the universes [62]. Unlike quantum optics, we do not have an experimental device to measure other universes rather than our own universe10. However, the extension of the complementary principle to the quantum description of the multiverse entails two main consequences. On the one hand, if the wave function of the universe has to be described in terms of 'particles', it means that in some appropriate representation we can formally distinguish the universal states as individual entities, giving rise therefore to the multiverse scenario. On the other hand, if it has to be complementary described in terms of waves, then, interference between the quantum states of two or more universes can generally be considered as well.

## **4.3. Thermodynamical properties of entangled universes**

Let us consider a multiverse made up of homogeneous and isotropic universes with a slow-varying scalar field *<sup>ϕ</sup>*, recalling that in the case for which *<sup>ϕ</sup>*˙ = 0 and *<sup>V</sup>*(*ϕ*0) ≡ <sup>Λ</sup>, the model effectively represents a multiverse formed by de-Sitter universes.

Let us consider one type of universes and describe the quantum state of the multiverse in terms of the annihilation and creation operators ˆ *b*(*a*) and ˆ *b*†(*a*) given in Eqs. (32-33). The vacuum state of the multiverse, |0¯�, is then defined as the eigenstate of the annihilation operator ˆ *b*(*a*) with eigenvalue zero, i.e. ˆ *b*(*a*)|0¯� ≡ 0. On the other hand, observers inhabiting a large parent universe would quantum mechanically describe the state of their respective universes in the asymptotic representation given by Eqs. (29-30), with a ground state |0� defined by, ˆ *<sup>b</sup>ω*(*a*)|0� ≡ 0.

We can consider therefore two representations: the one derived from a consistent formulation of the boundary condition of the whole multiverse, or *invariant representation*, given by the operators ˆ *b*(*a*) and ˆ *b*†(*a*), and the asymptotic representation given by the operators ˆ *<sup>b</sup>ω*(*a*) and ˆ *b*† *<sup>ω</sup>*(*a*), which might be called the *observer representation*. They both are related by the squeezing transformation

$$
\delta = \mu\_{\omega} \,\delta\_{\omega} + \nu\_{\omega} \,\delta\_{\omega \prime}^{\dagger} \tag{88}
$$

where the positive sign corresponds to the choice of the no-boundary condition and the negative sign to the tunneling boundary condition. Let us further assume that the multiverse is in the invariant vacuum state |0¯�. The density matrix that represents the quantum state of

where |0102�≡|01�|02�, with |01� and |02� being the ground states of a pair of entangled universes in their respective observer representations. Similarly to Eq. (54), the squeezing

> *r*(*a*,*ϕ*)ˆ *b*1 ˆ *b*2−*r*(*a*,*ϕ*)ˆ *b*† 1 ˆ *b*†

with *νω*(*a*, *<sup>ϕ</sup>*) being given by Eq. (91). We can then follow the procedure of Sec. 3.3 to compute the reduced density matrix, *ρ*ˆ1, that represents the quantum state of one single

> ∞ ∑ *N*=0 *e* − *<sup>ω</sup>*(*a*,*ϕ*) *<sup>T</sup>* (*N*<sup>+</sup> <sup>1</sup> 2 )

observer representation of each single universe, in thermal equilibrium with a temperature

*<sup>T</sup>* <sup>≡</sup> *<sup>T</sup>*(*a*, *<sup>ϕ</sup>*) = *<sup>ω</sup>*(*a*, *<sup>ϕ</sup>*)

*E*(*a*) = *ω*(*a*)(�*N*� +

1 2 ) <sup>≈</sup> *∂ω*(*a*, *<sup>ϕ</sup>*)

*∂*�*N*� *∂a*

2 ln <sup>1</sup> Γ(*a*,*ϕ*)

> 1 2

2. The variation of the quantum informational analogues to the work, *W*,

*<sup>∂</sup><sup>a</sup>* (�*N*� <sup>+</sup>

1 2

*Z*

*<sup>S</sup>* <sup>|</sup>0102��0102|U<sup>ˆ</sup>

*<sup>r</sup>*(*a*, *<sup>ϕ</sup>*) ≡ arcsinh|*νω*(*a*, *<sup>ϕ</sup>*)|, (94)

<sup>2</sup>*<sup>T</sup>* . The two universes of the entangled pair evolve, in the

*<sup>S</sup>*, (92)

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 209

<sup>2</sup> , (93)


, (96)

), (97)

*da*, (99)

) *da*, (98)

*ρ*ˆ(*a*, *ϕ*) ≡ |0¯��0¯| = Uˆ †

*<sup>S</sup>*(*a*, *<sup>ϕ</sup>*) = *<sup>e</sup>*

universe of the entangled pair. It is given then by the thermal state [62]

*<sup>ρ</sup>*ˆ1(*a*, *<sup>ϕ</sup>*) <sup>≡</sup> Tr2*ρ*<sup>ˆ</sup> <sup>=</sup> <sup>1</sup>

Uˆ

the multiverse turns out to be then

*<sup>S</sup>* is given by [62]

with, <sup>|</sup>*N*�≡|*N*�<sup>2</sup> and *<sup>Z</sup>*−<sup>1</sup> <sup>=</sup> 2 sinh *<sup>ω</sup>*

where, Γ(*a*, *ϕ*) ≡ tanh *r*(*a*, *ϕ*). The total energy reads

*δW* = *δω* (�*N*� +

*δQ* = *ω δ*�*N*� ≈ *ω*(*a*, *ϕ*)

of entanglement given by

where, �*N*�≡|*νω*|

and heat, *Q*, now read

where the squeezing parameter, *r*(*a*, *ϕ*), reads

operator Uˆ

$$
\delta^\dagger = \mu\_\omega^\* \delta\_\omega^\dagger + \nu\_\omega^\* \delta\_{\omega \prime} \tag{89}
$$

where, *µω* ≡ *µω*(*a*, *<sup>ϕ</sup>*) and *νω* ≡ *νω*(*a*, *<sup>ϕ</sup>*), are given by

$$\mu\_{\omega}(a,\rho) = \frac{1}{2\sqrt{\mathcal{M}(a)\omega(a,\rho)}} \left(\frac{1}{\mathcal{R}} + R\mathcal{M}(a)\omega(a,\rho) - i\dot{\mathcal{R}}\right),\tag{90}$$

$$\nu\_{\omega}(a,\varphi) = \frac{1}{2\sqrt{\mathcal{M}(a)\omega(a,\varphi)}} \left(\frac{1}{\mathcal{R}} - \mathcal{R}\mathcal{M}(a)\omega(a,\varphi) - i\dot{\mathcal{R}}\right),\tag{91}$$

,

with |*µω*| <sup>2</sup> − |*νω*| <sup>2</sup> = 1, and *R* ≡ *R*(*a*, *ϕ*) is given, in the semiclassical regime, by Eq. (34),

$$R \approx \frac{e^{\pm \frac{1}{\mathcal{W}(q)}}}{\sqrt{\mathcal{M}(a)\omega(a,q)}}$$

<sup>10</sup> In some sense, we are the 'measuring device' of our universe.

where the positive sign corresponds to the choice of the no-boundary condition and the negative sign to the tunneling boundary condition. Let us further assume that the multiverse is in the invariant vacuum state |0¯�. The density matrix that represents the quantum state of the multiverse turns out to be then

$$\rho(a,\varphi) \equiv |\mathbf{0}\rangle\langle\mathbf{0}| = \mathcal{J}\_{\mathbb{S}}^{\dagger}|0\_10\_2\rangle\langle 0\_10\_2|\mathcal{J}\_{\mathbb{S}}.\tag{92}$$

where |0102�≡|01�|02�, with |01� and |02� being the ground states of a pair of entangled universes in their respective observer representations. Similarly to Eq. (54), the squeezing operator Uˆ *<sup>S</sup>* is given by [62]

$$\mathcal{U}\_{\mathcal{S}}(a,\varphi) = e^{r(a,\varphi)\hat{b}\_1\hat{b}\_2 - r(a,\varphi)\hat{b}\_1^\dagger\hat{b}\_2^\dagger},\tag{93}$$

where the squeezing parameter, *r*(*a*, *ϕ*), reads

24 Open Questions in Cosmology

considered as well.

operator ˆ

defined by, ˆ

operators ˆ

with |*µω*|

and ˆ *b*†

universes [62]. Unlike quantum optics, we do not have an experimental device to measure other universes rather than our own universe10. However, the extension of the complementary principle to the quantum description of the multiverse entails two main consequences. On the one hand, if the wave function of the universe has to be described in terms of 'particles', it means that in some appropriate representation we can formally distinguish the universal states as individual entities, giving rise therefore to the multiverse scenario. On the other hand, if it has to be complementary described in terms of waves, then, interference between the quantum states of two or more universes can generally be

Let us consider a multiverse made up of homogeneous and isotropic universes with a slow-varying scalar field *<sup>ϕ</sup>*, recalling that in the case for which *<sup>ϕ</sup>*˙ = 0 and *<sup>V</sup>*(*ϕ*0) ≡ <sup>Λ</sup>,

Let us consider one type of universes and describe the quantum state of the multiverse in

vacuum state of the multiverse, |0¯�, is then defined as the eigenstate of the annihilation

a large parent universe would quantum mechanically describe the state of their respective universes in the asymptotic representation given by Eqs. (29-30), with a ground state |0�

We can consider therefore two representations: the one derived from a consistent formulation of the boundary condition of the whole multiverse, or *invariant representation*, given by the

*<sup>ω</sup>*(*a*), which might be called the *observer representation*. They both are related by the

*<sup>b</sup><sup>ω</sup>* + *νω* <sup>ˆ</sup> *b*†

1

1

± <sup>1</sup> 3*V*(*ϕ*) M(*a*)*ω*(*a*, *ϕ*)

<sup>2</sup> = 1, and *R* ≡ *R*(*a*, *ϕ*) is given, in the semiclassical regime, by Eq. (34),

,

*b*(*a*) and ˆ

*b*†(*a*), and the asymptotic representation given by the operators ˆ

*<sup>R</sup>* <sup>+</sup> *<sup>R</sup>*M(*a*)*ω*(*a*, *<sup>ϕ</sup>*) <sup>−</sup> *iR*˙

*<sup>R</sup>* <sup>−</sup> *<sup>R</sup>*M(*a*)*ω*(*a*, *<sup>ϕ</sup>*) <sup>−</sup> *iR*˙

*b*†(*a*) given in Eqs. (32-33). The

*<sup>ω</sup>*, (88)

*bω*, (89)

, (90)

, (91)

*<sup>b</sup>ω*(*a*)

*b*(*a*)|0¯� ≡ 0. On the other hand, observers inhabiting

**4.3. Thermodynamical properties of entangled universes**

terms of the annihilation and creation operators ˆ

*b*(*a*) with eigenvalue zero, i.e. ˆ

where, *µω* ≡ *µω*(*a*, *<sup>ϕ</sup>*) and *νω* ≡ *νω*(*a*, *<sup>ϕ</sup>*), are given by

*µω*(*a*, *<sup>ϕ</sup>*) = <sup>1</sup> 2

*νω*(*a*, *<sup>ϕ</sup>*) = <sup>1</sup> 2

<sup>10</sup> In some sense, we are the 'measuring device' of our universe.

*<sup>b</sup>ω*(*a*)|0� ≡ 0.

*b*(*a*) and ˆ

squeezing transformation

<sup>2</sup> − |*νω*|

the model effectively represents a multiverse formed by de-Sitter universes.

ˆ *<sup>b</sup>* = *µω* <sup>ˆ</sup>

M(*a*)*ω*(*a*, *ϕ*)

M(*a*)*ω*(*a*, *ϕ*)

*<sup>R</sup>* <sup>≈</sup> *<sup>e</sup>*

ˆ *<sup>b</sup>*† <sup>=</sup> *<sup>µ</sup>*<sup>∗</sup> *ω* ˆ *b*† *<sup>ω</sup>* <sup>+</sup> *<sup>ν</sup>*<sup>∗</sup> *ω* ˆ

$$r(a,\varphi) \equiv \arcsin \mathbf{h} |\upsilon\_{\omega}(a,\varphi)|\,\prime\tag{94}$$

with *νω*(*a*, *<sup>ϕ</sup>*) being given by Eq. (91). We can then follow the procedure of Sec. 3.3 to compute the reduced density matrix, *ρ*ˆ1, that represents the quantum state of one single universe of the entangled pair. It is given then by the thermal state [62]

$$\hat{\rho}\_1(a,\rho) \equiv \text{Tr}\_2 \hat{\rho} = \frac{1}{Z} \sum\_{N=0}^{\infty} e^{-\frac{\omega(a,\rho)}{T} (N + \frac{1}{2})} |N\rangle\langle N| \,\tag{95}$$

with, <sup>|</sup>*N*�≡|*N*�<sup>2</sup> and *<sup>Z</sup>*−<sup>1</sup> <sup>=</sup> 2 sinh *<sup>ω</sup>* <sup>2</sup>*<sup>T</sup>* . The two universes of the entangled pair evolve, in the observer representation of each single universe, in thermal equilibrium with a temperature of entanglement given by

$$T \equiv T(a, \varphi) = \frac{\omega(a, \varphi)}{2 \ln \frac{1}{\Gamma(a, \varphi)}},\tag{96}$$

where, Γ(*a*, *ϕ*) ≡ tanh *r*(*a*, *ϕ*). The total energy reads

$$E(a) = \omega(a)(\langle N \rangle + \frac{1}{2})\_\prime \tag{97}$$

where, �*N*�≡|*νω*| 2. The variation of the quantum informational analogues to the work, *W*, and heat, *Q*, now read

$$
\delta \mathcal{W} = \delta \omega \left( \langle \mathcal{N} \rangle + \frac{1}{2} \right) \approx \frac{\partial \omega(a, \varphi)}{\partial a} (\langle \mathcal{N} \rangle + \frac{1}{2}) \, da,\tag{98}
$$

$$
\delta \mathcal{Q} = \omega \,\delta \langle N \rangle \approx \omega \,(a, \,\varphi) \frac{\partial \langle N \rangle}{\partial a} \, da,\tag{99}
$$

observer inside the universe, its quantum state has been obtained from a highly non-classical state. Thus, it would not be expected that the classical constraint *σ* ≥ 0 would impose any

The second principle of entanglement thermodynamics [58] does provide us with an arrow of time for single universes. In the quantum multiverse, it can be reparaphrased as follows: by local operations and classical communications alone, the amount of entanglement between the universes cannot increase. Let us recall that by *local* operations we mean in the multiverse anything that happens within a single universe, i.e. everything we can observe. Therefore, the growth of cosmic structures, particle interactions and even the presence of life in the universe cannot increase the amount of entanglement between a pair of entangled universes provided that all these features are due to local interactions. They should decrease the rate

The amount of entanglement between the pair of universes only decreases for growing values of the scale factor (see Fig. 7). Thus, the second law of the entanglement thermodynamics implies that the universe has to expand once it is created in an entangled pair, as seen by an observer inside the universe. Furthermore, if the classical thermodynamics and the thermodynamics of entanglement were related, it could be followed that the negative change of entropy would be balanced by the creation of cosmic structures and other local processes that increase the local (classical) entropy. The decrease of the entropy of entanglement is larger for a small value of the scale factor. Then, the growth of local structures in the universe

*dS* <sup>=</sup> *<sup>δ</sup><sup>Q</sup>*

Eq. (101) can be compared with the equation that is customary used to define the energy of entanglement [42, 51–53], *dEent* = *TdS*. Then, in the case being considered, we can identify the energy of entanglement, *Eent*, with the informational heat, *Q*, and interpret it as a vacuum energy for each single universe of an entangled pair. It is given by the integral of Eq. (99),

1

*da* <sup>=</sup> <sup>−</sup> <sup>9</sup> *<sup>e</sup>*

2

*<sup>a</sup>*<sup>8</sup> <sup>+</sup> sinh<sup>2</sup> <sup>1</sup>

± <sup>2</sup> <sup>3</sup>*V*(*ϕ*0)

*<sup>V</sup>*(*ϕ*0)

3*V*(*ϕ*)

*a*2*V*(*ϕ*) − 1 ≈ *a*<sup>2</sup>

*<sup>T</sup>* =0, and thus, the variation of the entropy of

*<sup>T</sup>* . (101)

*<sup>∂</sup><sup>a</sup> da*, and therefore

, (102)

*<sup>a</sup>*−7*da*, (103)

*V*(*ϕ*), in units for

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 211

of entanglement in a non-reversible universe with dissipative processes, actually.

would be favored in the earliest phases of the universe, as it is expected.

*4.3.2. Energy of entanglement and the vacuum energy of the universe*

entanglement is related to the quantum informational heat, *Q*, by

�*N*� ≈ <sup>9</sup> *<sup>e</sup>*

which ¯*<sup>h</sup>* <sup>=</sup> 1. For a slow-varying field, *<sup>ϕ</sup>* <sup>≈</sup> *<sup>ϕ</sup>*<sup>0</sup> and *<sup>δ</sup>*�*N*� ≈ *<sup>∂</sup>*�*N*�

*δQ* ≈ *ω*

where it has been used that, M(*a*) = *a* and *ω* = *a*

± <sup>2</sup> 3*V*(*ϕ*) 16*V*(*ϕ*)

*∂*�*N*� *∂a*

In the model being considered, *σ* ≡ *dS* − *<sup>δ</sup><sup>Q</sup>*

with

arrow of time in the model.

**Figure 7.** Parameter of squeezing, *r* (dashed line), and entropy of entanglement, *Sent* (continuous line), with respect to the value of the scale factor, *a*.

where in the last equalities it has been taken into account that for a slow-varying field, *δω* ≈ *ω*˙ *da* and *δ*�*N*� ≈ ˙ �*N*� *da*. From Eqs. (97-99) it can be checked that the first principle of thermodynamics, *δE* = *δW* + *δQ*, is directly satisfied. The entropy of entanglement, Eq. (75), reads

$$S\_{\rm ent}(a,\varphi) = |\mu\_{\omega}(a,\varphi)|^2 \ln |\mu\_{\omega}(a,\varphi)|^2 - |\nu\_{\omega}(a,\varphi)|^2 \ln |\nu\_{\omega}(a,\varphi)|^2,\tag{100}$$

with, |*µω*(*a*, *<sup>ϕ</sup>*)| = cosh *<sup>r</sup>*(*a*, *<sup>ϕ</sup>*) and |*νω*(*a*, *<sup>ϕ</sup>*)| = sinh *<sup>r</sup>*(*a*, *<sup>ϕ</sup>*). Therefore, like in Sec. 3.3, the second principle of thermodynamics is also satisfied because the entropy production vanishes for any values of the scale factor and the scalar field, i.e. *σ* ≡ *σ*(*a*, *ϕ*) = 0.

#### *4.3.1. Entropy of entanglement as an arrow of time for single universes*

Let us summarize the general picture described so far. The multiverse stays in a squeezed vacuum state which is the product state of the wave functions that correspond to the state of pairs of entangled *i*-universes (see Eq. (39)), where the index *i* labels all the species of universes considered in the multiverse. The multiverse stays therefore in a highly non-classical state. Furthermore, the quantum entropy of a pure state is zero and, therefore, there is no thermodynamical arrow of time in the multiverse. Let us recall that, in the third quantization formalism, the scale factor was just taken as a formal time-like variable given by the Lorentzian structure of the minisupermetric. However, the minisuperspace is not space-time and, therefore, the scale factor has no meaning of a physical (i.e. a measurable) time, a priori, in the multiverse. It might well be said that (physical) time and (physical) evolution are concepts that really make sense within a single universe.

For an observer inside a universe, this is described by a thermal state which is indistinguishable from a classical mixture (see Eq. (73), and the comments thereafter), i.e. it is seen as a classical universe. The entropy of entanglement for a single universe is a monotonic function of the scale factor. However, the entropy production identically vanishes for any increasing or decreasing rate of the scale factor so that the customary formulation of the second principle of thermodynamics does not impose any arrow of time in the universe within the present approach. Although the universe can be seen as a classical mixture by an observer inside the universe, its quantum state has been obtained from a highly non-classical state. Thus, it would not be expected that the classical constraint *σ* ≥ 0 would impose any arrow of time in the model.

The second principle of entanglement thermodynamics [58] does provide us with an arrow of time for single universes. In the quantum multiverse, it can be reparaphrased as follows: by local operations and classical communications alone, the amount of entanglement between the universes cannot increase. Let us recall that by *local* operations we mean in the multiverse anything that happens within a single universe, i.e. everything we can observe. Therefore, the growth of cosmic structures, particle interactions and even the presence of life in the universe cannot increase the amount of entanglement between a pair of entangled universes provided that all these features are due to local interactions. They should decrease the rate of entanglement in a non-reversible universe with dissipative processes, actually.

The amount of entanglement between the pair of universes only decreases for growing values of the scale factor (see Fig. 7). Thus, the second law of the entanglement thermodynamics implies that the universe has to expand once it is created in an entangled pair, as seen by an observer inside the universe. Furthermore, if the classical thermodynamics and the thermodynamics of entanglement were related, it could be followed that the negative change of entropy would be balanced by the creation of cosmic structures and other local processes that increase the local (classical) entropy. The decrease of the entropy of entanglement is larger for a small value of the scale factor. Then, the growth of local structures in the universe would be favored in the earliest phases of the universe, as it is expected.

#### *4.3.2. Energy of entanglement and the vacuum energy of the universe*

26 Open Questions in Cosmology

value of the scale factor, *a*.

*Sent*(*a*, *<sup>ϕ</sup>*) = |*µω*(*a*, *<sup>ϕ</sup>*)|

reads

**Figure 7.** Parameter of squeezing, *r* (dashed line), and entropy of entanglement, *Sent* (continuous line), with respect to the

where in the last equalities it has been taken into account that for a slow-varying field, *δω* ≈ *ω*˙ *da* and *δ*�*N*� ≈ ˙ �*N*� *da*. From Eqs. (97-99) it can be checked that the first principle of thermodynamics, *δE* = *δW* + *δQ*, is directly satisfied. The entropy of entanglement, Eq. (75),

with, |*µω*(*a*, *<sup>ϕ</sup>*)| = cosh *<sup>r</sup>*(*a*, *<sup>ϕ</sup>*) and |*νω*(*a*, *<sup>ϕ</sup>*)| = sinh *<sup>r</sup>*(*a*, *<sup>ϕ</sup>*). Therefore, like in Sec. 3.3, the second principle of thermodynamics is also satisfied because the entropy production vanishes

Let us summarize the general picture described so far. The multiverse stays in a squeezed vacuum state which is the product state of the wave functions that correspond to the state of pairs of entangled *i*-universes (see Eq. (39)), where the index *i* labels all the species of universes considered in the multiverse. The multiverse stays therefore in a highly non-classical state. Furthermore, the quantum entropy of a pure state is zero and, therefore, there is no thermodynamical arrow of time in the multiverse. Let us recall that, in the third quantization formalism, the scale factor was just taken as a formal time-like variable given by the Lorentzian structure of the minisupermetric. However, the minisuperspace is not space-time and, therefore, the scale factor has no meaning of a physical (i.e. a measurable) time, a priori, in the multiverse. It might well be said that (physical) time and (physical)

For an observer inside a universe, this is described by a thermal state which is indistinguishable from a classical mixture (see Eq. (73), and the comments thereafter), i.e. it is seen as a classical universe. The entropy of entanglement for a single universe is a monotonic function of the scale factor. However, the entropy production identically vanishes for any increasing or decreasing rate of the scale factor so that the customary formulation of the second principle of thermodynamics does not impose any arrow of time in the universe within the present approach. Although the universe can be seen as a classical mixture by an

<sup>2</sup> − |*νω*(*a*, *<sup>ϕ</sup>*)|

<sup>2</sup> ln |*νω*(*a*, *<sup>ϕ</sup>*)|

2, (100)

<sup>2</sup> ln |*µω*(*a*, *<sup>ϕ</sup>*)|

for any values of the scale factor and the scalar field, i.e. *σ* ≡ *σ*(*a*, *ϕ*) = 0.

evolution are concepts that really make sense within a single universe.

*4.3.1. Entropy of entanglement as an arrow of time for single universes*

In the model being considered, *σ* ≡ *dS* − *<sup>δ</sup><sup>Q</sup> <sup>T</sup>* =0, and thus, the variation of the entropy of entanglement is related to the quantum informational heat, *Q*, by

$$d\mathcal{S} = \frac{\delta \mathcal{Q}}{T}.\tag{101}$$

Eq. (101) can be compared with the equation that is customary used to define the energy of entanglement [42, 51–53], *dEent* = *TdS*. Then, in the case being considered, we can identify the energy of entanglement, *Eent*, with the informational heat, *Q*, and interpret it as a vacuum energy for each single universe of an entangled pair. It is given by the integral of Eq. (99), with

$$
\langle N \rangle \approx \frac{9}{16V(\varphi)} \frac{e^{\pm \frac{2}{3V(\varphi)}}}{d^8} + \sinh^2 \frac{1}{3V(\varphi)'} \tag{102}
$$

where it has been used that, M(*a*) = *a* and *ω* = *a a*2*V*(*ϕ*) − 1 ≈ *a*<sup>2</sup> *V*(*ϕ*), in units for which ¯*<sup>h</sup>* <sup>=</sup> 1. For a slow-varying field, *<sup>ϕ</sup>* <sup>≈</sup> *<sup>ϕ</sup>*<sup>0</sup> and *<sup>δ</sup>*�*N*� ≈ *<sup>∂</sup>*�*N*� *<sup>∂</sup><sup>a</sup> da*, and therefore

$$
\delta Q \approx \omega \frac{\partial \langle N \rangle}{\partial a} da = -\frac{9}{2\sqrt{V(\varphi\_0)}} a^{-7} da,\tag{103}
$$

whose integration yields

$$E\_{\rm ent} = Q(a, \varphi\_0) = \frac{3}{4} \frac{e^{\pm \frac{2}{\mathcal{W}(\varphi\_0)}}}{\sqrt{V(\varphi\_0)}} a^{-6}. \tag{104}$$

The entropy of entanglement decreases for an increasing value of the scale factor. The second principle of thermodynamics is however satisfied because the process is non-adiabatic, in the quantum informational sense, and the entropy production is zero. In fact, the entropy production is zero for any increasing or decreasing rate of the scale factor, imposing therefore no correlation between the cosmic arrow of time, which is given by expansion or contraction rate of a single universe, and the customary formulation of the second principle of thermodynamics. This is in contrast to what it happens inside a single universe, where there is a correlation between the cosmic arrow of time and the entropy of matter fields [27, 40]. Let us recall that the entropy of entanglement is a quantum feature having no classical analogue and, thus, it is not expected that it imposes an arrow of time through the

Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 213

The second principle of entanglement thermodynamics, which states [58] that the entropy of entanglement cannot be increased by any *local* operation and any classical communication alone, does impose an arrow of time on single universes [60]. It should be noticed that by *local* we mean in the multiverse anything that happens in a single universe. Therefore, everything that we observe, i.e. the creation of particles, the growth of cosmic structures, and even life, cannot make the inter-universal entanglement to grow provided that all these processes are internal to a single universe. In an actual and non-reversible universe they should induce a decreasing of the entropy of inter-universal entanglement, enhancing therefore the expansion of the universe that would induce a correlation between the growth of cosmic structures and

In the model presented in this chapter, the energy of entanglement between a pair of entangled universes provides us with a vacuum energy for each single universes. The energy of entanglement of the universe is high in the early stage of the universes becoming very small at later times. That behavior might be compatible with an initial inflationary universe, for which a high value of the vacuum energy is assumed, that would eventually evolve to a state with a very small value of the cosmological constant, like the current state of the universe. However, it is not expected that such a simple model of the universe would fit with actual observational tests. A more realistic model of the universes that form the multiverse, in which genuine matter fields were considered, is needed to make a serious attempt of observational fitting. However, the fact that inter-universal entanglement provides us with testable properties of our universe opens the door to future developments that would make falseable the multiverse proposal giving an observational support to the quantum multiverse. In conclusion, the question of whether the multiverse is a physical theory or just a mathematical construction, derived however from the general laws of physics, holds on whether the existence of other universes may affect the properties of the observable universe. Inter-universal entanglement is a novel feature that supplies us with new explanations for unexpected cosmic phenomena and it might allow us to test the whole multiverse proposal. It will be the future theoretical developments and the improved observational tests what will

make us to decide whether to adopt or deny such a cosmological scenario.

for the revision of the text and their valuable comments.

In memory of Prof. González Díaz, who always encouraged us to adopt a fearless attitude as well in science as in everyday life. The author wishes to thank I. Garay and P. V. Moniz

customary, i.e. classical, formulation of the second principle of thermodynamics.

the entanglement arrow of time.

**Acknowledgements**

The energy of entanglement (104) provides us with a curve that might be compared with the evolution of the vacuum energy of the universe. From Eq. (104), it can be seen that the vacuum energy would follow a different curve depending on whether the tunneling condition or the no-boundary condition is imposed on the state of a single universe. The boundary condition imposed on a single universe might therefore be discriminated from observational data, at least in principle. However, the model being considered is unrealistic for at least two reasons. First, after the inflationary stage the universe becomes hot [46, 47] and the slow-roll approximation is no longer valid. Secondly, if the energy of entanglement is to be considered as a vacuum energy, it should have been considered as a variable of the model from the beginning. More realistic matter fields and the backreaction should be taken into account to make a first serious attempt to observational fitting. However, the important thing that is worth noticing is that the vacuum energy of entanglement might thus be tested as well as the whole multiverse proposal. Furthermore, different boundary conditions would provide us with different curves for the energy of entanglement along the entire evolution of the universe. Therefore, the boundary conditions of the whole multiverse might be tested as well by direct observation, which is a completely novel feature in quantum cosmology.

## **5. Conclusions: The physical multiverse**

In this chapter, we have presented a quantum mechanical description of a multiverse made up of large and disconnected regions of the space-time, called universes, with a high degree of symmetry. We have obtained, within the framework of a third quantization formalism, a wave function that quantum mechanically represents the state of the whole multiverse, and an appropriate boundary condition for the state of the multiverse has allowed us to interpret it as formed by entangled pairs of universes.

If universes were entangled to each other, then, the violation of classical inequalities like the Bell's inequalities could no longer be associated to the concepts of locality or non-locality because there is not generally a common space-time among the universes of the quantum multiverse. It would rather be related to the independence or interdependence of the quantum states that represent different universes. Furthermore, the complementary principle of quantum mechanics, being applied to the space-time as a whole, enhances us to: i) look for an appropriate boundary condition for which universes should be described as individual entities forcing us to consider a multiverse; and, ii) take into account as well interference effects between the quantum states of two or more universes.

For a pair of entangled universes, the quantum thermodynamical properties of each single universe have been computed. In the scenario of a multiverse made up of entangled pairs of universes, the picture is the following: the multiverse may state in the pure state that corresponds to the product state of the ground states derived from the boundary condition imposed on the multiverse, for each type of single universes. Then, the entropy of the whole multiverse vanishes and there is thus no physical arrow of time in the multiverse. For single universes, however, it appears an arrow of time derived from the entropy of entanglement with their partner universes.

The entropy of entanglement decreases for an increasing value of the scale factor. The second principle of thermodynamics is however satisfied because the process is non-adiabatic, in the quantum informational sense, and the entropy production is zero. In fact, the entropy production is zero for any increasing or decreasing rate of the scale factor, imposing therefore no correlation between the cosmic arrow of time, which is given by expansion or contraction rate of a single universe, and the customary formulation of the second principle of thermodynamics. This is in contrast to what it happens inside a single universe, where there is a correlation between the cosmic arrow of time and the entropy of matter fields [27, 40]. Let us recall that the entropy of entanglement is a quantum feature having no classical analogue and, thus, it is not expected that it imposes an arrow of time through the customary, i.e. classical, formulation of the second principle of thermodynamics.

The second principle of entanglement thermodynamics, which states [58] that the entropy of entanglement cannot be increased by any *local* operation and any classical communication alone, does impose an arrow of time on single universes [60]. It should be noticed that by *local* we mean in the multiverse anything that happens in a single universe. Therefore, everything that we observe, i.e. the creation of particles, the growth of cosmic structures, and even life, cannot make the inter-universal entanglement to grow provided that all these processes are internal to a single universe. In an actual and non-reversible universe they should induce a decreasing of the entropy of inter-universal entanglement, enhancing therefore the expansion of the universe that would induce a correlation between the growth of cosmic structures and the entanglement arrow of time.

In the model presented in this chapter, the energy of entanglement between a pair of entangled universes provides us with a vacuum energy for each single universes. The energy of entanglement of the universe is high in the early stage of the universes becoming very small at later times. That behavior might be compatible with an initial inflationary universe, for which a high value of the vacuum energy is assumed, that would eventually evolve to a state with a very small value of the cosmological constant, like the current state of the universe. However, it is not expected that such a simple model of the universe would fit with actual observational tests. A more realistic model of the universes that form the multiverse, in which genuine matter fields were considered, is needed to make a serious attempt of observational fitting. However, the fact that inter-universal entanglement provides us with testable properties of our universe opens the door to future developments that would make falseable the multiverse proposal giving an observational support to the quantum multiverse.

In conclusion, the question of whether the multiverse is a physical theory or just a mathematical construction, derived however from the general laws of physics, holds on whether the existence of other universes may affect the properties of the observable universe. Inter-universal entanglement is a novel feature that supplies us with new explanations for unexpected cosmic phenomena and it might allow us to test the whole multiverse proposal. It will be the future theoretical developments and the improved observational tests what will make us to decide whether to adopt or deny such a cosmological scenario.

## **Acknowledgements**

28 Open Questions in Cosmology

whose integration yields

**5. Conclusions: The physical multiverse**

it as formed by entangled pairs of universes.

with their partner universes.

effects between the quantum states of two or more universes.

*Eent* <sup>=</sup> *<sup>Q</sup>*(*a*, *<sup>ϕ</sup>*0) = <sup>3</sup>

4 *e* ± <sup>2</sup> <sup>3</sup>*V*(*ϕ*0) *<sup>V</sup>*(*ϕ*0)

The energy of entanglement (104) provides us with a curve that might be compared with the evolution of the vacuum energy of the universe. From Eq. (104), it can be seen that the vacuum energy would follow a different curve depending on whether the tunneling condition or the no-boundary condition is imposed on the state of a single universe. The boundary condition imposed on a single universe might therefore be discriminated from observational data, at least in principle. However, the model being considered is unrealistic for at least two reasons. First, after the inflationary stage the universe becomes hot [46, 47] and the slow-roll approximation is no longer valid. Secondly, if the energy of entanglement is to be considered as a vacuum energy, it should have been considered as a variable of the model from the beginning. More realistic matter fields and the backreaction should be taken into account to make a first serious attempt to observational fitting. However, the important thing that is worth noticing is that the vacuum energy of entanglement might thus be tested as well as the whole multiverse proposal. Furthermore, different boundary conditions would provide us with different curves for the energy of entanglement along the entire evolution of the universe. Therefore, the boundary conditions of the whole multiverse might be tested as well by direct observation, which is a completely novel feature in quantum cosmology.

In this chapter, we have presented a quantum mechanical description of a multiverse made up of large and disconnected regions of the space-time, called universes, with a high degree of symmetry. We have obtained, within the framework of a third quantization formalism, a wave function that quantum mechanically represents the state of the whole multiverse, and an appropriate boundary condition for the state of the multiverse has allowed us to interpret

If universes were entangled to each other, then, the violation of classical inequalities like the Bell's inequalities could no longer be associated to the concepts of locality or non-locality because there is not generally a common space-time among the universes of the quantum multiverse. It would rather be related to the independence or interdependence of the quantum states that represent different universes. Furthermore, the complementary principle of quantum mechanics, being applied to the space-time as a whole, enhances us to: i) look for an appropriate boundary condition for which universes should be described as individual entities forcing us to consider a multiverse; and, ii) take into account as well interference

For a pair of entangled universes, the quantum thermodynamical properties of each single universe have been computed. In the scenario of a multiverse made up of entangled pairs of universes, the picture is the following: the multiverse may state in the pure state that corresponds to the product state of the ground states derived from the boundary condition imposed on the multiverse, for each type of single universes. Then, the entropy of the whole multiverse vanishes and there is thus no physical arrow of time in the multiverse. For single universes, however, it appears an arrow of time derived from the entropy of entanglement

*<sup>a</sup>*<sup>−</sup>6. (104)

In memory of Prof. González Díaz, who always encouraged us to adopt a fearless attitude as well in science as in everyday life. The author wishes to thank I. Garay and P. V. Moniz for the revision of the text and their valuable comments.

## **Author details**

Salvador J. Robles Pérez

Grupo de Relatividad y Cosmología, Instituto de Física Fundamental - Consejo Superior de Investigaciones Científicas (IFF-CSIC), Madrid, Spain, and Estación Ecológica de Biocosmología (EEBM), Medellín, Spain

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**Chapter 9**

**Provisional chapter**

**Quintom Potential from Quantum Anisotropic**

**Quintom Potential from Quantum Anisotropic**

At the present time, there are some paradigms to explain the observations for the accelerated expansion of the universe. Most of these paradigms are based on the dynamics of a scalar (quintessence) or multiscalar field (quintom) cosmological models of dark energy, (see the review [1–3]). The main discussion yields over the evolution of these models in the *<sup>ω</sup>*-*ω*′ plane (*ω* is the equation of state parameter of the dark energy) [4–12, 14–19, 21–25]). In the present study we desire to perform our investigation in the case of quintom cosmology, constructed using both quintessence (*σ*) and phantom (*φ*) fields, mantaining a nonspecific potential form V(*φ*, *σ*). There are many works in the literature that deals with this type of problems, but in a general way, and not with a particular ansatz, one that considers dynamical systems [12, 13, 20]. One special class of potentials used to study this behaviour corresponds to the case of the exponential potentials [4, 6, 9, 26, 28] for each field, where the corresponding energy density of a scalar field has the range of scaling behaviors [29, 30], i.e, it scales exactly as a power of the scale factor like, *ρφ* <sup>∝</sup> *<sup>a</sup>*−*m*, when the dominant component has an energy density which scales in a similar way. There are other works where other type

How come that we claim that the analysis of general potentials using dynamical systems was made considering particular structures of them, in other words, how can we introduce this mathematical structure within a physical context?. We can partially answer this question, when the Bohmian formalism is introduced, i.e, many of them can be constructed using the Bohm formalism [32–34] of the quantum mechanics under the integral systems premise, which is known as the quantum potential approach. This approach makes possible to identify trajectories associated with the wave function of the universe [32] when we choose the superpotential function as the momenta associated to the coordinate field q*µ*. This

and reproduction in any medium, provided the original work is properly cited.

©2012 Socorro et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Socorro et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

J. Socorro, Paulo A. Rodríguez, O. Núñez-Soltero, Rafael Hernández and Abraham Espinoza-García

Additional information is available at the end of the chapter

of potentials are analyzed [1, 9, 15, 19, 20, 23, 24, 31].

Additional information is available at the end of the chapter

J. Socorro, Paulo A. Rodríguez , O. Núñez-Soltero, Rafael Hernández and Abraham Espinoza-García

**Cosmological Models**

**Cosmological Models**

http://dx.doi.org/10.5772/52054

**1. Introduction**


**Provisional chapter**

## **Quintom Potential from Quantum Anisotropic Cosmological Models Quintom Potential from Quantum Anisotropic Cosmological Models**

J. Socorro, Paulo A. Rodríguez, O. Núñez-Soltero, Rafael Hernández and Abraham Espinoza-García Rafael Hernández and Abraham Espinoza-García Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

J. Socorro, Paulo A. Rodríguez , O. Núñez-Soltero,

http://dx.doi.org/10.5772/52054

## **1. Introduction**

34 Open Questions in Cosmology

218 Open Questions in Cosmology

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At the present time, there are some paradigms to explain the observations for the accelerated expansion of the universe. Most of these paradigms are based on the dynamics of a scalar (quintessence) or multiscalar field (quintom) cosmological models of dark energy, (see the review [1–3]). The main discussion yields over the evolution of these models in the *<sup>ω</sup>*-*ω*′ plane (*ω* is the equation of state parameter of the dark energy) [4–12, 14–19, 21–25]). In the present study we desire to perform our investigation in the case of quintom cosmology, constructed using both quintessence (*σ*) and phantom (*φ*) fields, mantaining a nonspecific potential form V(*φ*, *σ*). There are many works in the literature that deals with this type of problems, but in a general way, and not with a particular ansatz, one that considers dynamical systems [12, 13, 20]. One special class of potentials used to study this behaviour corresponds to the case of the exponential potentials [4, 6, 9, 26, 28] for each field, where the corresponding energy density of a scalar field has the range of scaling behaviors [29, 30], i.e, it scales exactly as a power of the scale factor like, *ρφ* <sup>∝</sup> *<sup>a</sup>*−*m*, when the dominant component has an energy density which scales in a similar way. There are other works where other type of potentials are analyzed [1, 9, 15, 19, 20, 23, 24, 31].

How come that we claim that the analysis of general potentials using dynamical systems was made considering particular structures of them, in other words, how can we introduce this mathematical structure within a physical context?. We can partially answer this question, when the Bohmian formalism is introduced, i.e, many of them can be constructed using the Bohm formalism [32–34] of the quantum mechanics under the integral systems premise, which is known as the quantum potential approach. This approach makes possible to identify trajectories associated with the wave function of the universe [32] when we choose the superpotential function as the momenta associated to the coordinate field q*µ*. This

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Socorro et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Socorro et al., licensee InTech. This is an open access chapter distributed under the terms of the

investigation was undertaken within the framework of the minisuperspace approximation of quantum theory when we investigate the models with a finite number of degrees of freedom. Considering the anisotropic Bianchi Class A cosmological models from canonical quantum cosmology under determined conditions in the evolution of our universe, and employing the Bohmian formalism, and in particular the Bianchi type I to obtain a family of potentials that correspond to the most probable to model the present day cosmic acceleration. In our analysis, we found this special class of potentials, however these appear mixed.

−1 2

*∂φ* <sup>=</sup> 0, <sup>⇔</sup> *<sup>φ</sup>* <sup>+</sup>

here *<sup>ρ</sup>* is the energy density, P the pressure, and u*<sup>µ</sup>* the velocity, satisfying that u*µ*u*<sup>µ</sup>* = −1.

Let us recall here the canonical formulation in the ADM formalism of the diagonal Bianchi

where *<sup>β</sup>*ij(t) is a 3x3 diagonal matrix, *<sup>β</sup>*ij <sup>=</sup> diag(*β*<sup>+</sup> <sup>+</sup> <sup>√</sup>3*β*−, *<sup>β</sup>*<sup>+</sup> <sup>−</sup> <sup>√</sup>3*β*−, <sup>−</sup>2*β*+), *<sup>Ω</sup>*(*t*) is a scalar and *ω*<sup>i</sup> are one-forms that characterize each cosmological Bianchi type model, and

The corresponding metric of the Bianchi type I in Misner's parametrization has the following

<sup>3</sup>*β*<sup>−</sup> dx<sup>2</sup> <sup>+</sup> <sup>e</sup>2*Ω*+2*β*+−<sup>2</sup>

√

ds<sup>2</sup> <sup>=</sup> <sup>−</sup>N(t)dt<sup>2</sup> <sup>+</sup> e2*Ω*(t) (e2*β*(t)

jk*ω*<sup>j</sup> <sup>∧</sup> *<sup>ω</sup>*k, and Ci

√

the Bianchi type I is written as (where the overdot denotes time derivative),

*β*˙ 2 − <sup>N</sup> <sup>+</sup> <sup>6</sup> *<sup>ϕ</sup>*˙ <sup>2</sup>

<sup>3</sup>*β*<sup>−</sup> , R2 <sup>=</sup> <sup>e</sup>*Ω*+*β*+<sup>−</sup>

We use the Bianchi type I and IX cosmological models as toy model to apply the formalism, and write a master equation for all Bianchi Class A models. The lagrangian density (1) for

> <sup>N</sup> <sup>−</sup> <sup>6</sup> *ς*˙ 2

the fields were re-scaled as *<sup>φ</sup>* <sup>=</sup> <sup>√</sup>12*ϕ*, *<sup>σ</sup>* <sup>=</sup> <sup>√</sup>12*<sup>ς</sup>* for simplicity in the calculations.

*∂σ* <sup>=</sup> 0, <sup>⇔</sup> *<sup>σ</sup>* <sup>−</sup> *<sup>∂</sup>*<sup>V</sup>

<sup>g</sup>*µνφ*,*µν* − <sup>g</sup>*αβΓν*

<sup>g</sup>*µνσ*,*µν* − <sup>g</sup>*αβΓν*

**3. Hamiltonian approach**

obey the form d*ω*<sup>i</sup> = <sup>1</sup>

ds2

L<sup>I</sup> = e3*<sup>Ω</sup>*

 6 *Ω*˙ 2 <sup>N</sup> <sup>−</sup> <sup>6</sup>

The momenta are defined as *<sup>Π</sup>qi* <sup>=</sup> *<sup>∂</sup>*<sup>L</sup>

where the anisotropic radii are

model.

form

*αβ*∇*νφ* <sup>+</sup>

Class A cosmological models. The metric has the form

2Ci

<sup>I</sup> <sup>=</sup> <sup>−</sup>N2dt2 <sup>+</sup> <sup>e</sup>2*Ω*+2*β*++<sup>2</sup>

√

*β*˙ 2 + <sup>N</sup> <sup>−</sup> <sup>6</sup>

R1 <sup>=</sup> <sup>e</sup>*Ω*+*β*++

*αβ*∇*νσ* <sup>−</sup> *<sup>∂</sup>*<sup>V</sup>

*∂*V

T*µν*

<sup>g</sup>*αβ* <sup>V</sup>(*φ*, *<sup>σ</sup>*) <sup>−</sup> <sup>8</sup>*π*GT*αβ*, (2)

Quintom Potential from Quantum Anisotropic Cosmological Models

*∂*V *∂φ* <sup>=</sup> <sup>0</sup>

)ij *<sup>ω</sup>*<sup>i</sup> *<sup>ω</sup>*<sup>j</sup>

√

<sup>3</sup>*β*<sup>−</sup> , R3 <sup>=</sup> <sup>e</sup>*Ω*−2*β*<sup>+</sup> .

<sup>N</sup> <sup>+</sup> <sup>N</sup> (−V(*ϕ*, *<sup>ς</sup>*) + <sup>2</sup>*<sup>Λ</sup>* <sup>+</sup> <sup>16</sup>*π*G*ρ*)

*<sup>∂</sup>*q˙ <sup>i</sup> , where q<sup>i</sup> = (*β*±, *<sup>Ω</sup>*, *<sup>ϕ</sup>*, *<sup>ς</sup>*) are the coordinates fields.

jk are structure constants of the corresponding

, (4)

http://dx.doi.org/10.5772/52054

221

<sup>3</sup>*β*<sup>−</sup> dy<sup>2</sup> <sup>+</sup> e2*Ω*−4*β*<sup>+</sup> dz2, (5)

, (6)

*∂σ* <sup>=</sup> 0,

;*<sup>µ</sup>* <sup>=</sup> 0, with T*µν* <sup>=</sup> Pg*µν* + (<sup>P</sup> <sup>+</sup> *<sup>ρ</sup>*)u*µ*u*ν*, (3)

This work is arranged as follows. In section 2 we present the corresponding Einstein Klein Gordon equation for the quintom model. In section 3, we introduced the hamiltonian apparatus which is applied to Bianchi type I and the Bianchi Type IX in order to construct a master equation for all Bianchi Class A cosmological models with barotropic perfect fluid and cosmological constant. Furthermore, we present the classical equations for Bianchi type I, whose solutions are given in a quadrature form, which are presented in section 5 for particular scalar potentials. In section 4 we present the quantum scheme, where we use the Bohmian formalism and show its mathematical structure, also our approach is presented in a similar way. Our treatment is applied to build the mathematical structure of quintom scalar potentials using the integral systems formalism. For completeness we present the quantum solutions to the Wheeler-DeWitt equation. Moreover, this section represents our main objective for this work, and its where the utmost problem is treated. However it is important emphasize that the quantum potential from Bohm formalism will work as a constraint equation which restricts our family of potentials found. It is well known in the literature that in the Bohm formalism the imaginary part is never determined, however in this work such a problem is solved in order to find the quantum potentials, which is a more important matter for being able to find the classical trajectories, which is shown in section 5, that is devoted to obtain the classical solutions for particular scalar potentials, also we show through graphics how the classical trajectory is projected from its quantum counterpart. Finally we present the time dependence for the *Ω*, and quintom scalar fields (*ϕ* , *ς*).

## **2. The model**

We begin with the construction of the quintom cosmological paradigm, which requires the simultaneous consideration of two fields, namely one canonical *σ* and one phantom *φ* and the implication that dark energy will be attributed to their combination. The action of a universe with the constitution of such a two fields, the cosmological term contribution and the matter as perfect fluid content, is

$$\mathcal{L} = \sqrt{-\mathbf{g}} \left( \mathbf{R} - 2\Lambda - \frac{1}{2} \mathbf{g}^{\mu\nu} \nabla\_{\mu} \phi \nabla\_{\nu} \phi + \frac{1}{2} \mathbf{g}^{\mu\nu} \nabla\_{\mu} \sigma \nabla\_{\nu} \sigma - \mathcal{V}(\phi, \sigma) \right) + \mathcal{L}\_{\text{matter}} \tag{1}$$

and the corresponding field equations becomes

$$\begin{split} \mathbf{G}\_{\alpha\beta} + \mathbf{g}\_{\alpha\beta}\Lambda &= -\frac{1}{2} \left( \nabla\_{\alpha}\boldsymbol{\phi}\nabla\_{\beta}\boldsymbol{\phi} - \frac{1}{2} \mathbf{g}\_{\alpha\beta}\mathbf{g}^{\mu\nu}\nabla\_{\mu}\boldsymbol{\phi}\nabla\_{\nu}\boldsymbol{\phi} \right) \\ &+ \frac{1}{2} \left( \nabla\_{\alpha}\boldsymbol{\sigma}\nabla\_{\beta}\boldsymbol{\sigma} - \frac{1}{2}g\_{\alpha\beta}\mathbf{g}^{\mu\nu}\nabla\_{\mu}\boldsymbol{\sigma}\nabla\_{\nu}\boldsymbol{\sigma} \right) \end{split}$$

$$\begin{aligned} -\frac{1}{2}\mathbf{g}\_{a\beta}\mathbf{V}(\boldsymbol{\phi},\boldsymbol{\sigma}) - 8\pi\mathbf{C}\mathbf{T}\_{a\beta\prime} \end{aligned} \tag{2}$$

$$\begin{aligned} \mathbf{g}^{\mu\nu}\boldsymbol{\Phi}\_{,\mu\nu} - \mathbf{g}^{a\beta}\boldsymbol{I}^{\nu}\_{a\beta}\nabla\_{\boldsymbol{\nu}}\boldsymbol{\Phi} + \frac{\partial\mathbf{V}}{\partial\boldsymbol{\phi}} = 0, \qquad \Leftrightarrow \quad \square\boldsymbol{\phi} + \frac{\partial\mathbf{V}}{\partial\boldsymbol{\phi}} = 0 \end{aligned}$$

$$\begin{aligned} \mathbf{g}^{\mu\nu}\sigma\_{,\mu\nu} - \mathbf{g}^{a\beta}\boldsymbol{I}^{\nu}\_{a\beta}\nabla\_{\boldsymbol{\nu}}\nabla\_{\boldsymbol{\nu}}\sigma - \frac{\partial\mathbf{V}}{\partial\boldsymbol{\sigma}} = 0, \qquad \Leftrightarrow \quad \square\boldsymbol{\sigma} - \frac{\partial\mathbf{V}}{\partial\boldsymbol{\sigma}} = 0, \end{aligned}$$

$$\mathbf{T}^{\mu\nu}\_{;\mu} = 0, \quad \text{with} \quad \mathbf{T}\_{\mu\nu} = \mathbf{P}\mathbf{g}\_{\mu\nu} + (\mathbf{P} + \boldsymbol{\rho})\mathbf{u}\_{\mu}\mathbf{u}\_{\nu} \tag{3}$$

here *<sup>ρ</sup>* is the energy density, P the pressure, and u*<sup>µ</sup>* the velocity, satisfying that u*µ*u*<sup>µ</sup>* = −1.

## **3. Hamiltonian approach**

2 Open Questions in Cosmology

**2. The model**

the matter as perfect fluid content, is

<sup>R</sup> <sup>−</sup> <sup>2</sup>*<sup>Λ</sup>* <sup>−</sup> <sup>1</sup>

and the corresponding field equations becomes

2

L = −g

investigation was undertaken within the framework of the minisuperspace approximation of quantum theory when we investigate the models with a finite number of degrees of freedom. Considering the anisotropic Bianchi Class A cosmological models from canonical quantum cosmology under determined conditions in the evolution of our universe, and employing the Bohmian formalism, and in particular the Bianchi type I to obtain a family of potentials that correspond to the most probable to model the present day cosmic acceleration. In our

This work is arranged as follows. In section 2 we present the corresponding Einstein Klein Gordon equation for the quintom model. In section 3, we introduced the hamiltonian apparatus which is applied to Bianchi type I and the Bianchi Type IX in order to construct a master equation for all Bianchi Class A cosmological models with barotropic perfect fluid and cosmological constant. Furthermore, we present the classical equations for Bianchi type I, whose solutions are given in a quadrature form, which are presented in section 5 for particular scalar potentials. In section 4 we present the quantum scheme, where we use the Bohmian formalism and show its mathematical structure, also our approach is presented in a similar way. Our treatment is applied to build the mathematical structure of quintom scalar potentials using the integral systems formalism. For completeness we present the quantum solutions to the Wheeler-DeWitt equation. Moreover, this section represents our main objective for this work, and its where the utmost problem is treated. However it is important emphasize that the quantum potential from Bohm formalism will work as a constraint equation which restricts our family of potentials found. It is well known in the literature that in the Bohm formalism the imaginary part is never determined, however in this work such a problem is solved in order to find the quantum potentials, which is a more important matter for being able to find the classical trajectories, which is shown in section 5, that is devoted to obtain the classical solutions for particular scalar potentials, also we show through graphics how the classical trajectory is projected from its quantum counterpart.

analysis, we found this special class of potentials, however these appear mixed.

Finally we present the time dependence for the *Ω*, and quintom scalar fields (*ϕ* , *ς*).

<sup>g</sup>*µν*∇*µφ*∇*νφ* +

<sup>G</sup>*αβ* <sup>+</sup> <sup>g</sup>*αβΛ* <sup>=</sup> <sup>−</sup><sup>1</sup>

We begin with the construction of the quintom cosmological paradigm, which requires the simultaneous consideration of two fields, namely one canonical *σ* and one phantom *φ* and the implication that dark energy will be attributed to their combination. The action of a universe with the constitution of such a two fields, the cosmological term contribution and

> 1 2

2 

+ 1 2  <sup>g</sup>*µν*∇*µσ*∇*νσ* − <sup>V</sup>(*φ*, *<sup>σ</sup>*)

2

2

<sup>∇</sup>*αφ*∇*βφ* <sup>−</sup> <sup>1</sup>

<sup>∇</sup>*ασ*∇*βσ* <sup>−</sup> <sup>1</sup>

<sup>g</sup>*αβ*g*µν*∇*µφ*∇*νφ*

*<sup>g</sup>αβgµν*∇*µσ*∇*νσ*

+ Lmatter, (1)

Let us recall here the canonical formulation in the ADM formalism of the diagonal Bianchi Class A cosmological models. The metric has the form

$$\mathbf{ds}^2 = -\mathbf{N}(\mathbf{t})\mathbf{d}\mathbf{t}^2 + \mathbf{e}^{2\Omega(\mathbf{t})} \left(\mathbf{e}^{2\beta(\mathbf{t})}\right)\_{\ddot{\mathbf{n}}} \boldsymbol{\omega}^{\mathbf{i}} \boldsymbol{\omega}^{\mathbf{j}}.\tag{4}$$

where *<sup>β</sup>*ij(t) is a 3x3 diagonal matrix, *<sup>β</sup>*ij <sup>=</sup> diag(*β*<sup>+</sup> <sup>+</sup> <sup>√</sup>3*β*−, *<sup>β</sup>*<sup>+</sup> <sup>−</sup> <sup>√</sup>3*β*−, <sup>−</sup>2*β*+), *<sup>Ω</sup>*(*t*) is a scalar and *ω*<sup>i</sup> are one-forms that characterize each cosmological Bianchi type model, and obey the form d*ω*<sup>i</sup> = <sup>1</sup> 2Ci jk*ω*<sup>j</sup> <sup>∧</sup> *<sup>ω</sup>*k, and Ci jk are structure constants of the corresponding model.

The corresponding metric of the Bianchi type I in Misner's parametrization has the following form

$$\mathbf{ds}\_{\rm I}^{2} = -\mathbf{N}^{2}\mathbf{d}\mathbf{t}^{2} + \mathbf{e}^{2\Omega + 2\beta\_{+} + 2\sqrt{3}\theta\_{-}}\mathbf{dx}^{2} + \mathbf{e}^{2\Omega + 2\beta\_{+} - 2\sqrt{3}\theta\_{-}}\mathbf{dy}^{2} + \mathbf{e}^{2\Omega - 4\beta\_{+}}\mathbf{dz}^{2},\tag{5}$$

where the anisotropic radii are

$$\mathbf{R}\_1 = \mathbf{e}^{\Omega + \beta\_+ + \sqrt{3}\beta\_-} , \qquad \mathbf{R}\_2 = \mathbf{e}^{\Omega + \beta\_+ - \sqrt{3}\beta\_-} , \qquad \mathbf{R}\_3 = \mathbf{e}^{\Omega - 2\beta\_+} .$$

We use the Bianchi type I and IX cosmological models as toy model to apply the formalism, and write a master equation for all Bianchi Class A models. The lagrangian density (1) for the Bianchi type I is written as (where the overdot denotes time derivative),

$$\mathcal{L}\_{\text{I}} = \mathbf{e}^{3\boldsymbol{\Omega}} \left[ 6\frac{\dot{\boldsymbol{\Omega}}^{2}}{\mathbf{N}} - 6\frac{\dot{\boldsymbol{\beta}}^{2}\_{+}}{\mathbf{N}} - 6\frac{\dot{\boldsymbol{\beta}}^{2}\_{-}}{\mathbf{N}} + 6\frac{\dot{\boldsymbol{\phi}}^{2}}{\mathbf{N}} - 6\frac{\dot{\boldsymbol{\phi}}^{2}}{\mathbf{N}} + \mathbf{N} \left( -\mathbf{V}(\boldsymbol{\varphi}, \boldsymbol{\xi}) + 2\boldsymbol{\Lambda} + 16\pi \mathbf{G}\boldsymbol{\rho} \right) \right], \tag{6}$$

the fields were re-scaled as *<sup>φ</sup>* <sup>=</sup> <sup>√</sup>12*ϕ*, *<sup>σ</sup>* <sup>=</sup> <sup>√</sup>12*<sup>ς</sup>* for simplicity in the calculations. The momenta are defined as *<sup>Π</sup>qi* <sup>=</sup> *<sup>∂</sup>*<sup>L</sup> *<sup>∂</sup>*q˙ <sup>i</sup> , where qi = (*β*±, *<sup>Ω</sup>*, *<sup>ϕ</sup>*, *<sup>ς</sup>*) are the coordinates fields.

$$\begin{aligned} \boldsymbol{\varPi}\_{\Omega} &= \frac{\partial \mathcal{L}}{\partial \dot{\varOmega}} = \frac{12 \mathbf{e}^{3\Omega} \dot{\varOmega}}{\mathbf{N}}, \quad & \to \dot{\varOmega} = \frac{\mathbf{N} \boldsymbol{\varOmega}\_{\Omega}}{12} \mathbf{e}^{-3\Omega} \\ \boldsymbol{\varPi}\_{\pm} &= \frac{\partial \mathcal{L}}{\partial \dot{\varbeta}\_{\pm}} = -12 \frac{\mathbf{e}^{3\Omega} \dot{\varbeta}\_{\pm}}{\mathbf{N}}, \quad & \to \dot{\varbeta}\_{\pm} = -\frac{\mathbf{N} \boldsymbol{\varOmega}\_{\pm}}{12} \mathbf{e}^{-3\Omega} \\ \boldsymbol{\varPi}\_{\boldsymbol{\varTheta}} &= \frac{\partial \mathcal{L}}{\partial \dot{\varOmega}} = 12 \frac{\mathbf{e}^{3\Omega} \dot{\varbeta}}{\mathbf{N}}, \quad & \to \dot{\varbeta} = \frac{\mathbf{N} \boldsymbol{\varPi}\_{\theta}}{12} \mathbf{e}^{-3\Omega} \\ \boldsymbol{\varPi}\_{\xi} &= \frac{\partial \mathcal{L}}{\partial \dot{\xi}} = -12 \frac{\mathbf{e}^{3\Omega} \dot{\varxi}}{\mathbf{N}}, \quad & \to \dot{\boldsymbol{\varsigma}} = -\frac{\mathbf{N} \boldsymbol{\varPi}\_{\xi}}{12} \mathbf{e}^{-3\Omega}. \end{aligned} \tag{7}$$

<sup>H</sup>*<sup>A</sup>* <sup>=</sup> <sup>H</sup><sup>I</sup> <sup>−</sup> <sup>1</sup>

<sup>e</sup>4*β*++<sup>4</sup> √

Considering the inflationary phenomenon *γ* = −1, the Hamiltonian density is

*<sup>ς</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

− <sup>e</sup>−2*β*++<sup>2</sup>

−2 

UIX(*Ω*, *<sup>β</sup>*±) = 12e4*<sup>Ω</sup>*

<sup>H</sup>IX <sup>=</sup> <sup>e</sup>−3*<sup>Ω</sup>* 24 *Π*2 *<sup>Ω</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

where *<sup>λ</sup>*eff = <sup>48</sup>(*<sup>Λ</sup>* + <sup>8</sup>*π*GM<sup>−</sup><sup>1</sup> ).

−24*e* 4*Ω* 1 2 *e* 4*β*++4 √

under consideration, it can be read in table 1.

<sup>I</sup> <sup>e</sup>−3*<sup>Ω</sup>*

II <sup>e</sup>−3*<sup>Ω</sup>*

VI−<sup>1</sup> <sup>e</sup>−3*<sup>Ω</sup>*

VII0 *<sup>e</sup>*−3*<sup>Ω</sup>*

VIII <sup>e</sup>−3*<sup>Ω</sup>*

IX <sup>e</sup>−3*<sup>Ω</sup>*

**Bianchi type Hamiltonian density** H

<sup>24</sup> *Π*<sup>2</sup> *<sup>Ω</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

<sup>24</sup> *Π*<sup>2</sup> *<sup>Ω</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

<sup>24</sup> *Π*<sup>2</sup> *<sup>Ω</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

<sup>24</sup> *Π*<sup>2</sup> *<sup>Ω</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

<sup>24</sup> *Π*<sup>2</sup> *<sup>Ω</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

<sup>24</sup> *Π*<sup>2</sup> *<sup>Ω</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

where the gravitational potential can be seen in table I, in particular, the Bianchi Type IX is

e4*<sup>β</sup>*<sup>+</sup> <sup>+</sup> e2*<sup>β</sup>*+−<sup>2</sup>

<sup>+</sup> − *Π*<sup>2</sup>

<sup>3</sup>*β*<sup>−</sup> + *e*

The equation (13) can be considered as a master equation for all Bianchi Class A cosmological models in this formalism, where U(*Ω*, *β*±) is the potential term of the cosmological model

<sup>+</sup> − *Π*<sup>2</sup>

<sup>+</sup> − *Π*<sup>2</sup>

<sup>+</sup> − *Π*<sup>2</sup>

<sup>+</sup> − *Π*<sup>2</sup>

<sup>+</sup> − *Π*<sup>2</sup>

<sup>+</sup> − *Π*<sup>2</sup>

√

<sup>−</sup> + *Π*<sup>2</sup>

<sup>−</sup> + *Π*<sup>2</sup>

<sup>−</sup> + *Π*<sup>2</sup>

<sup>−</sup> + *Π*<sup>2</sup>

<sup>−</sup> + *Π*<sup>2</sup>

<sup>−</sup> + *Π*<sup>2</sup>

<sup>3</sup>*β*<sup>−</sup> + <sup>e</sup>−2*β*++<sup>2</sup>

<sup>3</sup>*β*<sup>−</sup> + *<sup>e</sup>*4*β*+−<sup>4</sup>

<sup>3</sup>*β*<sup>−</sup> + *<sup>e</sup>*4*β*+−<sup>4</sup>

√

<sup>3</sup>*β*<sup>−</sup> − *<sup>e</sup>*4*β*<sup>+</sup> + *<sup>e</sup>*4*β*+−<sup>4</sup>

*<sup>ϕ</sup>* <sup>+</sup> *<sup>e</sup>*6*<sup>Ω</sup>*

√

√

<sup>3</sup>*β*<sup>−</sup> − <sup>e</sup>−2*β*++<sup>2</sup>

√3*β*<sup>−</sup> 

<sup>3</sup>*β*<sup>−</sup> + *<sup>e</sup>*−8*β*<sup>+</sup>

√3*β*<sup>−</sup> 

<sup>3</sup>*β*<sup>−</sup> + *<sup>e</sup>*−8*β*<sup>+</sup>

√3*β*<sup>−</sup> 

*<sup>ς</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

*<sup>ς</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

*<sup>ς</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

*<sup>ς</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

*<sup>e</sup>*4*β*++<sup>4</sup> √

*<sup>ς</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

*<sup>e</sup>*4*β*++<sup>4</sup> √

e4*<sup>β</sup>*<sup>+</sup> − <sup>e</sup>−2*β*+−<sup>2</sup>

*<sup>ς</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

*<sup>e</sup>*4*β*++<sup>4</sup> √

e4*<sup>β</sup>*<sup>+</sup> + e2*<sup>β</sup>*+−<sup>2</sup>

**Table 1.** Hamiltonian density for the Bianchi Class A models in the quintom approach for the inflationary phenomenon.

√3*β*<sup>−</sup> 

−12e4*<sup>Ω</sup>*e4*β*++<sup>4</sup>

−48e4*<sup>Ω</sup>*e4*β*<sup>+</sup>

−12*e*4*<sup>Ω</sup>*

−12*e*4*<sup>Ω</sup>*

−12*e*4*<sup>Ω</sup>*

+2 

+2  √

<sup>3</sup>*β*<sup>−</sup> <sup>+</sup> e4*<sup>β</sup>*+−<sup>4</sup>

<sup>−</sup> + *Π*<sup>2</sup>

4*β*+−4 √

<sup>3</sup>*β*<sup>−</sup> <sup>+</sup> <sup>e</sup>−2*β*+−<sup>2</sup>

√

<sup>3</sup>*β*<sup>−</sup> + *e*

√

√

<sup>24</sup> <sup>e</sup>−3*<sup>Ω</sup>*UA(*Ω*, *<sup>β</sup>*±), (13)

Quintom Potential from Quantum Anisotropic Cosmological Models

√3*β*<sup>−</sup> .

<sup>3</sup>*β*<sup>−</sup> + e4*β*<sup>+</sup>

<sup>3</sup>*β*<sup>−</sup> <sup>+</sup> <sup>e</sup>−2*β*++<sup>2</sup>

*<sup>ϕ</sup>* <sup>+</sup> <sup>e</sup>6*<sup>Ω</sup>* {24V(*ϕ*, *<sup>ς</sup>*) <sup>−</sup> *<sup>λ</sup>*eff}

−8*β*<sup>+</sup> 

<sup>3</sup>*β*<sup>−</sup> + e4*<sup>β</sup>*<sup>+</sup>

*<sup>ϕ</sup>* <sup>+</sup> <sup>e</sup>6*<sup>Ω</sup>* {24V(*ϕ*, *<sup>ς</sup>*) <sup>−</sup> *<sup>λ</sup>*eff}

*<sup>ϕ</sup>* <sup>+</sup> <sup>e</sup>6*<sup>Ω</sup>* {24V(*ϕ*, *<sup>ς</sup>*) <sup>−</sup> *<sup>λ</sup>*eff}

*<sup>ϕ</sup>* <sup>+</sup> <sup>e</sup>6*<sup>Ω</sup>* {24V(*ϕ*, *<sup>ς</sup>*) <sup>−</sup> *<sup>λ</sup>*eff}

*<sup>ϕ</sup>* <sup>+</sup> <sup>e</sup>6*<sup>Ω</sup>* {24V(*ϕ*, *<sup>ς</sup>*) <sup>−</sup> *<sup>λ</sup>*eff}

*<sup>ϕ</sup>* <sup>+</sup> <sup>e</sup>6*<sup>Ω</sup>* {24V(*ϕ*, *<sup>ς</sup>*) <sup>−</sup> *<sup>λ</sup>*eff}

<sup>24</sup>*V*(*ϕ*, *<sup>ς</sup>*) <sup>−</sup> *<sup>λ</sup>eff*

, (14)

http://dx.doi.org/10.5772/52054

223

Writing (6) in canonical form, Lcanonical = *<sup>Π</sup>*qq˙ − <sup>N</sup>H and substituting the energy density for the barotropic fluid, we can find the Hamiltonian density H in the usual way

$$\mathcal{H}\_{\rm I} = \frac{\mathrm{e}^{-3\varOmega}}{24} \left[ \Pi\_{\Omega}^{2} - \Pi\_{\rm \xi}^{2} - \Pi\_{+}^{2} - \Pi\_{-}^{2} + \Pi\_{\rm \varphi}^{2} + \mathrm{e}^{6\varOmega} \left\{ 24 \mathrm{V}(\varphi, \mathfrak{g}) - 48 \left( \Lambda + 8\pi \mathrm{GM}\_{7} \mathrm{e}^{-3(\gamma+1)\varOmega} \right) \right\} \right]. \tag{8}$$

For the Bianchi type IX we have the Lagrangian and Hamiltonian density, respectively

$$\mathcal{L}\_{\text{IX}} = \mathbf{e}^{3\Omega} \left[ 6\frac{\dot{\Omega}^2}{\text{N}} - 6\frac{\dot{\mathcal{G}}\_+^2}{\text{N}} - 6\frac{\dot{\mathcal{G}}\_-^2}{\text{N}} + 6\frac{\dot{\mathcal{G}}^2}{\text{N}} - 6\frac{\dot{\mathcal{G}}^2}{\text{N}} + \text{N} \left( -\text{V}(\rho, \xi) + 2\Lambda + 16\pi \text{G}\rho \right) \right]$$

$$+ \text{Ne}^{-2\Omega} \left\{ \frac{1}{2} \left( \mathbf{e}^{4\mathcal{G}\_+ + 4\sqrt{3}\mathcal{G}\_-} + \mathbf{e}^{4\mathcal{G}\_+ - 4\sqrt{3}\mathcal{G}\_-} + \mathbf{e}^{-8\mathcal{G}\_+} \right) \right.$$

$$- \left( \mathbf{e}^{-2\mathcal{G}\_+ + 2\sqrt{3}\mathcal{G}\_-} + \mathbf{e}^{-2\mathcal{G}\_+ - 2\sqrt{3}\mathcal{G}\_-} + \mathbf{e}^{4\mathcal{G}\_+} \right) \right\}, \tag{9}$$

$$\mathcal{H}\_{\rm IX} = \frac{e^{-3\Omega}}{24} \left[ \Pi\_{\Omega}^{2} - \Pi\_{\xi}^{2} - \Pi\_{+}^{2} - \Pi\_{-}^{2} + \Pi\_{\theta}^{2} + \mathsf{e}^{6\Omega} \left\{ 24\mathcal{V}(\mathsf{p},\xi) - 48 \left( \Lambda + 8\pi \mathsf{GM}\_{\uparrow} \mathsf{e}^{-3(\gamma+1)\Omega} \right) \right\} \right],$$

$$-24e^{4\Omega} \left\{ \frac{1}{2} \left( e^{4\mathfrak{P}\_{+} + 4\sqrt{3}\mathfrak{\beta}\_{-}} + e^{4\mathfrak{P}\_{+} - 4\sqrt{3}\mathfrak{\beta}\_{-}} + e^{-8\mathfrak{P}\_{+}} \right)$$

$$- \left( \mathsf{e}^{-2\mathfrak{P}\_{+} + 2\sqrt{3}\mathfrak{\beta}\_{-}} + \mathsf{e}^{-2\mathfrak{P}\_{+} - 2\sqrt{3}\mathfrak{\beta}\_{-}} + \mathsf{e}^{4\mathfrak{P}\_{+}} \right) \right) \right\}, \tag{10}$$

where we have used the covariant derivative of (3), obtaining the relation

$$3\dot{\Omega}\dot{\Omega}p + 3\dot{\Omega}p + \dot{\rho} = 0,\tag{11}$$

whose solution becomes

$$\rho = \mathbf{M}\_{\gamma} \mathbf{e}^{-\Im(1+\gamma)\varOmega}.\tag{12}$$

where *M<sup>γ</sup>* is an integration constant, in this sense we have all Bianchi Class A cosmological models, and their corresponding Hamiltonian density becomes

$$\mathcal{H}\_{\rm A} = \mathcal{H}\_{\rm I} - \frac{1}{24} \mathbf{e}^{-3\Omega} \mathbf{U}\_{\rm A} (\Omega, \beta\_{\pm}), \tag{13}$$

where the gravitational potential can be seen in table I, in particular, the Bianchi Type IX is

$$\begin{split} \mathbf{U}\_{\mathrm{IX}}(\Omega, \mathfrak{\boldsymbol{\beta}}\_{\pm}) &= 12 \mathbf{e}^{4\Omega} \left( \mathbf{e}^{4\mathfrak{\boldsymbol{\beta}}\_{+} + 4\sqrt{3}\mathfrak{\boldsymbol{\beta}}\_{-}} + \mathbf{e}^{4\mathfrak{\boldsymbol{\beta}}\_{+} - 4\sqrt{3}\mathfrak{\boldsymbol{\beta}}\_{-}} + \mathbf{e}^{4\mathfrak{\boldsymbol{\beta}}\_{+}} \right. \\ & \left. - 2 \left\{ \mathbf{e}^{4\mathfrak{\boldsymbol{\beta}}\_{+}} + \mathbf{e}^{2\mathfrak{\boldsymbol{\beta}}\_{+} - 2\sqrt{3}\mathfrak{\boldsymbol{\beta}}\_{-}} + \mathbf{e}^{-2\mathfrak{\boldsymbol{\beta}}\_{+} + 2\sqrt{3}\mathfrak{\boldsymbol{\beta}}\_{-}} \right\} \right) . \end{split}$$

Considering the inflationary phenomenon *γ* = −1, the Hamiltonian density is

$$\mathcal{H}\_{\rm IX} = \frac{\mathbf{e}^{-3\varOmega}}{24} \left[ \Pi\_{\Omega}^{2} - \Pi\_{\xi}^{2} - \Pi\_{+}^{2} - \Pi\_{-}^{2} + \Pi\_{\varrho}^{2} + \mathbf{e}^{6\varOmega} \left\{ 24 \mathbf{V} (\varrho\_{\prime} \xi) - \lambda\_{\rm eff} \right\} \right]$$

$$-24e^{4\varOmega} \left\{ \frac{1}{2} \left( e^{4\not\upB\_{+} + 4\sqrt{3}\not\upB\_{-}} + e^{4\not\upB\_{+} - 4\sqrt{3}\not\upB\_{-}} + e^{-8\not\upB\_{+}} \right) \right.$$

$$-\left( \mathbf{e}^{-2\not\upB\_{+} + 2\sqrt{3}\not\upB\_{-}} + \mathbf{e}^{-2\not\upB\_{+} - 2\sqrt{3}\not\upB\_{-}} + \mathbf{e}^{4\not\upB\_{+}} \right) \right\}, \tag{14}$$

where *<sup>λ</sup>*eff = <sup>48</sup>(*<sup>Λ</sup>* + <sup>8</sup>*π*GM<sup>−</sup><sup>1</sup> ).

4 Open Questions in Cosmology

<sup>H</sup><sup>I</sup> <sup>=</sup> <sup>e</sup>−3*<sup>Ω</sup>* 24 *Π*2 *<sup>Ω</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

LIX = e3*<sup>Ω</sup>*

−24*e*4*<sup>Ω</sup>* <sup>1</sup>

whose solution becomes

<sup>H</sup>IX <sup>=</sup> <sup>e</sup>−3*<sup>Ω</sup>* 24 *Π*<sup>2</sup> *<sup>Ω</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

 6 *Ω*˙ 2 <sup>N</sup> <sup>−</sup> <sup>6</sup>

<sup>+</sup>Ne−2*<sup>Ω</sup>*

2 *<sup>e</sup>*4*β*++<sup>4</sup> √

> − <sup>e</sup>−2*β*++<sup>2</sup>

*ΠΩ* <sup>=</sup> *<sup>∂</sup>*<sup>L</sup>

*<sup>Π</sup>*<sup>±</sup> <sup>=</sup> *<sup>∂</sup>*<sup>L</sup> *∂β*˙ ±

*Πϕ* <sup>=</sup> *<sup>∂</sup>*<sup>L</sup>

*Πς* <sup>=</sup> *<sup>∂</sup>*<sup>L</sup>

*<sup>ς</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

*β*˙ 2 + <sup>N</sup> <sup>−</sup> <sup>6</sup>

1 2 <sup>e</sup>4*β*++<sup>4</sup> √

*<sup>ς</sup>* <sup>−</sup> *<sup>Π</sup>*<sup>2</sup>

<sup>+</sup> − *Π*<sup>2</sup>

√

models, and their corresponding Hamiltonian density becomes

− <sup>e</sup>−2*β*++<sup>2</sup>

<sup>+</sup> − *Π*<sup>2</sup>

*∂Ω*˙ <sup>=</sup> 12e3*ΩΩ*˙

*∂ϕ*˙ <sup>=</sup> <sup>12</sup> <sup>e</sup>3*<sup>Ω</sup> <sup>ϕ</sup>*˙

*∂ς*˙ <sup>=</sup> <sup>−</sup><sup>12</sup> <sup>e</sup><sup>3</sup>*Ως*˙

<sup>=</sup> <sup>−</sup><sup>12</sup> <sup>e</sup><sup>3</sup>*Ωβ*˙

<sup>N</sup> , <sup>→</sup> *<sup>Ω</sup>*˙ <sup>=</sup> <sup>N</sup>*ΠΩ*

<sup>N</sup> , <sup>→</sup> *<sup>ϕ</sup>*˙ <sup>=</sup> <sup>N</sup>*Πϕ*

Writing (6) in canonical form, Lcanonical = *<sup>Π</sup>*qq˙ − <sup>N</sup>H and substituting the energy density

*<sup>ϕ</sup>* <sup>+</sup> e6*<sup>Ω</sup>*

For the Bianchi type IX we have the Lagrangian and Hamiltonian density, respectively

<sup>3</sup>*β*<sup>−</sup> <sup>+</sup> e4*<sup>β</sup>*+−<sup>4</sup>

<sup>N</sup> <sup>−</sup> <sup>6</sup> *ς*˙ 2

<sup>3</sup>*β*<sup>−</sup> <sup>+</sup> <sup>e</sup>−2*β*+−<sup>2</sup>

*<sup>ϕ</sup>* <sup>+</sup> <sup>e</sup>6*<sup>Ω</sup>*

<sup>3</sup>*β*<sup>−</sup> + *<sup>e</sup>*−8*β*<sup>+</sup>

√

where *M<sup>γ</sup>* is an integration constant, in this sense we have all Bianchi Class A cosmological

√

<sup>3</sup>*β*<sup>−</sup> + <sup>e</sup>−2*β*+−<sup>2</sup>

where we have used the covariant derivative of (3), obtaining the relation

√

for the barotropic fluid, we can find the Hamiltonian density H in the usual way

<sup>−</sup> + *Π*<sup>2</sup>

*β*˙ 2 − <sup>N</sup> <sup>+</sup> <sup>6</sup> *<sup>ϕ</sup>*˙ <sup>2</sup>

√

<sup>−</sup> + *Π*<sup>2</sup>

<sup>3</sup>*β*<sup>−</sup> + *<sup>e</sup>*4*β*+−<sup>4</sup>

<sup>N</sup> , <sup>→</sup> *<sup>ς</sup>*˙ <sup>=</sup> <sup>−</sup> <sup>N</sup>*Πς*

24V(*ϕ*, *ς*) − 48

<sup>3</sup>*β*<sup>−</sup> <sup>+</sup> <sup>e</sup>−8*β*<sup>+</sup>

24V(*ϕ*, *ς*) − 48

<sup>3</sup>*β*<sup>−</sup> + e4*β*<sup>+</sup>

<sup>3</sup>*β*<sup>−</sup> + e4*<sup>β</sup>*<sup>+</sup>

√

± <sup>N</sup> , <sup>→</sup> *<sup>β</sup>*˙ <sup>12</sup> <sup>e</sup>−3*<sup>Ω</sup>*

<sup>±</sup> <sup>=</sup> <sup>−</sup> <sup>N</sup>*Π*<sup>±</sup>

<sup>12</sup> <sup>e</sup>−3*<sup>Ω</sup>*

<sup>12</sup> <sup>e</sup>−3*Ω*.

<sup>N</sup> <sup>+</sup> <sup>N</sup> (−V(*ϕ*, *<sup>ς</sup>*) + <sup>2</sup>*<sup>Λ</sup>* <sup>+</sup> <sup>16</sup>*π*G*ρ*)

3*Ωρ*˙ + 3*Ω*˙ *p* + *ρ*˙ = 0, (11)

*<sup>ρ</sup>* <sup>=</sup> <sup>M</sup>*γ*e−3(1+*γ*)*Ω*. (12)

<sup>12</sup> <sup>e</sup>−3*<sup>Ω</sup>* (7)

*<sup>Λ</sup>* <sup>+</sup> <sup>8</sup>*π*GM*γ*e−3(*γ*+1)*Ω*

. (8)

, (9)

*<sup>Λ</sup>* <sup>+</sup> <sup>8</sup>*π*GM*γ*e−3(*γ*+1)*Ω*

, (10)

The equation (13) can be considered as a master equation for all Bianchi Class A cosmological models in this formalism, where U(*Ω*, *β*±) is the potential term of the cosmological model under consideration, it can be read in table 1.


**Table 1.** Hamiltonian density for the Bianchi Class A models in the quintom approach for the inflationary phenomenon.

### **3.1. Classical field equation for Bianchi type I**

On the other hand, the Einstein field equations (2,3) for the Bianchi type I, are

$$3\frac{\dot{\Omega}^2}{\mathbf{N}^2} - 3\frac{\dot{\beta}\_+^2}{\mathbf{N}^2} - 3\frac{\dot{\beta}\_-^2}{\mathbf{N}^2} = 8\pi\text{G}\rho - 3\frac{\dot{\rho}^2}{\mathbf{N}^2} + 3\frac{\dot{\zeta}^2}{\mathbf{N}^2} + \Lambda + \frac{\mathbf{V}(\boldsymbol{\varphi}, \boldsymbol{\zeta})}{2},\tag{15}$$

$$2\frac{\ddot{\Omega}}{\mathbf{N}^2} + 3\frac{\dot{\Omega}^2}{\mathbf{N}^2} - 3\frac{\dot{\Omega}\dot{\beta}\_+}{\mathbf{N}^2} - 3\sqrt{3}\frac{\dot{\Omega}\dot{\beta}\_-}{\mathbf{N}^2} - 2\frac{\dot{\Omega}\dot{\mathbf{N}}}{\mathbf{N}^3} - \frac{\ddot{\beta}\_+}{\mathbf{N}^2} + 3\frac{\dot{\beta}\_+^2}{\mathbf{N}^2} + \frac{\dot{\beta}\_+\dot{\Lambda}}{\mathbf{N}^3} - \sqrt{3}\frac{\ddot{\beta}\_-}{\mathbf{N}^2} + 3\frac{\dot{\beta}\_-^2}{\mathbf{N}^2}$$

*ρϕ* <sup>=</sup> <sup>1</sup>

similar to equations (17,18) we find

equivalent, see equations (24,23), where

and the function h(*Ω*) is

where

16*π*G

<sup>−</sup>6*ϕ*′<sup>2</sup> <sup>+</sup> <sup>V</sup>(*ϕ*, *<sup>ς</sup>*)|*<sup>ς</sup>*

*<sup>ω</sup>DE* <sup>=</sup> <sup>6</sup>*ς*˙

*<sup>β</sup>*<sup>−</sup> = *<sup>β</sup>*<sup>0</sup>

*<sup>β</sup>*<sup>+</sup> = *<sup>β</sup>*<sup>1</sup>

then there is the relation between the anisotropic functions *<sup>β</sup>*<sup>−</sup> = *<sup>β</sup>*2*β*<sup>+</sup> with *<sup>β</sup>*<sup>2</sup> = *<sup>β</sup>*<sup>0</sup>

d*Ω*

*<sup>ς</sup>*′ <sup>=</sup> *<sup>ς</sup>*0e−3*∆Ω*, *<sup>ς</sup>*(*τ*) = *<sup>ς</sup>*<sup>0</sup>

*<sup>ϕ</sup>*′ <sup>=</sup> *<sup>ϕ</sup>*0e−3*∆Ω*, *<sup>ϕ</sup>*(*τ*) = *<sup>ϕ</sup>*<sup>0</sup>

where we obtain the relation between the functions *β*− and *Ω* as

introduce the total quintom energy density and pressure as:

which are useful when we study the behavior of dynamical systems. Additionally we can

To solve the set of differential equation (*β*±, *Ω*, *ϕ*, *ς*) we begin with the equations (16, 17)

For separable potentials, equations (24,23) can be solved in some cases in terms of the *Ω* function, then, using equation (15) we can obtain in a quadrature form, the structure of *Ω* as

where the function h(*Ω*) has the corresponding information of all functions presented in this equation (15). For instance, when the potential V(*ϕ*, *ς*) becomes null or constant, the formalism is like the one formulated by Sáez and Ballester in 1986 [35] because both field are

<sup>e</sup>−3*∆Ω*(*τ*)

<sup>e</sup>−3*∆Ω*(*τ*)

, *ρς* <sup>=</sup> <sup>1</sup>

<sup>2</sup> − 6*ϕ*˙ <sup>2</sup> − *V*(*ς*, *ϕ*)

16*π*G

*<sup>ρ</sup>DE* = *ρς* + *ρϕ*, *PDE* = *<sup>P</sup><sup>ς</sup>* + *<sup>P</sup>ϕ*, *PDE* = *<sup>ω</sup>DE <sup>ρ</sup>DE* (26)

Quintom Potential from Quantum Anisotropic Cosmological Models

<sup>6</sup>*ς*˙<sup>2</sup> <sup>−</sup> <sup>6</sup>*ϕ*˙ <sup>2</sup> <sup>+</sup> *<sup>V</sup>*(*ς*, *<sup>ϕ</sup>*) (27)

<sup>e</sup>−3*∆Ω*d*τ*, (28)

<sup>e</sup>−3*∆Ω*d*τ*, (29)

<sup>h</sup>(*Ω*) <sup>=</sup> *∆τ*, (30)

<sup>d</sup>*<sup>τ</sup>* + *<sup>ς</sup>*1, (31)

<sup>d</sup>*<sup>τ</sup>* + *<sup>ϕ</sup>*1, , (32)

*β*1 .

<sup>6</sup>*ς*′<sup>2</sup> <sup>+</sup> <sup>V</sup>(*ϕ*, *<sup>ς</sup>*)|*<sup>ϕ</sup>*

 ,

http://dx.doi.org/10.5772/52054

225

$$-\sqrt{3}\frac{\dot{\mathcal{B}}\\_\mathrm{N}}{\mathrm{N}^3} = -8\pi \mathrm{GP} + 3\frac{\dot{\mathcal{G}}^2}{\mathrm{N}^2} - 3\frac{\dot{\mathcal{G}}^2}{\mathrm{N}^2} + \Lambda + \frac{\mathrm{V}(\mathcal{op}, \emptyset)}{2},\tag{16}$$

$$\begin{split} 2\frac{\ddot{\Omega}}{\mathbf{N}^2} + 3\frac{\dot{\Omega}^2}{\mathbf{N}^2} - 3\frac{\dot{\Omega}\dot{\beta}\_+}{\mathbf{N}^2} + 3\sqrt{3}\frac{\dot{\Omega}\dot{\beta}\_-}{\mathbf{N}^2} - 2\frac{\dot{\Omega}\dot{\mathbf{N}}}{\mathbf{N}^3} - \frac{\ddot{\beta}\_+}{\mathbf{N}^2} + 3\frac{\dot{\beta}\_+^2}{\mathbf{N}^2} + \frac{\dot{\beta}\_+\dot{\Omega}}{\mathbf{N}^3} + \sqrt{3}\frac{\ddot{\beta}\_-}{\mathbf{N}^2} + 3\frac{\ddot{\beta}\_-}{\mathbf{N}^2} \\ - \sqrt{3}\frac{\dot{\beta}\_-\dot{\Omega}}{\mathbf{N}^2} = -8\pi\mathbf{GP} + 3\frac{\dot{\varphi}^2}{\mathbf{N}^2} - 3\frac{\dot{\xi}^2}{\mathbf{N}^2} + \Lambda + \frac{\mathbf{V}(\varphi,\xi)}{\mathbf{\tau}}. \end{split} \tag{17}$$

$$-\sqrt{3}\frac{\dot{\not p}\_{-}\dot{\text{N}}}{\text{N}^3} = -8\pi \text{GP} + 3\frac{\dot{\not p}^2}{\text{N}^2} - 3\frac{\dot{\not p}^2}{\text{N}^2} + \Lambda + \frac{\text{V}(\not p, \not q)}{2},\tag{17}$$

$$-\sqrt{2}\underline{\dot{\text{O}}} + 3\frac{\dot{\text{O}}^2}{\text{N}^2} + \kappa\dot{\text{I}}\dot{\text{P}}\dot{\text{\not}}\_{+} - 2\frac{\dot{\text{O}}\dot{\text{N}}}{\text{N}^2} + 2\frac{\ddot{\text{P}}\_{+} + 3\frac{\dot{\text{P}}\_{+}}{\text{N}^2} - 2\frac{\dot{\text{P}}\_{+} \dot{\text{N}}}{\text{N}}$$

$$\begin{split} 2\frac{\ddot{\Omega}}{\mathbf{N}^2} + 3\frac{\dot{\Omega}^2}{\mathbf{N}^2} + 6\frac{\dot{\Omega}\dot{\rho}\_+}{\mathbf{N}^2} - 2\frac{\dot{\Omega}\dot{\mathbf{N}}}{\mathbf{N}^3} + 2\frac{\ddot{\bar{\beta}}\_+}{\mathbf{N}^2} + 3\frac{\dot{\bar{\beta}}\_+^2}{\mathbf{N}^2} - 2\frac{\dot{\bar{\beta}}\_+\dot{\mathbf{N}}}{\mathbf{N}^3} \\ + 3\frac{\dot{\bar{\beta}}\_-^2}{\mathbf{N}^2} = -8\pi\text{GP} + 3\frac{\dot{\bar{\rho}}^2}{\mathbf{N}^2} - 3\frac{\dot{\xi}^2}{\mathbf{N}^2} + \Lambda + \frac{\mathbf{V}(\boldsymbol{\bar{\rho}},\boldsymbol{\xi})}{2}, \end{split} \tag{18}$$

$$-3\frac{\dot{\Omega}\ddagger}{\mathbf{N}^2} + \frac{\dot{\mathbf{N}}\xi}{\mathbf{N}^3} - \frac{\ddot{\xi}}{\mathbf{N}^2} + \frac{\partial \mathbf{V}(\xi, \boldsymbol{\varrho})}{\partial \xi} = 0,\tag{19}$$

$$-3\frac{\dot{\Omega}\dot{\varphi}}{\mathcal{N}^2} + \frac{\dot{\mathcal{N}}\dot{\varphi}}{\mathcal{N}^3} - \frac{\ddot{\mathcal{q}}}{\mathcal{N}^2} - \frac{\partial \mathcal{V}(\boldsymbol{\xi}, \boldsymbol{\varrho})}{\partial \boldsymbol{\varrho}} = 0,\tag{20}$$

which can be written as

$$8\pi \text{GP} - \Lambda + \frac{1}{2} \left( -6\varphi'^2 + 6\varphi'^2 - \mathcal{V}(\varphi, \mathfrak{g}) \right) = -\frac{2}{3} \frac{\text{a}''}{\text{a}} - 3\mathcal{H}^2,\tag{21}$$

$$8\pi \text{G}\rho + \Lambda + \frac{1}{2}\left(-6\rho'^2 + 6\varsigma'^2 + \mathcal{V}(\varphi, \varsigma)\right) = 3\mathcal{H}^2,\tag{22}$$

$$-3\Omega'\xi'-\xi''-\frac{\partial\mathcal{V}(\emptyset,\emptyset)}{\partial\xi}=0,\tag{23}$$

$$-\Im\Omega'\boldsymbol{\varrho}'-\boldsymbol{\varrho}''+\frac{\partial\mathcal{V}(\boldsymbol{\xi},\boldsymbol{\varrho})}{\partial\boldsymbol{\varrho}}=0,\tag{24}$$

where H2 is defined as H<sup>2</sup> <sup>=</sup> H1H2 <sup>+</sup> H1H3 <sup>+</sup> H2H3, a <sup>=</sup> R1R2R3, and Hi <sup>=</sup> R˙ <sup>i</sup> Ri . We have done the time transformation <sup>d</sup> <sup>d</sup>*<sup>τ</sup>* <sup>=</sup> <sup>d</sup> Ndt <sup>=</sup> ′. Adding (21) and (22) we arrive

$$-\frac{\mathbf{a}^{\prime\prime}}{\mathbf{a}} = 12\pi\mathbf{G}\left[\rho + \rho\_{\mathcal{P}} + \rho\_{\mathcal{G}} + \mathbf{P} + \mathbf{P}\_{\mathcal{P}} + \mathbf{P}\_{\mathcal{G}}\right],\tag{25}$$

where

$$\mathbf{P}\_{\boldsymbol{\theta}} = \frac{1}{16\pi \mathbf{G}} \left( -6\boldsymbol{\varrho}^{\prime 2} - \mathbf{V}(\boldsymbol{\varrho}, \boldsymbol{\varrho})|\_{\boldsymbol{\xi}} \right), \qquad \mathbf{P}\_{\boldsymbol{\xi}} = \frac{1}{16\pi \mathbf{G}} \left( 6\boldsymbol{\zeta}^{\prime 2} - \mathbf{V}(\boldsymbol{\varrho}, \boldsymbol{\varrho})|\_{\boldsymbol{\xi}} \right).$$

$$\rho\_{\mathcal{G}} = \frac{1}{16\pi \mathcal{G}} \left( -6\varphi'^2 + \mathcal{V}(\varphi,\boldsymbol{\zeta})|\_{\boldsymbol{\zeta}} \right) , \qquad \rho\_{\boldsymbol{\xi}} = \frac{1}{16\pi \mathcal{G}} \left( 6\boldsymbol{\zeta}'^2 + \mathcal{V}(\varphi,\boldsymbol{\zeta})|\_{\boldsymbol{\varphi}} \right) ,$$

which are useful when we study the behavior of dynamical systems. Additionally we can introduce the total quintom energy density and pressure as:

$$
\rho\_{\rm DE} = \rho\_{\xi} + \rho\_{\Psi\prime} \qquad P\_{\rm DE} = P\_{\xi} + P\_{\Psi\prime} \qquad P\_{\rm DE} = \omega\_{\rm DE} \rho\_{\rm DE} \tag{26}
$$

where

6 Open Questions in Cosmology

*β*˙ 2 + N2 <sup>−</sup> <sup>3</sup>

*Ω*˙ 2 N2 <sup>−</sup> <sup>3</sup>

*Ω*˙ 2 N2 <sup>−</sup> <sup>3</sup>

*Ω*˙ 2 N2 <sup>+</sup> <sup>6</sup>

*<sup>N</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>8*π*GP <sup>+</sup> <sup>3</sup> *<sup>ϕ</sup>*˙ <sup>2</sup>

N˙ *ς*˙ N3 <sup>−</sup> *<sup>ς</sup>*¨

N˙ *ϕ*˙ <sup>N</sup><sup>3</sup> <sup>−</sup> *<sup>ϕ</sup>*¨

done the time transformation <sup>d</sup>

<sup>P</sup>*<sup>ϕ</sup>* <sup>=</sup> <sup>1</sup>

16*π*G

where

3 *Ω*˙ 2 N2 <sup>−</sup> <sup>3</sup>

2 *Ω*¨ N2 <sup>+</sup> <sup>3</sup>

+ √ 3 *β*˙ <sup>−</sup>N˙

2 *Ω*¨ N2 <sup>+</sup> <sup>3</sup>

− √ 3 *β*˙ <sup>−</sup>N˙

2 *Ω*¨ N2 <sup>+</sup> <sup>3</sup>

+3 *β*˙ 2 −

−3 *Ω*˙ *ς*˙ N2 <sup>+</sup>

−3 *Ω*˙ *ϕ*˙ <sup>N</sup><sup>2</sup> <sup>+</sup>

which can be written as

**3.1. Classical field equation for Bianchi type I**

N2 <sup>=</sup> <sup>8</sup>*π*G*<sup>ρ</sup>* <sup>−</sup> <sup>3</sup> *<sup>ϕ</sup>*˙ <sup>2</sup>

√ 3 *Ω*˙ *β*˙ − N2 <sup>−</sup> <sup>2</sup>

√ 3 *Ω*˙ *β*˙ − N2 <sup>−</sup> <sup>2</sup>

<sup>N</sup><sup>2</sup> <sup>−</sup> <sup>3</sup> *<sup>ς</sup>*˙

<sup>N</sup><sup>2</sup> <sup>−</sup> <sup>3</sup> *<sup>ς</sup>*˙

2 <sup>N</sup><sup>2</sup> <sup>+</sup> *<sup>Λ</sup>* <sup>+</sup>

*Ω*˙ N˙ <sup>N</sup><sup>3</sup> <sup>+</sup> <sup>2</sup>

N2 <sup>−</sup> <sup>3</sup> *<sup>ς</sup>*˙

*∂*V(*ς*, *ϕ*)

<sup>N</sup><sup>2</sup> <sup>−</sup> *<sup>∂</sup>*V(*ς*, *<sup>ϕ</sup>*)

1 2 

1 2 

<sup>−</sup>3*Ω*′

<sup>−</sup>3*Ω*′

<sup>d</sup>*<sup>τ</sup>* <sup>=</sup> <sup>d</sup>

<sup>a</sup> <sup>=</sup> <sup>12</sup>*π*<sup>G</sup>

<sup>−</sup>6*ϕ*′<sup>2</sup> <sup>−</sup> <sup>V</sup>(*ϕ*, *<sup>ς</sup>*)|*<sup>ς</sup>*

<sup>−</sup> <sup>a</sup>′′

*β*˙ 2 −

*Ω*˙ *β*˙ + N2 <sup>−</sup> <sup>3</sup>

<sup>N</sup><sup>3</sup> <sup>=</sup> <sup>−</sup>8*π*GP <sup>+</sup> <sup>3</sup> *<sup>ϕ</sup>*˙ <sup>2</sup>

*Ω*˙ *β*˙ + N2 <sup>+</sup> <sup>3</sup>

<sup>N</sup><sup>3</sup> <sup>=</sup> <sup>−</sup>8*π*GP <sup>+</sup> <sup>3</sup> *<sup>ϕ</sup>*˙ <sup>2</sup>

*Ω*˙ *β*˙ + N2 <sup>−</sup> <sup>2</sup>

N2 <sup>+</sup>

8*π*GP − *Λ* +

8*π*G*ρ* + *Λ* +

On the other hand, the Einstein field equations (2,3) for the Bianchi type I, are

<sup>N</sup><sup>2</sup> <sup>+</sup> <sup>3</sup> *<sup>ς</sup>*˙

2 <sup>N</sup><sup>2</sup> <sup>+</sup> *<sup>Λ</sup>* <sup>+</sup>

2 <sup>N</sup><sup>2</sup> <sup>+</sup> *<sup>Λ</sup>* <sup>+</sup>

> *β*¨+ N2 <sup>+</sup> <sup>3</sup>

<sup>−</sup>6*ϕ*′<sup>2</sup> <sup>+</sup> <sup>6</sup>*ς*′<sup>2</sup> <sup>−</sup> <sup>V</sup>(*ϕ*, *<sup>ς</sup>*)

<sup>−</sup>6*ϕ*′<sup>2</sup> <sup>+</sup> <sup>6</sup>*ς*′<sup>2</sup> <sup>+</sup> <sup>V</sup>(*ϕ*, *<sup>ς</sup>*)

*<sup>ϕ</sup>*′ <sup>−</sup> *<sup>ϕ</sup>*′′ <sup>+</sup>

where H2 is defined as H<sup>2</sup> <sup>=</sup> H1H2 <sup>+</sup> H1H3 <sup>+</sup> H2H3, a <sup>=</sup> R1R2R3, and Hi <sup>=</sup> R˙ <sup>i</sup>

*<sup>ς</sup>*′ <sup>−</sup> *<sup>ς</sup>*′′ <sup>−</sup> *<sup>∂</sup>*V(*ς*, *<sup>ϕ</sup>*)

2 N2 <sup>+</sup> *<sup>Λ</sup>* <sup>+</sup>

*Ω*˙ N˙ <sup>N</sup><sup>3</sup> <sup>−</sup> *<sup>β</sup>*¨<sup>+</sup>

*Ω*˙ N˙ <sup>N</sup><sup>3</sup> <sup>−</sup> *<sup>β</sup>*¨<sup>+</sup>

> *β*˙ 2 + <sup>N</sup><sup>2</sup> <sup>−</sup> <sup>2</sup>

V(*ϕ*, *ς*)

V(*ϕ*, *ς*)

*β*˙ 2 + N2 <sup>+</sup> *<sup>β</sup>*˙

*β*˙ 2 + N2 <sup>+</sup> *<sup>β</sup>*˙

*∂ς* <sup>=</sup> 0, (19)

*∂ϕ* <sup>=</sup> 0, (20)

<sup>N</sup><sup>2</sup> <sup>+</sup> <sup>3</sup>

<sup>N</sup><sup>2</sup> <sup>+</sup> <sup>3</sup>

*β*˙ <sup>+</sup>N˙ N3

> = −2 3 a′′

*∂*V(*ς*, *ϕ*)

Ndt <sup>=</sup> ′. Adding (21) and (22) we arrive

16*π*G

*<sup>ρ</sup>* + *ρϕ* + *ρς* + <sup>P</sup> + <sup>P</sup>*<sup>ϕ</sup>* + <sup>P</sup>*<sup>ς</sup>*

, P*<sup>ς</sup>* <sup>=</sup> <sup>1</sup>

V(*ϕ*, *ς*)

V(*ϕ*, *ς*)

<sup>2</sup> , (15)

3 *β*¨− <sup>N</sup><sup>2</sup> <sup>+</sup> <sup>3</sup>

3 *β*¨− <sup>N</sup><sup>2</sup> <sup>+</sup> <sup>3</sup>

<sup>a</sup> <sup>−</sup> 3H2, (21)

Ri

, (25)

 , . We have

= 3H2, (22)

*∂ς* <sup>=</sup> 0, (23)

*∂ϕ* <sup>=</sup> 0, (24)

<sup>6</sup>*ς*′<sup>2</sup> <sup>−</sup> <sup>V</sup>(*ϕ*, *<sup>ς</sup>*)|*<sup>ϕ</sup>*

<sup>2</sup> , (16)

<sup>2</sup> , (17)

<sup>2</sup> , (18)

*β*˙ 2 − N2

*β*˙ 2 − N2

<sup>+</sup>N˙ <sup>N</sup><sup>3</sup> <sup>−</sup> <sup>√</sup>

<sup>+</sup>N˙ <sup>N</sup><sup>3</sup> <sup>+</sup> <sup>√</sup>

$$
\omega\_{\rm DE} = \frac{6\dot{\xi}^2 - 6\dot{\varphi}^2 - V(\xi, \varphi)}{6\dot{\xi}^2 - 6\dot{\varphi}^2 + V(\xi, \varphi)}\tag{27}
$$

To solve the set of differential equation (*β*±, *Ω*, *ϕ*, *ς*) we begin with the equations (16, 17) where we obtain the relation between the functions *β*− and *Ω* as

$$
\beta\_- = \beta\_0 \int \mathbf{e}^{-3\Delta\Omega} \mathbf{d}\tau\_\prime \tag{28}
$$

similar to equations (17,18) we find

$$
\beta\_+ = \beta\_1 \int \mathbf{e}^{-3\Delta\Omega} \mathbf{d}\tau\_\prime \tag{29}
$$

then there is the relation between the anisotropic functions *<sup>β</sup>*<sup>−</sup> = *<sup>β</sup>*2*β*<sup>+</sup> with *<sup>β</sup>*<sup>2</sup> = *<sup>β</sup>*<sup>0</sup> *β*1 .

For separable potentials, equations (24,23) can be solved in some cases in terms of the *Ω* function, then, using equation (15) we can obtain in a quadrature form, the structure of *Ω* as

$$\int \frac{\mathrm{d}\Omega}{\sqrt{\mathrm{h}(\Omega)}} = \Delta \mathrm{r},\tag{30}$$

where the function h(*Ω*) has the corresponding information of all functions presented in this equation (15). For instance, when the potential V(*ϕ*, *ς*) becomes null or constant, the formalism is like the one formulated by Sáez and Ballester in 1986 [35] because both field are equivalent, see equations (24,23), where

$$\boldsymbol{\xi}^{\prime} = \boldsymbol{\xi}\_{0} \mathbf{e}^{-3\Lambda\Omega}, \qquad \boldsymbol{\xi}(\boldsymbol{\tau}) = \boldsymbol{\xi}\_{0} \int \mathbf{e}^{-3\Lambda\Omega(\boldsymbol{\tau})} \mathbf{d}\boldsymbol{\tau} + \boldsymbol{\xi}\_{1\prime} \tag{31}$$

$$\boldsymbol{\varrho}^{\prime} = \boldsymbol{\varrho}\_{0} \mathbf{e}^{-3\Lambda\Omega}, \qquad \boldsymbol{\varrho}(\boldsymbol{\tau}) = \boldsymbol{\varrho}\_{0} \int \mathbf{e}^{-3\Lambda\Omega(\boldsymbol{\tau})} \mathbf{d}\boldsymbol{\tau} + \boldsymbol{\varrho}\_{1\prime\prime} \tag{32}$$

and the function h(*Ω*) is

$$h(\Omega) = \frac{\frac{8\pi GM\_{\gamma}}{3}e^{-3(1+\gamma)\Omega} + \left(-\wp\_{0}^{2} + \zeta\_{0}^{2}\right)e^{-6\Delta\Omega} + \frac{\Lambda\_{eff}}{3}}{1 - 9\left(1 + \beta\_{2}^{2}\right)\beta\_{1}^{2}e^{-6\Delta\Omega}},\tag{33}$$

H*ψ* =

and we consider that *ψ* has the following traditional decomposition

corresponding to the real and imaginary parts, respectively, which are

 1 ¯h<sup>2</sup> (∇S)

<sup>R</sup> <sup>+</sup> <sup>Q</sup>*∂*<sup>R</sup>

*<sup>∂</sup>*<sup>q</sup> <sup>−</sup> <sup>R</sup>

dq*<sup>µ</sup>*

R − R

the equations (38,39) are written as

where *q* is a single field coordinate.

this equation into (40), and we find (using *q*˙

<sup>R</sup> <sup>+</sup> <sup>Q</sup>*∂*<sup>R</sup>

*∂*q = R

function S, as *Π*q*<sup>µ</sup>* = *<sup>∂</sup>*<sup>S</sup>

previous section, thus,

*Q∂ψ*

*<sup>g</sup>µν*∇*µ*∇*<sup>ν</sup>* − *<sup>V</sup>*(*qµ*)

where the metric may be q*<sup>µ</sup>* dependent. The *ψ* is called the wave function of the universe,

with *R* and *S* as real functions. Inserting (37) into (36), we obtain two equations

when we consider the problem of factor ordering, usually in cosmological problems, as we indicated in the beginning of this section, equation (34), must be included as linear term of

*<sup>∂</sup><sup>q</sup>* , where Q is a real parameter that measures the ambiguity in this factor ordering. So,

<sup>2</sup>∇<sup>R</sup> · ∇<sup>S</sup> <sup>+</sup> <sup>R</sup><sup>S</sup> <sup>+</sup> <sup>R</sup>*∂*<sup>S</sup>

We assume that the wave function *ψ* is a solution of equation (36), and thus this equation is equally satisfied. Considering the Hamiton-Jacobi analysis, we can identify the equation (40) as the most important equation of this treatment, because with this equation we can derive the time dependence, and thus, it serves as the evolutionary equation in this formalism. Following the Hamilton-Jacobi procedure, the *Πq* momenta is related to the superpotential

dt <sup>=</sup> <sup>g</sup>*µν <sup>δ</sup>*H(S)

which defines the trajectory q*<sup>µ</sup>* in terms of the phase of the wave function S. We substitute

*<sup>µ</sup>* = *dq<sup>µ</sup>*

*δ <sup>∂</sup>*<sup>S</sup> *∂*q*<sup>ν</sup>*

<sup>2</sup> + V 

*<sup>∂</sup>*q*<sup>µ</sup>* , which are related with the classical momenta (8) written in the

*dt* and ¯*<sup>h</sup>* <sup>=</sup> 1),

<sup>g</sup>*µν*q˙ *<sup>µ</sup>*q˙ *<sup>ν</sup>* + <sup>V</sup>

 1 ¯h<sup>2</sup> (∇S)

i ¯h <sup>S</sup>(q*<sup>µ</sup>*)

*ψ* = R(q*µ*) e

<sup>2</sup> + V  *ψ* = 0, (36)

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227

Quintom Potential from Quantum Anisotropic Cosmological Models

, (37)

= 0, (40)

*<sup>∂</sup>*<sup>q</sup> <sup>=</sup> 0, (41)

, (42)

. (43)

= R − R [H(S)] = 0, (38)

2∇R · ∇S + RS = 0, (39)

where *<sup>Λ</sup>eff* = *<sup>Λ</sup>* + *<sup>V</sup>*0/2. For particular values in the *<sup>γ</sup>* parameter, the equation (30) has a solution. This formalism was studied by one of the author and collaborators, in the FRW and Bianchi type Class A cosmological models, [36–38].

## **4. Quantum approach**

On the Wheeler-DeWitt (WDW) equation there are a lot of papers dealing with different problems, for example in [39], they asked the question of what a typical wave function for the universe is. In Ref. [40] there appears an excellent summary of a paper on quantum cosmology where the problem of how the universe emerged from big bang singularity can no longer be neglected in the GUT epoch. On the other hand, the best candidates for quantum solutions become those that have a damping behavior with respect to the scale factor, in the sense that we obtain a good classical solution using the WKB approximation in any scenario in the evolution of our universe [41, 42]. Our goal in this paper deals with the problem to build the appropriate scalar potential in the inflationary scenario.

The Wheeler-DeWitt equation for this model is achieved by replacing *Π*q*<sup>µ</sup>* = −i*∂*q*<sup>µ</sup>* in (8). The factor e−3*<sup>Ω</sup>* may be a factor ordered with *<sup>Π</sup>*<sup>ˆ</sup> *<sup>Ω</sup>* in many ways. Hartle and Hawking [41] have suggested what might be called a semi-general factor ordering which in this case would order e−<sup>3</sup>*ΩΠ*<sup>ˆ</sup> <sup>2</sup> *<sup>Ω</sup>* as

$$-\mathbf{e}^{-(3-Q)\Omega}\,\partial\_{\Omega}\mathbf{e}^{-Q\Omega}\,\partial\_{\Omega} = -\mathbf{e}^{-3\Omega}\,\partial\_{\Omega}^2 + \mathbf{Q}\,\mathbf{e}^{-3\Omega}\partial\_{\Omega\prime} \tag{34}$$

where Q is any real constant that measure the ambiguity in the factor ordering in the variable *Ω*. In the following we will assume this factor ordering for the Wheeler-DeWitt equation, which becomes

$$
\Box \Psi + \mathcal{Q} \frac{\partial \Psi}{\partial \Omega} + \mathrm{e}^{\mathrm{f}\Omega} \mathcal{U}(\Omega, \mathcal{J}\_{\tau}, \mathcal{q}, \mathcal{\zeta}\_{\nu} \lambda\_{\mathrm{eff}}) \Psi = 0,\tag{35}
$$

where = − *<sup>∂</sup>*<sup>2</sup> *∂Ω*<sup>2</sup> <sup>+</sup> *<sup>∂</sup>*<sup>2</sup> *∂ς*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *∂ϕ*<sup>2</sup> <sup>+</sup> *<sup>∂</sup>*<sup>2</sup> *∂β*<sup>2</sup> − + *∂*<sup>2</sup> *∂β*<sup>2</sup> + is the d'Alambertian in the coordinates *q<sup>µ</sup>* = (*Ω*, *ς*, *β*±, *ϕ*). In the following, we introduce the main idea of the Bohm formalism, and why we choose the phase in the wave function to be real and not imaginary.

#### **4.1. Mathematical structure in the Bohm formalism**

In this section we will explain how the quantum potential approach or as is also known, the Bohm formalism [34], works in the context of quantum cosmology. For the cases that will be object of our investigation in the sections to come, it is sufficient to consider the simplest model, for which the whole quantum dynamics resides in the single equation,

$$\mathcal{H}\psi = \left(g^{\mu\nu}\nabla\_{\mu}\nabla\_{\nu} - V(q^{\mu})\right)\psi = 0,\tag{36}$$

where the metric may be q*<sup>µ</sup>* dependent. The *ψ* is called the wave function of the universe, and we consider that *ψ* has the following traditional decomposition

$$
\psi = \mathbf{R}(\mathbf{q}^{\mu}) \,\mathrm{e}^{\not p \mathbf{S}(\mathbf{q}^{\mu})},\tag{37}
$$

with *R* and *S* as real functions. Inserting (37) into (36), we obtain two equations corresponding to the real and imaginary parts, respectively, which are

$$
\Box \mathbf{R} - \mathbf{R} \left[ \frac{1}{\hbar^2} \left( \nabla \mathbf{S} \right)^2 + \mathbf{V} \right] = \Box \mathbf{R} - \mathbf{R} \left[ \mathbf{H}(\mathbf{S}) \right] = \mathbf{0},\tag{38}
$$

$$
\mathbf{2\nabla R} \cdot \nabla \mathbf{S} + \mathbf{R} \square \mathbf{S} = \mathbf{0},\tag{39}
$$

when we consider the problem of factor ordering, usually in cosmological problems, as we indicated in the beginning of this section, equation (34), must be included as linear term of *Q∂ψ <sup>∂</sup><sup>q</sup>* , where Q is a real parameter that measures the ambiguity in this factor ordering. So, the equations (38,39) are written as

$$
\Box \mathbf{R} + \mathbf{Q} \frac{\partial \mathbf{R}}{\partial \mathbf{q}} - \mathbf{R} \left[ \frac{1}{\hbar^2} \left( \nabla \mathbf{S} \right)^2 + \mathbf{V} \right] = \mathbf{0},\tag{40}
$$

$$2\nabla \mathbf{R} \cdot \nabla \mathbf{S} + \mathbf{R} \square \mathbf{S} + \mathbf{R} \frac{\partial \mathbf{S}}{\partial \mathbf{q}} = \mathbf{0},\tag{41}$$

where *q* is a single field coordinate.

8 Open Questions in Cosmology

**4. Quantum approach**

order e−<sup>3</sup>*ΩΠ*<sup>ˆ</sup> <sup>2</sup>

which becomes

where = − *<sup>∂</sup>*<sup>2</sup>

*∂Ω*<sup>2</sup> <sup>+</sup> *<sup>∂</sup>*<sup>2</sup>

*<sup>Ω</sup>* as

*h*(*Ω*) =

Bianchi type Class A cosmological models, [36–38].

8*πGM<sup>γ</sup>*

build the appropriate scalar potential in the inflationary scenario.

*<sup>Ψ</sup>* <sup>+</sup> <sup>Q</sup> *∂Ψ*

**4.1. Mathematical structure in the Bohm formalism**

*∂ϕ*<sup>2</sup> <sup>+</sup> *<sup>∂</sup>*<sup>2</sup> *∂β*<sup>2</sup> − + *∂*<sup>2</sup> *∂β*<sup>2</sup> +

we choose the phase in the wave function to be real and not imaginary.

model, for which the whole quantum dynamics resides in the single equation,

*∂ς*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup>

<sup>3</sup> *<sup>e</sup>*−3(1+*γ*)*<sup>Ω</sup>* <sup>+</sup>

1 − 9 1 + *β*<sup>2</sup> 2 *β*2

where *<sup>Λ</sup>eff* = *<sup>Λ</sup>* + *<sup>V</sup>*0/2. For particular values in the *<sup>γ</sup>* parameter, the equation (30) has a solution. This formalism was studied by one of the author and collaborators, in the FRW and

On the Wheeler-DeWitt (WDW) equation there are a lot of papers dealing with different problems, for example in [39], they asked the question of what a typical wave function for the universe is. In Ref. [40] there appears an excellent summary of a paper on quantum cosmology where the problem of how the universe emerged from big bang singularity can no longer be neglected in the GUT epoch. On the other hand, the best candidates for quantum solutions become those that have a damping behavior with respect to the scale factor, in the sense that we obtain a good classical solution using the WKB approximation in any scenario in the evolution of our universe [41, 42]. Our goal in this paper deals with the problem to

The Wheeler-DeWitt equation for this model is achieved by replacing *Π*q*<sup>µ</sup>* = −i*∂*q*<sup>µ</sup>* in (8). The factor e−3*<sup>Ω</sup>* may be a factor ordered with *<sup>Π</sup>*<sup>ˆ</sup> *<sup>Ω</sup>* in many ways. Hartle and Hawking [41] have suggested what might be called a semi-general factor ordering which in this case would

where Q is any real constant that measure the ambiguity in the factor ordering in the variable *Ω*. In the following we will assume this factor ordering for the Wheeler-DeWitt equation,

(*Ω*, *ς*, *β*±, *ϕ*). In the following, we introduce the main idea of the Bohm formalism, and why

In this section we will explain how the quantum potential approach or as is also known, the Bohm formalism [34], works in the context of quantum cosmology. For the cases that will be object of our investigation in the sections to come, it is sufficient to consider the simplest

<sup>−</sup> <sup>e</sup>−(3−Q)*<sup>Ω</sup> ∂Ω*e−<sup>Q</sup>*Ω∂Ω* <sup>=</sup> <sup>−</sup>e−3*<sup>Ω</sup> <sup>∂</sup>*<sup>2</sup>

−*ϕ*<sup>2</sup> <sup>0</sup> <sup>+</sup> *<sup>ς</sup>*<sup>2</sup> 0  *<sup>e</sup>*−6*∆Ω* + *<sup>Λ</sup>eff*

3

1*e*−6*∆Ω* , (33)

*<sup>Ω</sup>* <sup>+</sup> Q e−<sup>3</sup>*Ω∂Ω*, (34)

*∂Ω* <sup>+</sup> <sup>e</sup>6*Ω*U(*Ω*, *<sup>β</sup>*±, *<sup>ϕ</sup>*, *<sup>ς</sup>*, *<sup>λ</sup>*eff)*<sup>Ψ</sup>* <sup>=</sup> 0, (35)

is the d'Alambertian in the coordinates *q<sup>µ</sup>* =

We assume that the wave function *ψ* is a solution of equation (36), and thus this equation is equally satisfied. Considering the Hamiton-Jacobi analysis, we can identify the equation (40) as the most important equation of this treatment, because with this equation we can derive the time dependence, and thus, it serves as the evolutionary equation in this formalism. Following the Hamilton-Jacobi procedure, the *Πq* momenta is related to the superpotential function S, as *Π*q*<sup>µ</sup>* = *<sup>∂</sup>*<sup>S</sup> *<sup>∂</sup>*q*<sup>µ</sup>* , which are related with the classical momenta (8) written in the previous section, thus,

$$\frac{\mathbf{dq}^{\mu}}{\mathbf{dt}} = \mathbf{g}^{\mu\nu} \frac{\delta \mathbf{H}(\mathbf{S})}{\delta \frac{\partial \mathbf{S}}{\partial \mathbf{q}^{\nu}}} \, \tag{42}$$

which defines the trajectory q*<sup>µ</sup>* in terms of the phase of the wave function S. We substitute this equation into (40), and we find (using *q*˙ *<sup>µ</sup>* = *dq<sup>µ</sup> dt* and ¯*<sup>h</sup>* <sup>=</sup> 1),

$$\left[\Box \mathbf{R} + \mathbf{Q} \frac{\partial \mathbf{R}}{\partial \mathbf{q}}\right] = \mathbf{R} \left[\mathbf{g}\_{\mu \nu} \dot{\mathbf{q}}^{\mu} \dot{\mathbf{q}}^{\nu} + \mathbf{V}\right].\tag{43}$$

Therefore we see that the quantum evolution differs from the classical one only by the presence of the quantum potential term

$$\left[\Box\mathcal{R} + Q\frac{\partial\mathcal{R}}{\partial q}\right]$$

on the left-hand side of the equation of motion. Since we assume that the wave function is known, the quantum potential term is also known.

In the next subsection we will choose the *<sup>ψ</sup>* <sup>=</sup> *We*−*<sup>S</sup>* ansatz for the wave function, it was first remarked by Kodama [43, 44] that the solutions to the Wheeler-DeWitt (WDW) equation in the formulation of Arnowitt-Deser and Misner (ADM) and the Ashtekar formulation (in the connection representation) are related by *<sup>ψ</sup>ADM* <sup>=</sup> *<sup>ψ</sup>Ae*±*iΦ<sup>A</sup>* , where *<sup>Ψ</sup><sup>A</sup>* is the homogeneous specialization for the generating functional of the canonical transformation between ADM variables to Ashtekar's, [45]. This function was calculated explicitly for the diagonal Bianchi type IX model by Kodama, who also found *<sup>Ψ</sup><sup>A</sup>* = *const* as a solution, and *<sup>Ψ</sup><sup>A</sup>* is pure imaginary, for a certain factor ordering, one expects a solution of the form *<sup>ψ</sup>* <sup>=</sup> *We*±*Φ*, where W is a constant, and *<sup>Φ</sup>* = *<sup>i</sup>ΦA*. In fact this type of solution has been found for the diagonal Bianchi Class A cosmological models [46, 47], but W in some cases is a function, as we will see in our present study.

#### **4.2. Our treatment**

Using the ansatz

$$\Psi = \mathbf{e}^{\pm \mathbf{a}\_1 \theta\_+} \mathbf{e}^{\pm \mathbf{a}\_2 \theta\_-} \Xi(\Omega, \emptyset, \emptyset), \tag{44}$$

Eq (47) can be written as the following set of partial differential equations

*<sup>S</sup>* <sup>−</sup> *<sup>Q</sup> <sup>∂</sup><sup>S</sup>*

part, such as the equations presented in previous section (40, 41).

*∂Ω*

The first two equations correspond to the real part in a separated way, also, the first equation is called the Einstein-Hamilton-Jacobi equation (EHJ), and the third equation is the imaginary

Following the references [32, 33], first, we shall choose to solve Eqs. (48a) and (48c), whose solutions at the end will have to fulfill Eq. (48b), which will play the role of a constraint

where *∆ϕ* = *<sup>ϕ</sup>* − *<sup>ϕ</sup>*0, *∆ς* = *<sup>ς</sup>* − *<sup>ς</sup>*<sup>0</sup> with *<sup>ϕ</sup>*<sup>0</sup> and *<sup>ς</sup>*<sup>0</sup> as constant scalar fields, and bi as arbitrary

At this point we question ourselves how to solve this equation in relation to the constant c,

1. When we consider this equation as an expansion in powers of *eΩ*, then each term is null

b1gh +

these set of equations do not have solutions in closed form, because the first equation is not satisfied. So, it is necessary to take c=0, implying that the wave function in the

dh d*ς* 2

anisotropic coordinates have an oscilatory and hyperbolic behavior

b2 <sup>1</sup> <sup>−</sup> b2

b2 <sup>3</sup> <sup>h</sup> dg <sup>d</sup>*<sup>ϕ</sup>* <sup>−</sup> b3 3 g dh

<sup>3</sup> <sup>+</sup> b2

+ 9g2h2 − <sup>U</sup>(*ϕ*, *<sup>ς</sup>*, *<sup>λ</sup>*eff) = 0, (53)

dh d*ς* 2 + 9

*W* 

<sup>S</sup>(*Ω*, *<sup>ς</sup>*, *<sup>ϕ</sup>*) = e3*<sup>Ω</sup>*

constants. Then, Eq (48a) is transformed as

 dg d*ϕ*

 b1gh +

h2 dg d*ϕ*

2 <sup>−</sup> g2 *µ*2

> b2 <sup>3</sup> <sup>h</sup> dg <sup>d</sup>*<sup>ϕ</sup>* <sup>−</sup> b3 3 g dh d*ς* + c2 b2 <sup>1</sup> <sup>+</sup> b2

in a separated way, but maintaining that the constant c �= 0,

2 − g2

implying the behavior of the universe with the anisotropic parameter *β*±.

e6*<sup>Ω</sup>* h2 *µ*2

+ 6ce3*<sup>Ω</sup> µ*

equation.

Taking the ansatz

*<sup>W</sup>* <sup>−</sup> *<sup>Q</sup> <sup>∂</sup><sup>W</sup>*

(∇*S*)<sup>2</sup> − U = 0, (48a)

Quintom Potential from Quantum Anisotropic Cosmological Models

+ 2∇ *W* · ∇ *S* = 0, . (48c)

*<sup>µ</sup>* <sup>g</sup>(*ϕ*)h(*ς*) + <sup>c</sup> (b1*<sup>Ω</sup>* <sup>+</sup> b2*∆ϕ* <sup>+</sup> b3*∆ς*), (49)

= 0. (50)

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>=</sup> 0, (51)

<sup>d</sup>*<sup>ς</sup>* <sup>=</sup> 0, (52)

<sup>2</sup> <sup>−</sup> <sup>b</sup><sup>2</sup> <sup>3</sup> <sup>−</sup> <sup>1</sup> 

*<sup>µ</sup>*<sup>2</sup> g2h2 <sup>−</sup> <sup>U</sup>(*ϕ*, *<sup>ς</sup>*, *<sup>λ</sup>*eff)

*∂Ω* <sup>=</sup> <sup>0</sup> (48b)

http://dx.doi.org/10.5772/52054

229

the WDW equation is read as

$$
\left[\Box + \mathbf{Q}\frac{\partial}{\partial\Omega} + \mathbf{e}^{\mathbf{f}\circ\Omega}\mathbf{U}(\varphi\_{\prime}\boldsymbol{\varsigma}\_{\prime}\lambda\_{\mathrm{eff}}) + \mathbf{c}^{2}\right]\Sigma = \mathbf{0},\tag{45}
$$

where *c*<sup>2</sup> = *a*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> <sup>2</sup> and now is written in the reduced coordinates <sup>ℓ</sup>*<sup>µ</sup>* = (*Ω*, *<sup>ς</sup>*, *<sup>ϕ</sup>*)

We find that the WDW equation is solved when we choose an ansatz similar to the one employed in the Bohmian formalism of quantum mechanics [34], so we make the following Ansatz for the wave function

$$\Xi(\ell^{\mu}) = \mathbf{W}(\ell^{\mu}) \mathbf{e}^{-\mathbf{S}(\ell^{\mu})} \,\,\,\,\,\,\tag{46}$$

where *S*(ℓ*µ*) is known as the superpotential function, and W is the amplitude of probability that is employed in Bohmian formalism [34]. Then (45) transforms into

$$
\Box \mathcal{W} - \mathcal{W} \Box \mathcal{S} - 2 \nabla \mathcal{W} \cdot \nabla \mathcal{S} - Q \frac{\partial \mathcal{W}}{\partial \Omega} + Q \mathcal{W} \frac{\partial \mathcal{S}}{\partial \Omega} + \mathcal{W} \left[ \left( \nabla \mathcal{S} \right)^2 - \mathcal{U} \right] = 0,\tag{47}
$$

where now, = G*µν <sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>*ℓ*µ∂*ℓ*<sup>ν</sup>* , <sup>∇</sup> <sup>W</sup> · ∇ *<sup>Φ</sup>* <sup>=</sup> <sup>G</sup>*µν <sup>∂</sup>*<sup>W</sup> *<sup>∂</sup>*ℓ*<sup>µ</sup> ∂Φ <sup>∂</sup>*ℓ*<sup>ν</sup>* , (∇)<sup>2</sup> <sup>=</sup> <sup>G</sup>*µν <sup>∂</sup> <sup>∂</sup>*ℓ*<sup>µ</sup> <sup>∂</sup> <sup>∂</sup>*ℓ*<sup>ν</sup>* <sup>=</sup> <sup>−</sup>( *<sup>∂</sup> ∂ς* )<sup>2</sup> + ( *<sup>∂</sup> ∂Ω* )<sup>2</sup> <sup>+</sup> ( *∂ ∂ϕ* )2, with G*µν* <sup>=</sup> diag(−1, 1, 1), <sup>U</sup> <sup>=</sup> <sup>e</sup>6*Ω*U(*ς*, *<sup>ϕ</sup>*, *<sup>λ</sup>*eff) + <sup>c</sup><sup>2</sup> is the potential term of the cosmological model under consideration.

Eq (47) can be written as the following set of partial differential equations

$$(\nabla S)^2 - \mathcal{U} = 0,\tag{48a}$$

$$
\Box W - Q \frac{\partial W}{\partial \Omega} = 0 \tag{48b}
$$

$$W\left(\Box S - Q\frac{\partial S}{\partial \Omega}\right) + 2\nabla W \cdot \nabla S = 0\,,\tag{48c}$$

The first two equations correspond to the real part in a separated way, also, the first equation is called the Einstein-Hamilton-Jacobi equation (EHJ), and the third equation is the imaginary part, such as the equations presented in previous section (40, 41).

Following the references [32, 33], first, we shall choose to solve Eqs. (48a) and (48c), whose solutions at the end will have to fulfill Eq. (48b), which will play the role of a constraint equation.

Taking the ansatz

10 Open Questions in Cosmology

presence of the quantum potential term

we will see in our present study.

the WDW equation is read as

<sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup>

Ansatz for the wave function

where now, = G*µν <sup>∂</sup>*<sup>2</sup>

cosmological model under consideration.

( *∂*

**4.2. Our treatment**

Using the ansatz

where *c*<sup>2</sup> = *a*<sup>2</sup>

known, the quantum potential term is also known.

<sup>+</sup> <sup>Q</sup> *<sup>∂</sup>*

that is employed in Bohmian formalism [34]. Then (45) transforms into

*<sup>∂</sup>*ℓ*µ∂*ℓ*<sup>ν</sup>* , <sup>∇</sup> <sup>W</sup> · ∇ *<sup>Φ</sup>* <sup>=</sup> <sup>G</sup>*µν <sup>∂</sup>*<sup>W</sup>

*<sup>W</sup>* <sup>−</sup> *<sup>W</sup> <sup>S</sup>* <sup>−</sup> <sup>2</sup>∇*<sup>W</sup>* · ∇*<sup>S</sup>* <sup>−</sup> *<sup>Q</sup> <sup>∂</sup><sup>W</sup>*

Therefore we see that the quantum evolution differs from the classical one only by the

*<sup>R</sup>* <sup>+</sup> *<sup>Q</sup> <sup>∂</sup><sup>R</sup>*

on the left-hand side of the equation of motion. Since we assume that the wave function is

In the next subsection we will choose the *<sup>ψ</sup>* <sup>=</sup> *We*−*<sup>S</sup>* ansatz for the wave function, it was first remarked by Kodama [43, 44] that the solutions to the Wheeler-DeWitt (WDW) equation in the formulation of Arnowitt-Deser and Misner (ADM) and the Ashtekar formulation (in the connection representation) are related by *<sup>ψ</sup>ADM* <sup>=</sup> *<sup>ψ</sup>Ae*±*iΦ<sup>A</sup>* , where *<sup>Ψ</sup><sup>A</sup>* is the homogeneous specialization for the generating functional of the canonical transformation between ADM variables to Ashtekar's, [45]. This function was calculated explicitly for the diagonal Bianchi type IX model by Kodama, who also found *<sup>Ψ</sup><sup>A</sup>* = *const* as a solution, and *<sup>Ψ</sup><sup>A</sup>* is pure imaginary, for a certain factor ordering, one expects a solution of the form *<sup>ψ</sup>* <sup>=</sup> *We*±*Φ*, where W is a constant, and *<sup>Φ</sup>* = *<sup>i</sup>ΦA*. In fact this type of solution has been found for the diagonal Bianchi Class A cosmological models [46, 47], but W in some cases is a function, as

*∂Ω* <sup>+</sup> e6*<sup>Ω</sup>*U(*ϕ*, *<sup>ς</sup>*, *<sup>λ</sup>*eff) + <sup>c</sup><sup>2</sup>

We find that the WDW equation is solved when we choose an ansatz similar to the one employed in the Bohmian formalism of quantum mechanics [34], so we make the following

*<sup>Ξ</sup>*(ℓ*µ*) = <sup>W</sup>(ℓ*µ*)e−S(ℓ*µ*)

where *S*(ℓ*µ*) is known as the superpotential function, and W is the amplitude of probability

*∂Ω* <sup>+</sup> *QW <sup>∂</sup><sup>S</sup>*

*<sup>∂</sup>*ℓ*<sup>µ</sup> ∂Φ*

*∂ϕ* )2, with G*µν* <sup>=</sup> diag(−1, 1, 1), <sup>U</sup> <sup>=</sup> <sup>e</sup>6*Ω*U(*ς*, *<sup>ϕ</sup>*, *<sup>λ</sup>*eff) + <sup>c</sup><sup>2</sup> is the potential term of the

*∂Ω* <sup>+</sup> *<sup>W</sup>*

*<sup>∂</sup>*ℓ*<sup>ν</sup>* , (∇)<sup>2</sup> <sup>=</sup> <sup>G</sup>*µν <sup>∂</sup>*

 (∇*S*)

<sup>2</sup> and now is written in the reduced coordinates <sup>ℓ</sup>*<sup>µ</sup>* = (*Ω*, *<sup>ς</sup>*, *<sup>ϕ</sup>*)

*∂q* 

*<sup>Ψ</sup>* <sup>=</sup> <sup>e</sup>±a1*β*<sup>+</sup> <sup>e</sup>±a2*β*<sup>−</sup> *<sup>Ξ</sup>*(*Ω*, *<sup>ς</sup>*, *<sup>ϕ</sup>*), (44)

*Ξ* = 0, (45)

, (46)

= 0, (47)

*∂Ω* )<sup>2</sup> <sup>+</sup>

*∂ς* )<sup>2</sup> + ( *<sup>∂</sup>*

2 − U 

*<sup>∂</sup>*ℓ*<sup>ν</sup>* <sup>=</sup> <sup>−</sup>( *<sup>∂</sup>*

*<sup>∂</sup>*ℓ*<sup>µ</sup> <sup>∂</sup>*

$$\mathbf{S}(\Omega, \emptyset, \emptyset) = \frac{\mathbf{e}^{\mathfrak{J}\Omega}}{\mu} \mathbf{g}(\mathfrak{q}) \mathbf{h}(\mathfrak{q}) + \mathbf{c} \left( \mathbf{b}\_1 \Omega + \mathbf{b}\_2 \Delta \mathfrak{q} + \mathbf{b}\_3 \Delta \mathfrak{q} \right), \tag{49}$$

where *∆ϕ* = *<sup>ϕ</sup>* − *<sup>ϕ</sup>*0, *∆ς* = *<sup>ς</sup>* − *<sup>ς</sup>*<sup>0</sup> with *<sup>ϕ</sup>*<sup>0</sup> and *<sup>ς</sup>*<sup>0</sup> as constant scalar fields, and bi as arbitrary constants. Then, Eq (48a) is transformed as

$$\mathbf{e}^{6\Omega} \left[ \frac{\mathbf{h}^2}{\mu^2} \left( \frac{\mathbf{dg}}{\mathbf{d}\varphi} \right)^2 - \frac{\mathbf{g}^2}{\mu^2} \left( \frac{\mathbf{dh}}{\mathbf{d}\xi} \right)^2 + \frac{9}{\mu^2} \mathbf{g}^2 \mathbf{h}^2 - \mathbb{U}(\boldsymbol{\rho}, \boldsymbol{\xi}, \boldsymbol{\lambda}\_{\rm eff}) \right]$$

$$+ \frac{6\mathbf{ce}^{3\Omega}}{\mu} \left[ \mathbf{b}\_1 \mathbf{g} \mathbf{h} + \frac{\mathbf{b}\_2}{3} \mathbf{h} \frac{\mathbf{dg}}{\mathbf{d}\varphi} - \frac{\mathbf{b}\_3}{3} \mathbf{g} \frac{\mathbf{dh}}{\mathbf{d}\xi} \right] + \mathbf{c}^2 \left( \mathbf{b}\_1^2 + \mathbf{b}\_2^2 - \mathbf{b}\_3^2 - 1 \right) = 0. \tag{50}$$

At this point we question ourselves how to solve this equation in relation to the constant c, implying the behavior of the universe with the anisotropic parameter *β*±.

1. When we consider this equation as an expansion in powers of *eΩ*, then each term is null in a separated way, but maintaining that the constant c �= 0,

$$\mathbf{b}\_1^2 - \mathbf{b}\_3^2 + \mathbf{b}\_2^2 - 1 = 0,\tag{51}$$

$$\mathbf{b}\_1 \mathbf{g} \mathbf{h} + \frac{\mathbf{b}\_2}{3} \mathbf{h} \frac{\mathbf{dg}}{\mathbf{d}\boldsymbol{\varphi}} - \frac{\mathbf{b}\_3}{3} \mathbf{g} \frac{\mathbf{dh}}{\mathbf{d}\boldsymbol{\varsigma}} = \mathbf{0},\tag{52}$$

$$\mathrm{h}^{2}\left(\frac{\mathrm{dg}}{\mathrm{d}\varrho}\right)^{2} - \mathrm{g}^{2}\left(\frac{\mathrm{dh}}{\mathrm{d}\varrho}\right)^{2} + 9\mathrm{g}^{2}\mathrm{h}^{2} - \mathrm{U}(\varrho, \mathrm{g}, \lambda\_{\mathrm{eff}}) = 0,\tag{53}$$

these set of equations do not have solutions in closed form, because the first equation is not satisfied. So, it is necessary to take c=0, implying that the wave function in the anisotropic coordinates have an oscilatory and hyperbolic behavior

2. For the case c=0, we have the following.

The constants ai are related as a2 = ±ia1, hence the wave function corresponding to the anisotropic behavior becomes e±a1*β*+±ia1*β*<sup>−</sup> , i.e, one part goes as oscillatory in the anisotropic parameter.

#### **4.3. Mathematical structure of potential fields**

To solve the Hamilton-Jacobi equation (48a)

$$-\left(\frac{\partial \mathbf{S}}{\partial \boldsymbol{\zeta}}\right)^2 + \left(\frac{\partial \mathbf{S}}{\partial \boldsymbol{\Omega}}\right)^2 + \left(\frac{\partial \mathbf{S}}{\partial \boldsymbol{\varphi}}\right)^2 = \mathbf{e}^{\mathsf{f}\boldsymbol{\alpha}} \mathbf{U}(\boldsymbol{\varphi}, \boldsymbol{\zeta}, \boldsymbol{\lambda}\_{\mathrm{eff}})$$

we propose that the superpotential function has the form

$$\mathbf{S} = \frac{\mathbf{e}^{3\Omega}}{\mu} \mathbf{g}(\varphi) \mathbf{h}(\xi) \,\tag{54}$$

H(h) h(*ς*) G(g) g(*ϕ*) 0 h0e±ℓ*∆ς* 0 g0e±p*∆ϕ* H0 h0e±√ℓ2−c0 H0*∆ς* G0 g0e<sup>±</sup>

<sup>ℓ</sup> cosh [ℓ*∆ς*] G0g−<sup>2</sup> <sup>√</sup>c1 G0

G0g−<sup>n</sup> (n �<sup>=</sup> 2)

c0H0 <sup>v</sup>(*ϕ*) = <sup>−</sup>p2<sup>+</sup>

U(*ϕ*, *ς*) **Relation between all constants** 0 ℓ2(s − p<sup>2</sup> − 3k − 9)<sup>2</sup> − p2(s − ℓ2)<sup>2</sup>

b0H0r2 + a0G0*ω*2 quantum constraint is not satisfied

<sup>+</sup> <sup>6</sup> <sup>d</sup>*<sup>η</sup>*

<sup>d</sup>*<sup>Ω</sup>* <sup>−</sup> <sup>Q</sup> <sup>=</sup> k, (59)

<sup>d</sup>*<sup>ϕ</sup>* = [<sup>s</sup> <sup>−</sup> <sup>3</sup>(<sup>k</sup> <sup>+</sup> <sup>3</sup>)]g, (60)

<sup>d</sup>*<sup>ς</sup>* <sup>=</sup> sh, (61)

p2+c1G0*∆ϕ*] (ℓ<sup>2</sup> <sup>−</sup> c0H0)(<sup>s</sup> <sup>−</sup> <sup>p</sup><sup>2</sup> <sup>−</sup> 3k <sup>−</sup> <sup>9</sup> <sup>−</sup> c1G0)<sup>2</sup>

sin √c0H0 *∆ς <sup>ω</sup>*(*ϕ*) = <sup>p</sup>

H0h−<sup>2</sup> <sup>√</sup>c0H0

c0H0 <sup>ℓ</sup><sup>2</sup> cosh<sup>2</sup>

u(*ς*) =

ℓ c0H0

√

<sup>ℓ</sup><sup>2</sup> cosh2 <sup>n</sup>

<sup>ℓ</sup><sup>2</sup> cosh2 <sup>n</sup>

<sup>+</sup> <sup>9</sup> <sup>−</sup> <sup>1</sup> h d2h <sup>d</sup>*ς*<sup>2</sup> <sup>−</sup> <sup>2</sup> h d*λ* d*ς* dh d*ς*

<sup>2</sup> <sup>d</sup>*<sup>η</sup>*

dh d*ς* d*λ*

r(*ς*) = <sup>√</sup>

<sup>√</sup>ℓ2−c0H0*∆ς*<sup>+</sup>

 np*∆ϕ* 2 <sup>2</sup> <sup>n</sup> c0H0

 np*∆ϕ* 2

e2r(*ς*)+2*ω*(*ϕ*)

1 g d2g <sup>d</sup>*ϕ*<sup>2</sup> <sup>+</sup>

 <sup>2</sup>−<sup>n</sup> <sup>n</sup> c0H0

Also we present the relation between all constant that satisfy the eqn. (48b).

because it depend as we put the constants in the equations).

d2g <sup>d</sup>*ϕ*<sup>2</sup> <sup>+</sup> <sup>2</sup> dg d*ϕ* d*ξ*

d2h <sup>d</sup>*ς*<sup>2</sup> <sup>+</sup> <sup>2</sup>

2 g dg d*ϕ* d*ξ* d*ϕ*

<sup>n</sup>ℓ*∆ς* 2

ℓ2− c0H0 <sup>2</sup> *∆ς*<sup>2</sup>

1/n

H0 ln h <sup>e</sup>u(*ς*), G0 ln g ev(*ϕ*)

H0(ln h)<sup>2</sup> <sup>e</sup>r(*ς*) G0(ln g)<sup>2</sup> <sup>e</sup>*ω*(*ϕ*)

**Table 2.** Some exact solutions to eqs. (56a,56b), where n is any real number, G0 and H0 are arbitrary constants.

U0sinh<sup>2</sup>(p*∆ϕ*) + U1 cosh2(ℓ*∆ς*) <sup>k</sup>(<sup>k</sup> − <sup>6</sup>) = Q2, s = ℓ2,

<sup>2</sup> <sup>ℓ</sup>*∆ς* <sup>2</sup>−<sup>n</sup>

<sup>2</sup> <sup>ℓ</sup>*∆ς* <sup>2</sup> n e2u(*ς*)+2v(*ϕ*) [b0H0u(*ς*) + a0G0 <sup>v</sup>(*ϕ*)] quantum constraint is not satisfied

**Table 3.** The corresponding quintom potentials that emerge from quantum cosmology in direct relation with the table (2).

and using the method of separation of variables, we arrive to a set of ordinary differential equations for the functions *η*(*Ω*), *ξ*(*ϕ*) and *λ*(*ς*) (however, this decomposition is not unique,

H0h−<sup>n</sup> (n �<sup>=</sup> 2)

U0e±2[

b0H0

a0G0

 c1G0 p2 sinh<sup>2</sup>

 c1G0 <sup>p</sup><sup>2</sup> sinh<sup>2</sup> √

c1G0 p2 sinh<sup>2</sup>

Quintom Potential from Quantum Anisotropic Cosmological Models

√c1G0

+ℓ2p2(k2 − Q2) = 0

−(p<sup>2</sup> + c1G0)(<sup>s</sup> − ℓ<sup>2</sup> + c0H0)<sup>2</sup> +(ℓ<sup>2</sup> − c0H0)(p<sup>2</sup> + c1G0)(k2 − Q2) = <sup>0</sup>

6k(9 + p2) + 9Q<sup>2</sup> − p<sup>4</sup> + (ℓ<sup>2</sup> − 9)<sup>2</sup> = 0

<sup>d</sup>*<sup>Ω</sup>* <sup>−</sup> 3Q <sup>=</sup> 0, (58)

<sup>n</sup> + quantum constraint is not satisfied

p2+c1 G0*∆ϕ*

http://dx.doi.org/10.5772/52054

<sup>p</sup> sinh [p*∆ϕ*]

np*∆ϕ* 2

> c1G0 <sup>2</sup> *∆ϕ*<sup>2</sup> c1G0

sinh √c1G0 *∆ϕ*

1/n

231

and the potential

$$\mathbf{U} = \mathbf{g}^2 \mathbf{h}^2 \left[ \mathbf{a}\_0 \mathbf{G}(\mathbf{g}) + \mathbf{b}\_0 \mathbf{H}(\mathbf{h}) \right],\tag{55}$$

where g(*ϕ*), h(*ς*), G(g) and H(h) are generic functions of the arguments, which will be determined under this process. When we introduce the ansatz in (48a) we find the following master equations for the fields (*ϕ*, *<sup>ς</sup>*), (here c1 = *<sup>µ</sup>*a0 and c0 = *<sup>µ</sup>*0b0)

$$\mathbf{d}\boldsymbol{\varrho} = \pm \frac{\mathbf{d}\mathbf{g}}{\mathbf{g}\sqrt{\mathbf{p}^2 + \mathbf{c}\_1 \mathbf{G}}}, \qquad \text{with} \quad \mathbf{p}^2 = \boldsymbol{\nu}^2 - \frac{\mathbf{g}}{\mathbf{\tilde{z}}},\tag{56a}$$

$$\mathbf{d}\boldsymbol{\xi} = \pm \frac{\mathbf{d}\mathbf{h}}{\mathbf{h}\sqrt{\ell^2 - \mathbf{c}\_0 \mathbf{H}}}, \qquad \text{with} \quad \ell^2 = \nu^2 + \frac{9}{2}\text{.}\tag{56b}$$

where *ν* is the separation constant.

For a particular structure of functions *G* and *H* we can solve the g(*ϕ*) and h(*ς*) functions, and then use the expression for the potential term (55) over again to find the corresponding scalar potential that leads to an exact solution to the Hamilton-Jacobi equation (48a). Some examples are shown in Tables 2 and 3.

Thereby, the superpotential S(*Ω*, *ϕ*) is known, and the possible quintom potentials are shown in table 3

To solve (48c) we assume that

$$\mathcal{W} = e^{\left[\eta(\mathcal{Q}) + \xi(\varphi) + \lambda(\varrho)\right]} \,, \tag{57}$$

and introducing the corresponding superpotential function S (54) into the equation (48c), it follow the equation


**Table 2.** Some exact solutions to eqs. (56a,56b), where n is any real number, G0 and H0 are arbitrary constants.

12 Open Questions in Cosmology

and the potential

in table 3

anisotropic parameter.

2. For the case c=0, we have the following.

To solve the Hamilton-Jacobi equation (48a)

− *∂*S *∂ς* 2 + *∂*S *∂Ω*

**4.3. Mathematical structure of potential fields**

we propose that the superpotential function has the form

master equations for the fields (*ϕ*, *<sup>ς</sup>*), (here c1 = *<sup>µ</sup>*a0 and c0 = *<sup>µ</sup>*0b0)

<sup>d</sup>*<sup>ϕ</sup>* <sup>=</sup> <sup>±</sup> dg g

<sup>d</sup>*<sup>ς</sup>* <sup>=</sup> <sup>±</sup> dh h

where *ν* is the separation constant.

examples are shown in Tables 2 and 3.

To solve (48c) we assume that

follow the equation

ℓ<sup>2</sup> − c0H

The constants ai are related as a2 = ±ia1, hence the wave function corresponding to the anisotropic behavior becomes e±a1*β*+±ia1*β*<sup>−</sup> , i.e, one part goes as oscillatory in the

2

= e6*<sup>α</sup>*U(*ϕ*, *<sup>ς</sup>*, *<sup>λ</sup>*eff)

<sup>U</sup> = g2h2 [a0G(g) + b0H(h)] , (55)

*<sup>µ</sup>* <sup>g</sup>(*ϕ*)h(*ς*), (54)

2

9 2 , (56a)

, (56b)

, (57)

2 + *∂*S *∂ϕ*

<sup>S</sup> <sup>=</sup> <sup>e</sup>3*<sup>Ω</sup>*

where g(*ϕ*), h(*ς*), G(g) and H(h) are generic functions of the arguments, which will be determined under this process. When we introduce the ansatz in (48a) we find the following

For a particular structure of functions *G* and *H* we can solve the g(*ϕ*) and h(*ς*) functions, and then use the expression for the potential term (55) over again to find the corresponding scalar potential that leads to an exact solution to the Hamilton-Jacobi equation (48a). Some

Thereby, the superpotential S(*Ω*, *ϕ*) is known, and the possible quintom potentials are shown

*<sup>W</sup>* <sup>=</sup> *<sup>e</sup>*[*η*(*Ω*)+*ξ*(*ϕ*)+*λ*(*ς*)]

and introducing the corresponding superpotential function S (54) into the equation (48c), it

<sup>p</sup><sup>2</sup> <sup>+</sup> c1G, with p<sup>2</sup> <sup>=</sup> *<sup>ν</sup>*<sup>2</sup> <sup>−</sup> <sup>9</sup>

, with ℓ<sup>2</sup> = *ν*<sup>2</sup> +


**Table 3.** The corresponding quintom potentials that emerge from quantum cosmology in direct relation with the table (2). Also we present the relation between all constant that satisfy the eqn. (48b).

$$\frac{1}{\mathbf{g}} \frac{\mathbf{d}^2 \mathbf{g}}{\mathbf{d}\varphi^2} + \frac{2}{\mathbf{g}} \frac{\mathbf{d}\mathbf{g}}{\mathbf{d}\varphi} \frac{\mathbf{d}\tilde{\mathbf{g}}}{\mathbf{d}\varphi} + \mathbf{9} - \frac{1}{\mathbf{h}} \frac{\mathbf{d}^2 \mathbf{h}}{\mathbf{d}\xi^2} - \frac{2}{\mathbf{h}} \frac{\mathbf{d}\lambda}{\mathbf{d}\xi} \frac{\mathbf{d}\mathbf{h}}{\mathbf{d}\xi} + 6 \frac{\mathbf{d}\eta}{\mathbf{d}\Omega} - 3\mathbf{Q} = \mathbf{0},\tag{58}$$

and using the method of separation of variables, we arrive to a set of ordinary differential equations for the functions *η*(*Ω*), *ξ*(*ϕ*) and *λ*(*ς*) (however, this decomposition is not unique, because it depend as we put the constants in the equations).

$$2\frac{\mathbf{d}\eta}{\mathbf{d}\Omega} - \mathbf{Q} = \mathbf{k}\_{\prime} \tag{59}$$

$$2\frac{\mathbf{d}^2 \mathbf{g}}{\mathbf{d}\rho^2} + 2\frac{\mathbf{dg}}{\mathbf{d}\rho}\frac{\mathbf{d}\tilde{\mathbf{g}}}{\mathbf{d}\rho} = [\mathbf{s} - 3(\mathbf{k} + 3)]\mathbf{g}.\tag{60}$$

$$2\frac{\text{d}^2\text{h}}{\text{d}\text{g}^2} + 2\frac{\text{dh}}{\text{d}\text{g}}\frac{\text{d}\lambda}{\text{d}\text{g}} = \text{sh}\,,\tag{61}$$

whose solutions in the generic fields g and h are

$$\begin{aligned} \eta(\Omega) &= \frac{\mathbb{Q} + \mathbf{k}}{2} \Omega, \\ \lambda(\boldsymbol{\xi}) &= \frac{\mathbf{s}}{2} \int \frac{\mathbf{d}\boldsymbol{\xi}}{\partial \boldsymbol{\xi} (\text{ln} \mathbf{h})} - \frac{1}{2} \int \frac{\frac{\mathbf{d}^{2} \mathbf{h}}{\mathbf{d} \boldsymbol{\xi}^{2}}}{\partial \boldsymbol{\xi} \mathbf{h}} \mathbf{d} \boldsymbol{\xi}. \end{aligned}$$
 
$$\begin{aligned} \boldsymbol{\xi}(\boldsymbol{\varrho}) &= \left(\frac{\mathbf{s}}{2} - \frac{3 \mathbf{k}}{2} - \frac{9}{2}\right) \int \frac{\mathbf{d} \boldsymbol{\varrho}}{\partial \boldsymbol{\varrho} (\text{ln} \mathbf{g})} - \frac{1}{2} \int \frac{\frac{\mathbf{d}^{2} \mathbf{g}}{\mathbf{d} \boldsymbol{\varrho}^{2}}}{\partial \boldsymbol{\varrho} \mathbf{g}} \mathbf{d} \boldsymbol{\varrho}. \end{aligned}$$

then

$$\mathbf{W} = \mathbf{e}^{\frac{\mathbf{s}}{2}} \int \begin{pmatrix} \frac{d\zeta}{\delta\zeta(\ln b)} + \frac{d\varrho}{\delta\varrho(\ln b)} \end{pmatrix} \mathbf{e}^{-\frac{1}{2} \int \left(\frac{d\zeta^2}{d\zeta^2 b} \mathrm{d}\zeta + \frac{d\varrho^2}{\delta\varrho g} \mathrm{d}\varrho\right)} \mathbf{e}^{\frac{\mathbf{k}}{2} \left(\Omega - 3 \int \frac{d\varrho}{\delta\varrho(\ln b)}\right)} \mathbf{e}^{\frac{1}{2} \left(\Omega \Omega - 9 \int \frac{d\varrho}{\delta\varrho(\ln b)}\right)} . \tag{62}$$

In a similar way, the constraint (48b) can be written as

$$
\partial^2\_{\varphi} \mathfrak{F} + \left(\partial\_{\varphi} \mathfrak{f}\right)^2 - \partial^2\_{\zeta} \lambda - \left(\partial\_{\zeta} \lambda\right)^2 + \frac{\mathbf{k}^2 - \mathbf{Q}^2}{4} = \mathbf{0},\tag{63}
$$

**5. Classical solutions a la WKB**

For our study, we shall make use of a semi-classical approximation to extract the dynamics of the WDW equation. The semi-classical limit of the WDW equation is achieved by taking *<sup>Ψ</sup>* <sup>=</sup> <sup>e</sup><sup>−</sup>S, and imposing the usual WKB conditions on the superpotential function S, namely

Hence, the WDW equation, under the particular factor ordering Q = 0, becomes exactly the afore-mentioned EHJ equation (48a) (this approximation is equivalent to a zero quantum potential in the Bohmian interpretation of quantum cosmology [53]). The EHJ equation is also obtained if we introduce the following transformation on the canonical momenta *Π*<sup>q</sup> → *∂*qS in Eq. (8) and then Eq. (8) provides the classical solutions of the Einstein.Klein.Gordon (EKG) equations. Moreover, for the particular cases shown in Table 1, the classical solutions

 *∂*S *∂*q 2 >> *∂*2S *∂*q2

of the EKG, in terms of q(*τ*), arising from Eqs. (8) and (50) are given by

d*Ω*

1 h d*ϕ ∂ϕ*g

*∂*S *∂β*±

(3)). The particular exact solution for the wave function *Ξ* becomes

function, thence the classical solution will be complete.

<sup>d</sup>*<sup>τ</sup>* , <sup>d</sup>*<sup>ϕ</sup>*

*∂ϕ*Lng <sup>+</sup>

, d*τ* = −12*µ*

the second equation appears in the W function (57), then the W is simplified by, we also have

In the following subsection, we will give details about the solutions corresponding to some

To recover the solutions for the anisotropic function *β*±, (28,29) we need to extend the superpotential function S = *<sup>S</sup>* − *<sup>a</sup>*1*β*<sup>+</sup> − *ia*2*β*−, remember that these functions were used in the ansatz for the wave function (44) in order to simplify the WDW equation (45). With

= −bi = constants

and using the corresponding momenta (8), we obtain the corresponding solutions written in quadrature form in equations (28,29). In this subsection we calculate the solution for the *Ω*

This particular case corresponds to an null potential function *U*(*ϕ*, *ς*), (see first line in Table

d*ς*

1 g d*ς*

*∂ς*Lnh <sup>=</sup> 0, (64)

Quintom Potential from Quantum Anisotropic Cosmological Models

http://dx.doi.org/10.5772/52054

233

*∂ς*h. (65)

gh = 4*µ*

d*τ* = 12*µ*

the corresponding relation with the time *τ*

of the scalar potential shown in Table (3).

this extension, we has

*5.0.1. Free wave function*

or in other words (here *<sup>µ</sup>*<sup>0</sup> = <sup>s</sup> − <sup>3</sup>(<sup>3</sup> + *<sup>κ</sup>*))

$$\begin{split} &2\frac{\partial\_{\boldsymbol{\xi}}^{3}\mathbf{h}}{\partial\_{\boldsymbol{\xi}}\mathbf{h}} - 2\frac{\partial\_{\boldsymbol{\varphi}}^{3}\mathbf{g}}{\partial\_{\boldsymbol{\varphi}}\mathbf{g}} + 4\text{sh}\frac{\partial\_{\boldsymbol{\xi}}^{2}\mathbf{h}}{(\partial\_{\boldsymbol{\xi}}\mathbf{h})^{2}} - 4\mu\_{0}\mathbf{g}\frac{\partial\_{\boldsymbol{\varphi}}^{2}\mathbf{g}}{(\partial\_{\boldsymbol{\varphi}}\mathbf{g})^{2}} - 3\frac{(\partial\_{\boldsymbol{\xi}}^{2}\mathbf{h})^{2}}{(\partial\_{\boldsymbol{\xi}}\mathbf{h})^{2}} + 3\frac{(\partial\_{\boldsymbol{\varphi}}^{2}\mathbf{g})^{2}}{(\partial\_{\boldsymbol{\varphi}}\mathbf{g})^{2}} - \frac{\mathbf{s}^{2}\mathbf{h}^{2}}{(\partial\_{\boldsymbol{\xi}}\mathbf{h})^{2}} \\ &+ \frac{\mu\_{0}^{2}\mathbf{g}^{2}}{(\partial\_{\boldsymbol{\varphi}}\mathbf{g})^{2}} - 2\mathbf{s} + 2\mu\_{0} + \mathbf{k}^{2} - \mathbf{Q}^{2} = 0. \end{split}$$

when we use the different cases presented in table (2), the following relations between all constants were found, which we present in the same table II with the quintom potentials. So, the quantum solutions for each potential scalar fields are presented in quadrature form, using the equations (46, 54) and (57).

Thereby, under canonical quantization we were able to determine a family of potentials that are the most probable to characterize the inflation phenomenon in the evolution of our universe. The exact quantum solutions to the Wheeler-DeWitt equation were found using the Bohmian scheme [34] of quantum mechanics where the ansatz to the wave function *Ψ*(ℓ*µ*) = <sup>e</sup>a1*β*++iai*β*<sup>−</sup>W(ℓ*µ*)e−S(ℓ*µ*) includes the superpotential function which plays an important role in solving the Hamilton-Jacobi equation. It was necessary to study the classical behavior in order to know when the Universe evolves from a quintessence dominated phase to a phantom dominated phase crossing the *weff* = −1 dividing line, as a transient stage. Also, this family of potentials can be studied within the dynamical systems framework to obtain useful information about the asymptotic properties of the model and give a classification of which ones are in agreement with the observational data [48]

#### **5. Classical solutions a la WKB**

14 Open Questions in Cosmology

W = e s 2 � � <sup>d</sup>*<sup>ς</sup> ∂ς* (lnh) <sup>+</sup> <sup>d</sup>*<sup>ϕ</sup> ∂ϕ*(lng) � e − <sup>1</sup> 2 � d2h d*ς*2 *∂ς*<sup>h</sup> <sup>d</sup>*ς*<sup>+</sup>

2 *∂*3 *ς*h *∂ς*<sup>h</sup> <sup>−</sup> <sup>2</sup>

<sup>+</sup> *<sup>µ</sup>*<sup>2</sup> 0*g*2

then

whose solutions in the generic fields g and h are

*<sup>η</sup>*(*Ω*) = <sup>Q</sup> <sup>+</sup> <sup>k</sup>

� s <sup>2</sup> <sup>−</sup> 3k

*<sup>λ</sup>*(*ς*) = <sup>s</sup> 2

*ξ*(*ϕ*) =

In a similar way, the constraint (48b) can be written as

+ 4sh

(*∂ϕg*)<sup>2</sup> <sup>−</sup> <sup>2</sup>*<sup>s</sup>* <sup>+</sup> <sup>2</sup>*µ*<sup>0</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup> <sup>−</sup> *<sup>Q</sup>*<sup>2</sup> <sup>=</sup> 0.

which ones are in agreement with the observational data [48]

*∂ϕξ*

*∂*2 *ς*h (*∂ς*h)<sup>2</sup> <sup>−</sup> <sup>4</sup>*µ*0g

�<sup>2</sup> − *<sup>∂</sup>*<sup>2</sup>

*∂*2 *ϕξ* <sup>+</sup> �

or in other words (here *<sup>µ</sup>*<sup>0</sup> = <sup>s</sup> − <sup>3</sup>(<sup>3</sup> + *<sup>κ</sup>*))

*∂*3 *ϕ*g *∂ϕ*g

using the equations (46, 54) and (57).

<sup>2</sup> *<sup>Ω</sup>*,

� d*ς*

*∂ς*(lnh) <sup>−</sup> <sup>1</sup>

<sup>2</sup> <sup>−</sup> <sup>9</sup> 2 2 � d2h d*ς*<sup>2</sup> *∂ς*hd*ς*,

d2g d*ϕ*2 *∂ϕ*<sup>g</sup> <sup>d</sup>*<sup>ϕ</sup>* e k 2 � *Ω*−3 � <sup>d</sup>*<sup>ϕ</sup> ∂ϕ*(lng) � e 1 2 � Q*Ω*−9

*ςλ* <sup>−</sup> (*∂ςλ*)

*∂*2 *ϕ*g (*∂ϕ*g)<sup>2</sup> <sup>−</sup> <sup>3</sup>

when we use the different cases presented in table (2), the following relations between all constants were found, which we present in the same table II with the quintom potentials. So, the quantum solutions for each potential scalar fields are presented in quadrature form,

Thereby, under canonical quantization we were able to determine a family of potentials that are the most probable to characterize the inflation phenomenon in the evolution of our universe. The exact quantum solutions to the Wheeler-DeWitt equation were found using the Bohmian scheme [34] of quantum mechanics where the ansatz to the wave function *Ψ*(ℓ*µ*) = <sup>e</sup>a1*β*++iai*β*<sup>−</sup>W(ℓ*µ*)e−S(ℓ*µ*) includes the superpotential function which plays an important role in solving the Hamilton-Jacobi equation. It was necessary to study the classical behavior in order to know when the Universe evolves from a quintessence dominated phase to a phantom dominated phase crossing the *weff* = −1 dividing line, as a transient stage. Also, this family of potentials can be studied within the dynamical systems framework to obtain useful information about the asymptotic properties of the model and give a classification of

2 +

k2 − Q2

(*∂*2 *ς*h)<sup>2</sup> (*∂ς*h)<sup>2</sup> <sup>+</sup> <sup>3</sup>

� � d*ϕ*

*∂ϕ*(lng) <sup>−</sup> <sup>1</sup>

2 � d2g d*ϕ*<sup>2</sup> *∂ϕ*g d*ϕ*,

> � <sup>d</sup>*<sup>ϕ</sup> ∂ϕ*(lng) �

<sup>4</sup> <sup>=</sup> 0 , (63)

(*∂ς*h)<sup>2</sup>

(*∂*2 *ϕ*g)<sup>2</sup> (*∂ϕ*g)<sup>2</sup> <sup>−</sup> <sup>s</sup>2h2 . (62)

For our study, we shall make use of a semi-classical approximation to extract the dynamics of the WDW equation. The semi-classical limit of the WDW equation is achieved by taking *<sup>Ψ</sup>* <sup>=</sup> <sup>e</sup><sup>−</sup>S, and imposing the usual WKB conditions on the superpotential function S, namely

$$
\left(\frac{\partial \mathbf{S}}{\partial \mathbf{q}}\right)^2 >> \frac{\partial^2 \mathbf{S}}{\partial \mathbf{q}^2}
$$

Hence, the WDW equation, under the particular factor ordering Q = 0, becomes exactly the afore-mentioned EHJ equation (48a) (this approximation is equivalent to a zero quantum potential in the Bohmian interpretation of quantum cosmology [53]). The EHJ equation is also obtained if we introduce the following transformation on the canonical momenta *Π*<sup>q</sup> → *∂*qS in Eq. (8) and then Eq. (8) provides the classical solutions of the Einstein.Klein.Gordon (EKG) equations. Moreover, for the particular cases shown in Table 1, the classical solutions of the EKG, in terms of q(*τ*), arising from Eqs. (8) and (50) are given by

$$\text{gh} = 4\mu \frac{\text{d}\Omega}{\text{d}\tau}, \qquad \frac{\text{d}\varrho}{\partial\_{\text{\textphi}}\text{L}\text{ng}} + \frac{\text{d}\varrho}{\partial\_{\text{\textphi}}\text{L}\text{nh}} = 0,\tag{64}$$

the second equation appears in the W function (57), then the W is simplified by, we also have the corresponding relation with the time *τ*

$$\mathbf{d}\tau = 12\mu \frac{1}{\mathbf{h}} \frac{\mathbf{d}\varrho}{\partial\_{\varrho}\mathbf{g}} \,, \qquad \mathbf{d}\tau = -12\mu \frac{1}{\mathbf{g}} \frac{\mathbf{d}\varrho}{\partial\_{\zeta}\mathbf{h}} . \tag{65}$$

In the following subsection, we will give details about the solutions corresponding to some of the scalar potential shown in Table (3).

To recover the solutions for the anisotropic function *β*±, (28,29) we need to extend the superpotential function S = *<sup>S</sup>* − *<sup>a</sup>*1*β*<sup>+</sup> − *ia*2*β*−, remember that these functions were used in the ansatz for the wave function (44) in order to simplify the WDW equation (45). With this extension, we has

$$\frac{\partial \mathcal{S}}{\partial \mathcal{\beta}\_{\pm}} = -\mathbf{b}\_{\mathrm{i}} = \text{constants}$$

and using the corresponding momenta (8), we obtain the corresponding solutions written in quadrature form in equations (28,29). In this subsection we calculate the solution for the *Ω* function, thence the classical solution will be complete.

#### *5.0.1. Free wave function*

This particular case corresponds to an null potential function *U*(*ϕ*, *ς*), (see first line in Table (3)). The particular exact solution for the wave function *Ξ* becomes

$$\Xi(\Omega,\mathfrak{p},\mathfrak{g}) = \mathbf{e}^{\pm \frac{\mathbf{s}}{2} \left( \frac{\Lambda \zeta}{\ell} + \frac{\Lambda \rho}{p} \right)} \mathbf{e}^{\pm \frac{1}{2} (\ell \Lambda \mathfrak{g} + \mathbf{p} \Lambda \mathfrak{g})} \mathbf{e}^{\frac{\Omega}{2}(\mathbf{k} + \mathbf{Q}) \pm (3\mathbf{k} - \mathbf{9}) \frac{\Lambda \rho}{2p}} \operatorname{Exp} \left[ -\mathbf{g} \rho \mathbf{h} \varrho \mathbf{e}^{3\Omega \pm \ell \Lambda \zeta \pm \mathbf{p} \Lambda \varrho} \right], \tag{66}$$

*∆ς* <sup>=</sup> Ln �

*∆Ω* <sup>=</sup> Ln �

of the scale factor correspond at stiff matter epoch in the evolution on the universe.

WDW equation is similar to the last one, only we redefine the constants,

p2 Ln|cosh(p*∆ϕ*)<sup>|</sup>

Using the second equation in (64), we find the relation between the quintom fields

 

then, the time dependence of the quintom fields only were possible to write in quadrature

� d*ϕ*

p2 (p*∆ϕ*)

cosh <sup>9</sup>

F0

� F2<sup>ℓ</sup><sup>2</sup>

p2 (p*∆ϕ*)

 

<sup>0</sup> <sup>+</sup> cosh<sup>2</sup>ℓ<sup>2</sup>

p2 (p*∆ϕ*)

. (71)

ℓ2 ,

cosh <sup>1</sup>

ℓ → �

e k 2 � *∆Ω*− <sup>3</sup>

and the classical trajectory in the plane {*Ω*, *ϕ*} reads as

*∆Ω* <sup>−</sup> <sup>3</sup>

*∆ς* <sup>=</sup> <sup>1</sup> ℓ arcsinh

*∆τ* <sup>12</sup><sup>ℓ</sup> <sup>=</sup>

*5.0.2. Exponential scalar potential*

*5.0.3. Hyperbolic scalar potential*

<sup>2</sup> Ln|cosh(ℓ*∆ς*)sinh(p*∆ϕ*)<sup>|</sup>

form, having the following structure

� c0H0 c1G0

wave function for this is

*<sup>Υ</sup>* <sup>=</sup> <sup>e</sup><sup>−</sup> <sup>1</sup>

which corresponds to constant phase of second exponential in the W function. The behavior

For an exponential scalar potential, see second line in Table (3), the exact solution of the

This case corresponds to third line in Table ( U(*ϕ*, *<sup>ς</sup>*) = U0 sinh2(p*∆ϕ*) + U1 cosh2(ℓ*∆ς*), the

� e 1 2 � Q*∆Ω*− <sup>9</sup>

ℓ<sup>2</sup> − *<sup>c</sup>*1*G*0, *<sup>p</sup>* →

3g0h0 <sup>4</sup> *∆τ*�<sup>∓</sup> <sup>ℓ</sup>

3g0h0 <sup>4</sup> *∆τ*� <sup>1</sup> 3

�

*<sup>p</sup>*<sup>2</sup> + *<sup>c</sup>*0*H*<sup>0</sup>

p2 Ln|cosh(p*∆ϕ*)<sup>|</sup>

p2 Ln|cosh(p*∆ϕ*)<sup>|</sup> <sup>=</sup> *<sup>ǫ</sup>* <sup>=</sup> const. (70)

�

<sup>e</sup>[−g0h0e3*<sup>Ω</sup>*sinh(p*∆ϕ*)cosh(ℓ*∆ς*)]

(69)

9

Quintom Potential from Quantum Anisotropic Cosmological Models

, (68c)

http://dx.doi.org/10.5772/52054

235

, (68d)

**Figure 1.** Exact wave function for the free case, i.e. for *U*(*ϕ*, *ς*) = 0. The wave function (66) is peaked around the classical trajectory *∆Ω* ± <sup>3</sup> *<sup>p</sup>∆ϕ* <sup>=</sup> *<sup>ν</sup>*<sup>0</sup> <sup>=</sup> const, which is the solid line shown on the {*Ω*, *<sup>ϕ</sup>*} plane. For this case *<sup>ν</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> on equation (68a)

the classical trajectory implies that *∆ς* <sup>ℓ</sup> + *∆ϕ <sup>p</sup>* <sup>=</sup> 0, then there is the relation between the fields *ϕ* and *ς*, as

$$
\Delta \emptyset = -\frac{\ell}{\mathcal{P}} \Delta \varphi.
$$

So, this wavefunction can be written in terms of *ϕ* and *Ω* solely,

$$\Xi(\Omega,\boldsymbol{\varrho}) = \mathbf{e}^{\frac{\mathbf{k}}{2}\left(\Delta\Omega \pm \frac{3}{\mathrm{p}}\Delta\boldsymbol{\varrho}\right) + \frac{\mathbf{Q}}{2}\Delta\Omega} \times \left[ -\mathbf{g}\_{0}\mathbf{h}\_{0}\mathbf{e}^{\mathbf{j}\left(\Omega \pm \frac{3}{\mathrm{p}}\Delta\boldsymbol{\varrho}\right)} \right],\tag{67}$$

Using the equantion (65), we find the *classical trajectory* on the {*Ω*, *ϕ*} plane as

$$
\Delta\Omega \pm \frac{3}{p} \Delta\varphi = \nu\_0 = \text{const},\tag{68a}
$$

*∆ϕ* <sup>=</sup> Ln 3g0h0 <sup>4</sup> *∆τ*<sup>±</sup> <sup>p</sup> 9 , (68b)

$$
\Delta \mathbf{g} = \text{Ln} \left[ \frac{3 \mathbf{g}\_0 \mathbf{h}\_0}{4} \Delta \mathbf{\tau} \right]^{\mp \frac{\ell}{5}} \text{ .} \tag{68c}
$$

$$
\Delta\Omega = \text{Ln} \left[ \frac{3\text{g}\_0\text{h}\_0}{4} \Delta\tau \right]^{\frac{1}{5}} \text{ .} \tag{68d}
$$

which corresponds to constant phase of second exponential in the W function. The behavior of the scale factor correspond at stiff matter epoch in the evolution on the universe.

#### *5.0.2. Exponential scalar potential*

16 Open Questions in Cosmology

*Ξ*(*Ω*, *ϕ*, *ς*) = e

trajectory *∆Ω* ± <sup>3</sup>

*ϕ* and *ς*, as

the classical trajectory implies that *∆ς*

(68a)

± <sup>s</sup> 2 *∆ς* <sup>ℓ</sup> + *∆ϕ* p <sup>e</sup><sup>±</sup> <sup>1</sup>

<sup>2</sup> (ℓ*∆ς*+p*∆ϕ*)

e *Ω*

<sup>2</sup> (k+Q)±(3k−9) *∆ϕ*

**Figure 1.** Exact wave function for the free case, i.e. for *U*(*ϕ*, *ς*) = 0. The wave function (66) is peaked around the classical

*∆ς* <sup>=</sup> <sup>−</sup> <sup>ℓ</sup>

<sup>ℓ</sup> + *∆ϕ*

Using the equantion (65), we find the *classical trajectory* on the {*Ω*, *ϕ*} plane as

*∆ϕ* = Ln

So, this wavefunction can be written in terms of *ϕ* and *Ω* solely,

k 2 *∆Ω*± <sup>3</sup> <sup>p</sup>*∆ϕ* + <sup>Q</sup>

*∆Ω* <sup>±</sup> <sup>3</sup> *p*

*Ξ*(*Ω*, *ϕ*) = e

*<sup>p</sup>∆ϕ* <sup>=</sup> *<sup>ν</sup>*<sup>0</sup> <sup>=</sup> const, which is the solid line shown on the {*Ω*, *<sup>ϕ</sup>*} plane. For this case *<sup>ν</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> on equation

<sup>p</sup>*∆ϕ*.

<sup>2</sup> *∆Ω* Exp

 3g0h0 <sup>4</sup> *∆τ*

 −g0h0e 3 *Ω*± <sup>3</sup> <sup>p</sup>*∆ϕ* 

*<sup>p</sup>* <sup>=</sup> 0, then there is the relation between the fields

*∆ϕ* = *<sup>ν</sup>*<sup>0</sup> = const, (68a)

<sup>±</sup> <sup>p</sup> 9 , (67)

, (68b)

2p Exp 

<sup>−</sup>g0h0e<sup>3</sup>*Ω*±ℓ*∆ς*±p*∆ϕ*

, (66)

For an exponential scalar potential, see second line in Table (3), the exact solution of the WDW equation is similar to the last one, only we redefine the constants,

$$\ell \rightarrow \sqrt{\ell^2 - c\_1 G\_0} \qquad p \rightarrow \sqrt{p^2 + c\_0 H\_0}$$

#### *5.0.3. Hyperbolic scalar potential*

This case corresponds to third line in Table ( U(*ϕ*, *<sup>ς</sup>*) = U0 sinh2(p*∆ϕ*) + U1 cosh2(ℓ*∆ς*), the wave function for this is

*<sup>Υ</sup>* <sup>=</sup> <sup>e</sup><sup>−</sup> <sup>1</sup> <sup>2</sup> Ln|cosh(ℓ*∆ς*)sinh(p*∆ϕ*)<sup>|</sup> e k 2 � *∆Ω*− <sup>3</sup> p2 Ln|cosh(p*∆ϕ*)<sup>|</sup> � e 1 2 � Q*∆Ω*− <sup>9</sup> p2 Ln|cosh(p*∆ϕ*)<sup>|</sup> � <sup>e</sup>[−g0h0e3*<sup>Ω</sup>*sinh(p*∆ϕ*)cosh(ℓ*∆ς*)] (69)

and the classical trajectory in the plane {*Ω*, *ϕ*} reads as

$$
\Delta\Omega - \frac{3}{\mathbf{p}^2} \text{Ln} |\cosh(\mathbf{p}\Delta\phi)| = \epsilon = \text{const.}\tag{70}
$$

Using the second equation in (64), we find the relation between the quintom fields

$$\Delta \xi = \frac{1}{\ell} \text{arcsinh} \left[ \frac{\mathbf{F}\_0}{\cosh^{\frac{1}{\mathbb{P}^2}}(\mathbf{p} \Delta q)} \right]^{\ell^2} \lambda$$

then, the time dependence of the quintom fields only were possible to write in quadrature form, having the following structure

$$\sqrt{\frac{\mathbf{c}\_0 \mathbf{H}\_0}{\mathbf{c}\_1 \mathbf{G}\_0} \frac{\Delta \tau}{12\ell}} = \int \frac{\mathbf{d}\varrho}{\cosh^{\frac{\mathfrak{g}}{\mathfrak{r}^2}} (\mathbf{p}\Delta\varrho) \sqrt{\mathbf{F}\_0^{2\ell^2} + \cosh^{\frac{\mathfrak{g}^2}{\mathfrak{r}^2}} (\mathbf{p}\Delta\varrho)}}.\tag{71}$$

the set of equations that is necessary to solve. Also we give some explanation why this

Quintom Potential from Quantum Anisotropic Cosmological Models

http://dx.doi.org/10.5772/52054

237

Finally, we do the comment that solution to the equation (50), when we write this equation as a quadratic equation, C2x<sup>2</sup> + C1x + C0 = 0 with x = <sup>e</sup>*Ω*, is not possible because the set of equations that appear , only a subset have solution in closed form. The algebraic one is not fulfill, making that the c parameter become null. In forthcoming work we will analyze under the scheme of dynamical systems, the relation between the corresponding critical points and its stability properties with these classical solutions for separable potentials obtained in this

This work was partially supported by CONACYT 179881 grant. DAIP (2011-2012) and PROMEP grants UGTO-CA-3, UAM-I-43. PRB and MA were partially supported by UAEMex grant FEO1/2012 103.5/12/2126. This work is part of the collaboration within the Instituto Avanzado de Cosmología, and Red PROMEP: Gravitation and Mathematical Physics under project *Quantum aspects of gravity in cosmological models, phenomenology and geometry of space-time*. Many calculations were done by Symbolic Program REDUCE 3.8.

1 Departamento de Física de la DCeI de la Universidad de Guanajuato-Campus León,

2 Departamento de Física, Universidad Autónoma Metropolitana, Apartado Postal 55-534,

[1] E.J. Copeland, M. Sami and S. Tsujikawa *Dynamics of dark energy Int. J. Mod. Phys. D* 15

[3] Y.F. Cai, E.N. Saridakis, M.R. Setare and J.Q. Xia *Quintom cosmology: Theoretical*

[4] Z.K. Guo, Y.S. Piao, X. Zhang and Y.Z. Zhang *Cosmological evolution of a quintom model of*

[5] B. Feng, X. Wang and X. Zhang *Dark energy constraints from the cosmic age and supernova*

[6] H.M. Sadjadi and M. Alimohammadi *Transition from quintessence to the phantom phase in*

[2] B. Feng *The Quintom Model of Dark Energy* [ArXiv:astro-ph/0602156].

*implications and observations Phys. Rep.* 493, 1 (2010).

*the quintom model Phys. Rev. D* 74 (4), 043506 (2006).

decomposition in not unique.

**7. Acknowledgments**

**Author details**

Guanajuato, México

**References**

C.P. 09340 México, DF, México

1753, (2006) [arXiv:hep-th 0603057].

*dark energy Phys. Lett. B* 608, 177 (2005).

*Phys. Lett. B* 607, 35 (2005).

J. Socorro1,2, Paulo A. Rodríguez 1, O. Núñez-Soltero1, Rafael Hernández<sup>1</sup> and Abraham Espinoza-García<sup>1</sup>

work [54].

**Figure 2.** Wave Function for the potential <sup>U</sup>(*ϕ*, *<sup>ς</sup>*) = U0 sinh2(p*∆ϕ*) + U1 cosh2(ℓ*∆ς*). The wave function (69) is peaked around the classical trajectory *∆Ω* − <sup>3</sup> <sup>p</sup><sup>2</sup> Ln|cosh(p*∆ϕ*)<sup>|</sup> <sup>=</sup> *<sup>ǫ</sup>*, which is the solid line shown on the {*Ω*, *ϕ*} plane. For this case *ǫ* = 2.4 on equation (70)

## **6. Conclusions**

In summary, we presented the corresponding Einstein Klein Gordon equation for the quintom model, which is applied to the Bianchi Type I cosmological model including as a matter content a barotropic perfect fluid and cosmological constant, and the classical solutions are given in a quadrature form for null and constant scalar potentials, these solutions are related to the Sáez-Ballester formalism, [35–38].

The quantum scheme in the Bohmian formalism and its mathematical structure, and our approach were applied to the Bianchi type I cosmological model in order to build the mathematical structure of quintom scalar potentials using the integral systems formalism.

Also, we presented the quantum solutions to the Wheeler-DeWitt equation, which is the main equation to be solved and such a subject is our principal objective in this work to obtain the family of scalar potential in the inflation phenomenon.

We emphasize that the quantum potential from the Bohm formalism will work as a constraint equation which restricts our family of potentials found, see Table (3), [33].

It is well known in the literature that in the Bohm formalism the imaginary part is never determined, however in this work such a problem has been solved in order to find the quantum potentials, which was a more important matter for being able to find the classical trajectories, which were showed through graphics how the classical trajectory is projected from its quantum counterpart. We include some steps how we solve the imaginary like equation (48c) when we found the superpotential function S (54) and particular ansatz for the function W, being the equation (58), and using the separation variables method we find the set of equations that is necessary to solve. Also we give some explanation why this decomposition in not unique.

Finally, we do the comment that solution to the equation (50), when we write this equation as a quadratic equation, C2x<sup>2</sup> + C1x + C0 = 0 with x = <sup>e</sup>*Ω*, is not possible because the set of equations that appear , only a subset have solution in closed form. The algebraic one is not fulfill, making that the c parameter become null. In forthcoming work we will analyze under

the scheme of dynamical systems, the relation between the corresponding critical points and its stability properties with these classical solutions for separable potentials obtained in this work [54].

## **7. Acknowledgments**

18 Open Questions in Cosmology

(69) is peaked around the classical trajectory *∆Ω* − <sup>3</sup>

{*Ω*, *ϕ*} plane. For this case *ǫ* = 2.4 on equation (70)

solutions are related to the Sáez-Ballester formalism, [35–38].

family of scalar potential in the inflation phenomenon.

**6. Conclusions**

**Figure 2.** Wave Function for the potential <sup>U</sup>(*ϕ*, *<sup>ς</sup>*) = U0 sinh2(p*∆ϕ*) + U1 cosh2(ℓ*∆ς*). The wave function

In summary, we presented the corresponding Einstein Klein Gordon equation for the quintom model, which is applied to the Bianchi Type I cosmological model including as a matter content a barotropic perfect fluid and cosmological constant, and the classical solutions are given in a quadrature form for null and constant scalar potentials, these

The quantum scheme in the Bohmian formalism and its mathematical structure, and our approach were applied to the Bianchi type I cosmological model in order to build the mathematical structure of quintom scalar potentials using the integral systems formalism.

Also, we presented the quantum solutions to the Wheeler-DeWitt equation, which is the main equation to be solved and such a subject is our principal objective in this work to obtain the

We emphasize that the quantum potential from the Bohm formalism will work as a constraint

It is well known in the literature that in the Bohm formalism the imaginary part is never determined, however in this work such a problem has been solved in order to find the quantum potentials, which was a more important matter for being able to find the classical trajectories, which were showed through graphics how the classical trajectory is projected from its quantum counterpart. We include some steps how we solve the imaginary like equation (48c) when we found the superpotential function S (54) and particular ansatz for the function W, being the equation (58), and using the separation variables method we find

equation which restricts our family of potentials found, see Table (3), [33].

<sup>p</sup><sup>2</sup> Ln|cosh(p*∆ϕ*)<sup>|</sup> <sup>=</sup> *<sup>ǫ</sup>*, which is the solid line shown on the

This work was partially supported by CONACYT 179881 grant. DAIP (2011-2012) and PROMEP grants UGTO-CA-3, UAM-I-43. PRB and MA were partially supported by UAEMex grant FEO1/2012 103.5/12/2126. This work is part of the collaboration within the Instituto Avanzado de Cosmología, and Red PROMEP: Gravitation and Mathematical Physics under project *Quantum aspects of gravity in cosmological models, phenomenology and geometry of space-time*. Many calculations were done by Symbolic Program REDUCE 3.8.

## **Author details**

J. Socorro1,2, Paulo A. Rodríguez 1, O. Núñez-Soltero1, Rafael Hernández<sup>1</sup> and Abraham Espinoza-García<sup>1</sup>

1 Departamento de Física de la DCeI de la Universidad de Guanajuato-Campus León, Guanajuato, México

2 Departamento de Física, Universidad Autónoma Metropolitana, Apartado Postal 55-534, C.P. 09340 México, DF, México

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[10] Y.F. Cai, M. Li,J.X. Lu, Y.S. Piao, T. Qiu and X. Zhang *A string-inspired quintom model of*

[11] Y.F. Cai, T. Qiu, X. Zhang, Y.S. Piao and M. Li *Bouncing universe with Quintom matter J.*

[13] R. Lazkoz, G. León and I. Quiros *Quintom cosmologies with arbitrary potentials Phys. Lett.*

[14] Y.F. Cai, T. Qiu, R. Brandenberger, Y.S. Piao, and X. Zhang *On perturbations of a quintom*

[15] Y.F. Cai and J. Wang *Dark energy model with spinor matter and its quintom scenario Classical*

[16] S. Zhang and B. Chen *Reconstructing a string-inspired quintom model of dark energy Phys.*

[17] J. Sadeghi, M.R. Setare, A. Banijamali and F. Milani *Non-minimally coupled quintom model*

[18] K. Nozari. M.R. Setare, T. Azizi and N. Behrouz *A non-minimally coupled quintom dark*

[19] M.R. Setare and E.N. Saridakis *Quintom dark energy models with nearly flat potentials Phys.*

[20] M. R. Setare and E. N. Saridakis *Quintom Cosmology with general potentials*. *Int. Jour. of*

[21] J. Sadeghi, M.R. Setare and A. Banijamali *String inspired quintom model with non-minimally*

[23] E.N. Saridakis *Quintom evolution in power-law potentials Nuclear Phys. B* 830, 374 (2010).

[24] A.R. Amani *Stability of Quintom Model of Dark Energy in ( ω, ω') Phase Plane Int. J. of Theor.*

*energy model on the warped DGP brane Physica Scripta*, 80 (2), 025901 (2009).

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*inspired by string theory Phys. Lett. B* 662, 92, (2008).

*coupled modified gravity Phys. Lett. B* 678,164 (2009)


**Chapter 10**

**Provisional chapter**

**Leptogenesis and Neutrino Masses in an Inflationary**

One of the most promising and well-motivated mechanisms for the generation of the *Baryon Asymmetry of the Universe* (BAU) is via an initial generation of a lepton asymmetry, which can be subsequently converted to BAU through sphaleron effects – see e.g. Ref. [1, 2]. *Non-Thermal Leptogenesis* (nTL) [3, 4] is a variant of this proposal, in which the necessitated departure from equilibrium is achieved by construction. Namely, the *right-handed* (RH)

onset, since their masses are larger than the reheating temperature. Such a set-up can be

place out-of-equilibrium. Therefore, such a leptogenesis paradigm largely depends on the

In a recent paper [5] – for similar attempts, see Ref. [6–8] –, we investigate an inflationary model where a *Standard Model* (SM) singlet component of the Higgs fields involved in the spontaneous breaking of a *supersymmetric* (SUSY) *Pati-Salam* (PS) *Grand Unified Theory* (GUT) can produce inflation of chaotic-type, named *non-minimal Higgs Inflation* (nMHI), since there is a relatively strong non-minimal coupling of the inflaton field to gravity [9–12]. This GUT provides a natural framework to implement our leptogenesis scenario, since the

realization this GUT leads to third family *Yukawa unification* (YU), and does not suffer from the doublet-triplet splitting problem since both Higgs doublets are contained in a bidoublet other than the GUT scale Higgs fields. Although this GUT is not completely unified – as, e.g., a GUT based on the *SO*(10) gauge symmetry group – it emerges in standard weakly

The inflationary model relies on renormalizable superpotential terms and does not lead to overproduction of magnetic monopoles. It is largely independent of the one-loop radiative corrections [15], and it can become consistent with the fitting [16] of the seven-year data of

and reproduction in any medium, provided the original work is properly cited.

©2012 Pallis and Toumbas, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Pallis and Toumbas; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

presence of the *SU*(2)<sup>R</sup> gauge symmetry predicts the existence of three *<sup>ν</sup><sup>c</sup>*

coupled heterotic string models [13] and in recent D-brane constructions [14].

*<sup>i</sup>* , whose decay produces the lepton asymmetry, are out-of-equilibrium at the

*<sup>i</sup>* through the inflaton decay, which can also take

*<sup>i</sup>* . In its simplest

**SUSY Pati-Salam Model**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Leptogenesis and Neutrino Masses**

**in an Inflationary SUSY Pati-Salam Model**

C. Pallis and N. Toumbas

C. Pallis and N. Toumbas

http://dx.doi.org/10.5772/51888

achieved by the direct production of *ν<sup>c</sup>*

inflationary stage, which it follows.

**1. Introduction**

neutrinos, *ν<sup>c</sup>*


**Provisional chapter**

## **Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model in an Inflationary SUSY Pati-Salam Model**

C. Pallis and N. Toumbas Additional information is available at the end of the chapter

C. Pallis and N. Toumbas

Additional information is available at the end of the chapter

**Leptogenesis and Neutrino Masses**

http://dx.doi.org/10.5772/51888

## **1. Introduction**

22 Open Questions in Cosmology

240 Open Questions in Cosmology

[42] S.W. Hawking *Nucl. Phys. B* 239, 257 (1984).

[44] H. Kodama *Phys. Rev D* 42, 2548 (1990).

[45] A. Ashtekar *Phys. Rev. D* 36,1587 (1989).

[43] H. Kodama *Progress of Theor. Phys.* 80, 1024 (1988).

[46] V. Moncrief and M.P. Ryan *Phys. Rev. D* 44, 2375 (1991).

[49] M.P, Ryan *Hamiltonian cosmology*, (Springer, Berlin, (1992)).

*models Int. J. of Theor. Phys.* 35 (7), 1381 (1995).

in Physics 803 & 804, (Springer, Berlin) (2010).

[54] León, G., Leyva, Y., and Socorro, J.: [gr-qc.1208-0061]

70 103512 (2004).

[47] O. Obregón and J. Socorro *<sup>Ψ</sup>* <sup>=</sup> *We*±*<sup>Φ</sup> quantum cosmological solutions for Class A Bianchi*

[50] Andrei C. Polyanin & Valentin F. Zaitsev *Handbook of Exact solutions for ordinary*

[52] P.V. Moniz *Quantum cosmology -the supersymmetric perspective- Vol. 1 & 2*, Lecture Notes

[53] G. D. Barbosa and N. Pinto-Neto, *Noncommutative geometry and cosmology*, *Phys. Rev. D*

[51] M.P. M.P. & L.C. Shepley *Homogeneous Relativistic Cosmologies* (Princenton (1985))

[48] Private communication with Luis Ureña, DCI-Universidad de Guanajuato.

*differential equations*, *Second edition*, Chapman & Hall/CRC (2003).

One of the most promising and well-motivated mechanisms for the generation of the *Baryon Asymmetry of the Universe* (BAU) is via an initial generation of a lepton asymmetry, which can be subsequently converted to BAU through sphaleron effects – see e.g. Ref. [1, 2]. *Non-Thermal Leptogenesis* (nTL) [3, 4] is a variant of this proposal, in which the necessitated departure from equilibrium is achieved by construction. Namely, the *right-handed* (RH) neutrinos, *ν<sup>c</sup> <sup>i</sup>* , whose decay produces the lepton asymmetry, are out-of-equilibrium at the onset, since their masses are larger than the reheating temperature. Such a set-up can be achieved by the direct production of *ν<sup>c</sup> <sup>i</sup>* through the inflaton decay, which can also take place out-of-equilibrium. Therefore, such a leptogenesis paradigm largely depends on the inflationary stage, which it follows.

In a recent paper [5] – for similar attempts, see Ref. [6–8] –, we investigate an inflationary model where a *Standard Model* (SM) singlet component of the Higgs fields involved in the spontaneous breaking of a *supersymmetric* (SUSY) *Pati-Salam* (PS) *Grand Unified Theory* (GUT) can produce inflation of chaotic-type, named *non-minimal Higgs Inflation* (nMHI), since there is a relatively strong non-minimal coupling of the inflaton field to gravity [9–12]. This GUT provides a natural framework to implement our leptogenesis scenario, since the presence of the *SU*(2)<sup>R</sup> gauge symmetry predicts the existence of three *<sup>ν</sup><sup>c</sup> <sup>i</sup>* . In its simplest realization this GUT leads to third family *Yukawa unification* (YU), and does not suffer from the doublet-triplet splitting problem since both Higgs doublets are contained in a bidoublet other than the GUT scale Higgs fields. Although this GUT is not completely unified – as, e.g., a GUT based on the *SO*(10) gauge symmetry group – it emerges in standard weakly coupled heterotic string models [13] and in recent D-brane constructions [14].

The inflationary model relies on renormalizable superpotential terms and does not lead to overproduction of magnetic monopoles. It is largely independent of the one-loop radiative corrections [15], and it can become consistent with the fitting [16] of the seven-year data of

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Pallis and Toumbas; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Pallis and Toumbas, licensee InTech. This is an open access chapter distributed under the terms of the

the *Wilkinson Microwave Anisotropy Probe Satellite* (WMAP7) combined with the *baryon-acoustic oscillation* (BAO) and the measurement of the *Hubble constant* (*H*0). At the same time the GUT symmetry breaking scale attains its SUSY value and the *µ* problem of the *Minimal SUSY SM* (MSSM) is resolved via a Peccei-Quinn (PQ) symmetry, solving also the strong CP problem. Inflation can be followed by non-thermal leptogenesis, compatible with the gravitino (*<sup>G</sup>*) limit [17–19] on the reheating temperature, leading to efficient baryogenesis. In Ref. [5] we connect non-thermal leptogenesis with neutrino data, implementing a two-generation formulation of the see-saw [20–22] mechanism and imposing extra restrictions from the data on the light neutrino masses and the GUT symmetry on the heaviest Dirac neutrino mass. There we [5] assume that the mixing angle between the first and third generation, *θ*13, vanishes. However, the most updated [23, 24] analyses of the low energy neutrino data suggest that non-zero values for *θ*<sup>13</sup> are now preferred, while the zero value can be excluded at 8 standard deviations. Therefore, a revision of our results, presented in Ref. [5], is worth pursuing.

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model 3

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

<sup>L</sup>*U*<sup>C</sup> *Qi*a(**3**, **<sup>2</sup>**, 1/6) <sup>1</sup> <sup>−</sup><sup>1</sup> <sup>−</sup>

*<sup>i</sup>*a(**3¯**, **<sup>1</sup>**, <sup>−</sup>2/3) 1 0 <sup>−</sup>

*<sup>H</sup>*a(**3¯**, **<sup>1</sup>**, <sup>−</sup>2/3) 0 0 <sup>+</sup>

*<sup>H</sup>*a(**3**, **<sup>1</sup>**, 2/3) 0 0 <sup>+</sup>

<sup>a</sup>(**3**, **<sup>1</sup>**, <sup>−</sup>1/3) 2 0 <sup>+</sup>

<sup>2</sup> symmetry ('matter parity') under which *<sup>F</sup>*, *<sup>F</sup><sup>c</sup>*

*<sup>i</sup>* , which generate

2

243

http://dx.doi.org/10.5772/51888

**Super- Represe- Trasfor- Decompo- Global fields ntations mations sitions Charges**

> C*U*<sup>∗</sup> R*F<sup>c</sup>*

C*U*<sup>∗</sup>

*Fi* (**4**, **<sup>2</sup>**, **<sup>1</sup>**) *FiU*†

*<sup>i</sup>* (**4¯**, **<sup>1</sup>**, **<sup>2</sup>**) *<sup>U</sup>*<sup>∗</sup>

*<sup>H</sup><sup>c</sup>* (**4¯**, **<sup>1</sup>**, **<sup>2</sup>**) *<sup>U</sup>*<sup>∗</sup>

*<sup>H</sup>*¯ *<sup>c</sup>* (**4**, **<sup>1</sup>**, **<sup>2</sup>**) *<sup>H</sup>*¯ *cU*<sup>R</sup> *<sup>U</sup>*<sup>C</sup> *<sup>u</sup>*¯*<sup>c</sup>*

*<sup>G</sup>* (**6**, **<sup>1</sup>**, **<sup>1</sup>**) *<sup>U</sup>*C*GU*<sup>C</sup> *<sup>g</sup>*¯*<sup>c</sup>*

transpose, the hermitian conjugate and the complex conjugate of a matrix respectively.

an R symmetry, as well as a discrete **Z**mp

*Fc*

**under** *<sup>G</sup>*PS **under** *<sup>G</sup>*PS **under** *<sup>G</sup>*SM *<sup>R</sup>* PQ **<sup>Z</sup>**mp

*Li*(**1**, **<sup>2</sup>**, −1/2)

*<sup>i</sup>*a(**3¯**, **<sup>1</sup>**, 1/3)

*<sup>H</sup>*a(**3¯**, **<sup>1</sup>**, 1/3)

*<sup>H</sup>*a(**3**, **<sup>1</sup>**, <sup>−</sup>1/3)

*<sup>H</sup>*(**1**, **<sup>1</sup>**, <sup>−</sup>1)

<sup>a</sup>(**3¯**, **<sup>1</sup>**, 1/3)

*Hd*(**1**, **<sup>2</sup>**, −1/2)

Matter Superfields

*dc*

Higgs Superfields

*dc*

¯*dc*

*νc <sup>H</sup>*(**1**, **<sup>1</sup>**, 0)

*ec <sup>H</sup>*(**1**, **<sup>1</sup>**, 1)

*ν*¯*c <sup>H</sup>*(**1**, **<sup>1</sup>**, 0)

*e*¯ *c*

*gc*

*S* (**1**, **1**, **1**) *S S*(**1**, **1**, 0) 2 0 +

*IH* (**1**, **<sup>2</sup>**, **<sup>2</sup>**) *<sup>U</sup>*L*IHU*<sup>R</sup> *Hu*(**1**, **<sup>2</sup>**, 1/2) 0 1 <sup>+</sup>

*P* (**1**, **1**, **1**) *P P*(**1**, **1**, 0) 1 −1 + *P*¯ (**1**, **1**, **1**) *P*¯ *P*¯(**1**, **1**, 0) 0 1 +

**Table 1.** The representations, the transformations under *G*PS, the decompositions under *G*SM as well as the extra global charges of the superfields of our model. Here *U*<sup>C</sup> ∈ *SU*(4)C, *U*<sup>L</sup> ∈ *SU*(2)L, *U*<sup>R</sup> ∈ *SU*(2)<sup>R</sup> and , † and ∗ stand for the

realization of this model [13, 30], the electroweak doublets *Hu* and *Hd*, which couple to the up and down quarks respectively, are exclusively contained in the bidoublet superfield *IH*. In addition to *<sup>G</sup>*PS, the model possesses two global *<sup>U</sup>*(1) symmetries, namely a PQ and

change sign. The last symmetry forbids undesirable mixings of *F* and *IH* and/or *F<sup>c</sup>* and *H<sup>c</sup>* and ensures the stability of the *lightest SUSY particle* (LSP). The imposed *U*(1) R symmetry, *<sup>U</sup>*(1)*R*, guarantees the linearity of the superpotential w.r.t the singlet *<sup>S</sup>*. Finally the *<sup>U</sup>*(1) PQ symmetry, *<sup>U</sup>*(1)PQ, assists us to generate the *<sup>µ</sup>*-term of the MSSM. The PQ breaking occurs at an intermediate scale through the v.e.vs of *P*, *P*¯, and the *µ*-term is generated via a non-renormalizable coupling of *P* and *IH*. Following Ref. [30], we introduce into the

scheme quartic (non-renormalizable) superpotential couplings of *H*¯ *<sup>c</sup>* to *F<sup>c</sup>*

R*H<sup>c</sup> <sup>u</sup><sup>c</sup>*

*νc <sup>i</sup>* (**1**, **<sup>1</sup>**, 0)

*ec <sup>i</sup>*(**1**, **<sup>1</sup>**, 1)

*<sup>i</sup> <sup>u</sup><sup>c</sup>*

The three-generation implementation of the see-saw mechanism is here adopted, following a bottom-up approach, along the lines of Ref. [25–28]. In particular, we use as input parameters the low energy neutrino observables considering several schemes of neutrino masses. Using also the third generation Dirac neutrino mass predicted by the PS GUT, assuming a mild hierarchy for the two residual generations and imposing the restriction from BAU, we constrain the masses of *ν<sup>c</sup> <sup>i</sup>* 's and the residual neutrino Dirac mass spectrum. Renormalization group effects [28, 29] are also incorporated in our analysis.

We present the basic ingredients of our model in Sec. 2. In Sec. 3 we describe the inflationary potential and derive the inflationary observables. In Sec. 4 we outline the mechanism of non-thermal leptogenesis, while in Sec. 5 we exhibit the relevant imposed constraints and restrict the parameters of our model. Our conclusions are summarized in Sec. 6. Throughout the text, we use natural units for Planck's and Boltzmann's constants and the speed of light (¯*<sup>h</sup>* = *<sup>c</sup>* = *<sup>k</sup>*<sup>B</sup> = 1); the subscript of type , *<sup>χ</sup>* denotes derivation *with respect to* (w.r.t) the field *<sup>χ</sup>* (e.g., ,*χχ* = *<sup>∂</sup>*2/*∂χ*2); charge conjugation is denoted by a star and log [ln] stands for logarithm with basis 10 [*e*].

## **2. The Pati-Salam SUSY GUT model**

In this section, we present the particle content (Sec. 2.1), the structure of the superpotential and the Kähler potential(Sec. 2.2) and describe the SUSY limit (Sec. 2.3) of our model.

#### **2.1. Particle content**

We focus on a SUSY PS GUT model described in detail in Ref. [5, 30]. The representations and the transformation properties of the various superfields under *<sup>G</sup>*PS = *SU*(4)<sup>C</sup> × *SU*(2)<sup>L</sup> × *SU*(2)R, their decomposition under *<sup>G</sup>*SM = *SU*(3)<sup>C</sup> × *SU*(2)<sup>L</sup> × *<sup>U</sup>*(1)*Y*, as well as their extra global charges are presented in Table 1.

The *<sup>i</sup>*th generation (*<sup>i</sup>* = 1, 2, 3) *left-handed* (LH) quark and lepton superfields, *ui*a, *di*<sup>a</sup> (a = 1, 2, 3 is a color index), *ei* and *ν<sup>i</sup>* are accommodated in a superfield *Fi*. The LH antiquark and antilepton superfields *u<sup>c</sup> <sup>i</sup>*a, *<sup>d</sup><sup>c</sup> <sup>i</sup>*a, *<sup>e</sup><sup>c</sup> <sup>i</sup>* and *<sup>ν</sup><sup>c</sup> <sup>i</sup>* are arranged in another superfield *<sup>F</sup><sup>c</sup> <sup>i</sup>* . The gauge symmetry *G*PS can be spontaneously broken down to *G*SM through v.e.vs which the superfields *H<sup>c</sup>* and *H*¯ *<sup>c</sup>* acquire in the directions *ν<sup>c</sup> <sup>H</sup>* and *<sup>ν</sup>*¯*<sup>c</sup> <sup>H</sup>*. The model also contains a gauge singlet *<sup>S</sup>*, which triggers the breaking of *<sup>G</sup>*PS, as well as an *SU*(4)<sup>C</sup> **<sup>6</sup>**-plet *<sup>G</sup>*, which splits into *g<sup>c</sup>* <sup>a</sup> and *g*¯*<sup>c</sup>* <sup>a</sup> under *GSM* and gives [13] superheavy masses to *d<sup>c</sup> <sup>H</sup>*<sup>a</sup> and ¯*d<sup>c</sup> <sup>H</sup>*a. In the simplest


2 Open Questions in Cosmology

242 Open Questions in Cosmology

pursuing.

constrain the masses of *ν<sup>c</sup>*

with basis 10 [*e*].

**2.1. Particle content**

group effects [28, 29] are also incorporated in our analysis.

**2. The Pati-Salam SUSY GUT model**

global charges are presented in Table 1.

*<sup>i</sup>*a, *<sup>d</sup><sup>c</sup> <sup>i</sup>*a, *<sup>e</sup><sup>c</sup>*

superfields *H<sup>c</sup>* and *H*¯ *<sup>c</sup>* acquire in the directions *ν<sup>c</sup>*

and antilepton superfields *u<sup>c</sup>*

into *g<sup>c</sup>*

<sup>a</sup> and *g*¯*<sup>c</sup>*

the *Wilkinson Microwave Anisotropy Probe Satellite* (WMAP7) combined with the *baryon-acoustic oscillation* (BAO) and the measurement of the *Hubble constant* (*H*0). At the same time the GUT symmetry breaking scale attains its SUSY value and the *µ* problem of the *Minimal SUSY SM* (MSSM) is resolved via a Peccei-Quinn (PQ) symmetry, solving also the strong CP problem. Inflation can be followed by non-thermal leptogenesis, compatible with the gravitino (*<sup>G</sup>*) limit [17–19] on the reheating temperature, leading to efficient baryogenesis. In Ref. [5] we connect non-thermal leptogenesis with neutrino data, implementing a two-generation formulation of the see-saw [20–22] mechanism and imposing extra restrictions from the data on the light neutrino masses and the GUT symmetry on the heaviest Dirac neutrino mass. There we [5] assume that the mixing angle between the first and third generation, *θ*13, vanishes. However, the most updated [23, 24] analyses of the low energy neutrino data suggest that non-zero values for *θ*<sup>13</sup> are now preferred, while the zero value can be excluded at 8 standard deviations. Therefore, a revision of our results, presented in Ref. [5], is worth

The three-generation implementation of the see-saw mechanism is here adopted, following a bottom-up approach, along the lines of Ref. [25–28]. In particular, we use as input parameters the low energy neutrino observables considering several schemes of neutrino masses. Using also the third generation Dirac neutrino mass predicted by the PS GUT, assuming a mild hierarchy for the two residual generations and imposing the restriction from BAU, we

We present the basic ingredients of our model in Sec. 2. In Sec. 3 we describe the inflationary potential and derive the inflationary observables. In Sec. 4 we outline the mechanism of non-thermal leptogenesis, while in Sec. 5 we exhibit the relevant imposed constraints and restrict the parameters of our model. Our conclusions are summarized in Sec. 6. Throughout the text, we use natural units for Planck's and Boltzmann's constants and the speed of light (¯*<sup>h</sup>* = *<sup>c</sup>* = *<sup>k</sup>*<sup>B</sup> = 1); the subscript of type , *<sup>χ</sup>* denotes derivation *with respect to* (w.r.t) the field *<sup>χ</sup>* (e.g., ,*χχ* = *<sup>∂</sup>*2/*∂χ*2); charge conjugation is denoted by a star and log [ln] stands for logarithm

In this section, we present the particle content (Sec. 2.1), the structure of the superpotential and the Kähler potential(Sec. 2.2) and describe the SUSY limit (Sec. 2.3) of our model.

We focus on a SUSY PS GUT model described in detail in Ref. [5, 30]. The representations and the transformation properties of the various superfields under *<sup>G</sup>*PS = *SU*(4)<sup>C</sup> × *SU*(2)<sup>L</sup> × *SU*(2)R, their decomposition under *<sup>G</sup>*SM = *SU*(3)<sup>C</sup> × *SU*(2)<sup>L</sup> × *<sup>U</sup>*(1)*Y*, as well as their extra

The *<sup>i</sup>*th generation (*<sup>i</sup>* = 1, 2, 3) *left-handed* (LH) quark and lepton superfields, *ui*a, *di*<sup>a</sup> (a = 1, 2, 3 is a color index), *ei* and *ν<sup>i</sup>* are accommodated in a superfield *Fi*. The LH antiquark

gauge symmetry *G*PS can be spontaneously broken down to *G*SM through v.e.vs which the

singlet *<sup>S</sup>*, which triggers the breaking of *<sup>G</sup>*PS, as well as an *SU*(4)<sup>C</sup> **<sup>6</sup>**-plet *<sup>G</sup>*, which splits

*<sup>H</sup>* and *<sup>ν</sup>*¯*<sup>c</sup>*

*<sup>i</sup>* are arranged in another superfield *<sup>F</sup><sup>c</sup>*

*<sup>H</sup>*. The model also contains a gauge

*<sup>H</sup>*<sup>a</sup> and ¯*d<sup>c</sup>*

*<sup>i</sup>* . The

*<sup>H</sup>*a. In the simplest

*<sup>i</sup>* and *<sup>ν</sup><sup>c</sup>*

<sup>a</sup> under *GSM* and gives [13] superheavy masses to *d<sup>c</sup>*

*<sup>i</sup>* 's and the residual neutrino Dirac mass spectrum. Renormalization

**Table 1.** The representations, the transformations under *G*PS, the decompositions under *G*SM as well as the extra global charges of the superfields of our model. Here *U*<sup>C</sup> ∈ *SU*(4)C, *U*<sup>L</sup> ∈ *SU*(2)L, *U*<sup>R</sup> ∈ *SU*(2)<sup>R</sup> and , † and ∗ stand for the transpose, the hermitian conjugate and the complex conjugate of a matrix respectively.

realization of this model [13, 30], the electroweak doublets *Hu* and *Hd*, which couple to the up and down quarks respectively, are exclusively contained in the bidoublet superfield *IH*.

In addition to *<sup>G</sup>*PS, the model possesses two global *<sup>U</sup>*(1) symmetries, namely a PQ and an R symmetry, as well as a discrete **Z**mp <sup>2</sup> symmetry ('matter parity') under which *<sup>F</sup>*, *<sup>F</sup><sup>c</sup>* change sign. The last symmetry forbids undesirable mixings of *F* and *IH* and/or *F<sup>c</sup>* and *H<sup>c</sup>* and ensures the stability of the *lightest SUSY particle* (LSP). The imposed *U*(1) R symmetry, *<sup>U</sup>*(1)*R*, guarantees the linearity of the superpotential w.r.t the singlet *<sup>S</sup>*. Finally the *<sup>U</sup>*(1) PQ symmetry, *<sup>U</sup>*(1)PQ, assists us to generate the *<sup>µ</sup>*-term of the MSSM. The PQ breaking occurs at an intermediate scale through the v.e.vs of *P*, *P*¯, and the *µ*-term is generated via a non-renormalizable coupling of *P* and *IH*. Following Ref. [30], we introduce into the scheme quartic (non-renormalizable) superpotential couplings of *H*¯ *<sup>c</sup>* to *F<sup>c</sup> <sup>i</sup>* , which generate intermediate-scale masses for the *ν<sup>c</sup> <sup>i</sup>* and, thus, masses for the light neutrinos, *νi*, via the seesaw mechanism [20–22]. Moreover, these couplings allow for the decay of the inflaton into *ν<sup>c</sup> <sup>i</sup>* , leading to a reheating temperature consistent with the *<sup>G</sup>*� constraint with more or less natural values of the parameters. As shown finally in Ref. [30], the proton turns out to be practically stable in this model.

#### **2.2. Superpotential and Kähler potential**

The superpotential *W* of our model splits into three parts:

$$W = W\_{\rm MSSM} + W\_{\rm PQ} + W\_{\rm HPS} \tag{1}$$

where *<sup>M</sup>*<sup>S</sup> ≃ <sup>5</sup> · 1017 GeV is the String scale. The scalar potential, which is generated by the first term in the RHS of Eq. (3), after gravity-mediated SUSY breaking, is studied in Ref. [30, 35]. For a suitable choice of parameters, the minimum lies at |�*P*�| = |�*P*¯�| ∼ *<sup>m</sup>*3/2*M*S. Hence, the PQ symmetry breaking scale is of order

10<sup>10</sup> − 1011 GeV. The *µ*-term of the MSSM is generated from the second

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

= *<sup>µ</sup>Hd <sup>ε</sup>Hu* ⇒ *<sup>µ</sup>* ≃ *λµ*

�*P*�<sup>2</sup> *M*<sup>S</sup>

*<sup>i</sup>* [*d<sup>c</sup>*

*<sup>G</sup>*¯ *<sup>ε</sup>H*¯ *<sup>c</sup>* <sup>+</sup> *<sup>λ</sup>iν<sup>c</sup>*

*<sup>H</sup>* and ¯*d<sup>c</sup>*

 *H*¯ *cFc i* 2

*M*<sup>S</sup>

(*H*¯ *cHc* + h.c.)

*<sup>H</sup>* <sup>+</sup> h.c.

(7a)

*<sup>H</sup>*]. It takes

, (5)

 ,

(6)

, (4)

http://dx.doi.org/10.5772/51888

245

*<sup>m</sup>*3/2*M*<sup>S</sup> ≃

the form

*K* = −3*m*<sup>2</sup>

*K* = −3*m*<sup>2</sup>

<sup>P</sup> ln

*<sup>W</sup>*HPS = *<sup>λ</sup><sup>S</sup>*

<sup>1</sup> <sup>−</sup> *<sup>H</sup>c*†*H<sup>c</sup>* 3*m*<sup>2</sup> P

the various fields, *K* in Eq. (6) reads

<sup>1</sup> <sup>−</sup> *<sup>φ</sup>αφ*<sup>∗</sup>*α*¯ 3*m*<sup>2</sup> P

+ *kS* |*S*| 4 3*m*<sup>4</sup> P + *kH* 2*m*<sup>2</sup> P *νc Hν*¯ *c <sup>H</sup>* <sup>+</sup> *<sup>e</sup> c He*¯ *c <sup>H</sup>* <sup>+</sup> *<sup>u</sup><sup>c</sup> <sup>H</sup>u*¯ *c <sup>H</sup>* <sup>+</sup> *<sup>d</sup><sup>c</sup> <sup>H</sup>* ¯*d<sup>c</sup>*

<sup>P</sup> ln

*H*¯ *cHc* − *M*<sup>2</sup>

the adoption of a Kähler potential, *K*, of the following type

<sup>−</sup> *<sup>H</sup>*¯ *cH*¯ *<sup>c</sup>*† 3*m*<sup>2</sup> P

−

term of the RHS of Eq. (3) as follows:

− *λµ*

�*P*�<sup>2</sup> 2*M*<sup>S</sup>

*IHεIH ε*

the generation of intermediate Majorana [superheavy] masses for *ν<sup>c</sup>*

PS

which is of the right magnitude if *λµ* ∼ (0.001 − 0.01). Let us note that *<sup>V</sup>*PQ has an additional local minimum at *P* = *P*¯ = 0, which is separated from the global PQ minimum by a sizable potential barrier, thus preventing transitions from the trivial to the PQ vacuum. Since this situation persists at all cosmic temperatures after reheating, we are obliged to assume that, after the termination of nMHI, the system emerges with the appropriate combination of initial conditions so that it is led [36] in the PQ vacuum. • *W*HPS, is the part of *W* which is relevant for nMHI, the spontaneous breaking of *G*PS and

<sup>+</sup> *<sup>λ</sup><sup>H</sup> <sup>H</sup><sup>c</sup> <sup>G</sup>εH<sup>c</sup>* <sup>+</sup> *<sup>λ</sup>H*¯ *<sup>H</sup>*¯ *<sup>c</sup>*

<sup>−</sup> <sup>|</sup>*S*<sup>|</sup> 2 3*m*<sup>2</sup> P + *kS* |*S*| 4 3*m*<sup>4</sup> P + *kH* 2*m*<sup>2</sup> P

where *M*PS is a superheavy mass scale related to *M*GUT – see Sec. 3.2 – and *G*¯ is the dual tensor of *G*. The parameters *λ* and *M*PS can be made positive by field redefinitions.

According to the general recipe [11, 12], the implementation of nMHI within SUGRA requires

 *G*†*G* 

6*m*<sup>2</sup> P

where *<sup>m</sup>*<sup>P</sup> = 2.44 · <sup>10</sup><sup>18</sup> GeV is the reduced Planck scale and the complex scalar components of the superfields *Hc*, *H*¯ *<sup>c</sup>*, *G* and *S* are denoted by the same symbol. The coefficients *kS* and *kH* are taken real. From Eq. (6) we can infer that we adopt the standard quadratic non-minimal coupling for Higgs-inflaton, which respects the gauge and global symmetries of the model. This non-minimal coupling of the Higgs fields to gravity is transparent in the Jordan frame. We also added the fifth term in the RHS of Eq. (6) in order to cure the tachyonic mass problem encountered in similar models [10–12] – see Sec. 3.1. In terms of the components of

which are analyzed in the following.

• *W*MSSM is the part of *W* which contains the usual terms – except for the *µ* term – of the MSSM, supplemented by Yukawa interactions among the left-handed leptons and *ν<sup>c</sup> i* :

$$\mathbf{W\_{MSSM}} = y\_{ij}\mathbf{F\_i}\mathbf{H}\mathbf{F\_j^\varepsilon} = \mathbf{0}$$

$$\mathbf{S} = y\_{ij} \left(\mathbf{H\_d}^\mathsf{T}\boldsymbol{\varepsilon}\boldsymbol{L}\_i\boldsymbol{\varepsilon}\_j^\varepsilon - \mathbf{H\_u}^\mathsf{T}\boldsymbol{\varepsilon}\boldsymbol{L}\_i\boldsymbol{\upsilon\_j^\varepsilon} + \mathbf{H\_d}^\mathsf{T}\boldsymbol{\varepsilon}\boldsymbol{Q\_{ia}}\mathbf{d\_{ja}^\varepsilon} - \mathbf{H\_u}^\mathsf{T}\boldsymbol{\varepsilon}\boldsymbol{Q\_{ia}}\mathbf{u\_{ja}^\varepsilon}\right), \text{ with } \boldsymbol{\varepsilon} = \begin{Bmatrix} 0 & 1\\ -1 & 0 \end{Bmatrix}. \tag{2}$$

Here *Qi*<sup>a</sup> = *ui*<sup>a</sup> *di*<sup>a</sup> and *Li* <sup>=</sup> *ν<sup>i</sup> ei* are the *<sup>i</sup>*-th generation *SU*(2)<sup>L</sup> doublet LH quark and lepton superfields respectively. Summation over repeated color and generation indices is assumed. Obviously the model predicts YU at *M*GUT since the fermion masses per family originate from a unique term of the PS GUT. It is shown [31, 32] that exact third family YU combined with non-universalities in the gaugino sector and/or the scalar sector can become consistent with a number of phenomenological and cosmological low-energy requirements. On the other hand, it is expected on generic grounds that the predictions of this simple model for the fermion masses of the two lighter generations are not valid. Usually this difficulty can be avoided by introducing [33] an abelian symmetry which establishes a hierarchy between the flavor dependent couplings. Alternatively, the present model can be augmented [34] with other Higgs fields so that *Hu* and *Hd* are not exclusively contained in *IH*, but receive subdominant contributions from other representations too. As a consequence, a moderate violation of exact YU can be achieved, allowing for an acceptable low-energy phenomenology, even with universal boundary conditions for the soft SUSY breaking terms. However, we prefer here to work with the simplest version of the PS model, using the prediction of the third family YU in order to determine the corresponding Dirac neutrino mass – see Sec. 5.1.

• *<sup>W</sup>*PQ, is the part of *<sup>W</sup>* which is relevant for the spontaneous breaking of *<sup>U</sup>*(1)PQ and the generation of the *µ* term of the MSSM. It is given by

$$\mathcal{W}\_{\rm PQ} = \lambda\_{\rm PQ} \frac{P^2 \vec{P}^2}{M\_\odot} - \lambda\_\mu \frac{P^2}{2M\_\odot} \mathsf{Tr} \left( \mathsf{H} \varepsilon \boldsymbol{H}^\mathsf{T} \mathfrak{e} \right), \tag{3}$$

where *<sup>M</sup>*<sup>S</sup> ≃ <sup>5</sup> · 1017 GeV is the String scale. The scalar potential, which is generated by the first term in the RHS of Eq. (3), after gravity-mediated SUSY breaking, is studied in Ref. [30, 35]. For a suitable choice of parameters, the minimum lies at |�*P*�| = |�*P*¯�| ∼ *<sup>m</sup>*3/2*M*S. Hence, the PQ symmetry breaking scale is of order *<sup>m</sup>*3/2*M*<sup>S</sup> ≃ 10<sup>10</sup> − 1011 GeV. The *µ*-term of the MSSM is generated from the second term of the RHS of Eq. (3) as follows:

4 Open Questions in Cosmology

into *ν<sup>c</sup>*

intermediate-scale masses for the *ν<sup>c</sup>*

which are analyzed in the following.

*Hd εLie c*

> *ui*<sup>a</sup> *di*<sup>a</sup>

= *yij* �

Here *Qi*<sup>a</sup> =

**2.2. Superpotential and Kähler potential**

The superpotential *W* of our model splits into three parts:

*<sup>j</sup>* <sup>−</sup> *Hu <sup>ε</sup>Liν<sup>c</sup>*

and *Li* <sup>=</sup>

determine the corresponding Dirac neutrino mass – see Sec. 5.1.

generation of the *µ* term of the MSSM. It is given by

*<sup>W</sup>*PQ = *<sup>λ</sup>*PQ

practically stable in this model.

*<sup>i</sup>* and, thus, masses for the light neutrinos, *νi*, via the

*<sup>W</sup>* = *<sup>W</sup>*MSSM + *<sup>W</sup>*PQ + *<sup>W</sup>*HPS, (1)

*j*a � , with *ε* =

are the *<sup>i</sup>*-th generation *SU*(2)<sup>L</sup> doublet LH

 0 1 −1 0 · (2)

*i* :

seesaw mechanism [20–22]. Moreover, these couplings allow for the decay of the inflaton

• *W*MSSM is the part of *W* which contains the usual terms – except for the *µ* term – of the MSSM, supplemented by Yukawa interactions among the left-handed leptons and *ν<sup>c</sup>*

*j* =

quark and lepton superfields respectively. Summation over repeated color and generation indices is assumed. Obviously the model predicts YU at *M*GUT since the fermion masses per family originate from a unique term of the PS GUT. It is shown [31, 32] that exact third family YU combined with non-universalities in the gaugino sector and/or the scalar sector can become consistent with a number of phenomenological and cosmological low-energy requirements. On the other hand, it is expected on generic grounds that the predictions of this simple model for the fermion masses of the two lighter generations are not valid. Usually this difficulty can be avoided by introducing [33] an abelian symmetry which establishes a hierarchy between the flavor dependent couplings. Alternatively, the present model can be augmented [34] with other Higgs fields so that *Hu* and *Hd* are not exclusively contained in *IH*, but receive subdominant contributions from other representations too. As a consequence, a moderate violation of exact YU can be achieved, allowing for an acceptable low-energy phenomenology, even with universal boundary conditions for the soft SUSY breaking terms. However, we prefer here to work with the simplest version of the PS model, using the prediction of the third family YU in order to

• *<sup>W</sup>*PQ, is the part of *<sup>W</sup>* which is relevant for the spontaneous breaking of *<sup>U</sup>*(1)PQ and the

− *λµ*

*P*2 2*M*<sup>S</sup>

�

*IHεIH ε*

�

, (3)

*P*2*P*¯2 *M*<sup>S</sup>

*<sup>j</sup>*<sup>a</sup> <sup>−</sup> *Hu <sup>ε</sup>Qi*a*u<sup>c</sup>*

*<sup>W</sup>*MSSM = *yijFiIHF<sup>c</sup>*

 *ν<sup>i</sup> ei* 

*<sup>j</sup>* <sup>+</sup> *Hd <sup>ε</sup>Qi*a*d<sup>c</sup>*

*<sup>i</sup>* , leading to a reheating temperature consistent with the *<sup>G</sup>*� constraint with more or less natural values of the parameters. As shown finally in Ref. [30], the proton turns out to be

$$-\lambda\_{\mu}\frac{\langle P \rangle^{2}}{2M\_{\text{S}}}\mathsf{Tr}\left(H\varepsilon H^{\mathsf{T}}\varepsilon\right) = \mu H\_{d}\,^{\mathsf{T}}\varepsilon H\_{\text{u}} \quad \Rightarrow \quad \mu \simeq \lambda\_{\mu}\frac{\langle P \rangle^{2}}{M\_{\text{S}}},\tag{4}$$

which is of the right magnitude if *λµ* ∼ (0.001 − 0.01). Let us note that *<sup>V</sup>*PQ has an additional local minimum at *P* = *P*¯ = 0, which is separated from the global PQ minimum by a sizable potential barrier, thus preventing transitions from the trivial to the PQ vacuum. Since this situation persists at all cosmic temperatures after reheating, we are obliged to assume that, after the termination of nMHI, the system emerges with the appropriate combination of initial conditions so that it is led [36] in the PQ vacuum.

• *W*HPS, is the part of *W* which is relevant for nMHI, the spontaneous breaking of *G*PS and the generation of intermediate Majorana [superheavy] masses for *ν<sup>c</sup> <sup>i</sup>* [*d<sup>c</sup> <sup>H</sup>* and ¯*d<sup>c</sup> <sup>H</sup>*]. It takes the form

$$\mathcal{W}\_{\rm HPS} = \lambda S \left( \bar{H}^{\rm c} H^{\rm c} - M\_{\rm PS}^2 \right) + \lambda\_H H^{\rm c} \bar{\sf T} \,\mathrm{c} \varepsilon H^{\rm c} + \lambda\_H \bar{H}^{\rm c} \bar{\sf G} \varepsilon \bar{H}^{\rm c} \overline{\sf T} + \lambda\_{\rm ii^c} \frac{\left( \bar{H}^{\rm c} F\_{\bar{i}}^{\rm c} \right)^2}{M\_{\rm S}}, \tag{5}$$

where *M*PS is a superheavy mass scale related to *M*GUT – see Sec. 3.2 – and *G*¯ is the dual tensor of *G*. The parameters *λ* and *M*PS can be made positive by field redefinitions.

According to the general recipe [11, 12], the implementation of nMHI within SUGRA requires the adoption of a Kähler potential, *K*, of the following type

$$K = -3m\_{\rm P}^2 \ln\left(1 - \frac{H^{\rm \dagger}H^{\rm c}}{3m\_{\rm P}^2} - \frac{\bar{H}^{\rm c}\bar{H}^{\rm c\dagger}}{3m\_{\rm P}^2} - \frac{\text{Tr}\left(\bar{G}^{\dagger}G\right)}{6m\_{\rm P}^2} - \frac{|S|^2}{3m\_{\rm P}^2} + k\_S \frac{|S|^4}{3m\_{\rm P}^4} + \frac{k\_H}{2m\_{\rm P}^2} \left(\bar{H}^{\rm c}H^{\rm c} + \text{h.c.}\right)\right),\tag{6}$$

where *<sup>m</sup>*<sup>P</sup> = 2.44 · <sup>10</sup><sup>18</sup> GeV is the reduced Planck scale and the complex scalar components of the superfields *Hc*, *H*¯ *<sup>c</sup>*, *G* and *S* are denoted by the same symbol. The coefficients *kS* and *kH* are taken real. From Eq. (6) we can infer that we adopt the standard quadratic non-minimal coupling for Higgs-inflaton, which respects the gauge and global symmetries of the model. This non-minimal coupling of the Higgs fields to gravity is transparent in the Jordan frame. We also added the fifth term in the RHS of Eq. (6) in order to cure the tachyonic mass problem encountered in similar models [10–12] – see Sec. 3.1. In terms of the components of the various fields, *K* in Eq. (6) reads

$$K = -3m\_{\rm P}^2 \ln\left(1 - \frac{\phi^a \phi^{\*3}}{3m\_{\rm P}^2} + k\_{\rm S} \frac{|\mathcal{S}|^4}{3m\_{\rm P}^4} + \frac{k\_H}{2m\_{\rm P}^2} \left(\nu\_H^\varepsilon \bar{\nu}\_H^\varepsilon + e\_H^\varepsilon \bar{e}\_H^\varepsilon + \nu\_H^\varepsilon \bar{\mu}\_H^\varepsilon + d\_H^\varepsilon \bar{d}\_H^\varepsilon + \text{h.c.}\right)\right) \tag{7a}$$

with

$$\boldsymbol{\upmu}^{\mu} = \boldsymbol{\upnu}^{\varepsilon}\_{\mathrm{H}^{\prime}} \boldsymbol{\upnu}^{\varepsilon}\_{\mathrm{H}^{\prime}} \boldsymbol{e}^{\varepsilon}\_{\mathrm{H}^{\prime}} \boldsymbol{u}^{\varepsilon}\_{\mathrm{H}^{\prime}} \boldsymbol{u}^{\varepsilon}\_{\mathrm{H}^{\prime}} \boldsymbol{u}^{\varepsilon}\_{\mathrm{H}^{\prime}} \boldsymbol{d}^{\varepsilon}\_{\mathrm{H}^{\prime}} \boldsymbol{d}^{\varepsilon}\_{\mathrm{H}^{\prime}} \boldsymbol{g}^{\varepsilon}\_{\mathrm{}^{\prime}} \boldsymbol{g}^{\varepsilon} \text{ and } \mathbf{S} \tag{7b}$$

where *g* is the unified gauge coupling constant and the summation is applied over the 21

*<sup>γ</sup>*¯ , F*<sup>α</sup>* <sup>=</sup> *<sup>W</sup>*HPS,*φ<sup>α</sup>* <sup>+</sup> *<sup>K</sup>*,*φ<sup>α</sup>W*HPS/*m*<sup>2</sup>

The *φα*'s are given in Eq. (7b). If we parameterize the SM singlet components of *H<sup>c</sup>* and *H*¯ *<sup>c</sup>*

2 and *ν*¯

Along this direction, the D-terms in Eq. (10a) – and, also, *<sup>V</sup>*<sup>D</sup> in Eq. (8b) – vanish, and so *<sup>V</sup>*�HI

, *xh* <sup>=</sup> *<sup>h</sup>*

From Eq. (13), we can verify that for *<sup>c</sup>*<sup>R</sup> <sup>≫</sup> 1 and *<sup>m</sup>*PS <sup>≪</sup> 1, *<sup>V</sup>*�HI takes a form suitable for the realization of nMHI, since it develops a plateau – see also Sec. 3.2. The (almost) constant potential energy density *<sup>V</sup>*�HI0 and the corresponding Hubble parameter *<sup>H</sup>*�HI (along

We next proceed to check the stability of the trajectory in Eq. (12) w.r.t the fluctuations of the

where *X* = *e*

Notice that the field *S* can be rotated to the real axis via a suitable R transformation. Along

with *MK* =

*c <sup>H</sup>* <sup>=</sup> *he<sup>i</sup>* ¯

*<sup>H</sup>* <sup>=</sup> *<sup>u</sup>*¯ *c <sup>H</sup>* <sup>=</sup> *<sup>d</sup><sup>c</sup>*

*<sup>h</sup>* <sup>−</sup> <sup>4</sup>*m*<sup>2</sup>

*m*<sup>P</sup>

and *<sup>H</sup>*�HI <sup>=</sup> *<sup>V</sup>*�1/2 √ HI0 3*m*<sup>P</sup>

> *c <sup>H</sup>*, *<sup>u</sup><sup>c</sup> <sup>H</sup>*, *<sup>d</sup><sup>c</sup>*

 *κ κ*¯ *κ κ* ¯ 

, *<sup>κ</sup>*¯ <sup>=</sup> <sup>3</sup>*c*<sup>2</sup>

<sup>R</sup>*x*<sup>2</sup>

PS)<sup>2</sup>

*<sup>θ</sup>* sin *θν*/

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

*<sup>H</sup>* <sup>=</sup> ¯*d<sup>c</sup>*

and *<sup>c</sup>*<sup>R</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

≃ *λm*<sup>P</sup> 4 √3*c*R

√

√

*c <sup>H</sup>* <sup>=</sup> *<sup>e</sup>*¯ *c <sup>H</sup>* <sup>=</sup> *<sup>u</sup><sup>c</sup>*

*<sup>V</sup>*�HI <sup>=</sup> *<sup>m</sup>*<sup>4</sup> P *λ*2(*x*<sup>2</sup>

*m*<sup>P</sup>

P 16*c*<sup>2</sup> R

various fields. To this end, we expand them in real and imaginary parts as follows

*<sup>h</sup>*, *<sup>m</sup>*PS <sup>=</sup> *<sup>M</sup>*PS

<sup>16</sup> *<sup>f</sup>* <sup>2</sup> <sup>≃</sup> *<sup>λ</sup>*2*m*<sup>4</sup>

<sup>P</sup> and *Da* <sup>=</sup> *φα* (*Ta*)

*<sup>H</sup>* <sup>=</sup> *<sup>g</sup><sup>c</sup>* <sup>=</sup> *<sup>g</sup>*¯

<sup>16</sup> *<sup>f</sup>* <sup>2</sup> (13)

6 + *kH* *α*

http://dx.doi.org/10.5772/51888

2, (11)

*<sup>β</sup> K*,*φ<sup>β</sup>* (10b)

247

*<sup>c</sup>* = 0. (12)

<sup>4</sup> · (14)

· (15)

*<sup>H</sup>*, *<sup>g</sup><sup>c</sup>* and *<sup>x</sup>* <sup>=</sup> *<sup>e</sup>*, *<sup>u</sup>*, *<sup>d</sup>*, *<sup>g</sup>* . (16)

*<sup>h</sup>* and *<sup>κ</sup>* <sup>=</sup> *<sup>f</sup>* <sup>+</sup> *<sup>κ</sup>*¯. (17)

generators *Ta* of the PS gauge group – see Ref. [5]. Also, we have

*<sup>H</sup>* <sup>=</sup> *hei<sup>θ</sup>* cos *θν*/

*β*¯

*νc*

we can easily deduce that a D-flat direction occurs at

*<sup>θ</sup>* = 0, *θν* = *<sup>π</sup>*/4 and *<sup>e</sup>*

*<sup>K</sup>αβ*¯ <sup>=</sup> *<sup>K</sup>*,*φαφ*<sup>∗</sup>*β*¯ , *<sup>K</sup>βα*¯ *<sup>K</sup>αγ*¯ <sup>=</sup> *<sup>δ</sup>*

*θ* = ¯

with *<sup>f</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>c</sup>*R*x*<sup>2</sup>

*<sup>X</sup>* <sup>=</sup> *<sup>x</sup>*<sup>1</sup> <sup>+</sup> *ix*<sup>2</sup> √

the trajectory in Eq. (12) we find

 *MK <sup>f</sup>* <sup>2</sup> , <sup>1</sup> *f* , ..., <sup>1</sup> *f* � �� � 3+6·3 times

� *Kαβ*¯ � = diag

the trajectory in Eq. (12)) are given by

*<sup>V</sup>*�HI0 <sup>=</sup> *<sup>λ</sup>*2*h*<sup>4</sup>

<sup>2</sup> , *<sup>X</sup>*¯ <sup>=</sup> *<sup>x</sup>*¯1 <sup>+</sup> *ix*¯2

√2

takes the form

by

and summation over the repeated Greek indices is implied.

#### **2.3. The SUSY limit**

In the limit where *m*<sup>P</sup> tends to infinity, we can obtain the SUSY limit of the SUGRA potential. Assuming that the SM non-singlet components vanish, the F-term potential in this limit, *V*F, turns out to be

$$V\_{\rm F} = \lambda^2 \left| \overline{\nu}\_H^c \nu\_H^c - M\_{\rm PS}^2 \right|^2 + \lambda^2 |\mathbf{S}|^2 \left( |\nu\_H^c|^2 + |\overline{\nu}\_H^c|^2 \right),\tag{8a}$$

while the D-term potential is

$$V\_{\rm D} = \frac{5g^2}{16} \left( |\nu\_H^{\varepsilon}|^2 - |\bar{\nu}\_H^{\varepsilon}|^2 \right)^2. \tag{8b}$$

Restricting ourselves to the D-flat direction |*ν<sup>c</sup> <sup>H</sup>*<sup>|</sup> <sup>=</sup> <sup>|</sup>*ν*¯*<sup>c</sup> <sup>H</sup>*|, we find from *<sup>V</sup>*<sup>F</sup> that the SUSY vacuum lies at

$$|\langle S \rangle \simeq 0 \quad \text{and} \quad |\langle \boldsymbol{\upsilon}\_H^c \rangle| = |\langle \boldsymbol{\overline{\upsilon}}\_H^c \rangle| = M\_{\rm PS}. \tag{9}$$

Therefore, *W*HPS leads to spontaneous breaking of *G*PS. As we shall see in Sec. 3, the same superpotential, *W*HPS, gives rise to a stage of nMHI . Indeed, along the D-flat direction |*νc <sup>H</sup>*<sup>|</sup> <sup>=</sup> <sup>|</sup>*ν*¯*<sup>c</sup> <sup>H</sup>*| ≫ *<sup>M</sup>*PS and *<sup>S</sup>* <sup>=</sup> 0, *<sup>V</sup>*SUSY tends to a quartic potential, which can be employed in conjunction with *K* in Eq. (6) for the realization of nMHI along the lines of Ref. [12].

It should be mentioned that soft SUSY breaking and instanton effects explicitly break *<sup>U</sup>*(1)*<sup>R</sup>* × *<sup>U</sup>*(1)PQ to **<sup>Z</sup>**<sup>2</sup> × **<sup>Z</sup>**6. The latter symmetry is spontaneously broken by �*P*� and �*P*¯�. This would lead to a domain wall problem if the PQ transition took place after nMHI. However, as we already mentioned above, *<sup>U</sup>*(1)PQ is assumed already broken before or during nMHI. The final unbroken symmetry of the model is *<sup>G</sup>*SM <sup>×</sup> **<sup>Z</sup>**mp <sup>2</sup> .

#### **3. The inflationary scenario**

Next we outline the salient features of our inflationary scenario (Sec. 3.1) and calculate a number of observable quantities in Sec. 3.2.

#### **3.1. Structure of the inflationary potential**

At tree-level the *Einstein Frame* (EF) SUGRA potential, *<sup>V</sup>*HI, is given by [11]

$$\hat{V}\_{\rm HI} = e^{K/m\_{\rm P}^2} \left( K^{a\bar{\beta}} \mathcal{F}\_a \mathcal{F}\_{\bar{\beta}}^\* - 3 \frac{|\mathcal{W}\_{\rm HFS}|^2}{m\_{\rm P}^2} \right) + \frac{1}{2} g^2 \sum\_a D\_a D\_{a\nu} \tag{10a}$$

where *g* is the unified gauge coupling constant and the summation is applied over the 21 generators *Ta* of the PS gauge group – see Ref. [5]. Also, we have

$$K\_{a\bar{\beta}} = K\_{\rho\bar{\rho}^a\phi^{\bar{\rho}}}, \; K^{\bar{\theta}a}K\_{a\bar{\gamma}} = \delta^{\bar{\beta}}\_{\bar{\gamma}\prime}, \; \mathcal{F}\_{\mathfrak{a}} = W\_{\mathrm{HFS},\phi^a} + K\_{\rho\theta}W\_{\mathrm{HFS}}/m\_{\mathrm{P}}^2 \text{ and } \; D\_{\mathfrak{a}} = \phi\_{\mathfrak{a}}\left(\mathcal{T}\_{\mathfrak{a}}\right)^{\underline{\mathfrak{a}}}\_{\bar{\beta}}\mathcal{K}\_{\phi^{\bar{\theta}}} \tag{10b}$$

The *φα*'s are given in Eq. (7b). If we parameterize the SM singlet components of *H<sup>c</sup>* and *H*¯ *<sup>c</sup>* by

$$\upsilon\_H^\varepsilon = h e^{i\theta} \cos \theta\_\upsilon / \sqrt{2} \quad \text{and} \quad \overline{\upsilon}\_H^\varepsilon = h e^{i\overline{\theta}} \sin \theta\_\upsilon / \sqrt{2},\tag{11}$$

we can easily deduce that a D-flat direction occurs at

$$
\theta = \overline{\theta} = 0, \; \theta\_V = \pi/4 \text{ and } \; \underline{e}\_H^{\underline{c}} = \overline{e}\_H^{\underline{c}} = \mathfrak{u}\_H^{\underline{c}} = \overline{\mathfrak{u}}\_H^{\underline{c}} = \mathfrak{d}\_H^{\underline{c}} = \overline{\mathfrak{d}}\_H^{\underline{c}} = \mathfrak{g}^{\underline{c}} = \overline{\mathfrak{g}}^{\underline{c}} = 0. \tag{12}
$$

Along this direction, the D-terms in Eq. (10a) – and, also, *<sup>V</sup>*<sup>D</sup> in Eq. (8b) – vanish, and so *<sup>V</sup>*�HI takes the form

$$
\hat{V}\_{\rm HI} = m\_{\rm P}^4 \frac{\lambda^2 (\chi\_h^2 - 4m\_{\rm PS}^2)^2}{16f^2} \tag{13}
$$

6 Open Questions in Cosmology

**2.3. The SUSY limit**

while the D-term potential is

**3. The inflationary scenario**

number of observable quantities in Sec. 3.2.

**3.1. Structure of the inflationary potential**

*<sup>V</sup>*HI <sup>=</sup> *<sup>e</sup>*

*K*/*m*<sup>2</sup> P *<sup>K</sup>αβ*¯ <sup>F</sup>*α*F<sup>∗</sup> *<sup>β</sup>*¯ <sup>−</sup> <sup>3</sup>

turns out to be

vacuum lies at


*<sup>V</sup>*<sup>F</sup> = *<sup>λ</sup>*<sup>2</sup>

Restricting ourselves to the D-flat direction |*ν<sup>c</sup>*

*<sup>H</sup>*, *ν*¯ *c <sup>H</sup>*,*e c <sup>H</sup>*,*e*¯ *c <sup>H</sup>*, *<sup>u</sup><sup>c</sup> <sup>H</sup>*, *u*¯ *c <sup>H</sup>*, *<sup>d</sup><sup>c</sup> <sup>H</sup>*, ¯*d<sup>c</sup> <sup>H</sup>*, *<sup>g</sup><sup>c</sup>* , *g*¯

In the limit where *m*<sup>P</sup> tends to infinity, we can obtain the SUSY limit of the SUGRA potential. Assuming that the SM non-singlet components vanish, the F-term potential in this limit, *V*F,

> + *λ*2|*S*| 2 |*νc H*| <sup>2</sup> + |*ν*¯ *c H*| 2

*<sup>H</sup>*<sup>|</sup> <sup>=</sup> <sup>|</sup>*ν*¯*<sup>c</sup>*

*c*

*<sup>H</sup>*�| <sup>=</sup> |�*ν*¯

*<sup>H</sup>*| ≫ *<sup>M</sup>*PS and *<sup>S</sup>* <sup>=</sup> 0, *<sup>V</sup>*SUSY tends to a quartic potential, which can be employed in

Therefore, *W*HPS leads to spontaneous breaking of *G*PS. As we shall see in Sec. 3, the same superpotential, *W*HPS, gives rise to a stage of nMHI . Indeed, along the D-flat direction

It should be mentioned that soft SUSY breaking and instanton effects explicitly break *<sup>U</sup>*(1)*<sup>R</sup>* × *<sup>U</sup>*(1)PQ to **<sup>Z</sup>**<sup>2</sup> × **<sup>Z</sup>**6. The latter symmetry is spontaneously broken by �*P*� and �*P*¯�. This would lead to a domain wall problem if the PQ transition took place after nMHI. However, as we already mentioned above, *<sup>U</sup>*(1)PQ is assumed already broken before or

Next we outline the salient features of our inflationary scenario (Sec. 3.1) and calculate a


*m*2 P

conjunction with *K* in Eq. (6) for the realization of nMHI along the lines of Ref. [12].

and summation over the repeated Greek indices is implied.

 *ν*¯ *c Hν<sup>c</sup> <sup>H</sup>* <sup>−</sup> *<sup>M</sup>*<sup>2</sup> PS 2

> *<sup>V</sup>*<sup>D</sup> <sup>=</sup> <sup>5</sup>*g*<sup>2</sup> 16 |*νc H*| <sup>2</sup> − |*ν*¯ *c H*| 2 2

�*S*� ≃ 0 and |�*ν<sup>c</sup>*

during nMHI. The final unbroken symmetry of the model is *<sup>G</sup>*SM <sup>×</sup> **<sup>Z</sup>**mp

At tree-level the *Einstein Frame* (EF) SUGRA potential, *<sup>V</sup>*HI, is given by [11]

*<sup>c</sup>* and *S* (7b)

, (8a)

. (8b)

*<sup>H</sup>*|, we find from *<sup>V</sup>*<sup>F</sup> that the SUSY

*<sup>H</sup>*�| <sup>=</sup> *<sup>M</sup>*PS. (9)

<sup>2</sup> .

*DaDa*, (10a)

with

$$f = 1 + c\chi x\_{h'}^2 \quad m\_{\rm PS} = \frac{M\_{\rm PS}}{m\_{\rm P}} \quad x\_h = \frac{h}{m\_{\rm P}} \quad \text{and} \quad c\chi = -\frac{1}{6} + \frac{k\_H}{4} \tag{14}$$

From Eq. (13), we can verify that for *<sup>c</sup>*<sup>R</sup> <sup>≫</sup> 1 and *<sup>m</sup>*PS <sup>≪</sup> 1, *<sup>V</sup>*�HI takes a form suitable for the realization of nMHI, since it develops a plateau – see also Sec. 3.2. The (almost) constant potential energy density *<sup>V</sup>*�HI0 and the corresponding Hubble parameter *<sup>H</sup>*�HI (along the trajectory in Eq. (12)) are given by

$$
\hat{V}\_{\rm HI} = \frac{\lambda^2 \hbar^4}{16f^2} \simeq \frac{\lambda^2 m\_{\rm P}^4}{16c\_{\mathcal{R}}^2} \text{ and } \hat{H}\_{\rm HI} = \frac{\hat{V}\_{\rm HI}^{1/2}}{\sqrt{3}m\_{\rm P}} \simeq \frac{\lambda m\_{\rm P}}{4\sqrt{3}c\_{\mathcal{R}}} \text{.} \tag{15}
$$

We next proceed to check the stability of the trajectory in Eq. (12) w.r.t the fluctuations of the various fields. To this end, we expand them in real and imaginary parts as follows

$$X = \frac{\mathbf{x}\_1 + i\mathbf{x}\_2}{\sqrt{2}}, \quad \bar{X} = \frac{\bar{\mathbf{x}}\_1 + i\bar{\mathbf{x}}\_2}{\sqrt{2}} \text{ where } \bar{X} = \mathfrak{e}\_{H'}^\varepsilon u\_{H'}^\varepsilon d\_{H'}^\varepsilon \mathfrak{g}^\varepsilon \text{ and } \mathbf{x} = \mathbf{e}\_\prime u\_\prime d\_\prime \mathfrak{g} \text{ .}\tag{16}$$

Notice that the field *S* can be rotated to the real axis via a suitable R transformation. Along the trajectory in Eq. (12) we find

$$\left(\mathbf{K}\_{a\tilde{\mathbb{R}}}\right) = \text{diag}\left(\underbrace{M\_{\mathbf{K}}}\_{\mathbf{3}^{+}}, \underbrace{\mathbf{1}\_{\mathbf{3}^{+}}}\_{\mathbf{3}^{+} + \mathbf{6} \text{ times}}, \frac{\mathbf{1}\_{\mathbf{3}^{+}}}{\mathbf{1}\_{\mathbf{3}^{+}}}\right) \text{ with } M\_{\mathbf{K}} = \begin{cases} \text{x} & \text{x} \\ \text{\(\tilde{\mathbf{x}} \cdot \mathbf{\tilde{x}}\)}, \text{ \(\tilde{\mathbf{x}} = 3c\_{\mathbf{R}}^{2} x\_{\mathbf{h}}^{2}\text{ and } \mathbf{x} = \mathbf{f} + \tilde{\mathbf{x}}. & \text{(17)} \end{cases}$$

To canonically normalize the fields *ν<sup>c</sup> <sup>H</sup>* and *<sup>ν</sup>*¯*<sup>c</sup> <sup>H</sup>*, we first diagonalize the matrix *MK*. This can be achieved via a similarity transformation involving an orthogonal matrix *UK* as follows:

$$\mathbf{U}\_{\mathbf{K}} \mathbf{M}\_{\mathbf{K}} \mathbf{U}\_{\mathbf{K}}^{\mathsf{T}} = \mathsf{diag} \left( \vec{f}, f \right), \text{ where } \vec{f} = f + 6c\_{\mathsf{R}}^{2} \mathbf{x}\_{\mathsf{H}}^{2} \text{ and } \mathbf{U}\_{\mathbf{K}} = \frac{1}{\sqrt{2}} \begin{Bmatrix} 1 & 1 \\ -1 & 1 \end{Bmatrix}. \tag{18}$$

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model 9

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

(6*kS <sup>f</sup>* − <sup>1</sup>) /6 *<sup>f</sup>* <sup>2</sup> ¯

*<sup>H</sup>* <sup>S</sup>ectors

/24 *f* <sup>2</sup> *θ*

/12 *<sup>f</sup>* <sup>2</sup> *<sup>u</sup>*<sup>a</sup>

*<sup>u</sup>*<sup>−</sup> *<sup>e</sup>*1−, *<sup>e</sup>*2<sup>+</sup>

/24 *<sup>f</sup>* <sup>2</sup> *<sup>g</sup>*<sup>a</sup>

/24 *<sup>f</sup>* <sup>2</sup> *<sup>g</sup>*¯

/4 *f* <sup>2</sup> *d*

/12 *f* <sup>2</sup> *d*

HI *θ* +

*<sup>f</sup> <sup>S</sup>*

*ν*

http://dx.doi.org/10.5772/51888

249

<sup>1</sup>−, *<sup>u</sup>*<sup>a</sup> <sup>2</sup>+,

1, *g*a 2

a 1, *g*¯ a 2

a <sup>1</sup>+, *d* a 2−

a <sup>1</sup>−, *d* a 2+

*<sup>H</sup>* fields which are related with

*<sup>H</sup>* and the *<sup>g</sup>c*, *<sup>g</sup>*¯*<sup>c</sup>* fields. Upon

*<sup>S</sup>* <sup>&</sup>gt; <sup>0</sup>

*θ ν* and

with *x* = *u*,*e*, *d* and *g*. (24)

1,2 acquire from the second and third

<sup>−</sup> for *<sup>λ</sup><sup>H</sup>* of order unity. We have also

**Fields Masses Squared Eigenstates**

*<sup>H</sup>* <sup>S</sup>ector

*<sup>h</sup>* <sup>−</sup> <sup>6</sup>) + <sup>15</sup>*g*<sup>2</sup> *<sup>f</sup>*

*<sup>h</sup>* (<sup>1</sup> <sup>+</sup> <sup>6</sup>*c*R) /12*J*<sup>2</sup> *<sup>f</sup>* <sup>3</sup> <sup>≃</sup> <sup>4</sup>*<sup>H</sup>*<sup>2</sup>

*<sup>H</sup>* – *e*¯ *c*

<sup>a</sup> (a <sup>=</sup> 1, 2, 3) Sectors

*H f* 

PS and we assume *<sup>λ</sup><sup>H</sup>* <sup>≃</sup> *<sup>λ</sup>H*¯ for the derivation of the masses of the scalars in the

*<sup>H</sup>*,*e*¯ *c*

*<sup>H</sup>*, ¯*d<sup>c</sup>*

*<sup>H</sup>* and the *<sup>e</sup><sup>c</sup>*

*<sup>h</sup>* <sup>−</sup> <sup>3</sup>) + <sup>3</sup>*g*<sup>2</sup> *<sup>f</sup>*

*<sup>h</sup>* <sup>+</sup> <sup>24</sup>*λ*<sup>2</sup> *H*¯ *f* 

*<sup>h</sup>* <sup>+</sup> <sup>24</sup>*λ*<sup>2</sup> *H f* 

*λ*<sup>2</sup> + 4*λ*<sup>2</sup>

*<sup>H</sup>* – *<sup>ν</sup>*¯*<sup>c</sup>*

The *S* – *ν<sup>c</sup>*

P*x*2

*<sup>H</sup>*<sup>a</sup> (a <sup>=</sup> 1, 2, 3) and *<sup>e</sup><sup>c</sup>*

*m*2 *<sup>e</sup>*<sup>−</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup>

<sup>a</sup> – *g*¯*<sup>c</sup>*

P*x*2 *h* 

**Table 2.** The scalar mass spectrum of our model along the inflationary trajectory of Eq. (12). To avoid very lengthy formulas

involved in the expansion of *<sup>V</sup>*HI. We arrange our findings into three groups: the SM singlet

√2 *<sup>x</sup>*¯2 <sup>∓</sup> *<sup>x</sup>*<sup>2</sup>

As we observe from the relevant eigenvalues, no instability – as the one found in Ref. [37] – arises in the spectrum. In particular, it is evident that *kS* 1 assists us to achieve *m*<sup>2</sup>

*<sup>u</sup>*<sup>−</sup> – proportional to the gauge coupling constant *<sup>g</sup>* <sup>≃</sup> 0.7 – ensure the positivity of these

numerically verified that the masses of the various scalars remain greater than the Hubble parameter during the last 50 − 60 e-foldings of nMHI, and so any inflationary perturbations

The 8 Goldstone bosons, associated with the modes *<sup>x</sup>*<sup>1</sup><sup>+</sup> and *<sup>x</sup>*<sup>2</sup><sup>−</sup> with *<sup>x</sup>* <sup>=</sup> *<sup>u</sup>*<sup>a</sup> and *<sup>e</sup>*, are not exactly massless since *<sup>V</sup>*HI,*<sup>h</sup>* �<sup>=</sup> 0 – contrary to the situation of Ref. [30] where the

*mx*<sup>0</sup> = *<sup>λ</sup>m*P*xh*/2 *<sup>f</sup>* . On the contrary, the angular parametrization in Eq. (11) assists us

*d*

*<sup>H</sup>*� minimizes the potential. These masses turn out to be

– in accordance with the results of Ref. [12]. Moreover, the D-term contributions to *m*<sup>2</sup>

*<sup>H</sup>*, *<sup>u</sup>*¯*<sup>c</sup>*

diagonalization of the relevant matrices we obtain the following mass eigenstates:

and *<sup>x</sup>*<sup>2</sup><sup>∓</sup> <sup>=</sup> <sup>1</sup>

P*x*2 *h* 12 + *x*<sup>2</sup> *h* ¯ *f* 

P*x*2 *h λ*2(*x*<sup>2</sup>

*<sup>H</sup>*<sup>a</sup> and *<sup>g</sup><sup>c</sup>*

P*x*2 *h λ*<sup>2</sup> *x*2 *<sup>h</sup>* <sup>−</sup> <sup>3</sup> + 12*λ*<sup>2</sup> *H f* 

*<sup>g</sup>* <sup>=</sup> *<sup>m</sup>*<sup>2</sup> P*x*2 *h λ*2*x*<sup>2</sup>

*m*2 *<sup>g</sup>*¯ <sup>=</sup> *<sup>m</sup>*<sup>2</sup> P*x*2 *h λ*2*x*<sup>2</sup>

*m*2 *d* <sup>+</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup>

= *m*<sup>2</sup> P*x*2 *h* 2*λ*2(*x*<sup>2</sup>

<sup>+</sup> <sup>=</sup> *<sup>λ</sup>*2*m*<sup>4</sup>

*<sup>S</sup>* <sup>=</sup> *<sup>λ</sup>*2*m*<sup>2</sup>

*<sup>u</sup>*<sup>−</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup>

*θ ν*

*m*2 *θ*

*<sup>H</sup>*<sup>a</sup> – *<sup>u</sup>*¯*<sup>c</sup>*

*<sup>H</sup>*<sup>a</sup> – ¯*d<sup>c</sup>*

*m*2 *d* <sup>−</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup>

*<sup>H</sup>*, the sector with the *<sup>u</sup><sup>c</sup>*

the broken generators of *G*PS and the sector with the *d<sup>c</sup>*

masses squared. Finally the masses that the scalars *d*

term of the RHS of Eq. (5) lead to the positivity of *m*<sup>2</sup>

of the fields other than the inflaton are safely eliminated.

2 real scalars *m*<sup>2</sup>

1 complex scalar *m*<sup>2</sup>

2(3 + 1) real scalars *m*<sup>2</sup>

The *u<sup>c</sup>*

The *d<sup>c</sup>*

3 · 8 real scalars *m*<sup>2</sup>

we neglect terms proportional to *m*<sup>2</sup>

*<sup>H</sup>* <sup>−</sup> *<sup>ν</sup>*¯*<sup>c</sup>*

*<sup>x</sup>*<sup>1</sup><sup>±</sup> <sup>=</sup> <sup>1</sup> √2 *<sup>x</sup>*¯1 <sup>±</sup> *<sup>x</sup>*<sup>1</sup>

direction with non vanishing �*ν<sup>c</sup>*

*<sup>H</sup>* and ¯*d<sup>c</sup> H*.

superfields *d<sup>c</sup>*

*m*2

sector, *S* − *ν<sup>c</sup>*

Utilizing *UK*, the kinetic terms of the various fileds can be brought into the following form

$$\mathcal{K}\_{\mathbf{a}\overline{\boldsymbol{\beta}}}\dot{\boldsymbol{\phi}}^{\mathbf{a}}\dot{\boldsymbol{\phi}}^{\mathbf{a}\overline{\boldsymbol{\phi}}} = \frac{\overline{f}}{2f^{2}}\left(\dot{\boldsymbol{h}}^{2} + \frac{1}{2}h^{2}\dot{\theta}\_{+}^{2}\right) + \frac{h^{2}}{2f}\left(\frac{1}{2}\dot{\theta}\_{-}^{2} + \dot{\theta}\_{\mathbf{v}}^{2}\right) + \frac{1}{2f}\dot{\chi}\_{\mathbf{a}}\dot{\chi}\_{\mathbf{a}} = \frac{1}{2}\dot{\hat{\boldsymbol{h}}}^{2} + \frac{1}{2}\dot{\hat{\psi}}\_{\mathbf{a}}\dot{\hat{\psi}}\_{\mathbf{a}\prime} \tag{19}$$

where *θ*<sup>±</sup> = � ¯ *θ* ± *θ* � / √2, *χα* = *<sup>x</sup>*1, *<sup>x</sup>*2, *<sup>x</sup>*¯1, *<sup>x</sup>*¯2, *<sup>S</sup>* and *ψα* = *<sup>θ</sup>*+, *<sup>θ</sup>*−, *θν*, *χα* and the dot denotes derivation w.r.t the cosmic time, *t*. In the last line, we introduce the EF canonically normalized fields, �*<sup>h</sup>* and *<sup>ψ</sup>*�, which can be obtained as follows – cf. Ref. [5, 11, 12, 37]:

$$\frac{d\widehat{\boldsymbol{h}}}{d\boldsymbol{h}} = \boldsymbol{I} = \frac{\sqrt{\boldsymbol{f}}}{f}, \ \widehat{\boldsymbol{\theta}}\_{+} = \frac{\boldsymbol{f}h\boldsymbol{\theta}\_{+}}{\sqrt{2}}, \ \widehat{\boldsymbol{\theta}}\_{-} = \frac{h\boldsymbol{\theta}\_{-}}{\sqrt{2}f}, \ \widehat{\boldsymbol{\theta}}\_{\boldsymbol{v}} = \frac{h}{\sqrt{f}}\left(\boldsymbol{\theta}\_{\boldsymbol{v}} - \frac{\pi}{4}\right) \text{ and } \widehat{\chi}\_{\boldsymbol{a}} = \frac{\chi\_{\boldsymbol{a}}}{\sqrt{f}}.\tag{20}$$

Taking into account the approximate expressions for ˙ *<sup>h</sup>*, *<sup>J</sup>* and the slow-roll parameters �*ǫ*, *<sup>η</sup>*�, which are displayed in Sec. 3.2, we can verify that, during a stage of slow-roll inflation, �˙ *θ*<sup>+</sup> ≃ *Jh* ˙ *θ*+/ √2 since *Jh* <sup>≃</sup> <sup>√</sup>6*m*P, �˙ *θ*<sup>−</sup> ≃ *h* ˙ *θ*−/ �<sup>2</sup> *<sup>f</sup>* and �˙ *θν* ≃ *<sup>h</sup>* ˙ *θν*/ �*f* since *h*/ �*<sup>f</sup>* ≃ *<sup>m</sup>*P/ <sup>√</sup>*c*R. On the other hand, we can show that *<sup>χ</sup>*�˙ *<sup>α</sup>* ≃ *<sup>χ</sup>*˙ *<sup>α</sup>*/ �*f* , since the quantity ˙ *f* /2 *f* 3/2*χα*, involved in relating *<sup>χ</sup>*˙ *<sup>α</sup>* to *<sup>χ</sup>*�˙ *<sup>α</sup>*, turns out to be negligibly small compared with *<sup>χ</sup>*�˙ *<sup>α</sup>*. Indeed, the *<sup>χ</sup>*�*α*'s acquire effective masses *<sup>m</sup>χ*�*<sup>α</sup>* <sup>≫</sup> *<sup>H</sup>*�HI – see below – and therefore enter a phase of oscillations about *<sup>χ</sup>*�*<sup>α</sup>* <sup>=</sup> 0 with decreasing amplitude. Neglecting the oscillatory part of the relevant solutions, we find

$$\chi \simeq \hat{\chi}\_{\rm a0} \sqrt{f} e^{-2\hat{N}/3} \text{ and } \dot{\hat{\chi}}\_{\rm a} \simeq -2\chi\_{\rm a0} \sqrt{f} \hat{H}\_{\rm HI} \hat{\eta}\_{\chi\_{\rm a}} e^{-2\hat{N}/3},\tag{21}$$

where *<sup>χ</sup>*�*α*<sup>0</sup> represents the initial amplitude of the oscillations, *<sup>η</sup>*�*χα* <sup>=</sup> *<sup>m</sup>*<sup>2</sup> *χ*�*α* /3*H*�HI and we assume *<sup>χ</sup>*�˙ *<sup>α</sup>*(*<sup>t</sup>* = <sup>0</sup>) = 0. Taking into account the approximate expressions for ˙ *h* and the slow-roll parameter �*<sup>ǫ</sup>* in Sec. 3.2, we find

$$-f/2f^{3/2}\chi\_{\mathfrak{a}} = \left(c\_{\mathfrak{R}}\hat{\varepsilon}\hat{H}\_{\text{HI}}^2/m\_{\hat{\chi}\_{\mathfrak{a}}}^2\right)\hat{\chi}\_{\mathfrak{a}} \ll \hat{\chi}\_{\mathfrak{a}}.\tag{22}$$

Having defined the canonically normalized scalar fields, we can proceed in investigating the stability of the inflationary trajectory of Eq. (12). To this end, we expand *<sup>V</sup>*�HI in Eq. (10a) to quadratic order in the fluctuations around the direction of Eq. (12), as described in detail in Ref. [5]. In Table 2 we list the eigenvalues of the mass-squared matrices

$$M\_{a\beta}^2 = \left. \frac{\partial^2 \hat{V}\_{\text{HI}}}{\partial \hat{\psi}\_a \partial \hat{\psi}\_\beta} \right|\_{\text{Eq. (12)}} \quad \text{with} \quad \psi\_\text{\"\!\/} = \theta\_+, \theta\_-, \theta\_{\text{V}}, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{\tilde{x}}\_1, \mathbf{\tilde{x}}\_2 \text{ and } \mathbf{S} \tag{23}$$


8 Open Questions in Cosmology

248 Open Questions in Cosmology

*<sup>K</sup>αβ*¯*φ*˙ *αφ*˙ <sup>∗</sup>*β*¯ <sup>=</sup> ¯

where *θ*<sup>±</sup> = � ¯

*d*�*h dh* <sup>=</sup> *<sup>J</sup>* <sup>=</sup>

relating *<sup>χ</sup>*˙ *<sup>α</sup>* to *<sup>χ</sup>*�˙

*Jh* ˙ *θ*+/ √

we find

assume *<sup>χ</sup>*�˙

To canonically normalize the fields *ν<sup>c</sup>*

*UKMKUK* <sup>=</sup> diag � ¯

*f* 2 *f* <sup>2</sup> � ˙ *h*<sup>2</sup> + 1 2 *h*<sup>2</sup> ˙ *θ*2 + � + *h*2 2 *f*

*θ* ± *θ* � / √

> � ¯ *f <sup>f</sup>* , *<sup>θ</sup>*

2 since *Jh* <sup>≃</sup> <sup>√</sup>6*m*P, �˙

the other hand, we can show that *<sup>χ</sup>*�˙

*<sup>χ</sup>* <sup>≃</sup> *<sup>χ</sup>*�*α*<sup>0</sup>

− ˙

� � � � � Eq. (12)

slow-roll parameter �*<sup>ǫ</sup>* in Sec. 3.2, we find

*M*<sup>2</sup>

*αβ* <sup>=</sup> *<sup>∂</sup>*2*V*�HI *∂ψ*�*α∂ψ*�*<sup>β</sup>*

�<sup>+</sup> <sup>=</sup> *Jhθ*<sup>+</sup> √<sup>2</sup> , *<sup>θ</sup>*

Taking into account the approximate expressions for ˙

*θ*<sup>−</sup> ≃ *h* ˙

*θ*−/

�*f e*−2*N*�/3 and *<sup>χ</sup>*�˙

*<sup>f</sup>* /2 *<sup>f</sup>* 3/2*χα* =

Ref. [5]. In Table 2 we list the eigenvalues of the mass-squared matrices

where *<sup>χ</sup>*�*α*<sup>0</sup> represents the initial amplitude of the oscillations, *<sup>η</sup>*�*χα* <sup>=</sup> *<sup>m</sup>*<sup>2</sup>

*<sup>α</sup>* ≃ *<sup>χ</sup>*˙ *<sup>α</sup>*/

*<sup>α</sup>*, turns out to be negligibly small compared with *<sup>χ</sup>*�˙

*f* , *f* � *<sup>H</sup>* and *<sup>ν</sup>*¯*<sup>c</sup>*

, where ¯

be achieved via a similarity transformation involving an orthogonal matrix *UK* as follows:

Utilizing *UK*, the kinetic terms of the various fileds can be brought into the following form

�1 2 ˙ *θ*2 <sup>−</sup> + ˙ *θ*2 *ν* � + 1 2 *f*

derivation w.r.t the cosmic time, *t*. In the last line, we introduce the EF canonically

, *θ* �*<sup>ν</sup>* <sup>=</sup> *<sup>h</sup>* �*f* � *θν* <sup>−</sup> *<sup>π</sup>* 4 �

which are displayed in Sec. 3.2, we can verify that, during a stage of slow-roll inflation, �˙

�<sup>2</sup> *<sup>f</sup>* and �˙

effective masses *<sup>m</sup>χ*�*<sup>α</sup>* <sup>≫</sup> *<sup>H</sup>*�HI – see below – and therefore enter a phase of oscillations about *<sup>χ</sup>*�*<sup>α</sup>* <sup>=</sup> 0 with decreasing amplitude. Neglecting the oscillatory part of the relevant solutions,

*<sup>α</sup>*(*<sup>t</sup>* = <sup>0</sup>) = 0. Taking into account the approximate expressions for ˙

Having defined the canonically normalized scalar fields, we can proceed in investigating the stability of the inflationary trajectory of Eq. (12). To this end, we expand *<sup>V</sup>*�HI in Eq. (10a) to quadratic order in the fluctuations around the direction of Eq. (12), as described in detail in

� *<sup>c</sup>*R�*ǫH*�<sup>2</sup>

*<sup>α</sup>* ≃ −2*χα*<sup>0</sup>

HI/*m*<sup>2</sup> *χ*�*α* � *χ*�˙ *<sup>α</sup>* <sup>≪</sup> *<sup>χ</sup>*�˙

*θν* ≃ *<sup>h</sup>* ˙ *θν*/

�*f* , since the quantity ˙

�*<sup>f</sup> <sup>H</sup>*�HI*η*�*χα <sup>e</sup>*

with *ψα* = *<sup>θ</sup>*+, *<sup>θ</sup>*−, *θν*, *<sup>x</sup>*1, *<sup>x</sup>*2, *<sup>x</sup>*¯1, *<sup>x</sup>*¯2 and *<sup>S</sup>* (23)

normalized fields, �*<sup>h</sup>* and *<sup>ψ</sup>*�, which can be obtained as follows – cf. Ref. [5, 11, 12, 37]:

�<sup>−</sup> <sup>=</sup> *<sup>h</sup>θ*<sup>−</sup> �<sup>2</sup> *<sup>f</sup>*

*f* = *f* + 6*c*<sup>2</sup>

<sup>R</sup>*x*<sup>2</sup>

2, *χα* = *<sup>x</sup>*1, *<sup>x</sup>*2, *<sup>x</sup>*¯1, *<sup>x</sup>*¯2, *<sup>S</sup>* and *ψα* = *<sup>θ</sup>*+, *<sup>θ</sup>*−, *θν*, *χα* and the dot denotes

*<sup>H</sup>*, we first diagonalize the matrix *MK*. This can

√2 1 1 −1 1 

and *<sup>χ</sup>*�*<sup>α</sup>* <sup>=</sup> *χα* �*<sup>f</sup>*

�*<sup>f</sup>* ≃ *<sup>m</sup>*P/

*f* /2 *f* 3/2*χα*, involved in

*<sup>α</sup>*. Indeed, the *<sup>χ</sup>*�*α*'s acquire

<sup>−</sup>2*N*�/3, (21)

*<sup>α</sup>*. (22)

/3*H*�HI and we

*h* and the

*χ*�*α*

*<sup>h</sup>*, *<sup>J</sup>* and the slow-roll parameters �*ǫ*, *<sup>η</sup>*�,

. (18)

�˙ *ψα* �˙ *ψα*, (19)

· (20)

*θ*<sup>+</sup> ≃

<sup>√</sup>*c*R. On

*<sup>h</sup>* and *UK* <sup>=</sup> <sup>1</sup>

*<sup>χ</sup>*˙ *αχ*˙ *<sup>α</sup>* <sup>=</sup> <sup>1</sup> 2 �˙ *h* 2 + 1 2

�*f* since *h*/

**Table 2.** The scalar mass spectrum of our model along the inflationary trajectory of Eq. (12). To avoid very lengthy formulas we neglect terms proportional to *m*<sup>2</sup> PS and we assume *<sup>λ</sup><sup>H</sup>* <sup>≃</sup> *<sup>λ</sup>H*¯ for the derivation of the masses of the scalars in the superfields *d<sup>c</sup> <sup>H</sup>* and ¯*d<sup>c</sup> H*.

involved in the expansion of *<sup>V</sup>*HI. We arrange our findings into three groups: the SM singlet sector, *S* − *ν<sup>c</sup> <sup>H</sup>* <sup>−</sup> *<sup>ν</sup>*¯*<sup>c</sup> <sup>H</sup>*, the sector with the *<sup>u</sup><sup>c</sup> <sup>H</sup>*, *<sup>u</sup>*¯*<sup>c</sup> <sup>H</sup>* and the *<sup>e</sup><sup>c</sup> <sup>H</sup>*,*e*¯ *c <sup>H</sup>* fields which are related with the broken generators of *G*PS and the sector with the *d<sup>c</sup> <sup>H</sup>*, ¯*d<sup>c</sup> <sup>H</sup>* and the *<sup>g</sup>c*, *<sup>g</sup>*¯*<sup>c</sup>* fields. Upon diagonalization of the relevant matrices we obtain the following mass eigenstates:

$$
\hat{\mathfrak{x}}\_{1\pm} = \frac{1}{\sqrt{2}} \left( \hat{\mathfrak{x}}\_1 \pm \hat{\mathfrak{x}}\_1 \right) \text{ and } \hat{\mathfrak{x}}\_{2\mp} = \frac{1}{\sqrt{2}} \left( \hat{\mathfrak{x}}\_2 \mp \hat{\mathfrak{x}}\_2 \right) \text{ with } \ge = u, e, d \text{ and } \mathfrak{g}. \tag{24}
$$

As we observe from the relevant eigenvalues, no instability – as the one found in Ref. [37] – arises in the spectrum. In particular, it is evident that *kS* 1 assists us to achieve *m*<sup>2</sup> *<sup>S</sup>* <sup>&</sup>gt; <sup>0</sup> – in accordance with the results of Ref. [12]. Moreover, the D-term contributions to *m*<sup>2</sup> *θ ν* and *m*2 *<sup>u</sup>*<sup>−</sup> – proportional to the gauge coupling constant *<sup>g</sup>* <sup>≃</sup> 0.7 – ensure the positivity of these masses squared. Finally the masses that the scalars *d* 1,2 acquire from the second and third term of the RHS of Eq. (5) lead to the positivity of *m*<sup>2</sup> *d* <sup>−</sup> for *<sup>λ</sup><sup>H</sup>* of order unity. We have also numerically verified that the masses of the various scalars remain greater than the Hubble parameter during the last 50 − 60 e-foldings of nMHI, and so any inflationary perturbations of the fields other than the inflaton are safely eliminated.

The 8 Goldstone bosons, associated with the modes *<sup>x</sup>*<sup>1</sup><sup>+</sup> and *<sup>x</sup>*<sup>2</sup><sup>−</sup> with *<sup>x</sup>* <sup>=</sup> *<sup>u</sup>*<sup>a</sup> and *<sup>e</sup>*, are not exactly massless since *<sup>V</sup>*HI,*<sup>h</sup>* �<sup>=</sup> 0 – contrary to the situation of Ref. [30] where the direction with non vanishing �*ν<sup>c</sup> <sup>H</sup>*� minimizes the potential. These masses turn out to be *mx*<sup>0</sup> = *<sup>λ</sup>m*P*xh*/2 *<sup>f</sup>* . On the contrary, the angular parametrization in Eq. (11) assists us to isolate the massless mode *θ* <sup>−</sup>, in agreement with the analysis of Ref. [11]. Employing the well-known Coleman-Weinberg formula [15], we can compute the one-loop radiative corrections to the potential in our model. However, these have no significant effect on the inflationary dynamics and predictions, since the slope of the inflationary path is generated at the classical level – see the expressions for *<sup>ǫ</sup>* and *<sup>η</sup>* below.

#### **3.2. The inflationary observables**

Based on the potential of Eq. (13) and keeping in mind that the EF canonically inflaton *<sup>h</sup>* is related to *h* via Eq. (20), we can proceed to the analysis of nMHI in the EF, employing the standard slow-roll approximation. Namely, a stage of slow-roll nMHI is determined by the condition – see e.g. Ref. [38, 39]:

$$\mathsf{max}\{\widehat{\mathfrak{e}}(h), |\widehat{\mathfrak{y}}(h)|\} \le 1\_{\mathsf{w}}$$

where

$$\hat{\varepsilon} = \frac{m\_{\text{P}}^2}{2} \left( \frac{\hat{V}\_{\text{HI},\hat{h}}}{\hat{V}\_{\text{HI}}} \right)^2 = \frac{m\_{\text{P}}^2}{2f^2} \left( \frac{\hat{V}\_{\text{HI},h}}{\hat{V}\_{\text{HI}}} \right)^2 \simeq \frac{4f\_0^2 m\_{\text{P}}^4}{3c\_{\mathcal{R}}^2 h^4} \tag{25a}$$

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model 11

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

*<sup>N</sup>*<sup>∗</sup> *<sup>f</sup>*0/3*c*R· (29)

http://dx.doi.org/10.5772/51888

251

· (30)

∗ (32b)

*<sup>N</sup>*<sup>∗</sup> <sup>≃</sup> <sup>3</sup>*c*<sup>R</sup> 4 *f*<sup>0</sup>

<sup>∆</sup><sup>R</sup> <sup>=</sup> <sup>1</sup> 2 √3 *πm*<sup>3</sup> P

> *<sup>α</sup>*<sup>s</sup> <sup>=</sup> <sup>2</sup> 3 4*η*2

HI,*<sup>h</sup><sup>h</sup><sup>h</sup>*/*<sup>V</sup>*<sup>2</sup>

HI <sup>=</sup> *<sup>m</sup>*<sup>P</sup>

evaluated at *h* = *h*<sup>∗</sup> and Eqs. (25a) and (25b) have been employed.

The power spectrum ∆<sup>2</sup>

is estimated as follows

through the relations:

and

where *ξ*

<sup>=</sup> *<sup>m</sup>*<sup>4</sup>

P*<sup>V</sup>*HI,*hV*

**4. Non-thermal leptogenesis**

*h*2 <sup>∗</sup> − *h*<sup>2</sup> f *m*2 P

*<sup>V</sup>*HI(*<sup>h</sup>*∗)3/2

HI,*<sup>h</sup>*(*<sup>h</sup>*∗)<sup>|</sup>

Since the scalars listed in Table 2 are massive enough during nMHI, ∆<sup>R</sup> can be identified with its central observational value – see Sec. 5 – with almost constant *<sup>N</sup>*<sup>∗</sup>. The resulting

The (scalar) spectral index *n*s, its running *a*s, and the scalar-to-tensor ratio *r* can be estimated

 − 2*ξ*

*<sup>r</sup>* <sup>=</sup> <sup>16</sup>*<sup>ǫ</sup>*<sup>∗</sup> <sup>≃</sup> 12/*<sup>N</sup>*<sup>2</sup>

In this section, we specify how the SUSY inflationary scenario makes a transition to the radiation dominated era (Sec. 4.1) and give an explanation of the origin of the observed BAU (Sec. 4.2) consistently with the *<sup>G</sup>* constraint and the low energy neutrino data (Sec. 4.3).

<sup>∗</sup> − (*n*<sup>s</sup> − 1)<sup>2</sup>

√


relation reveals that *λ* is to be proportional to *c*R. Indeed we find

⇒ *<sup>h</sup>*<sup>∗</sup> = <sup>2</sup>*m*<sup>P</sup>

<sup>≃</sup> *<sup>λ</sup>h*<sup>2</sup>

R of the curvature perturbations generated by *h* at the pivot scale *k*∗

∗ <sup>16</sup>√2*<sup>π</sup> <sup>f</sup>*0*m*<sup>2</sup>

*<sup>λ</sup>* <sup>≃</sup> 8.4 · <sup>10</sup>−4*πc*R/*<sup>N</sup>*<sup>∗</sup> <sup>⇒</sup> *<sup>c</sup>*<sup>R</sup> <sup>≃</sup> <sup>20925</sup>*<sup>λ</sup>* for *<sup>N</sup>*<sup>∗</sup> <sup>≃</sup> 55. (31)

*<sup>n</sup>*<sup>s</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>6</sup>*<sup>ǫ</sup>*<sup>∗</sup> <sup>+</sup> <sup>2</sup>*<sup>η</sup>*<sup>∗</sup> <sup>≃</sup> <sup>1</sup> <sup>−</sup> 2/*<sup>N</sup>*<sup>∗</sup>, (32a)

<sup>∗</sup> ≃ −2/*<sup>N</sup>*<sup>2</sup>

<sup>2</sup>*<sup>ǫ</sup> <sup>η</sup>*,*h*/*<sup>J</sup>* <sup>+</sup> <sup>2</sup>*<sup>η</sup><sup>ǫ</sup>*. The variables with subscript <sup>∗</sup> are

∗ , (32c)

<sup>∗</sup> ≃ −2*<sup>ξ</sup>*

P

≃ *<sup>λ</sup><sup>N</sup>*<sup>∗</sup> <sup>12</sup>√2*πc*<sup>R</sup>

and

$$
\hat{\eta} = m\_{\text{P}}^2 \frac{\dot{V}\_{\text{HI}, \hat{h}\hat{h}}}{\hat{V}\_{\text{HI}}} = \frac{m\_{\text{P}}^2}{J^2} \left( \frac{\hat{V}\_{\text{HI}, hh}}{\hat{V}\_{\text{HI}}} - \frac{\hat{V}\_{\text{HI}, h}}{\hat{V}\_{\text{HI}}} \frac{J\_h}{J} \right) \simeq -\frac{4f\_0 m\_{\text{P}}^2}{3c\eta h^2} \,, \tag{25b}
$$

are the slow-roll parameters and *<sup>f</sup>*<sup>0</sup> = *<sup>f</sup>* (�*h*� = <sup>2</sup>*M*PS) = <sup>1</sup> + <sup>4</sup>*c*R*m*<sup>2</sup> PS – see Sec. 4.1. Here we employ Eq. (15) and the following approximate relations:

$$J \simeq \sqrt{6} \frac{m\_{\rm P}}{h}, \ \hat{V}\_{\rm HI} \hat{\mu} \simeq \frac{4 \hat{V}\_{\rm HI}}{c\_{\rm R} h^3} f\_0 m\_{\rm P}^2 \text{ and } \ \hat{V}\_{\rm HI, hl} \simeq -\frac{12 \hat{V}\_{\rm HI}}{c\_{\rm R} h^4} f\_0 m\_{\rm P}^2. \tag{26}$$

The numerical computation reveals that nMHI terminates due to the violation of the *<sup>ǫ</sup>* criterion at a value of *h* equal to *h*f, which is calculated to be

$$
\widehat{\varepsilon}\left(h\_{\rm f}\right) = 1 \Rightarrow h\_{\rm f} = \left(4/3\right)^{1/4} m\_{\rm P} \sqrt{f\_0/c\_{\mathcal{R}}}.\tag{27}
$$

The number of e-foldings, *<sup>N</sup>*<sup>∗</sup>, that the scale *<sup>k</sup>*<sup>∗</sup> <sup>=</sup> 0.002/Mpc suffers during nMHI can be calculated through the relation:

$$
\hat{N}\_{\*} = \frac{1}{m\_{\text{P}}^{2}} \int\_{\hat{h}\_{\text{fl}}}^{\hat{h}\_{\*}} d\hat{h} \,\frac{\hat{V}\_{\text{HI}}}{\hat{V}\_{\text{HI},\hat{h}}} = \frac{1}{m\_{\text{P}}^{2}} \int\_{h\_{\text{fl}}}^{h\_{\*}} d\hbar \, f^{2} \frac{\hat{V}\_{\text{HI}}}{\hat{V}\_{\text{HI},\hat{h}}} \, \tag{28}
$$

where *<sup>h</sup>*<sup>∗</sup> [*<sup>h</sup>*∗] is the value of *<sup>h</sup>* [*<sup>h</sup>*] when *<sup>k</sup>*<sup>∗</sup> crosses the inflationary horizon. Given that *<sup>h</sup>*<sup>f</sup> <sup>≪</sup> *<sup>h</sup>*∗, we can write *<sup>h</sup>*<sup>∗</sup> as a function of *<sup>N</sup>*<sup>∗</sup> as follows

$$
\hat{N}\_{\*} \simeq \frac{3c\_{\mathcal{R}}}{4f\_{0}} \frac{h\_{\*}^{2} - h\_{\text{f}}^{2}}{m\_{\text{P}}^{2}} \Rightarrow h\_{\*} = 2m\_{\text{P}} \sqrt{\hat{N}\_{\*} f\_{0} / 3c\_{\mathcal{R}}}.\tag{29}
$$

The power spectrum ∆<sup>2</sup> R of the curvature perturbations generated by *h* at the pivot scale *k*∗ is estimated as follows

$$\Delta\_{\mathcal{R}} = \frac{1}{2\sqrt{3}\,\pi m\_{\text{P}}^{3}} \frac{\hat{V}\_{\text{HI}}(\hat{h}\_{\ast})^{3/2}}{|\hat{V}\_{\text{HI}\hat{h}}(\hat{h}\_{\ast})|} \simeq \frac{\lambda h\_{\ast}^{2}}{16\sqrt{2}\,\pi f\_{0} m\_{\text{P}}^{2}} \simeq \frac{\lambda \hat{N}\_{\ast}}{12\sqrt{2}\pi c\_{\text{R}}} \cdot \tag{30}$$

Since the scalars listed in Table 2 are massive enough during nMHI, ∆<sup>R</sup> can be identified with its central observational value – see Sec. 5 – with almost constant *<sup>N</sup>*<sup>∗</sup>. The resulting relation reveals that *λ* is to be proportional to *c*R. Indeed we find

$$
\lambda \simeq 8.4 \cdot 10^{-4} \pi c\_{\mathcal{R}} / \hat{N}\_\* \Rightarrow c\_{\mathcal{R}} \simeq 20925 \lambda \quad \text{for} \quad \hat{N}\_\* \simeq 55. \tag{31}
$$

The (scalar) spectral index *n*s, its running *a*s, and the scalar-to-tensor ratio *r* can be estimated through the relations:

$$n\_{\hat{\mathbf{e}}} = 1 - 6\hat{\mathbf{e}}\_{\*} + 2\hat{\eta}\_{\*} \simeq 1 - 2/\hat{\mathcal{N}}\_{\*}.\tag{32a}$$

$$\mathfrak{a}\_{\mathsf{s}} = \frac{2}{3} \left( 4\widehat{\eta}\_{\mathsf{s}}^{2} - (n\_{\mathsf{s}} - 1)^{2} \right) - 2\widehat{\xi}\_{\mathsf{s}} \simeq -2\widehat{\xi}\_{\mathsf{s}} \simeq -2/\widehat{\mathsf{N}}\_{\mathsf{s}}^{2} \tag{32b}$$

and

10 Open Questions in Cosmology

250 Open Questions in Cosmology

to isolate the massless mode *θ*

**3.2. The inflationary observables**

condition – see e.g. Ref. [38, 39]:

where

and

at the classical level – see the expressions for *<sup>ǫ</sup>* and *<sup>η</sup>* below.

*<sup>ǫ</sup>* <sup>=</sup> *<sup>m</sup>*<sup>2</sup> P 2

*<sup>η</sup>* <sup>=</sup> *<sup>m</sup>*<sup>2</sup> P *V* HI,*<sup>h</sup><sup>h</sup> <sup>V</sup>*HI

*<sup>J</sup>* <sup>≃</sup> <sup>√</sup> 6 *m*<sup>P</sup>

calculated through the relation:

*<sup>V</sup>* HI,*<sup>h</sup> <sup>V</sup>*HI

> <sup>=</sup> *<sup>m</sup>*<sup>2</sup> P *J*2

are the slow-roll parameters and *<sup>f</sup>*<sup>0</sup> = *<sup>f</sup>* (�*h*� = <sup>2</sup>*M*PS) = <sup>1</sup> + <sup>4</sup>*c*R*m*<sup>2</sup>

*<sup>h</sup>* , *<sup>V</sup>*HI,*<sup>h</sup>* <sup>≃</sup> <sup>4</sup>*<sup>V</sup>*HI

criterion at a value of *h* equal to *h*f, which is calculated to be

*<sup>N</sup>*<sup>∗</sup> <sup>=</sup> <sup>1</sup> *m*2 P

*<sup>h</sup>*<sup>f</sup> <sup>≪</sup> *<sup>h</sup>*∗, we can write *<sup>h</sup>*<sup>∗</sup> as a function of *<sup>N</sup>*<sup>∗</sup> as follows

employ Eq. (15) and the following approximate relations:

2

 *<sup>V</sup>*HI,*hh <sup>V</sup>*HI

*<sup>c</sup>*R*h*<sup>3</sup> *<sup>f</sup>*0*m*<sup>2</sup>

*<sup>ǫ</sup>* (*h*f) <sup>=</sup> <sup>1</sup> <sup>⇒</sup> *<sup>h</sup>*<sup>f</sup> <sup>=</sup> (4/3)

 *<sup>h</sup>*<sup>∗</sup> *h*f

The numerical computation reveals that nMHI terminates due to the violation of the *<sup>ǫ</sup>*

The number of e-foldings, *<sup>N</sup>*<sup>∗</sup>, that the scale *<sup>k</sup>*<sup>∗</sup> <sup>=</sup> 0.002/Mpc suffers during nMHI can be

where *<sup>h</sup>*<sup>∗</sup> [*<sup>h</sup>*∗] is the value of *<sup>h</sup>* [*<sup>h</sup>*] when *<sup>k</sup>*<sup>∗</sup> crosses the inflationary horizon. Given that

= 1 *m*2 P

*<sup>d</sup><sup>h</sup> <sup>V</sup>*HI *V* HI,*<sup>h</sup>*

<sup>−</sup>, in agreement with the analysis of Ref. [11]. Employing

the well-known Coleman-Weinberg formula [15], we can compute the one-loop radiative corrections to the potential in our model. However, these have no significant effect on the inflationary dynamics and predictions, since the slope of the inflationary path is generated

Based on the potential of Eq. (13) and keeping in mind that the EF canonically inflaton *<sup>h</sup>* is related to *h* via Eq. (20), we can proceed to the analysis of nMHI in the EF, employing the standard slow-roll approximation. Namely, a stage of slow-roll nMHI is determined by the

max{*<sup>ǫ</sup>*(*h*), <sup>|</sup>*<sup>η</sup>*(*h*)|} ≤ 1,

<sup>=</sup> *<sup>m</sup>*<sup>2</sup> P 2*J*<sup>2</sup>  *<sup>V</sup>*HI,*<sup>h</sup> <sup>V</sup>*HI

− *<sup>V</sup>*HI,*<sup>h</sup> <sup>V</sup>*HI

2

*J*,*h J* 

<sup>P</sup> and *<sup>V</sup>*HI,*hh* ≃ −12*<sup>V</sup>*HI

1/4 *<sup>m</sup>*<sup>P</sup>

 *<sup>h</sup>*<sup>∗</sup> *h*f

*dh J*<sup>2</sup> *<sup>V</sup>*HI *<sup>V</sup>*HI,*<sup>h</sup>*

<sup>≃</sup> <sup>4</sup> *<sup>f</sup>* <sup>2</sup> <sup>0</sup> *<sup>m</sup>*<sup>4</sup> P 3*c*<sup>2</sup>

≃ −<sup>4</sup> *<sup>f</sup>*0*m*<sup>2</sup>

P

*<sup>c</sup>*R*h*<sup>4</sup> *<sup>f</sup>*0*m*<sup>2</sup>

*f*0/*c*R. (27)

, (28)

<sup>R</sup>*h*<sup>4</sup> (25a)

<sup>3</sup>*c*R*h*<sup>2</sup> , (25b)

PS – see Sec. 4.1. Here we

<sup>P</sup>. (26)

$$r = 16\hat{\mathbf{e}}\_{\*} \simeq 12/\hat{\mathbf{N}}\_{\*}^{2} \tag{32c}$$

where *ξ* <sup>=</sup> *<sup>m</sup>*<sup>4</sup> P*<sup>V</sup>*HI,*hV* HI,*<sup>h</sup><sup>h</sup><sup>h</sup>*/*<sup>V</sup>*<sup>2</sup> HI <sup>=</sup> *<sup>m</sup>*<sup>P</sup> √ <sup>2</sup>*<sup>ǫ</sup> <sup>η</sup>*,*h*/*<sup>J</sup>* <sup>+</sup> <sup>2</sup>*<sup>η</sup><sup>ǫ</sup>*. The variables with subscript <sup>∗</sup> are evaluated at *h* = *h*<sup>∗</sup> and Eqs. (25a) and (25b) have been employed.

#### **4. Non-thermal leptogenesis**

In this section, we specify how the SUSY inflationary scenario makes a transition to the radiation dominated era (Sec. 4.1) and give an explanation of the origin of the observed BAU (Sec. 4.2) consistently with the *<sup>G</sup>* constraint and the low energy neutrino data (Sec. 4.3).

#### **4.1. The inflaton's decay**

When nMHI is over, the inflaton continues to roll down towards the SUSY vacuum, Eq. (9). There is a brief stage of tachyonic preheating [40] which does not lead to significant particle production [41]. Soon after, the inflaton settles into a phase of damped oscillations initially around zero – where *<sup>V</sup>*HI0 has a maximum – and then around one of the minima of *<sup>V</sup>*HI0. Whenever the inflaton passes through zero, particle production may occur creating mostly superheavy bosons via the mechanism of instant preheating [42]. This process becomes more efficient as *λ* decreases, and further numerical investigation is required in order to check the viability of the non-thermal leptogenesis scenario for small values of *λ*. For this reason, we restrict to *λ*'s larger than 0.001, which ensures a less frequent passage of the inflaton through zero, weakening thereby the effects from instant preheating and other parametric resonance effects – see Appendix B of Ref. [5]. Intuitively the reason is that larger *λ*'s require larger *c*R's, see Eq. (31), diminishing therefore *h*<sup>f</sup> given by Eq. (29), which sets the amplitude of the very first oscillations.

Nonetheless the standard perturbative approach to the inflaton decay provides a very efficient decay rate. Namely, at the SUSY vacuum *ν<sup>c</sup> <sup>H</sup>* and *<sup>ν</sup>*¯*<sup>c</sup> <sup>H</sup>* acquire the v.e.vs shown in Eq. (9) giving rise to the masses of the (canonically normalized) inflaton *δ <sup>h</sup>* <sup>=</sup> (*<sup>h</sup>* <sup>−</sup> <sup>2</sup>*M*PS) /*J*<sup>0</sup> and RH neutrinos, *<sup>ν</sup><sup>c</sup> <sup>i</sup>* , which are given, respectively, by

$$\text{(a)}\quad m\_{\text{I}} = \sqrt{2} \frac{\lambda M\_{\text{PS}}}{\langle l \rangle f\_{0}} \quad \text{and} \quad \text{(b)}\quad M\_{\text{i}\hat{v}^{\text{c}}} = 2 \frac{\lambda\_{\text{i}\hat{v}^{\text{c}}} M\_{\text{PS}}^2}{M\_{\text{S}} \sqrt{f\_{0}}},\tag{33}$$

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model 13

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

*<sup>i</sup>* <sup>+</sup> h.c. . (36)

http://dx.doi.org/10.5772/51888

253

PS

, (37)

*<sup>i</sup>* is induced by two lagrangian terms.

1 − 12*c*R*m*<sup>2</sup>

*<sup>j</sup>* 's into which the inflaton can decay. The

*<sup>j</sup>* with *<sup>j</sup>* <sup>=</sup> 1 and 2. Note that the decay of the

*M*PS *m*2 P

*f* 3/2 0 *J*0

, (39)

+ h.c. , (38)

<sup>3</sup>) is also disfavored by the *<sup>G</sup>* constraint – see below.

<sup>L</sup>I*ν<sup>c</sup>*

From Eq. (36) we deduce that the decay of *δ*

I*jνc* 64*π m*I

where *Mj<sup>ν</sup><sup>c</sup>* is the Majorana mass of the *<sup>ν</sup><sup>c</sup>*

<sup>Γ</sup>I*j<sup>ν</sup><sup>c</sup>* <sup>=</sup> *<sup>c</sup>*<sup>2</sup>

allowed decay channels of *δ*

inflaton to the heaviest of the *<sup>ν</sup><sup>c</sup>*

LI*<sup>y</sup>* = <sup>6</sup>*yc*<sup>R</sup>

to the following 3-body decay width

<sup>Γ</sup>I*<sup>y</sup>* <sup>=</sup> <sup>14</sup>*c*<sup>2</sup>

I*y* <sup>512</sup>*π*<sup>3</sup> *<sup>m</sup>*<sup>3</sup>

decay width

*<sup>i</sup>* <sup>=</sup> <sup>−</sup>*λiν<sup>c</sup>*

<sup>1</sup> <sup>−</sup> <sup>4</sup>*M*<sup>2</sup>

*jνc m*2 I

*<sup>h</sup>* are those into *<sup>ν</sup><sup>c</sup>*

*<sup>j</sup>* 's (*<sup>ν</sup><sup>c</sup>*

obtain the effective interactions described by the langrangian part

*f* 3/2 0 2*J*<sup>0</sup> *δ h* 

*M*PS *m*2 P

<sup>I</sup> <sup>≃</sup> <sup>3</sup>*y*<sup>2</sup> 33 <sup>64</sup>*π*<sup>3</sup> *<sup>f</sup>* <sup>3</sup> 0 *m*<sup>I</sup> *m*<sup>P</sup>

of freedom, in conjunction with the assumption that *<sup>m</sup>*<sup>I</sup> <sup>&</sup>lt; <sup>2</sup>*M*3*<sup>ν</sup><sup>c</sup>* .

*M*PS *M*<sup>S</sup>

*f*0 *J*0 

1 − 12*c*R*m*<sup>2</sup>

*<sup>h</sup>* into *<sup>ν</sup><sup>c</sup>*

with *<sup>c</sup>*I*j<sup>ν</sup><sup>c</sup>* <sup>=</sup> *Mj<sup>ν</sup><sup>c</sup>*

implementation – see Sec. 4.3 – of the seesaw mechanism for the derivation of the light-neutrinos masses, in conjunction with the *<sup>G</sup>*PS prediction *<sup>m</sup>*3D ≃ *mt* and our assumption that *<sup>m</sup>*1D <sup>&</sup>lt; *<sup>m</sup>*2D <sup>≪</sup> *<sup>m</sup>*3D – see Sec. 5.1 – results to 2*M*3*<sup>ν</sup><sup>c</sup>* <sup>&</sup>gt; *<sup>m</sup>*I. Therefore, the kinematically

In addition, there are SUGRA-induced [43] – i.e., even without direct superpotential couplings – decay channels of the inflaton to the MSSM particles via non-renormalizable interaction terms. For a typical trilinear superpotential term of the form *Wy* = *yXYZ*, we

where *y* is a Yukawa coupling constant and *ψX*, *ψ<sup>Y</sup>* and *ψ<sup>Z</sup>* are the chiral fermions associated with the superfields *X*,*Y* and *Z*. Their scalar components are denoted with the superfield symbol. Taking into account the terms of Eq. (2) and the fact that the adopted SUSY GUT predicts YU for the 3rd generation at *M*PS, we conclude that the interaction above gives rise

2

with *<sup>y</sup>*<sup>33</sup> ≃ (0.55 − 0.7) being the common Yukawa coupling constant of the third generation computed at the *m*<sup>I</sup> scale, and summation is taken over color, weak and hypercharge degrees

The first one originates exclusively from the non-renormalizable term of Eq. (5) – as in the case of a similar model in Ref. [30]. The second term is a higher order decay channel due to the SUGRA lagrangian – cf. Ref. [43]. The interaction in Eq. (36) gives rise to the following

PS *δ hνc i νc*

*M*PS

*<sup>X</sup><sup>ψ</sup><sup>Y</sup><sup>ψ</sup><sup>Z</sup>* <sup>+</sup> *<sup>Y</sup><sup>ψ</sup><sup>X</sup><sup>ψ</sup><sup>Z</sup>* <sup>+</sup> *<sup>Z</sup><sup>ψ</sup><sup>X</sup><sup>ψ</sup><sup>Y</sup>*

*<sup>m</sup>*<sup>I</sup> where *<sup>c</sup>*I*<sup>y</sup>* = <sup>6</sup>*y*33*c*<sup>R</sup>

*f* 3/2 0 *J*0 

where *f*<sup>0</sup> is defined below Eq. (25b) and ¯ *<sup>f</sup>*<sup>0</sup> = *<sup>f</sup>*<sup>0</sup> + <sup>24</sup>*c*<sup>2</sup> <sup>R</sup>*m*<sup>2</sup> PS <sup>≃</sup> *<sup>J</sup>*<sup>2</sup> <sup>0</sup> . Here, we assume the existence of a term similar to the second one inside ln of Eq. (7a) for *ν<sup>c</sup> <sup>i</sup>* too.

For larger *<sup>λ</sup>*'s �*J*� = *<sup>J</sup>*(*<sup>h</sup>* = <sup>2</sup>*M*PS) ranges from 3 to 90 and so *<sup>m</sup>*<sup>I</sup> is kept independent of *<sup>λ</sup>* and almost constant at the level of 1013 GeV. Indeed, if we express *δ <sup>h</sup>* as a function of *<sup>δ</sup><sup>h</sup>* through the relation

$$\frac{\delta\hbar}{\delta\hbar} \simeq J\_0 \text{ where } J\_0 = \sqrt{1 + \frac{3}{2}m\_\text{P}^2 f\_{,\hbar}^2(\langle\hbar\rangle)} = \sqrt{1 + 24c\_\text{\mathcal{R}}^2 m\_\text{\text{PS}}^2} \tag{34}$$

we find

$$\rho\_{\rm I} \simeq \frac{\sqrt{2}\lambda M\_{\rm PS}}{f\_{\rm I}l\_{\rm I}} \simeq \frac{\lambda m\_{\rm P}}{2\sqrt{3}c\_{\rm R}} \simeq \frac{10^{-4}m\_{\rm P}}{4.2\sqrt{3}} \simeq 3 \cdot 10^{13} \text{ GeV} \text{ for } \lambda \gtrsim \frac{10^{-4}}{4.2\sqrt{6}m\_{\rm PS}} \simeq 1.3 \cdot 10^{-3} \tag{35}$$

where we make use of Eq. (31) – note that *<sup>f</sup>*<sup>0</sup> ≃ 1. The derivation of the (s)particle spectrum, listed in Table 2, at the SUSY vaccum of the model reveals [5] that perturbative decays of *δ <sup>h</sup>* into these massive particles are kinematically forbidden and therefore, narrow parametric resonance [40] effects are absent. Also *δ <sup>h</sup>* can not decay via renormalizable interaction terms to SM particles.

The inflaton can decay into a pair of *<sup>ν</sup><sup>c</sup> <sup>i</sup>* 's through the following lagrangian terms:

$$\mathcal{L}\_{\text{lv}\_{i}^{\varepsilon}} = -\lambda\_{\text{iv}^{\varepsilon}} \frac{M\_{\text{PS}}}{M\_{\text{S}}} \frac{f\_{0}}{f\_{0}} \left(1 - 12c\_{\text{\mathcal{R}}}m\_{\text{PS}}^{2}\right) \delta \hat{h} \hat{v}\_{i}^{\varepsilon} \hat{v}\_{i}^{\varepsilon} + \text{h.c.}\tag{36}$$

From Eq. (36) we deduce that the decay of *δ <sup>h</sup>* into *<sup>ν</sup><sup>c</sup> <sup>i</sup>* is induced by two lagrangian terms. The first one originates exclusively from the non-renormalizable term of Eq. (5) – as in the case of a similar model in Ref. [30]. The second term is a higher order decay channel due to the SUGRA lagrangian – cf. Ref. [43]. The interaction in Eq. (36) gives rise to the following decay width

12 Open Questions in Cosmology

252 Open Questions in Cosmology

very first oscillations.

and RH neutrinos, *<sup>ν</sup><sup>c</sup>*

the relation

we find

*<sup>m</sup>*<sup>I</sup> ≃

*δ*

√

to SM particles.

2*λM*PS *f*<sup>0</sup> *J*<sup>0</sup>

efficient decay rate. Namely, at the SUSY vacuum *ν<sup>c</sup>*

where *f*<sup>0</sup> is defined below Eq. (25b) and ¯

≃ *λm*<sup>P</sup> 2 √3*c*R

resonance [40] effects are absent. Also *δ*

The inflaton can decay into a pair of *<sup>ν</sup><sup>c</sup>*

*δ h* *<sup>m</sup>*<sup>I</sup> <sup>=</sup> <sup>√</sup> 2 *λM*PS �*J*�*f*<sup>0</sup>

Eq. (9) giving rise to the masses of the (canonically normalized) inflaton *δ*

existence of a term similar to the second one inside ln of Eq. (7a) for *ν<sup>c</sup>*

 1 + 3 2 *m*2 P *f* 2

almost constant at the level of 1013 GeV. Indeed, if we express *δ*

<sup>≃</sup> <sup>10</sup>−4*m*<sup>P</sup>

*<sup>δ</sup><sup>h</sup>* <sup>≃</sup> *<sup>J</sup>*<sup>0</sup> where *<sup>J</sup>*<sup>0</sup> <sup>=</sup>

*<sup>i</sup>* , which are given, respectively, by

**4.1. The inflaton's decay**

When nMHI is over, the inflaton continues to roll down towards the SUSY vacuum, Eq. (9). There is a brief stage of tachyonic preheating [40] which does not lead to significant particle production [41]. Soon after, the inflaton settles into a phase of damped oscillations initially around zero – where *<sup>V</sup>*HI0 has a maximum – and then around one of the minima of *<sup>V</sup>*HI0. Whenever the inflaton passes through zero, particle production may occur creating mostly superheavy bosons via the mechanism of instant preheating [42]. This process becomes more efficient as *λ* decreases, and further numerical investigation is required in order to check the viability of the non-thermal leptogenesis scenario for small values of *λ*. For this reason, we restrict to *λ*'s larger than 0.001, which ensures a less frequent passage of the inflaton through zero, weakening thereby the effects from instant preheating and other parametric resonance effects – see Appendix B of Ref. [5]. Intuitively the reason is that larger *λ*'s require larger *c*R's, see Eq. (31), diminishing therefore *h*<sup>f</sup> given by Eq. (29), which sets the amplitude of the

Nonetheless the standard perturbative approach to the inflaton decay provides a very

and *Mi<sup>ν</sup><sup>c</sup>* <sup>=</sup> <sup>2</sup>

*<sup>f</sup>*<sup>0</sup> = *<sup>f</sup>*<sup>0</sup> + <sup>24</sup>*c*<sup>2</sup>

For larger *<sup>λ</sup>*'s �*J*� = *<sup>J</sup>*(*<sup>h</sup>* = <sup>2</sup>*M*PS) ranges from 3 to 90 and so *<sup>m</sup>*<sup>I</sup> is kept independent of *<sup>λ</sup>* and

where we make use of Eq. (31) – note that *<sup>f</sup>*<sup>0</sup> ≃ 1. The derivation of the (s)particle spectrum, listed in Table 2, at the SUSY vaccum of the model reveals [5] that perturbative decays of

*<sup>h</sup>* into these massive particles are kinematically forbidden and therefore, narrow parametric

*<sup>H</sup>* and *<sup>ν</sup>*¯*<sup>c</sup>*

*<sup>λ</sup>iν<sup>c</sup> <sup>M</sup>*<sup>2</sup> PS

PS <sup>≃</sup> *<sup>J</sup>*<sup>2</sup>

*M*<sup>S</sup> *f*<sup>0</sup>

<sup>R</sup>*m*<sup>2</sup>

1 + 24*c*<sup>2</sup>

4.2√6*m*PS

*<sup>h</sup>* can not decay via renormalizable interaction terms

*<sup>i</sup>* 's through the following lagrangian terms:

,*<sup>h</sup>* (�*h*�) <sup>=</sup>

4.2√<sup>3</sup> <sup>≃</sup> <sup>3</sup> · <sup>1013</sup> GeV for *<sup>λ</sup>* <sup>10</sup>−<sup>4</sup>

*<sup>H</sup>* acquire the v.e.vs shown in

*<sup>h</sup>* <sup>=</sup> (*<sup>h</sup>* <sup>−</sup> <sup>2</sup>*M*PS) /*J*<sup>0</sup>

, (33)

<sup>0</sup> . Here, we assume the

PS (34)

<sup>≃</sup> 1.3 · <sup>10</sup>−<sup>3</sup> (35)

*<sup>h</sup>* as a function of *<sup>δ</sup><sup>h</sup>* through

*<sup>i</sup>* too.

<sup>R</sup>*m*<sup>2</sup>

$$\Gamma\_{\rm Ij\hat{\nu}} = \frac{c\_{\rm Ij\hat{\nu}}^2}{64\pi} m\_{\rm I} \sqrt{1 - \frac{4M\_{\hat{\mathcal{V}}^{\hat{\nu}}}}{m\_{\rm I}^2}} \text{ with } c\_{\rm Ij\hat{\nu}^c} = \frac{M\_{\hat{\mathcal{V}}^c}}{M\_{\rm PS}} \frac{f\_0^{3/2}}{J\_0} \left(1 - 12c\_{\mathcal{R}}m\_{\rm PS}^2\right) \tag{37}$$

where *Mj<sup>ν</sup><sup>c</sup>* is the Majorana mass of the *<sup>ν</sup><sup>c</sup> <sup>j</sup>* 's into which the inflaton can decay. The implementation – see Sec. 4.3 – of the seesaw mechanism for the derivation of the light-neutrinos masses, in conjunction with the *<sup>G</sup>*PS prediction *<sup>m</sup>*3D ≃ *mt* and our assumption that *<sup>m</sup>*1D <sup>&</sup>lt; *<sup>m</sup>*2D <sup>≪</sup> *<sup>m</sup>*3D – see Sec. 5.1 – results to 2*M*3*<sup>ν</sup><sup>c</sup>* <sup>&</sup>gt; *<sup>m</sup>*I. Therefore, the kinematically allowed decay channels of *δ <sup>h</sup>* are those into *<sup>ν</sup><sup>c</sup> <sup>j</sup>* with *<sup>j</sup>* <sup>=</sup> 1 and 2. Note that the decay of the inflaton to the heaviest of the *<sup>ν</sup><sup>c</sup> <sup>j</sup>* 's (*<sup>ν</sup><sup>c</sup>* <sup>3</sup>) is also disfavored by the *<sup>G</sup>* constraint – see below.

In addition, there are SUGRA-induced [43] – i.e., even without direct superpotential couplings – decay channels of the inflaton to the MSSM particles via non-renormalizable interaction terms. For a typical trilinear superpotential term of the form *Wy* = *yXYZ*, we obtain the effective interactions described by the langrangian part

$$\mathcal{L}\_{\rm Iy} = 6y c\_{\mathcal{R}} \frac{M\_{\rm PS}}{m\_{\rm P}^2} \frac{f\_0^{3/2}}{2J\_0} \delta h \left( \hat{X} \hat{\boldsymbol{\psi}}\_Y \hat{\boldsymbol{\psi}}\_Z + \hat{Y} \hat{\boldsymbol{\psi}}\_X \hat{\boldsymbol{\psi}}\_Z + \hat{Z} \hat{\boldsymbol{\psi}}\_X \hat{\boldsymbol{\psi}}\_Y \right) + \text{h.c.},\tag{38}$$

where *y* is a Yukawa coupling constant and *ψX*, *ψ<sup>Y</sup>* and *ψ<sup>Z</sup>* are the chiral fermions associated with the superfields *X*,*Y* and *Z*. Their scalar components are denoted with the superfield symbol. Taking into account the terms of Eq. (2) and the fact that the adopted SUSY GUT predicts YU for the 3rd generation at *M*PS, we conclude that the interaction above gives rise to the following 3-body decay width

$$\Gamma\_{\rm Iy} = \frac{14c\_{\rm Iy}^2}{512\pi^3} m\_{\rm I}^3 \simeq \frac{3y\_{33}^2}{64\pi^3} f\_0^3 \left(\frac{m\_{\rm I}}{m\_{\rm P}}\right)^2 m\_{\rm I} \text{ where } c\_{\rm Iy} = 6y\_{33}c\_{\mathcal{R}} \frac{M\_{\rm PS}}{m\_{\rm P}^2} \frac{f\_0^{3/2}}{l\_{\rm I}},\tag{39}$$

with *<sup>y</sup>*<sup>33</sup> ≃ (0.55 − 0.7) being the common Yukawa coupling constant of the third generation computed at the *m*<sup>I</sup> scale, and summation is taken over color, weak and hypercharge degrees of freedom, in conjunction with the assumption that *<sup>m</sup>*<sup>I</sup> <sup>&</sup>lt; <sup>2</sup>*M*3*<sup>ν</sup><sup>c</sup>* .

Since the decay width of the produced *<sup>ν</sup><sup>c</sup> <sup>j</sup>* is much larger than <sup>Γ</sup>I– see below – the reheating temperature, *T*rh, is exclusively determined by the inflaton decay and is given by [44]

$$T\_{\rm rh} = \left(\frac{72}{5\pi^2 g\_\*}\right)^{1/4} \sqrt{\Gamma\_{\rm I} m\_{\rm P}} \text{ with } \begin{array}{l} \Gamma\_{\rm I} = \Gamma\_{\rm I\hat{1}\hat{v}^c} + \Gamma\_{\rm I\hat{2}\hat{v}^c} + \Gamma\_{\rm Iy} \end{array} \tag{40}$$

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model 15

http://dx.doi.org/10.5772/51888

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

2, and a higgsino, *<sup>ψ</sup>*<sup>+</sup> = (*ψP*¯ <sup>+</sup> *<sup>ψ</sup>P*)/

*<sup>h</sup>* decay, can generate a

, (44a)

*<sup>i</sup>* -*basis*

√2, 255

√2,

*YG* <sup>≃</sup> *cG<sup>T</sup>*rh with *cG* <sup>=</sup> 1.9 · <sup>10</sup><sup>−</sup>22/GeV, (43)

The required for successful nTL *<sup>T</sup>*rh must be compatible with constraints on the *<sup>G</sup>* abundance,

where we assume that *<sup>G</sup>* is much heavier than the gauginos. Let us note that non-thermal *<sup>G</sup>* production within SUGRA is [43] also possible. However, we here prefer to adopt the conservative approach based on the estimation of *YG* via Eq. (43) since the latter *<sup>G</sup>* production

Both Eqs. (41) and (43) yield the correct values of the *<sup>B</sup>* and *<sup>G</sup>* abundances provided that no entropy production occurs for *T* < *T*rh. This fact can be easily achieved within our setting. The mass spectrum of the *P*-*P*¯ system is comprised by axion and saxion *P*<sup>−</sup> = (*P*¯ − *P*)/

with mass of order 1 TeV and *ψ* denoting a Weyl spinor. The higgs and higgsinos can decay to lighter higgs and higgsinos before domination [36]. Regarding the saxion, *P*−, we can assume that its decay mode to axions is suppressed (w.r.t the ones to gluons, higgses and higgsinos [47, 48]) and the initial amplitude of its oscillations is equal to *fa* ≃ 1012 GeV. Under these circumstances, it can [47] decay before domination too, and evades [48] the constraints from the effective number of neutrinos for the *fa*'s and *T*rh's encountered in our model. As a consequence of its relatively large decay temperature, the LSPs produced by the saxion decay are likely to be thermalized and therefore, no upper bound on the saxion abundance is [48] to be imposed. Finally, axino can not play the role of LSP due to its large expected mass and the relatively high *T*rh's encountered in our set-up which result to a large *Cold Dark Matter* (CDM) abundance. Nonetheless, it may enhance non-thermally the abundance of a higgsino-like neutralino-LSP, rendering it a successful CDM candidate.

√

<sup>1</sup>, emerging from the *δ*

*xij*, *yi*, *yj*

<sup>=</sup> (*m*†

<sup>D</sup>*m*D)*ii*

+ *<sup>F</sup>*V(*xij*)

<sup>8</sup>*π*�*Hu*�<sup>2</sup> (44b)

2, a higgs, *P*<sup>+</sup> = (*P*¯ + *P*)/

<sup>2</sup> and *<sup>ν</sup><sup>c</sup>*

<sup>D</sup>*m*D)*ii*

*<sup>i</sup>* are mass eigenstates, as follows:

lepton asymmetry, *<sup>ε</sup><sup>i</sup>* (with *<sup>i</sup>* = 1, 2) caused by the interference between the tree and one-loop decay diagrams, provided that a CP-violation occurs in *hNij*'s. The produced *ε<sup>i</sup>* can be expressed in terms of the Dirac mass matrix of *νi*, *m*D, defined in a basis (called *ν<sup>c</sup>*

> *F*S

and *yi* :<sup>=</sup> <sup>Γ</sup>*iν<sup>c</sup>*

(with *<sup>i</sup>*, *<sup>j</sup>* = 1, 2, 3). Also *<sup>F</sup>*<sup>V</sup> and *<sup>F</sup>*<sup>S</sup> represent, respectively, the contributions from vertex and

*Mi<sup>ν</sup><sup>c</sup>*

*YG*, at the onset of *nucleosynthesis* (BBN). This is estimated to be [19]:

depends on the mechanism of SUSY breaking.

√

**4.3. Lepton-number asymmetry and neutrino masses**

Im (*m*† D*m*D)<sup>2</sup> *ij*

*xij* :<sup>=</sup> *Mj<sup>ν</sup><sup>c</sup> Mi<sup>ν</sup><sup>c</sup>*

8*π*�*Hu*�2(*m*†

As mentioned above, the decay of *<sup>ν</sup><sup>c</sup>*

*<sup>ε</sup><sup>i</sup>* <sup>=</sup> ∑ *i*�=*j*

where we take �*Hu*� ≃ 174 GeV, for large tan *β* and

self-energy diagrams which in SUSY theories read [49]

henceforth) where *ν<sup>c</sup>*

axino *<sup>ψ</sup>*<sup>−</sup> = (*ψP*¯ <sup>−</sup> *<sup>ψ</sup>P*)/

where *g*<sup>∗</sup> counts the effective number of relativistic degrees of freedom at temperature *T*rh. For the MSSM spectrum plus the particle content of the superfields *P* and *P*¯ we find *g*<sup>∗</sup> ≃ 228.75 + 4(1 + 7/8) = 236.25.

#### **4.2. Lepton-number and gravitino abundances**

If *<sup>T</sup>*rh <sup>≪</sup> *Mi<sup>ν</sup><sup>c</sup>* , the out-of-equilibrium condition [2] for the implementation of nTL is automatically satisfied. Subsequently *<sup>ν</sup><sup>c</sup> <sup>i</sup>* decay into *Hu* and *<sup>L</sup>*<sup>∗</sup> *<sup>i</sup>* via the tree-level couplings derived from the second term in the RHS of Eq. (2). Interference between tree-level and one-loop diagrams generates a lepton-number asymmetry (per *<sup>ν</sup><sup>c</sup> <sup>j</sup>* decay) *ε<sup>j</sup>* [2], when CP conservation is violated. The resulting lepton-number asymmetry after reheating can be partially converted through sphaleron effects into baryon-number asymmetry. In particular, the *B* yield can be computed as

$$\text{(a)}\quad Y\_{\text{B}} = -0.35Y\_{L}\text{ with (b)}\quad Y\_{L} = 2\frac{5}{4}\frac{T\_{\text{rh}}}{m\_{\text{I}}}\sum\_{j=1}^{2}\frac{\Gamma\_{\text{I}\hat{j}\hat{r}^{\circ}}}{\Gamma\_{\text{I}}}\varepsilon\_{\text{j}}\cdot\tag{41}$$

The numerical factor in the RHS of Eq. (41 ) comes from the sphaleron effects, whereas the one (5/4) in the RHS of Eq. (41 ) is due to the slightly different calculation [44] of *T*rh – cf. Ref. [1]. In the major part of our allowed parameter space – see Sec. 5.2 – <sup>Γ</sup><sup>I</sup> ≃ <sup>Γ</sup>I*<sup>y</sup>* and so the involved branching ratio of the produced *<sup>ν</sup><sup>c</sup> <sup>i</sup>* is given by

$$\frac{\Gamma\_{\text{I1\hat{}}\hat{\imath}^{\text{c}}} + \Gamma\_{\text{I2\hat{}}\hat{\imath}^{\text{c}}}}{\Gamma\_{\text{I}}} \simeq \frac{\Gamma\_{\text{I2\hat{}}\hat{\imath}^{\text{c}}}}{\Gamma\_{\text{Iy}}} = \frac{\pi^{2} \left(1 - 12c\_{\mathcal{R}}m\_{\text{PS}}^{2}\right)^{2}}{72c\_{\mathcal{R}}^{2}y\_{\text{33}}^{2}m\_{\text{PS}}^{4}} \frac{M\_{\text{2\hat{}}\hat{\imath}^{\text{c}}}^{2}}{m\_{\text{I}}^{2}} \cdot \tag{42}$$

For *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* <sup>≃</sup> <sup>10</sup><sup>11</sup> − <sup>10</sup>12 GeV the ratio above takes adequately large values so that *YL* is sizable. Therefore, the presence of more than one inflaton decay channels does not invalidate the scenario of nTL.

It is worth emphasizing, however, that if *M*1*ν<sup>c</sup>* 10*T*rh, part of the *YL* can be washed out due to *<sup>ν</sup><sup>c</sup>* <sup>1</sup> mediated inverse decays and <sup>∆</sup>*<sup>L</sup>* <sup>=</sup> 1 scatterings – this possibility is analyzed in Ref. [27]. Trying to avoid the relevant computational complications we limit ourselves to cases with *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* <sup>10</sup>*T*rh, so as any washout of the non-thermally produced *YL* is evaded. On the other hand, *YL* is not erased by the <sup>∆</sup>*<sup>L</sup>* = 2 scattering processes [45] at all temperatures *<sup>T</sup>* with 100 GeV *T T*rh since *YL* is automatically protected by SUSY [46] for 107 GeV *T T*rh and for *T* 10<sup>7</sup> GeV these processes are well out of equilibrium provided that that mass of the heaviest light neutrino is 10 eV. This constraint, however, is overshadowed by a more stringent one induced by WMAP7 data [16] – see Sec. 5.1.

The required for successful nTL *<sup>T</sup>*rh must be compatible with constraints on the *<sup>G</sup>* abundance, *YG*, at the onset of *nucleosynthesis* (BBN). This is estimated to be [19]:

$$Y\_{\tilde{G}} \simeq c\_{\tilde{G}} T\_{\text{rh}} \text{ with } c\_{\tilde{G}} = 1.9 \cdot 10^{-22} / \text{GeV} \tag{43}$$

where we assume that *<sup>G</sup>* is much heavier than the gauginos. Let us note that non-thermal *<sup>G</sup>* production within SUGRA is [43] also possible. However, we here prefer to adopt the conservative approach based on the estimation of *YG* via Eq. (43) since the latter *<sup>G</sup>* production depends on the mechanism of SUSY breaking.

Both Eqs. (41) and (43) yield the correct values of the *<sup>B</sup>* and *<sup>G</sup>* abundances provided that no entropy production occurs for *T* < *T*rh. This fact can be easily achieved within our setting. The mass spectrum of the *P*-*P*¯ system is comprised by axion and saxion *P*<sup>−</sup> = (*P*¯ − *P*)/ √2, axino *<sup>ψ</sup>*<sup>−</sup> = (*ψP*¯ <sup>−</sup> *<sup>ψ</sup>P*)/ √2, a higgs, *P*<sup>+</sup> = (*P*¯ + *P*)/ √2, and a higgsino, *<sup>ψ</sup>*<sup>+</sup> = (*ψP*¯ <sup>+</sup> *<sup>ψ</sup>P*)/ √2, with mass of order 1 TeV and *ψ* denoting a Weyl spinor. The higgs and higgsinos can decay to lighter higgs and higgsinos before domination [36]. Regarding the saxion, *P*−, we can assume that its decay mode to axions is suppressed (w.r.t the ones to gluons, higgses and higgsinos [47, 48]) and the initial amplitude of its oscillations is equal to *fa* ≃ 1012 GeV. Under these circumstances, it can [47] decay before domination too, and evades [48] the constraints from the effective number of neutrinos for the *fa*'s and *T*rh's encountered in our model. As a consequence of its relatively large decay temperature, the LSPs produced by the saxion decay are likely to be thermalized and therefore, no upper bound on the saxion abundance is [48] to be imposed. Finally, axino can not play the role of LSP due to its large expected mass and the relatively high *T*rh's encountered in our set-up which result to a large *Cold Dark Matter* (CDM) abundance. Nonetheless, it may enhance non-thermally the abundance of a higgsino-like neutralino-LSP, rendering it a successful CDM candidate.

#### **4.3. Lepton-number asymmetry and neutrino masses**

14 Open Questions in Cosmology

254 Open Questions in Cosmology

Since the decay width of the produced *<sup>ν</sup><sup>c</sup>*

*<sup>T</sup>*rh =

automatically satisfied. Subsequently *<sup>ν</sup><sup>c</sup>*

the involved branching ratio of the produced *<sup>ν</sup><sup>c</sup>*

<sup>Γ</sup>I1*<sup>ν</sup><sup>c</sup>* <sup>+</sup> <sup>Γ</sup>I2*<sup>ν</sup><sup>c</sup>* ΓI

stringent one induced by WMAP7 data [16] – see Sec. 5.1.

228.75 + 4(1 + 7/8) = 236.25.

the *B* yield can be computed as

For *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* <sup>≃</sup>

due to *<sup>ν</sup><sup>c</sup>*

the scenario of nTL.

 72 5*π*2*g*<sup>∗</sup>

**4.2. Lepton-number and gravitino abundances**

1/4

one-loop diagrams generates a lepton-number asymmetry (per *<sup>ν</sup><sup>c</sup>*

(a) *YB* = −0.35*YL* with (b) *YL* = <sup>2</sup>

<sup>≃</sup> <sup>Γ</sup>I2*<sup>ν</sup><sup>c</sup>* ΓI*y*

temperature, *T*rh, is exclusively determined by the inflaton decay and is given by [44]

where *g*<sup>∗</sup> counts the effective number of relativistic degrees of freedom at temperature *T*rh. For the MSSM spectrum plus the particle content of the superfields *P* and *P*¯ we find *g*<sup>∗</sup> ≃

If *<sup>T</sup>*rh <sup>≪</sup> *Mi<sup>ν</sup><sup>c</sup>* , the out-of-equilibrium condition [2] for the implementation of nTL is

derived from the second term in the RHS of Eq. (2). Interference between tree-level and

conservation is violated. The resulting lepton-number asymmetry after reheating can be partially converted through sphaleron effects into baryon-number asymmetry. In particular,

The numerical factor in the RHS of Eq. (41 ) comes from the sphaleron effects, whereas the one (5/4) in the RHS of Eq. (41 ) is due to the slightly different calculation [44] of *T*rh – cf. Ref. [1]. In the major part of our allowed parameter space – see Sec. 5.2 – <sup>Γ</sup><sup>I</sup> ≃ <sup>Γ</sup>I*<sup>y</sup>* and so

<sup>=</sup> *<sup>π</sup>*<sup>2</sup>

sizable. Therefore, the presence of more than one inflaton decay channels does not invalidate

It is worth emphasizing, however, that if *M*1*ν<sup>c</sup>* 10*T*rh, part of the *YL* can be washed out

Ref. [27]. Trying to avoid the relevant computational complications we limit ourselves to cases with *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* <sup>10</sup>*T*rh, so as any washout of the non-thermally produced *YL* is evaded. On the other hand, *YL* is not erased by the <sup>∆</sup>*<sup>L</sup>* = 2 scattering processes [45] at all temperatures *<sup>T</sup>* with 100 GeV *T T*rh since *YL* is automatically protected by SUSY [46] for 107 GeV *T T*rh and for *T* 10<sup>7</sup> GeV these processes are well out of equilibrium provided that that mass of the heaviest light neutrino is 10 eV. This constraint, however, is overshadowed by a more

*<sup>i</sup>* decay into *Hu* and *<sup>L</sup>*<sup>∗</sup>

5 4 *T*rh *m*I

*<sup>i</sup>* is given by

72*c*<sup>2</sup> <sup>R</sup>*y*<sup>2</sup> 33*m*<sup>4</sup> PS

<sup>1</sup> mediated inverse decays and <sup>∆</sup>*<sup>L</sup>* <sup>=</sup> 1 scatterings – this possibility is analyzed in

<sup>10</sup><sup>11</sup> − <sup>10</sup>12 GeV the ratio above takes adequately large values so that *YL* is

1 − 12*c*R*m*<sup>2</sup>

PS 2

*M*<sup>2</sup> 2*νc m*2 I

2 ∑ *j*=1 ΓI*jνc* ΓI

*<sup>j</sup>* is much larger than <sup>Γ</sup>I– see below – the reheating

<sup>Γ</sup>I*m*<sup>P</sup> with <sup>Γ</sup><sup>I</sup> <sup>=</sup> <sup>Γ</sup>I1*<sup>ν</sup><sup>c</sup>* <sup>+</sup> <sup>Γ</sup>I2*<sup>ν</sup><sup>c</sup>* <sup>+</sup> <sup>Γ</sup>I*y*, (40)

*<sup>i</sup>* via the tree-level couplings

*<sup>j</sup>* decay) *ε<sup>j</sup>* [2], when CP

*<sup>ε</sup>j*· (41)

· (42)

As mentioned above, the decay of *<sup>ν</sup><sup>c</sup>* <sup>2</sup> and *<sup>ν</sup><sup>c</sup>* <sup>1</sup>, emerging from the *δ <sup>h</sup>* decay, can generate a lepton asymmetry, *<sup>ε</sup><sup>i</sup>* (with *<sup>i</sup>* = 1, 2) caused by the interference between the tree and one-loop decay diagrams, provided that a CP-violation occurs in *hNij*'s. The produced *ε<sup>i</sup>* can be expressed in terms of the Dirac mass matrix of *νi*, *m*D, defined in a basis (called *ν<sup>c</sup> <sup>i</sup>* -*basis* henceforth) where *ν<sup>c</sup> <sup>i</sup>* are mass eigenstates, as follows:

$$\varepsilon\_{i} = \sum\_{i \neq j} \frac{\text{lm}\left[ (m\_{\text{D}}^{\dagger} m\_{\text{D}})\_{ij}^{2} \right]}{8\pi \langle H\_{\text{l}} \rangle^{2} (m\_{\text{D}}^{\dagger} m\_{\text{D}})\_{\text{li}}} \left( F\_{\text{S}} \left( \mathbf{x}\_{i \rangle \prime} y\_{i \rangle} y\_{j} \right) + F\_{\text{V}} (\mathbf{x}\_{i \overline{j}}) \right) \tag{44a}$$

where we take �*Hu*� ≃ 174 GeV, for large tan *β* and

$$\text{tr}\_{\hat{\imath}\hat{\jmath}} := \frac{M\_{\hat{\jmath}\hat{\imath}^{c}}}{M\_{\hat{\imath}\hat{\imath}^{c}}} \text{ and } \ y\_{\hat{\imath}} := \frac{\Gamma\_{\hat{\imath}\hat{\imath}^{c}}}{M\_{\hat{\imath}\hat{\imath}^{c}}} = \frac{(m\_{\text{D}}^{\dagger}m\_{\text{D}})\_{\hat{\imath}\hat{\imath}}}{8\pi\langle H\_{\text{u}}\rangle^{2}}\tag{44b}$$

(with *<sup>i</sup>*, *<sup>j</sup>* = 1, 2, 3). Also *<sup>F</sup>*<sup>V</sup> and *<sup>F</sup>*<sup>S</sup> represent, respectively, the contributions from vertex and self-energy diagrams which in SUSY theories read [49]

$$F\_{\mathbf{V}}(\mathbf{x}) = -\mathbf{x} \ln \left( 1 + \mathbf{x}^{-2} \right) \quad \text{and} \quad F\_{\mathbf{S}}(\mathbf{x}, y, z) = \frac{-2\mathbf{x}(\mathbf{x}^{2} - 1)}{\left(\mathbf{x}^{2} - 1\right)^{2} + \left(\mathbf{x}^{2}z - y\right)^{2}} \cdot \tag{44c}$$

Note that for strongly hierarchical *<sup>M</sup><sup>ν</sup><sup>c</sup>* 's with *xij* <sup>≫</sup> 1 and *xij* <sup>≫</sup> *yi*, *yj*, we obtain the well-known approximate result [26, 27]

$$F\_{\mathsf{V}} + F\_{\mathsf{S}} \simeq -\mathsf{3} / \mathfrak{x}\_{\mathsf{ij}}^2. \tag{45}$$

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model 17

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

 · 

*<sup>e</sup>*−*iϕ*1/2

*dνcU<sup>c</sup>* (53)

. (54)

1 <sup>2</sup> ln *<sup>f</sup>*(*h*f) *f*(*h*∗)

· (55)

*<sup>e</sup>*−*iϕ*2/2

http://dx.doi.org/10.5772/51888

1

, (52)

257

and parameterized as follows:

CP-violating Majorana phases.

*<sup>c</sup>*12*c*<sup>13</sup> *<sup>s</sup>*12*c*<sup>13</sup> *<sup>s</sup>*13*e*−*i<sup>δ</sup>*

with *cij* := cos *<sup>θ</sup>ij*, *sij* := sin *<sup>θ</sup>ij*, *<sup>δ</sup>* the CP-violating Dirac phase and *<sup>ϕ</sup>*<sup>1</sup> and *<sup>ϕ</sup>*<sup>2</sup> the two

Following a bottom-up approach, along the lines of Ref. [26–28], we can find *m*¯ *<sup>ν</sup>* via Eq. (51) using as input parameters the low energy neutrino observables, the CP violating phases and adopting the normal or inverted hierarchical scheme of neutrino masses. Taking also *mi*<sup>D</sup> as

<sup>D</sup> *<sup>m</sup>*¯ *<sup>ν</sup>d*−<sup>1</sup>

*<sup>ν</sup><sup>c</sup>* <sup>=</sup> *<sup>U</sup>c*†*WW*†*U<sup>c</sup>*

Note that *WW*† is a 3 × 3 complex, hermitian matrix and can be diagonalized following the algorithm described in Ref. [50]. Having determined the elements of *<sup>U</sup><sup>c</sup>* and the *Mi<sup>ν</sup>*�*<sup>c</sup>* 's we

We exhibit the constraints that we impose on our cosmological set-up in Sec. 5.1, and

The parameters of our model can be restricted once we impose the following requirements:

According to the inflationary paradigm, the horizon and flatness problems of the standard Big Bang cosmology can be successfully resolved provided that the number of e-foldings, *<sup>N</sup>*�∗, that the scale *<sup>k</sup>*<sup>∗</sup> <sup>=</sup> 0.002/Mpc suffers during nMHI takes a certain value, which depends on the details of the cosmological model. Employing standard methods [51], we can easily derive the required *<sup>N</sup>*�<sup>∗</sup> for our model, consistently with the fact that the *<sup>P</sup>* <sup>−</sup> *<sup>P</sup>*¯ system remains subdominant during the post-inflationary era. Namely we obtain

> <sup>3</sup> ln *<sup>V</sup>*HI(*h*f)1/4 1 GeV <sup>+</sup>

1 <sup>3</sup> ln *<sup>T</sup>*rh 1 GeV <sup>+</sup>

<sup>D</sup> <sup>=</sup> *<sup>U</sup><sup>c</sup>*

−*c*23*s*<sup>12</sup> − *<sup>s</sup>*23*c*12*s*13*ei<sup>δ</sup> <sup>c</sup>*23*c*<sup>12</sup> − *<sup>s</sup>*23*s*12*s*13*ei<sup>δ</sup> <sup>s</sup>*23*c*<sup>13</sup> *<sup>s</sup>*23*s*<sup>12</sup> − *<sup>c</sup>*23*c*12*s*13*ei<sup>δ</sup>* −*s*23*c*<sup>12</sup> − *<sup>c</sup>*23*s*12*s*13*ei<sup>δ</sup> <sup>c</sup>*23*c*<sup>13</sup>

input parameters we can construct the complex symmetric matrix

can compute *m*<sup>D</sup> through Eq. (48) and the *εi*'s through Eq. (44a).

delineate the allowed parameter space of our model in Sec. 5.2.

**5. Constraining the model parameters**

*<sup>N</sup>*�<sup>∗</sup> <sup>≃</sup> 22.5 <sup>+</sup> 2 ln *<sup>V</sup>*HI(*h*∗)1/4

1 GeV <sup>−</sup> <sup>4</sup>

**5.1. Imposed constraints**

– see Eq. (50) – from which we can extract *dν<sup>c</sup>* as follows:

*<sup>W</sup>* <sup>=</sup> <sup>−</sup>*d*−<sup>1</sup>

*<sup>d</sup>*−<sup>2</sup>

*<sup>U</sup><sup>ν</sup>* =

The involved in Eq. (44a) *m*<sup>D</sup> can be diagonalized if we define a basis – called *weak basis* henceforth – in which the lepton Yukawa couplings and the *SU*(2)<sup>L</sup> interactions are diagonal in the space of generations. In particular we have

$$
\Delta U^{\dagger} m\_{\rm D} \mathcal{U}^{\dagger} = d\_{\rm D} = \text{diag} \left( m\_{\rm ID}, m\_{\rm 2D}, m\_{\rm 3D} \right) \,. \tag{46}
$$

where *<sup>U</sup>* and *<sup>U</sup><sup>c</sup>* are 3 × 3 unitary matrices which relate *Li* and *<sup>ν</sup><sup>c</sup> <sup>i</sup>* (in the *<sup>ν</sup><sup>c</sup> <sup>i</sup>* -basis) with the ones *<sup>L</sup>*′ *<sup>i</sup>* and *<sup>ν</sup>c*′ *<sup>i</sup>* in the weak basis as follows:

$$L' = LU \quad \text{and} \quad \nu^{\mathcal{C}} = \mathcal{U}^{\mathcal{C}} \nu^{\mathcal{C}}.\tag{47}$$

Here, we write LH lepton superfields, i.e. *SU*(2)<sup>L</sup> doublet leptons, as row 3-vectors in family space and RH anti-lepton superfields, i.e. *SU*(2)<sup>L</sup> singlet anti-leptons, as column 3-vectors. Consequently, the combination *m*† <sup>D</sup>*m*<sup>D</sup> appeared in Eq. (44a) turns out to be a function just of *d*<sup>D</sup> and *Uc*. Namely,

$$
\mathcal{U}m\_{\rm D}^{\dagger}m\_{\rm D} = \mathcal{U}^{c\dagger}d\_{\rm D}^{\dagger}d\_{\rm D}\mathcal{U}^{c}.\tag{48}
$$

The connection of the nTL scenario with the low energy neutrino data can be achieved through the seesaw formula, which gives the light-neutrino mass matrix *m<sup>ν</sup>* in terms of *mi*<sup>D</sup> and *Mi<sup>ν</sup><sup>c</sup>* . Working in the *<sup>ν</sup><sup>c</sup> <sup>i</sup>* -basis, we have

$$m\_{\boldsymbol{\nu}} = -m\_{\mathrm{D}} \, d\_{\boldsymbol{\nu}^{\boldsymbol{\epsilon}}}^{-1} m\_{\mathrm{D}^{\boldsymbol{\prime}}}^{\mathsf{T}} \quad \text{where} \; d\_{\boldsymbol{\nu}^{\boldsymbol{\epsilon}}} = \mathsf{diag}\left(M\_{1\widehat{\boldsymbol{\nu}}^{\boldsymbol{\epsilon}}}, M\_{2\widehat{\boldsymbol{\nu}}^{\boldsymbol{\epsilon}}}, M\_{\widehat{3}\widehat{\boldsymbol{\epsilon}}^{\boldsymbol{\epsilon}}}\right) \tag{49}$$

with *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* <sup>≤</sup> *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* <sup>≤</sup> *<sup>M</sup>*3*<sup>ν</sup><sup>c</sup>* real and positive. Solving Eq. (46) w.r.t *<sup>m</sup>*<sup>D</sup> and inserting the resulting expression in Eq. (49) we extract the mass matrix

$$
\bar{m}\_{\vee} = \mathcal{U}^{\dagger} m\_{\vee} \mathcal{U}^{\*} = -d\_{\mathcal{D}} \mathcal{U}^{c} d\_{\nu^{c}}^{-1} \mathcal{U}^{c} \overline{\mathcal{U}}\_{\mathcal{D}\prime} \tag{50}
$$

which can be diagonalized by the unitary PMNS matrix satisfying

$$\mathfrak{m}\_{\vee} = \mathcal{U}\_{\vee}^\* \operatorname{diag} \left( m\_{1\nu\prime} m\_{2\nu\prime} m\_{3\nu} \right) \mathcal{U}\_{\vee}^\dagger \tag{51}$$

and parameterized as follows:

16 Open Questions in Cosmology

256 Open Questions in Cosmology

ones *<sup>L</sup>*′

*<sup>i</sup>* and *<sup>ν</sup>c*′

of *d*<sup>D</sup> and *Uc*. Namely,

Consequently, the combination *m*†

*mi*<sup>D</sup> and *Mi<sup>ν</sup><sup>c</sup>* . Working in the *<sup>ν</sup><sup>c</sup>*

*<sup>m</sup><sup>ν</sup>* <sup>=</sup> <sup>−</sup>*m*<sup>D</sup> *<sup>d</sup>*−<sup>1</sup>

resulting expression in Eq. (49) we extract the mass matrix

which can be diagonalized by the unitary PMNS matrix satisfying

*<sup>m</sup>*¯ *<sup>ν</sup>* <sup>=</sup> *<sup>U</sup>*<sup>∗</sup>

*<sup>F</sup>*<sup>V</sup> (*x*) = −*<sup>x</sup>* ln

well-known approximate result [26, 27]

in the space of generations. In particular we have

where *<sup>U</sup>* and *<sup>U</sup><sup>c</sup>* are 3 × 3 unitary matrices which relate *Li* and *<sup>ν</sup><sup>c</sup>*

*m*†

*<sup>i</sup>* -basis, we have

*<sup>m</sup>*¯ *<sup>ν</sup>* <sup>=</sup> *<sup>U</sup>*†*mνU*<sup>∗</sup> <sup>=</sup> <sup>−</sup>*d*D*U<sup>c</sup>*

*<sup>i</sup>* in the weak basis as follows:

 <sup>1</sup> <sup>+</sup> *<sup>x</sup>*−<sup>2</sup>  and *<sup>F</sup>*<sup>S</sup> (*x*, *<sup>y</sup>*, *<sup>z</sup>*) <sup>=</sup> <sup>−</sup>2*x*(*x*<sup>2</sup> <sup>−</sup> <sup>1</sup>)

Note that for strongly hierarchical *<sup>M</sup><sup>ν</sup><sup>c</sup>* 's with *xij* <sup>≫</sup> 1 and *xij* <sup>≫</sup> *yi*, *yj*, we obtain the

*<sup>F</sup>*<sup>V</sup> + *<sup>F</sup>*<sup>S</sup> ≃ −3/*x*<sup>2</sup>

The involved in Eq. (44a) *m*<sup>D</sup> can be diagonalized if we define a basis – called *weak basis* henceforth – in which the lepton Yukawa couplings and the *SU*(2)<sup>L</sup> interactions are diagonal

*<sup>L</sup>*′ <sup>=</sup> *LU* and *<sup>ν</sup>c*′ <sup>=</sup> *<sup>U</sup><sup>c</sup>*

Here, we write LH lepton superfields, i.e. *SU*(2)<sup>L</sup> doublet leptons, as row 3-vectors in family space and RH anti-lepton superfields, i.e. *SU*(2)<sup>L</sup> singlet anti-leptons, as column 3-vectors.

The connection of the nTL scenario with the low energy neutrino data can be achieved through the seesaw formula, which gives the light-neutrino mass matrix *m<sup>ν</sup>* in terms of

with *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* <sup>≤</sup> *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* <sup>≤</sup> *<sup>M</sup>*3*<sup>ν</sup><sup>c</sup>* real and positive. Solving Eq. (46) w.r.t *<sup>m</sup>*<sup>D</sup> and inserting the

<sup>D</sup>*m*<sup>D</sup> <sup>=</sup> *<sup>U</sup>c*†*d*†

(*x*<sup>2</sup> − 1)

*<sup>U</sup>*†*m*D*Uc*† = *<sup>d</sup>*<sup>D</sup> = diag (*m*1D, *<sup>m</sup>*2D, *<sup>m</sup>*3D), (46)

*νc*

D*d*D*U<sup>c</sup>*

*<sup>d</sup>*−<sup>1</sup>

*<sup>ν</sup>* diag (*m*1*ν*, *<sup>m</sup>*2*ν*, *<sup>m</sup>*3*ν*) *<sup>U</sup>*†

<sup>D</sup>*m*<sup>D</sup> appeared in Eq. (44a) turns out to be a function just

*<sup>ν</sup><sup>c</sup> <sup>m</sup>*D, where *<sup>d</sup>ν<sup>c</sup>* <sup>=</sup> diag (*M*1*<sup>ν</sup><sup>c</sup>* , *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* , *<sup>M</sup>*3*<sup>ν</sup><sup>c</sup>* ) (49)

<sup>2</sup> + (*x*2*z* − *y*)

*ij*. (45)

*<sup>i</sup>* (in the *<sup>ν</sup><sup>c</sup>*

. (47)

. (48)

*<sup>ν</sup><sup>c</sup> <sup>U</sup><sup>c</sup> <sup>d</sup>*D, (50)

*<sup>ν</sup>* (51)

<sup>2</sup> · (44c)

*<sup>i</sup>* -basis) with the

$$\mathbf{M}\_{\boldsymbol{\nu}} = \begin{Bmatrix} c\mathbf{1}2\mathbf{\mathcal{C}}\mathbf{1}3 & s\_{12}\mathbf{\mathcal{C}}\mathbf{1}3 & s\_{13}e^{-i\delta} \\ -c\mathbf{2}s\_{12} - s\_{23}c\_{12}s\_{13}e^{i\delta} & c\_{23}c\_{12} - s\_{23}s\_{12}s\_{13}e^{i\delta} & s\_{23}c\_{13} \\ s\_{23}s\_{12} - c\_{23}c\_{12}s\_{13}e^{i\delta} & -s\_{23}c\_{12} - c\_{23}s\_{12}s\_{13}e^{i\delta} & c\_{23}c\_{13} \end{Bmatrix} \cdot \begin{Bmatrix} e^{-i\boldsymbol{\rho}\_{1}/2} & & \\ & e^{-i\boldsymbol{\rho}\_{2}/2} & \\ & & 1 \end{Bmatrix} \tag{52}$$

with *cij* := cos *<sup>θ</sup>ij*, *sij* := sin *<sup>θ</sup>ij*, *<sup>δ</sup>* the CP-violating Dirac phase and *<sup>ϕ</sup>*<sup>1</sup> and *<sup>ϕ</sup>*<sup>2</sup> the two CP-violating Majorana phases.

Following a bottom-up approach, along the lines of Ref. [26–28], we can find *m*¯ *<sup>ν</sup>* via Eq. (51) using as input parameters the low energy neutrino observables, the CP violating phases and adopting the normal or inverted hierarchical scheme of neutrino masses. Taking also *mi*<sup>D</sup> as input parameters we can construct the complex symmetric matrix

$$\mathcal{W} = -d\_{\rm D}^{-1} \vec{m}\_{\nu} d\_{\rm D}^{-1} = \mathcal{U}^{c} d\_{\nu^c} \mathcal{U}^{c \top} \tag{53}$$

– see Eq. (50) – from which we can extract *dν<sup>c</sup>* as follows:

$$d\_{\nu^{\varepsilon}}^{-2} = \mathcal{U}^{\varepsilon \dagger} WW^{\dagger} \mathcal{U}^{\varepsilon}. \tag{54}$$

Note that *WW*† is a 3 × 3 complex, hermitian matrix and can be diagonalized following the algorithm described in Ref. [50]. Having determined the elements of *<sup>U</sup><sup>c</sup>* and the *Mi<sup>ν</sup>*�*<sup>c</sup>* 's we can compute *m*<sup>D</sup> through Eq. (48) and the *εi*'s through Eq. (44a).

## **5. Constraining the model parameters**

We exhibit the constraints that we impose on our cosmological set-up in Sec. 5.1, and delineate the allowed parameter space of our model in Sec. 5.2.

#### **5.1. Imposed constraints**

The parameters of our model can be restricted once we impose the following requirements:

According to the inflationary paradigm, the horizon and flatness problems of the standard Big Bang cosmology can be successfully resolved provided that the number of e-foldings, *<sup>N</sup>*�∗, that the scale *<sup>k</sup>*<sup>∗</sup> <sup>=</sup> 0.002/Mpc suffers during nMHI takes a certain value, which depends on the details of the cosmological model. Employing standard methods [51], we can easily derive the required *<sup>N</sup>*�<sup>∗</sup> for our model, consistently with the fact that the *<sup>P</sup>* <sup>−</sup> *<sup>P</sup>*¯ system remains subdominant during the post-inflationary era. Namely we obtain

$$\hat{N}\_{\*} \simeq 22.5 + 2 \ln \frac{V\_{\rm HI}(h\_{\*})^{1/4}}{1 \,\text{GeV}} - \frac{4}{3} \ln \frac{V\_{\rm HI}(h\_{\rm f})^{1/4}}{1 \,\text{GeV}} + \frac{1}{3} \ln \frac{T\_{\rm rh}}{1 \,\text{GeV}} + \frac{1}{2} \ln \frac{f(h\_{\rm f})}{f(h\_{\*})} \,\text{s} \tag{55}$$

The inflationary observables derived in Sec. 3.2 are to be consistent with the fitting [16] of the WMAP7, BAO and *H*<sup>0</sup> data. As usual, we adopt the central value of ∆R, whereas we allow the remaining quantities to vary within the 95% *confidence level* (c.l.) ranges. Namely,

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259

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**Normal Inverted Hierarchy**

<sup>−</sup>0.10 <sup>−</sup>2.4+0.10

−0.017

<sup>−</sup>0.004 0.027+0.003

<sup>−</sup>0.05 0.53+0.05

<sup>−</sup>0.64 0.07

−0.07

−0.004

−0.07

<sup>31</sup>, for *normally ordered* (NO) *mν*'s

�, for *invertedly ordered* (IO) *<sup>m</sup>ν*'s

� <sup>≤</sup> 2.3 eV. (63)

<sup>∑</sup>*imi<sup>ν</sup>* <sup>≤</sup> 0.58 eV (62)

21

(61)

**Parameter Best Fit** ±1*σ*

<sup>21</sup>/10<sup>−</sup>3eV2 7.62 <sup>±</sup> 0.19

sin2 *<sup>θ</sup>*<sup>12</sup> 0.320+0.015

**Table 3.** Low energy experimental neutrino data for normal or inverted hierarchical neutrino masses. In the second case the

where *mt* is the top quark mass and the numerical values correspond to *<sup>y</sup>*33(*m*I)=(0.55 − 0.7) – cf. Ref. [55] – found [32, 56] working in the context of several MSSM versions with tan *β* ≃ 50 and taking into account the SUSY threshold corrections. As regards the lighter generation, we limit ourselves in imposing just a mild hierarchy between *m*1D and *m*2D, i.e., *<sup>m</sup>*1D < *<sup>m</sup>*2D ≪ *<sup>m</sup>*3D since it is not possible to achieve a simultaneous fulfilment of all the residual constraints if we impose relations similar to Eq. (60) – cf. Ref. [25–27].

From the solar, atmospheric, accelerator and reactor neutrino experiments we take into account the inputs listed in Table 3 on the neutrino mass-squared differences ∆*m*<sup>2</sup>

[inverted] neutrino mass hierarchy [23] – see also Ref. [24]. In particular, *miν*'s can be

or

at 95% c.l. This is more restrictive than the 95% c.l. upper bound arising from the effective

<sup>1</sup>*iνmi<sup>ν</sup>* � �

However, in the future, the KATRIN experiment [58] expects to reach the sensitivity of

� *m*2

� *m*2 <sup>3</sup>*<sup>ν</sup>* <sup>+</sup> � �∆*m*<sup>2</sup> 31 �

<sup>31</sup>, on the mixing angles *θij* and on the CP-violating Dirac phase, *δ* for normal

<sup>1</sup>*<sup>ν</sup>* <sup>+</sup> <sup>∆</sup>*m*<sup>2</sup>

31/10<sup>−</sup>3eV2 2.53+0.08

sin2 *<sup>θ</sup>*<sup>13</sup> 0.026+0.003

sin<sup>2</sup> *<sup>θ</sup>*<sup>23</sup> 0.49+0.08

*<sup>δ</sup>*/*<sup>π</sup>* 0.83+0.54

∆*m*<sup>2</sup>

full range (0 − 2*π*) is allowed at 1*σ* for the phase *δ*.

determined via the relations:

<sup>1</sup>*<sup>ν</sup>* <sup>+</sup> <sup>∆</sup>*m*<sup>2</sup>

electron neutrino mass in *β*-decay [57]:

<sup>21</sup> and

 

*<sup>m</sup>*3*<sup>ν</sup>* <sup>=</sup>

*<sup>m</sup>*1*<sup>ν</sup>* <sup>=</sup>

The sum of *miν*'s can be bounded from above by the WMAP7 data [16]

*<sup>m</sup><sup>β</sup>* :<sup>=</sup> � � �∑*iU*<sup>2</sup>

and ∆*m*<sup>2</sup>

*<sup>m</sup>*2*<sup>ν</sup>* <sup>=</sup>

� *m*2

*<sup>m</sup><sup>β</sup>* <sup>≃</sup> 0.2 eV at 90% c.l.

∆*m*<sup>2</sup>

$$\text{(a) } \Delta\_{\overline{\mathcal{R}}} \simeq 4.93 \cdot 10^{-5} \text{ (b) } n\_{\circ} = 0.968 \pm 0.024 \text{ (c) } -0.062 \le a\_{\circ} \le 0.018 \text{ and (d) } r < 0.24 \text{ (e) } -0.062 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ (f) } -0.02 \le a\_{\circ} \le 0.24 \text{ ($$

The scale *M*PS can be determined by requiring that the v.e.vs of the Higgs fields take the values dictated by the unification of the gauge couplings within the MSSM. As we now recognize – cf. Ref. [5] – the unification scale *<sup>M</sup>*GUT ≃ <sup>2</sup> · <sup>10</sup><sup>16</sup> GeV is to be identified with the *lowest* mass scale of the model in the SUSY vacuum, Eq. (9), in order to avoid any extra contribution to the running of the MSSM gauge couplings, i.e.,

$$\text{I(a)}\ \frac{\text{g}M\_{\text{PS}}}{\sqrt{f\_0}} = M\_{\text{GUT}} \Rightarrow m\_{\text{PS}} = \frac{1}{2\sqrt{2c\_{\text{\mathcal{R}}}^{\text{max}} - c\_{\text{\mathcal{R}}}}} \text{ with (b)}\ c\_{\text{\mathcal{R}}}^{\text{max}} = \frac{g^2 m\_{\text{P}}^2}{8M\_{\text{GUT}}^2} \qquad (57)$$

The requirement 2*c*max <sup>R</sup> − *c*<sup>R</sup> > 0 sets an upper bound *c*<sup>R</sup> < 2*c*max <sup>R</sup> ≃ 1.8 · 103, which however can be significantly lowered if we combine Eqs. (55) and (28) – see Sec. 5.2.1.

For the realization of nMHI , we assume that *c*R takes relatively large values – see e.g. Eq. (17). This assumption may [52, 53] jeopardize the validity of the classical approximation, on which the analysis of the inflationary behavior is based. To avoid this inconsistency – which is rather questionable [11, 54] though – we have to check the hierarchy between the ultraviolet cut-off, <sup>Λ</sup> = *<sup>m</sup>*P/*c*R, of the effective theory and the inflationary scale, which is represented by *<sup>V</sup>*HI(*h*∗)1/4 or, less restrictively, by the corresponding Hubble parameter, *<sup>H</sup>*<sup>∗</sup> <sup>=</sup> *<sup>V</sup>*HI(*h*∗)1/2/ √3*m*P. In particular, the validity of the effective theory implies [52, 53]

$$
\text{(a) } \hat{V}\_{\text{HI}}(\hbar\_\*)^{1/4} \le \Lambda \text{ or (b) } \hat{H}\_\* \le \Lambda \text{ for (c) } c\_{\mathcal{R}} \ge 1. \tag{58}
$$

As discussed in Sec. 4.2, to avoid any erasure of the produced *YL* and to ensure that the inflaton decay to *<sup>ν</sup>*<sup>2</sup> is kinematically allowed we have to bound *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* and *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* respectively as follows:

$$(\text{a) } M\_{\text{l}\tilde{\text{e}}} \gtrsim 10T\_{\text{rh}} \text{ and } (\text{b) } m\_{\text{l}} \ge 2M\_{\text{2\tilde{v}}^{\text{c}}} \Rightarrow M\_{\text{2\tilde{v}}^{\text{c}}} \lesssim \frac{\lambda m\_{\text{P}}}{4\sqrt{3}c\_{\text{\mathcal{R}}}} \simeq 1.5 \cdot 10^{13} \text{ GeV},\tag{59}$$

where we make use of Eq. (35). Recall that we impose also the restriction *λ* ≥ 0.001 which allows us to ignore effects of instant preheating [5, 42].

As discussed below Eq. (2), the adopted GUT predicts YU at *M*GUT. Assuming negligible running of *<sup>m</sup>*3D from *<sup>M</sup>*GUT until the scale of nTL, <sup>Λ</sup>*L*, which is taken to be <sup>Λ</sup>*<sup>L</sup>* = *<sup>m</sup>*I, we end up with the requirement:

$$m\_{\rm 3D}(m\_{\rm I}) = m\_{\rm l}(m\_{\rm I}) \simeq (100 - 120) \text{ GeV.} \tag{60}$$


18 Open Questions in Cosmology

258 Open Questions in Cosmology

(a) *gM* PS *f*0

The requirement 2*c*max

as follows:

Namely,

The inflationary observables derived in Sec. 3.2 are to be consistent with the fitting [16] of the WMAP7, BAO and *H*<sup>0</sup> data. As usual, we adopt the central value of ∆R, whereas we allow the remaining quantities to vary within the 95% *confidence level* (c.l.) ranges.

<sup>∆</sup><sup>R</sup> <sup>≃</sup> 4.93 · <sup>10</sup><sup>−</sup>5, *<sup>n</sup>*<sup>s</sup> <sup>=</sup> 0.968 <sup>±</sup> 0.024, <sup>−</sup> 0.062 <sup>≤</sup> *<sup>a</sup>*<sup>s</sup> <sup>≤</sup> 0.018 and *<sup>r</sup>* <sup>&</sup>lt; 0.24

<sup>R</sup> − *c*<sup>R</sup>

√

*<sup>V</sup>*HI(*h*∗)1/4 <sup>≤</sup> <sup>Λ</sup> or *<sup>H</sup>*<sup>∗</sup> <sup>≤</sup> <sup>Λ</sup> for *<sup>c</sup>*<sup>R</sup> <sup>≥</sup> 1. (58)

4 √3*c*R

*<sup>m</sup>*3D(*m*I) = *mt*(*m*I) ≃ (<sup>100</sup> − <sup>120</sup>) GeV. (60)

with (b) *c*max

<sup>R</sup> <sup>=</sup> *<sup>g</sup>*2*m*<sup>2</sup>

3*m*P. In particular, the validity of

≃ 1.5 · 1013 GeV, (59)

P 8*M*<sup>2</sup> GUT

<sup>R</sup> ≃ 1.8 · 103, which

The scale *M*PS can be determined by requiring that the v.e.vs of the Higgs fields take the values dictated by the unification of the gauge couplings within the MSSM. As we now recognize – cf. Ref. [5] – the unification scale *<sup>M</sup>*GUT ≃ <sup>2</sup> · <sup>10</sup><sup>16</sup> GeV is to be identified with the *lowest* mass scale of the model in the SUSY vacuum, Eq. (9), in order to avoid any

<sup>R</sup> − *c*<sup>R</sup> > 0 sets an upper bound *c*<sup>R</sup> < 2*c*max

As discussed in Sec. 4.2, to avoid any erasure of the produced *YL* and to ensure that the inflaton decay to *<sup>ν</sup>*<sup>2</sup> is kinematically allowed we have to bound *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* and *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* respectively

where we make use of Eq. (35). Recall that we impose also the restriction *λ* ≥ 0.001 which

As discussed below Eq. (2), the adopted GUT predicts YU at *M*GUT. Assuming negligible running of *<sup>m</sup>*3D from *<sup>M</sup>*GUT until the scale of nTL, <sup>Λ</sup>*L*, which is taken to be <sup>Λ</sup>*<sup>L</sup>* = *<sup>m</sup>*I, we

however can be significantly lowered if we combine Eqs. (55) and (28) – see Sec. 5.2.1. For the realization of nMHI , we assume that *c*R takes relatively large values – see e.g. Eq. (17). This assumption may [52, 53] jeopardize the validity of the classical approximation, on which the analysis of the inflationary behavior is based. To avoid this inconsistency – which is rather questionable [11, 54] though – we have to check the hierarchy between the ultraviolet cut-off, <sup>Λ</sup> = *<sup>m</sup>*P/*c*R, of the effective theory and the inflationary scale, which is represented by *<sup>V</sup>*HI(*h*∗)1/4 or, less restrictively, by the

extra contribution to the running of the MSSM gauge couplings, i.e.,

<sup>=</sup> *<sup>M</sup>*GUT <sup>⇒</sup> *<sup>m</sup>*PS <sup>=</sup> <sup>1</sup>

corresponding Hubble parameter, *<sup>H</sup>*<sup>∗</sup> <sup>=</sup> *<sup>V</sup>*HI(*h*∗)1/2/

(a) *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* <sup>10</sup>*T*rh and (b) *<sup>m</sup>*<sup>I</sup> <sup>≥</sup> <sup>2</sup>*M*2*<sup>ν</sup><sup>c</sup>* <sup>⇒</sup> *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup> <sup>λ</sup>m*<sup>P</sup>

allows us to ignore effects of instant preheating [5, 42].

the effective theory implies [52, 53]

end up with the requirement:

2 2*c*max (56)

(57)

**Table 3.** Low energy experimental neutrino data for normal or inverted hierarchical neutrino masses. In the second case the full range (0 − 2*π*) is allowed at 1*σ* for the phase *δ*.

where *mt* is the top quark mass and the numerical values correspond to *<sup>y</sup>*33(*m*I)=(0.55 − 0.7) – cf. Ref. [55] – found [32, 56] working in the context of several MSSM versions with tan *β* ≃ 50 and taking into account the SUSY threshold corrections. As regards the lighter generation, we limit ourselves in imposing just a mild hierarchy between *m*1D and *m*2D, i.e., *<sup>m</sup>*1D < *<sup>m</sup>*2D ≪ *<sup>m</sup>*3D since it is not possible to achieve a simultaneous fulfilment of all the residual constraints if we impose relations similar to Eq. (60) – cf. Ref. [25–27].

From the solar, atmospheric, accelerator and reactor neutrino experiments we take into account the inputs listed in Table 3 on the neutrino mass-squared differences ∆*m*<sup>2</sup> 21 and ∆*m*<sup>2</sup> <sup>31</sup>, on the mixing angles *θij* and on the CP-violating Dirac phase, *δ* for normal [inverted] neutrino mass hierarchy [23] – see also Ref. [24]. In particular, *miν*'s can be determined via the relations:

$$m\_{2\nu} = \sqrt{m\_{1\nu}^2 + \Delta m\_{21}^2} \text{ and } \begin{cases} m\_{3\nu} = \sqrt{m\_{1\nu}^2 + \Delta m\_{31}^2}, \text{ for normally ordered (NO) } m\_{\nu}\text{'s} \\ \text{or} \\ m\_{1\nu} = \sqrt{m\_{3\nu}^2 + |\Delta m\_{31}^2|}, \text{ for invertedly ordered (IO) } m\_{\nu}\text{'s} \end{cases} \tag{61}$$

The sum of *miν*'s can be bounded from above by the WMAP7 data [16]

$$
\Sigma\_i m\_{i\nu} \le 0.58 \text{ eV} \tag{62}
$$

at 95% c.l. This is more restrictive than the 95% c.l. upper bound arising from the effective electron neutrino mass in *β*-decay [57]:

$$\mathcal{I}m\_{\beta} := \left| \sum\_{i} \mathcal{U}\_{1i\nu}^{2} m\_{i\nu} \right| \le 2.3 \text{ eV}. \tag{63}$$

However, in the future, the KATRIN experiment [58] expects to reach the sensitivity of *<sup>m</sup><sup>β</sup>* <sup>≃</sup> 0.2 eV at 90% c.l.

The interpretation of BAU through nTL dictates [16] at 95% c.l.

$$Y\_B = (8.74 \pm 0.42) \cdot 10^{-11} \implies 8.32 \le 10^{11} \\ Y\_B \le 9.16. \tag{64}$$

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261

As can be seen from the relevant expressions in Secs. 2 and 4, our cosmological set-up

*λ*, *λH*, *λH*¯ , *kS*, *g*, *y*33, *m*ℓ*ν*, *mi*D, *ϕ*<sup>1</sup> and *ϕ*2,

where *<sup>m</sup>*ℓ*<sup>ν</sup>* is the low scale mass of the lightest of *<sup>ν</sup>i*'s and can be identified with *<sup>m</sup>*1*<sup>ν</sup>* [*m*3*ν*] for NO [IO] neutrino mass spectrum. Recall that we determine *<sup>M</sup>*PS via Eq. (57) with *<sup>g</sup>* = 0.7. We do not consider *c*<sup>R</sup> and *λiν<sup>c</sup>* as independent parameters since *c*<sup>R</sup> is related to *m* via Eq. (31) while *λiν<sup>c</sup>* can be derived from the last six parameters above which affect exclusively the *YL* calculation and can be constrained through the requirements 5 - 9 of Sec. 5.1. Note that the *<sup>λ</sup>iν<sup>c</sup>* 's can be replaced by *Mi<sup>ν</sup><sup>c</sup>* 's given in Eq. (33 ) keeping in mind that perturbativity requires *<sup>λ</sup>iν<sup>c</sup>* <sup>≤</sup> <sup>√</sup>4*<sup>π</sup>* or *Mi<sup>ν</sup><sup>c</sup>* <sup>≤</sup> <sup>10</sup><sup>16</sup> GeV. Note that if we replace *<sup>M</sup>*<sup>S</sup> with *<sup>m</sup>*<sup>P</sup> in Eq. (5), we obtain a tighter bound, i.e., *Mi<sup>ν</sup><sup>c</sup>* <sup>≤</sup> 2.3 · <sup>1015</sup> GeV. Our results are essentially independent of *<sup>λ</sup>H*, *<sup>λ</sup>H*¯

*<sup>S</sup>* in Table 2 are positive for *<sup>λ</sup>* <sup>&</sup>lt; 1. We therefore set *<sup>λ</sup><sup>H</sup>* <sup>=</sup> *<sup>λ</sup>H*¯ <sup>=</sup> 0.5 and *kS* <sup>=</sup> 1 throughout our calculation. Finally *T*rh can be calculated self-consistently in our model as a function of *<sup>m</sup>*I, *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* <sup>≫</sup> *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* and the unified Yukawa coupling constant *<sup>y</sup>*<sup>33</sup> – see Sec. 4.1 – for which we

The selected values for the above quantities give us a wide and natural allowed region for the remaining fundamental parameters of our model, as we show below concentrating separately

In this part of our numerical code, we use as input parameters *<sup>h</sup>*∗, *<sup>m</sup>*2D ≫ *<sup>m</sup>*1D and *<sup>c</sup>*R. For every chosen *<sup>c</sup>*<sup>R</sup> ≥ 1 and *<sup>m</sup>*2D, we restrict *<sup>λ</sup>* and *<sup>h</sup>*<sup>∗</sup> so that the conditions Eq. (55) and (56 ) are satisfied. In our numerical calculations, we use the complete formulas for the slow-roll parameters and ∆<sup>R</sup> in Eqs. (25a), (25b) and (30) and not the approximate relations listed in Sec. 3.2 for the sake of presentation. Our results are displayed in Fig. 1, where we draw the allowed values of *c*<sup>R</sup> (solid line), *T*rh (dashed line), the inflaton mass, *m*<sup>I</sup> (dot-dashed line) and *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* (dotted line) – see Sec. 4.1 – [*h*<sup>f</sup> (solid line) and *<sup>h</sup>*<sup>∗</sup> (dashed line)] versus *<sup>λ</sup>* (a) [(b)] for the *m*2D's required from Eq. (64) and for the parameters adopted along the black dashed line of Fig. 2 – see Sec. 5.2.2. The required via Eq. (55) *<sup>N</sup>*<sup>∗</sup> remains almost constant and close

The lower bound of the depicted lines comes from the saturation of the Eq. (58 ). The constraint of Eq. (58 ) is satisfied along the various curves whereas Eq. (58 ) is valid only along the gray and light gray segments of these. Along the light gray segments, though, we obtain *<sup>h</sup>*<sup>∗</sup> ≥ *<sup>m</sup>*P. The latter regions of parameter space are not necessarily excluded, since the energy density of the inflaton remains sub-Planckian and so, corrections from quantum

in the inflationary period (Sec. 5.2.1) and in the stage of nTL (Sec. 5.2.2).

*kS* <sup>=</sup> 1, *<sup>λ</sup><sup>H</sup>* <sup>=</sup> *<sup>λ</sup>H*¯ <sup>=</sup> 0.5, *<sup>g</sup>* <sup>=</sup> 0.7 and *<sup>y</sup>*<sup>33</sup> <sup>=</sup> 0.6. (68)

and *kS*, provided that we choose some relatively large values for these so as *m*<sup>2</sup>

**5.2. Numerical results**

depends on the parameters:

*m*2

take *<sup>y</sup>*<sup>33</sup> = 0.6.

*5.2.1. The Inflationary Stage*

to 54.5.

Summarizing, we set throughout our calculation:

In order to avoid spoiling the success of the BBN, an upper bound on *YG*� is to be imposed depending on the *<sup>G</sup>*� mass, *mG*�, and the dominant *<sup>G</sup>*� decay mode. For the conservative case where *<sup>G</sup>*� decays with a tiny hadronic branching ratio, we have [19]

$$Y\_{\widetilde{G}} \lesssim \begin{cases} 10^{-14} \\ 10^{-13} & \text{for } \ m\_{\widetilde{G}} \simeq \begin{cases} 0.69 \text{ TeV} \\ 10.6 \text{ TeV} \\ 13.5 \text{ TeV} \end{cases} \tag{65}$$

As we see below, this bound is achievable within our model only for *mG*� 10 TeV. Taking into account that the soft masses of the scalars are not necessarily equal to *mG*�, we do not consider such a restriction as a very severe tuning of the SUSY parameter space. Using Eq. (43) the bounds on *YG*� can be translated into bounds on *<sup>T</sup>*rh. Specifically we take *<sup>T</sup>*rh <sup>≃</sup> (0.53 <sup>−</sup> 5.3) · <sup>108</sup> GeV [*T*rh <sup>≃</sup> (0.53 <sup>−</sup> 5.3) · <sup>109</sup> GeV] for *YG*� <sup>≃</sup> (0.1 <sup>−</sup> <sup>1</sup>) · <sup>10</sup>−<sup>13</sup> [*YG*� <sup>≃</sup> (0.1 <sup>−</sup> <sup>1</sup>) · <sup>10</sup><sup>−</sup>12].

Let us, finally, comment on the axion isocurvature perturbations generated in our model. Indeed, since the PQ symmetry is broken during nMHI, the axion acquires quantum fluctuations as all the almost massless degrees of freedom. At the QCD phase transition, these fluctuations turn into isocurvature perturbations in the axion energy density, which means that the partial curvature perturbation in axions is different than the one in photons. The results of WMAP put stringent bounds on the possible CDM isocurvature perturbation. Namely, taking into account the WMAP7, BAO and *H*<sup>0</sup> data on the parameter *α*<sup>0</sup> we find the following bound for the amplitude of the CDM isocurvature perturbation

$$|\mathcal{S}\_{\mathbb{C}}| = \Delta\_{\mathbb{R}} \sqrt{\frac{\mathfrak{a}\_{0}}{1 - \mathfrak{a}\_{0}}} \lesssim 1.5 \cdot 10^{-5} \text{ at } 95\% \text{ c.l.} \tag{66}$$

On the other, |Sc| due to axion, can be estimated by

$$|\mathcal{S}\_{\mathbb{C}}| = \frac{\Omega\_{\mathbb{d}}}{\Omega\_{\mathbb{c}}} \frac{\hat{H}\_{\text{HI}}}{\pi |\theta\_{\mathbb{I}}| \hat{\phi}\_{P\*}} \text{ with } \frac{\Omega\_{\mathbb{d}}}{\Omega\_{\mathbb{c}}} \simeq \theta\_{\mathbb{I}}^2 \left(\frac{f\_{\text{a}}}{1.56 \cdot 10^{11} \text{ GeV}}\right)^{1.175} \tag{67}$$

where <sup>Ω</sup>*<sup>a</sup>* [Ωc] is the axion [CDM] density parameter, *<sup>φ</sup>*�*P*<sup>∗</sup> <sup>∼</sup> <sup>1016</sup> GeV [36] denotes the field value of the PQ scalar when the cosmological scales exit the horizon and *θ*<sup>I</sup> is the initial misalignment angle which lies [36] in the interval [−*π*/6, *π*/6]. Satisfying Eq. (66) requires |*θ*I| *<sup>π</sup>*/70 which is a rather low but not unacceptable value. Therefore, a large axion contribution to CDM is disfavored within our model.

#### **5.2. Numerical results**

20 Open Questions in Cosmology

260 Open Questions in Cosmology

[*YG*� <sup>≃</sup> (0.1 <sup>−</sup> <sup>1</sup>) · <sup>10</sup><sup>−</sup>12].

The interpretation of BAU through nTL dictates [16] at 95% c.l.

case where *<sup>G</sup>*� decays with a tiny hadronic branching ratio, we have [19]

<sup>10</sup>−<sup>14</sup> <sup>10</sup>−<sup>13</sup> <sup>10</sup>−<sup>12</sup>

 

following bound for the amplitude of the CDM isocurvature perturbation

� *α*<sup>0</sup> <sup>1</sup> − *<sup>α</sup>*<sup>0</sup>

> with <sup>Ω</sup>*<sup>a</sup>* Ωc

where <sup>Ω</sup>*<sup>a</sup>* [Ωc] is the axion [CDM] density parameter, *<sup>φ</sup>*�*P*<sup>∗</sup> <sup>∼</sup> <sup>1016</sup> GeV [36] denotes the field value of the PQ scalar when the cosmological scales exit the horizon and *θ*<sup>I</sup> is the initial misalignment angle which lies [36] in the interval [−*π*/6, *π*/6]. Satisfying Eq. (66) requires |*θ*I| *<sup>π</sup>*/70 which is a rather low but not unacceptable value. Therefore, a large axion

≃ *θ*<sup>2</sup> I

� *fa*

1.56 · 1011 GeV


*<sup>H</sup>*�HI *<sup>π</sup>*|*θ*I|*φ*�*P*<sup>∗</sup>

On the other, |Sc| due to axion, can be estimated by

contribution to CDM is disfavored within our model.


*YG*�

*YB* <sup>=</sup> (8.74 <sup>±</sup> 0.42) · <sup>10</sup>−<sup>11</sup> <sup>⇒</sup> 8.32 <sup>≤</sup> <sup>10</sup><sup>11</sup>*YB* <sup>≤</sup> 9.16. (64)

 

0.69 TeV 10.6 TeV 13.5 TeV.

1.5 · <sup>10</sup>−<sup>5</sup> at 95% c.l. (66)

�1.175

(65)

(67)

In order to avoid spoiling the success of the BBN, an upper bound on *YG*� is to be imposed depending on the *<sup>G</sup>*� mass, *mG*�, and the dominant *<sup>G</sup>*� decay mode. For the conservative

for *mG*� <sup>≃</sup>

As we see below, this bound is achievable within our model only for *mG*� 10 TeV. Taking into account that the soft masses of the scalars are not necessarily equal to *mG*�, we do not consider such a restriction as a very severe tuning of the SUSY parameter space. Using Eq. (43) the bounds on *YG*� can be translated into bounds on *<sup>T</sup>*rh. Specifically we take *<sup>T</sup>*rh <sup>≃</sup> (0.53 <sup>−</sup> 5.3) · <sup>108</sup> GeV [*T*rh <sup>≃</sup> (0.53 <sup>−</sup> 5.3) · <sup>109</sup> GeV] for *YG*� <sup>≃</sup> (0.1 <sup>−</sup> <sup>1</sup>) · <sup>10</sup>−<sup>13</sup>

Let us, finally, comment on the axion isocurvature perturbations generated in our model. Indeed, since the PQ symmetry is broken during nMHI, the axion acquires quantum fluctuations as all the almost massless degrees of freedom. At the QCD phase transition, these fluctuations turn into isocurvature perturbations in the axion energy density, which means that the partial curvature perturbation in axions is different than the one in photons. The results of WMAP put stringent bounds on the possible CDM isocurvature perturbation. Namely, taking into account the WMAP7, BAO and *H*<sup>0</sup> data on the parameter *α*<sup>0</sup> we find the As can be seen from the relevant expressions in Secs. 2 and 4, our cosmological set-up depends on the parameters:

*λ*, *λH*, *λH*¯ , *kS*, *g*, *y*33, *m*ℓ*ν*, *mi*D, *ϕ*<sup>1</sup> and *ϕ*2,

where *<sup>m</sup>*ℓ*<sup>ν</sup>* is the low scale mass of the lightest of *<sup>ν</sup>i*'s and can be identified with *<sup>m</sup>*1*<sup>ν</sup>* [*m*3*ν*] for NO [IO] neutrino mass spectrum. Recall that we determine *<sup>M</sup>*PS via Eq. (57) with *<sup>g</sup>* = 0.7. We do not consider *c*<sup>R</sup> and *λiν<sup>c</sup>* as independent parameters since *c*<sup>R</sup> is related to *m* via Eq. (31) while *λiν<sup>c</sup>* can be derived from the last six parameters above which affect exclusively the *YL* calculation and can be constrained through the requirements 5 - 9 of Sec. 5.1. Note that the *<sup>λ</sup>iν<sup>c</sup>* 's can be replaced by *Mi<sup>ν</sup><sup>c</sup>* 's given in Eq. (33 ) keeping in mind that perturbativity requires *<sup>λ</sup>iν<sup>c</sup>* <sup>≤</sup> <sup>√</sup>4*<sup>π</sup>* or *Mi<sup>ν</sup><sup>c</sup>* <sup>≤</sup> <sup>10</sup><sup>16</sup> GeV. Note that if we replace *<sup>M</sup>*<sup>S</sup> with *<sup>m</sup>*<sup>P</sup> in Eq. (5), we obtain a tighter bound, i.e., *Mi<sup>ν</sup><sup>c</sup>* <sup>≤</sup> 2.3 · <sup>1015</sup> GeV. Our results are essentially independent of *<sup>λ</sup>H*, *<sup>λ</sup>H*¯ and *kS*, provided that we choose some relatively large values for these so as *m*<sup>2</sup> *<sup>u</sup>*<sup>−</sup>, *<sup>m</sup>*<sup>2</sup> *d* <sup>−</sup> and *m*2 *<sup>S</sup>* in Table 2 are positive for *<sup>λ</sup>* <sup>&</sup>lt; 1. We therefore set *<sup>λ</sup><sup>H</sup>* <sup>=</sup> *<sup>λ</sup>H*¯ <sup>=</sup> 0.5 and *kS* <sup>=</sup> 1 throughout our calculation. Finally *T*rh can be calculated self-consistently in our model as a function of *<sup>m</sup>*I, *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* <sup>≫</sup> *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* and the unified Yukawa coupling constant *<sup>y</sup>*<sup>33</sup> – see Sec. 4.1 – for which we take *<sup>y</sup>*<sup>33</sup> = 0.6.

Summarizing, we set throughout our calculation:

$$k\_{\mathbb{S}} = 1, \; \lambda\_H = \lambda\_H = 0.5, \mathbf{g} = 0.7 \text{ and } y\_{\overline{3}\overline{3}} = 0.6. \tag{68}$$

The selected values for the above quantities give us a wide and natural allowed region for the remaining fundamental parameters of our model, as we show below concentrating separately in the inflationary period (Sec. 5.2.1) and in the stage of nTL (Sec. 5.2.2).

#### *5.2.1. The Inflationary Stage*

In this part of our numerical code, we use as input parameters *<sup>h</sup>*∗, *<sup>m</sup>*2D ≫ *<sup>m</sup>*1D and *<sup>c</sup>*R. For every chosen *<sup>c</sup>*<sup>R</sup> ≥ 1 and *<sup>m</sup>*2D, we restrict *<sup>λ</sup>* and *<sup>h</sup>*<sup>∗</sup> so that the conditions Eq. (55) and (56 ) are satisfied. In our numerical calculations, we use the complete formulas for the slow-roll parameters and ∆<sup>R</sup> in Eqs. (25a), (25b) and (30) and not the approximate relations listed in Sec. 3.2 for the sake of presentation. Our results are displayed in Fig. 1, where we draw the allowed values of *c*<sup>R</sup> (solid line), *T*rh (dashed line), the inflaton mass, *m*<sup>I</sup> (dot-dashed line) and *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* (dotted line) – see Sec. 4.1 – [*h*<sup>f</sup> (solid line) and *<sup>h</sup>*<sup>∗</sup> (dashed line)] versus *<sup>λ</sup>* (a) [(b)] for the *m*2D's required from Eq. (64) and for the parameters adopted along the black dashed line of Fig. 2 – see Sec. 5.2.2. The required via Eq. (55) *<sup>N</sup>*<sup>∗</sup> remains almost constant and close to 54.5.

The lower bound of the depicted lines comes from the saturation of the Eq. (58 ). The constraint of Eq. (58 ) is satisfied along the various curves whereas Eq. (58 ) is valid only along the gray and light gray segments of these. Along the light gray segments, though, we obtain *<sup>h</sup>*<sup>∗</sup> ≥ *<sup>m</sup>*P. The latter regions of parameter space are not necessarily excluded, since the energy density of the inflaton remains sub-Planckian and so, corrections from quantum

**Figure 1.** The allowed (by all the imposed constraints) values of *c*<sup>R</sup> (solid line), *T*rh – given by Eq. (40) – (dashed line), *m*<sup>I</sup> (dot-dashed line) and *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* (dotted line) [*h*<sup>f</sup> (solid line) and *<sup>h</sup>*<sup>∗</sup> (dashed line)] versus *<sup>λ</sup>* (a) [(b)] for *kS* <sup>=</sup> <sup>1</sup>, *<sup>λ</sup><sup>H</sup>* <sup>=</sup> *<sup>λ</sup>H*¯ <sup>=</sup> 0.5 and *y*<sup>33</sup> = 0.6. The light gray and gray segments denote values of the various quantities satisfying Eq. (58 ) too, whereas along the light gray segments we obtain *h*<sup>∗</sup> ≥ *m*P.

gravity can still be assumed to be small. As *c*R increases beyond 906, *f*<sup>0</sup> becomes much larger than 1, *<sup>N</sup>*<sup>∗</sup> derived by Eq. (28) starts decreasing and therefore, nMHI fails to fulfil Eq. (55). This can be understood by the observation that *<sup>N</sup>*<sup>∗</sup>, approximated fairly by Eq. (29), becomes monotonically decreasing function of *c*<sup>R</sup> for *c*<sup>R</sup> > *c*max <sup>R</sup> where *c*max R can be found by the condition

$$\frac{d\hat{N}\_\*}{dc\_{\mathcal{R}}} \simeq \frac{3h\_\*^2}{4m\_{\mathcal{P}}^2} \frac{\left(c\_{\mathcal{R}}^{\text{max}} - c\_{\mathcal{R}}\right)}{c\_{\mathcal{R}}^{\text{max}}} = 0 \Rightarrow c\_{\mathcal{R}} \simeq c\_{\mathcal{R}}^{\text{max}} \,\,\,\tag{69}$$

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model 23

http://dx.doi.org/10.5772/51888

263

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

AB C D E F G Normal Degenerate Inverted Hierarchy Masses Hierarchy

Low Scale Parameters *m*1*ν*/0.1 eV 0.01 0.1 0.5 1. 0.7 0.5 0.49 *m*2*ν*/0.1 eV 0.088 0.13 0.5 1. 0.7 0.51 0.5 *m*3*ν*/0.1 eV 0.5 0.5 0.71 1.1 0.5 0.1 0.05 ∑*<sup>i</sup> miν*/0.1 eV 0.6 0.74 1.7 3.1 1.9 1.1 1 *mβ*/0.1 eV 0.03 0.013 0.14 0.68 0.45 0.33 0.4 *<sup>ϕ</sup>*<sup>1</sup> *<sup>π</sup>*/2 *<sup>π</sup>*/2 *<sup>π</sup>*/2 −*π*/2 −*π*/2 −*π*/2 −*π*/4 *<sup>ϕ</sup>*<sup>2</sup> <sup>0</sup> −*π*/2 −*π*/2 *πππ* <sup>0</sup> Leptogenesis-Scale Parameters *m*1D/GeV 0.4 0.3 0.8 1 0.9 0.9 0.9 *m*2D/GeV 9.2 3 6.5 8.6 3.95 6.6 9.2 *m*3D/GeV 120 100 100 120 120 110 110 *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>*/1010 GeV 4.4 4.7 3.6 1.2 1.5 1.6 1.7 *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>*/1012 GeV 2.7 0.6 0.65 1.5 0.9 1.5 2.8 *<sup>M</sup>*3*<sup>ν</sup><sup>c</sup>*/1014 GeV 27 0.28 0.46 0.4 0.4 3.8 10 Resulting *B*-Yield

*<sup>B</sup>* 8.75 8.9 8.6 8.63 8.9 9. 8.76 1011*YB* 9.3 8.7 8.5 8.4 9.7 8.8 9.2

<sup>R</sup>esulting *<sup>T</sup>*rh and *YG T*rh/109 GeV 1.2 0.89 0.89 0.99 0.91 0.99 1.2

**Table 4.** Parameters yielding the correct BAU for various neutrino mass schemes for *λ* = 0.01 and *c*<sup>R</sup> = 220.

for *<sup>λ</sup>* 0.03, *<sup>T</sup>*rh remains almost constant since <sup>Γ</sup>I*<sup>y</sup>* dominates over <sup>Γ</sup>I2*<sup>ν</sup><sup>c</sup>* and *<sup>f</sup>* <sup>3</sup>

<sup>0</sup> <sup>≃</sup> <sup>1</sup> <sup>+</sup> <sup>12</sup>*c*R*m*<sup>2</sup>

<sup>10</sup><sup>13</sup>*YG* 2.4 1.7 1.7 1.9 1.7 1.88 1.88

with *<sup>c</sup>*<sup>R</sup> or *<sup>λ</sup>* as shown in Fig. 1. The required by Eq. (64) *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* follows the behavior of the

In this part of our numerical program, for a given neutrino mass scheme, we take as input parameters: *m*ℓ*ν*, *mi*D, *ϕ*1, *ϕ*<sup>2</sup> and the best-fit values of the neutrino parameters listed in

<sup>0</sup> <sup>≃</sup> 1 – see

PS starts to deviate from unity and so, *T*rh increases

**Parameters Cases**

1011*Y*<sup>0</sup>

Eq. (39). For *λ* 0.03, *f* <sup>3</sup>

required *m*2D – see Fig. 2- of Sec. 5.2.2.

*5.2.2. The Stage of non-Thermal Leptogenesis*

where *c*max R is defined in Eq. (57 ) and Eq. (57 ) is also taken into account. As a consequence, the embedding of nMHI in a SUSY GUT provides us with a clear upper bound of *c*R. All in all, we obtain

$$0.001 \lesssim \lambda \lesssim 0.0412 \text{ and } 1 \lesssim c\_{\mathcal{R}} \lesssim 907 \text{ for } 53.9 \lesssim \dot{N}\_\* \lesssim 54.7 \tag{70}$$

When *<sup>c</sup>*<sup>R</sup> ranges within its allowed region, we take *<sup>M</sup>*PS ≃ (2.87 − <sup>4</sup>) · <sup>10</sup><sup>16</sup> GeV.

From Fig. 1- , we can verify our analytical estimation in Eq. (31) according to which *λ* is proportional to *c*R. On the other hand, the variation of *h*<sup>f</sup> and *h*<sup>∗</sup> as a function of *c*<sup>R</sup> – drawn in Fig. 1- – is consistent with Eqs. (27) and (29). Letting *λ* or *c*R vary within its allowed region in Eq. (70), we obtain

$$n\_8 \simeq 0.964, \ -6.5 \lesssim \frac{n\_8}{10^{-4}} \lesssim -6.2 \text{ and } 4.2 \gtrsim \frac{r}{10^{-3}} \gtrsim 3.5. \tag{71}$$

Clearly, the predicted *α*s and *r* lie within the allowed ranges given in Eq. (56 ) and Eq. (56 ) respectively, whereas *n*s turns out to be impressively close to its central observationally favored value – see Eq. (56 ) and cf. Ref. [12].

From Fig. 1- we can conclude that *m*<sup>I</sup> is kept independent of *λ* and almost constant at the level of 1013 GeV, as anticipated in Eq. (35). From the same plot we also remark that


**Table 4.** Parameters yielding the correct BAU for various neutrino mass schemes for *λ* = 0.01 and *c*<sup>R</sup> = 220.

for *<sup>λ</sup>* 0.03, *<sup>T</sup>*rh remains almost constant since <sup>Γ</sup>I*<sup>y</sup>* dominates over <sup>Γ</sup>I2*<sup>ν</sup><sup>c</sup>* and *<sup>f</sup>* <sup>3</sup> <sup>0</sup> <sup>≃</sup> 1 – see Eq. (39). For *λ* 0.03, *f* <sup>3</sup> <sup>0</sup> <sup>≃</sup> <sup>1</sup> <sup>+</sup> <sup>12</sup>*c*R*m*<sup>2</sup> PS starts to deviate from unity and so, *T*rh increases with *<sup>c</sup>*<sup>R</sup> or *<sup>λ</sup>* as shown in Fig. 1. The required by Eq. (64) *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* follows the behavior of the required *m*2D – see Fig. 2- of Sec. 5.2.2.

#### *5.2.2. The Stage of non-Thermal Leptogenesis*

22 Open Questions in Cosmology

262 Open Questions in Cosmology

**M2**<sup>ν</sup> **c / 109 GeV**

^

**10**

condition

where *c*max

all, we obtain

region in Eq. (70), we obtain

**102**

**103**

**10-3 10-2**

light gray segments we obtain *h*<sup>∗</sup> ≥ *m*P.

**mI**

**c**R

**Trh / 108 GeV**

> λ (a)

 **/ 1012 GeV**

monotonically decreasing function of *c*<sup>R</sup> for *c*<sup>R</sup> > *c*max

<sup>≃</sup> <sup>3</sup>*h*<sup>2</sup> ∗ 4*m*<sup>2</sup> P

*<sup>n</sup>*<sup>s</sup> <sup>≃</sup> 0.964, <sup>−</sup>6.5 *<sup>α</sup>*<sup>s</sup>

favored value – see Eq. (56 ) and cf. Ref. [12].

 *c*max <sup>R</sup> − *c*<sup>R</sup>

> *c*max R

When *<sup>c</sup>*<sup>R</sup> ranges within its allowed region, we take *<sup>M</sup>*PS ≃ (2.87 − <sup>4</sup>) · <sup>10</sup><sup>16</sup> GeV.

*dN*<sup>∗</sup> *dc*R

**10-3 10-2**

<sup>R</sup> where *c*max

= 0 ⇒ *c*<sup>R</sup> ≃ *c*max

**i = f**

λ (b)

R can be found by the

R , (69)

<sup>10</sup>−<sup>3</sup> 3.5. (71)

**i = \***

**10-1**

**hi / mP**

**Figure 1.** The allowed (by all the imposed constraints) values of *c*<sup>R</sup> (solid line), *T*rh – given by Eq. (40) – (dashed line), *m*<sup>I</sup> (dot-dashed line) and *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* (dotted line) [*h*<sup>f</sup> (solid line) and *<sup>h</sup>*<sup>∗</sup> (dashed line)] versus *<sup>λ</sup>* (a) [(b)] for *kS* <sup>=</sup> <sup>1</sup>, *<sup>λ</sup><sup>H</sup>* <sup>=</sup> *<sup>λ</sup>H*¯ <sup>=</sup> 0.5 and *y*<sup>33</sup> = 0.6. The light gray and gray segments denote values of the various quantities satisfying Eq. (58 ) too, whereas along the

gravity can still be assumed to be small. As *c*R increases beyond 906, *f*<sup>0</sup> becomes much larger than 1, *<sup>N</sup>*<sup>∗</sup> derived by Eq. (28) starts decreasing and therefore, nMHI fails to fulfil Eq. (55). This can be understood by the observation that *<sup>N</sup>*<sup>∗</sup>, approximated fairly by Eq. (29), becomes

the embedding of nMHI in a SUSY GUT provides us with a clear upper bound of *c*R. All in

From Fig. 1- , we can verify our analytical estimation in Eq. (31) according to which *λ* is proportional to *c*R. On the other hand, the variation of *h*<sup>f</sup> and *h*<sup>∗</sup> as a function of *c*<sup>R</sup> – drawn in Fig. 1- – is consistent with Eqs. (27) and (29). Letting *λ* or *c*R vary within its allowed

Clearly, the predicted *α*s and *r* lie within the allowed ranges given in Eq. (56 ) and Eq. (56 ) respectively, whereas *n*s turns out to be impressively close to its central observationally

From Fig. 1- we can conclude that *m*<sup>I</sup> is kept independent of *λ* and almost constant at the level of 1013 GeV, as anticipated in Eq. (35). From the same plot we also remark that

R is defined in Eq. (57 ) and Eq. (57 ) is also taken into account. As a consequence,

0.001 *<sup>λ</sup>* 0.0412 and 1 *<sup>c</sup>*<sup>R</sup> 907 for 53.9 *<sup>N</sup>*<sup>∗</sup> 54.7 (70)

<sup>10</sup>−<sup>4</sup> <sup>−</sup>6.2 and 4.2 *<sup>r</sup>*

**1**

In this part of our numerical program, for a given neutrino mass scheme, we take as input parameters: *m*ℓ*ν*, *mi*D, *ϕ*1, *ϕ*<sup>2</sup> and the best-fit values of the neutrino parameters listed in Table 3. We then find the *renormalization group* (RG) evolved values of these parameters at the scale of nTL, <sup>Λ</sup>*L*, which is taken to be <sup>Λ</sup>*<sup>L</sup>* = *<sup>m</sup>*I, integrating numerically the complete expressions of the RG equations – given in Ref. [29] – for *miν*, *θij*, *δ*, *ϕ*<sup>1</sup> and *ϕ*2. In doing this, we consider the MSSM with tan *β* ≃ 50, favored by the preliminary LHC results – see, e.g., Ref. [32, 56] – as an effective theory between <sup>Λ</sup>*<sup>L</sup>* and a SUSY-breaking scale, *<sup>M</sup>*SUSY = 1.5 TeV. Following the procedure described in Sec. 4.3, we evaluate *Mi<sup>ν</sup>*�*<sup>c</sup>* at <sup>Λ</sup>*L*. We do not consider the running of *mi*<sup>D</sup> and *Mi<sup>ν</sup>*�*<sup>c</sup>* and therefore, we give their values at <sup>Λ</sup>*L*.

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model 25

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

**10-3 10-2**

**m3**<sup>ν</sup>

λ

 **/ eV m1D / GeV m3D / GeV** φ**<sup>1</sup>** φ**<sup>2</sup> 0.001 0.4 120** π**/2 0 0.01 0.3 100** π**/2 -**π**/2 0.05 0.8 100** π**/2 -**π**/2 0.1 1 120 -**π**/2** π

http://dx.doi.org/10.5772/51888

265

 **/ eV m1D / GeV m3D / GeV** φ**<sup>1</sup>** φ**<sup>2</sup> 0.05 0.9 120 -**π**/2** π **0.01 0.9 110 -**π**/2** π **0.005 0.9 110 -**π**/4 0** 

**(b)**

**m2D (GeV)**

*<sup>m</sup>*1*ν*(eV) *<sup>λ</sup>* (10−2) *<sup>m</sup>*2D(GeV) *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* (10<sup>10</sup> GeV) *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* (<sup>1012</sup> GeV) *<sup>M</sup>*3*<sup>ν</sup><sup>c</sup>* (10<sup>14</sup> GeV) 0.001 0.1 − 1.8 5.5 − 17 4.7 1 − 9.8 32

0.05 0.1 − 2. 3 − 28 3.5 − 4 2.1 − 11 0.5 − 0.6

0.1 0.1 − 1.9 4 − 23 1.3 0.4 − 8 0.4 − 0.5

*<sup>m</sup>*3*ν*(eV) *<sup>λ</sup>* (10−2) *<sup>m</sup>*2D(GeV) *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* (10<sup>10</sup> GeV) *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* (<sup>1012</sup> GeV) *<sup>M</sup>*3*<sup>ν</sup><sup>c</sup>* (10<sup>14</sup> GeV) 0.05 0.1 − 2 2.1 − 12 1.5 − 1.6 0.28 − 7.8 0.48 − 0.56

0.01 0.1 − 2 2.8 − 17 2.2 0.4 − 11 5.4 − 5.6

0.005 0.1 − 1.8 5 − 17 1.8 0.7 − 10.1 12

(black [gray] lines). The corresponding ranges of *Mi<sup>ν</sup><sup>c</sup>* 's are also shown in the table included.

2.4 − 4.1 17 − 10 4.7 17 − 10 32 0.01 0.1 − 2 1.5 − 13 3.5 − 6 0.26 − 8.75 0.33 − 0.4

2 − 4.1 15 − 3.4 6 − 5.3 11 − 0.84 0.42 − 0.34

2.1 − 4.1 28 − 7.3 4 − 3.9 11 − 0.8 0.6 − 0.5

2.2 − 4.1 23 − 9 1.3 10 − 2 0.5 − 0.4

2.1 − 4.1 13 − 4.4 1.6 8.7 − 1.2 0.57 − 0.49

2 − 4.1 17 − 7.3 2.2 11 − 2 5.6 − 5.4

1.8 − 4.1 17.7 − 10 1.8 10.5 − 3.5 12

**Figure 2.** Contours on the *λ* − *m*2D plane, yielding the central *YB* in Eq. (64), consistently with the inflationary requirements, for *<sup>λ</sup><sup>H</sup>* = *<sup>λ</sup>H*¯ = 0.5, *kS* = <sup>1</sup> and *<sup>y</sup>*<sup>33</sup> = 0.6 and various (*m*ℓ*ν*, *<sup>m</sup>*1D, *<sup>ϕ</sup>*1, *<sup>ϕ</sup>*2)'s indicated next to the graph (c) and NO [IO] *miν*'s

variation of the drawn lines is negligible. The resulting *Mj<sup>ν</sup><sup>c</sup>* 's are displayed in the table included. The conventions adopted for the types and the color of the various lines are also

**10-3 10-2**

**10-3 10-2**

λ

**(c) m1**<sup>ν</sup>

λ

**m2D (GeV)**

**(a)**

**m2D (GeV)**

We start the exposition of our results arranging in Table 4 some representative values of the parameters leading to the correct BAU for *λ* = 0.01 and *c*<sup>R</sup> = 220 and normally hierarchical (cases A and B), degenerate (cases C, D and E) and invertedly hierarchical (cases F and G) *mν*'s. For comparison we display the *B*-yield with (*YB*) or without (*Y*<sup>0</sup> *<sup>B</sup>*) taking into account the RG effects. We observe that the two results are more or less close with each other. In all cases the current limit of Eq. (62) is safely met – the case D approaches it –, while *m<sup>β</sup>* turns out to be well below the projected sensitivity of KATRIN [58]. Shown are also the obtained *<sup>T</sup>*rh's, which are close to 109 GeV in all cases, and the corresponding *YG*�'s, which are consistent with Eq. (65) for *mG*� 11 TeV.

From Table 4 we also remark that the achievement of *YB* within the range of Eq. (64) dictates a clear hierarchy between the *Mi<sup>ν</sup>*�*<sup>c</sup>* 's, which follows the imposed hierarchy in the sector of *mi*D's – see paragraph 6 of Sec. 5.1. This is expected since, in the limit of hierarchical *mi*D's, the *Mi<sup>ν</sup>*�*<sup>c</sup>* 's can be approximated by the following expressions [25, 26]

$$(M\_{1\hat{0}^{\circ}\prime}M\_{2\hat{0}^{\circ}\prime}M\_{3\hat{0}^{\circ}}) \sim \begin{cases} \left(\frac{m\_{1\text{D}}^{2}}{m\_{2\text{D}}s\_{12}^{2}}, \frac{2m\_{2\text{D}}^{2}}{m\_{3\text{v}}}, \frac{m\_{3\text{D}}^{2}s\_{12}^{2}}{2m\_{1\text{v}}}\right) & \text{for NO } m\_{\text{v}}\text{'s} \\\\ \left(\frac{m\_{1\text{D}}^{2}}{\sqrt{|\Delta m\_{3\text{I}}|}}, \frac{2m\_{2\text{D}}^{2}}{\sqrt{|\Delta m\_{3\text{I}}|}}, \frac{m\_{3\text{D}}^{2}}{2m\_{3\text{v}}}\right) & \text{for IO } m\_{\text{v}}\text{'s} \end{cases} \tag{72a}$$

Indeed, we see e.g. that for fixed *<sup>j</sup>*, the *Mj<sup>ν</sup>*�*<sup>c</sup>* 's depends exclusively on the *mj*D's and *<sup>M</sup>*3*ν*�*<sup>c</sup>* increases when *m*ℓ*<sup>ν</sup>* decreases with fixed *m*3D. As a consequence, satisfying Eq. (59 ) pushes the *m*1D's well above the mass of the quark of the first generation. Similarly, the *m*2D's required by Eq. (64) turns out to be heavier than the quark of the second generation. Also, the required by seesaw *<sup>M</sup>*3*ν*�*<sup>c</sup>* 's are lower in the case of degenerate *<sup>ν</sup><sup>i</sup>* spectra and can be as low as 3 · <sup>1013</sup> GeV in sharp contrast to our findings in Ref. [5], where much larger *<sup>M</sup>*3*ν*�*<sup>c</sup>* 's are necessitated. An order of magnitude estimation for the derived *εL*'s can be achieved by [25, 26]

$$\varepsilon\_{2} \sim -\frac{3M\_{2\hat{\nu}}}{8\pi \langle H\_{\text{u}} \rangle^{2}} \begin{cases} \left(\frac{m\_{1\nu}}{s\_{12}^{2}}\right) & \text{for NO } m\_{\nu}\text{'s} \\\ m\_{3\nu} & \text{for IO } m\_{\nu}\text{'s} \end{cases} \tag{72b}$$

which is rather accurate, especially in the case of IO *mν*'s.

To highlight further our conclusions inferred from Table 4, we can fix *m*ℓ*<sup>ν</sup>* (*m*1*<sup>ν</sup>* for NO *miν*'s or *m*3*<sup>ν</sup>* for IO *miν*'s) *m*1D, *m*3D, *ϕ*<sup>1</sup> and *ϕ*<sup>2</sup> to their values shown in this table and vary *m*2D so that the central value of Eq. (64) is achieved. This is doable since, according Eq. (72a), variation of *<sup>m</sup>*2D induces an exclusive variation to *<sup>M</sup>*2*ν*�*<sup>c</sup>* which, in turn, heavily influences *ε<sup>L</sup>* – see Eqs. (44a) and (45) – and *YL* – see Eqs. (41) and (42). The resulting contours in the *<sup>λ</sup>* − *<sup>m</sup>*2D plane are presented in Fig. 2 – since the range of Eq. (64) is very narrow the possible

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model 25 Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model http://dx.doi.org/10.5772/51888 265


24 Open Questions in Cosmology

264 Open Questions in Cosmology

Table 3. We then find the *renormalization group* (RG) evolved values of these parameters at the scale of nTL, <sup>Λ</sup>*L*, which is taken to be <sup>Λ</sup>*<sup>L</sup>* = *<sup>m</sup>*I, integrating numerically the complete expressions of the RG equations – given in Ref. [29] – for *miν*, *θij*, *δ*, *ϕ*<sup>1</sup> and *ϕ*2. In doing this, we consider the MSSM with tan *β* ≃ 50, favored by the preliminary LHC results – see, e.g., Ref. [32, 56] – as an effective theory between <sup>Λ</sup>*<sup>L</sup>* and a SUSY-breaking scale, *<sup>M</sup>*SUSY = 1.5 TeV. Following the procedure described in Sec. 4.3, we evaluate *Mi<sup>ν</sup>*�*<sup>c</sup>* at <sup>Λ</sup>*L*. We do not consider

We start the exposition of our results arranging in Table 4 some representative values of the parameters leading to the correct BAU for *λ* = 0.01 and *c*<sup>R</sup> = 220 and normally hierarchical (cases A and B), degenerate (cases C, D and E) and invertedly hierarchical (cases F and G)

the RG effects. We observe that the two results are more or less close with each other. In all cases the current limit of Eq. (62) is safely met – the case D approaches it –, while *m<sup>β</sup>* turns out to be well below the projected sensitivity of KATRIN [58]. Shown are also the obtained *<sup>T</sup>*rh's, which are close to 109 GeV in all cases, and the corresponding *YG*�'s, which

From Table 4 we also remark that the achievement of *YB* within the range of Eq. (64) dictates a clear hierarchy between the *Mi<sup>ν</sup>*�*<sup>c</sup>* 's, which follows the imposed hierarchy in the sector of *mi*D's – see paragraph 6 of Sec. 5.1. This is expected since, in the limit of hierarchical *mi*D's,

Indeed, we see e.g. that for fixed *<sup>j</sup>*, the *Mj<sup>ν</sup>*�*<sup>c</sup>* 's depends exclusively on the *mj*D's and *<sup>M</sup>*3*ν*�*<sup>c</sup>* increases when *m*ℓ*<sup>ν</sup>* decreases with fixed *m*3D. As a consequence, satisfying Eq. (59 ) pushes the *m*1D's well above the mass of the quark of the first generation. Similarly, the *m*2D's required by Eq. (64) turns out to be heavier than the quark of the second generation. Also, the required by seesaw *<sup>M</sup>*3*ν*�*<sup>c</sup>* 's are lower in the case of degenerate *<sup>ν</sup><sup>i</sup>* spectra and can be as low as 3 · <sup>1013</sup> GeV in sharp contrast to our findings in Ref. [5], where much larger *<sup>M</sup>*3*ν*�*<sup>c</sup>* 's are necessitated. An order of magnitude estimation for the derived *εL*'s can be achieved by

> 

� *m*1*<sup>ν</sup> s*2 12 �

To highlight further our conclusions inferred from Table 4, we can fix *m*ℓ*<sup>ν</sup>* (*m*1*<sup>ν</sup>* for NO *miν*'s or *m*3*<sup>ν</sup>* for IO *miν*'s) *m*1D, *m*3D, *ϕ*<sup>1</sup> and *ϕ*<sup>2</sup> to their values shown in this table and vary *m*2D so that the central value of Eq. (64) is achieved. This is doable since, according Eq. (72a), variation of *<sup>m</sup>*2D induces an exclusive variation to *<sup>M</sup>*2*ν*�*<sup>c</sup>* which, in turn, heavily influences *ε<sup>L</sup>* – see Eqs. (44a) and (45) – and *YL* – see Eqs. (41) and (42). The resulting contours in the *<sup>λ</sup>* − *<sup>m</sup>*2D plane are presented in Fig. 2 – since the range of Eq. (64) is very narrow the possible

, <sup>2</sup>*m*<sup>2</sup> √ 2D |∆*m*31| , *<sup>m</sup>*<sup>2</sup> 3D 2*m*3*<sup>ν</sup>* �

for NO *mν*'s

*m*3*<sup>ν</sup>* for IO *mν*'s

*<sup>B</sup>*) taking into account

(72a)

(72b)

for NO *mν*'s

for IO *mν*'s

the running of *mi*<sup>D</sup> and *Mi<sup>ν</sup>*�*<sup>c</sup>* and therefore, we give their values at <sup>Λ</sup>*L*.

*mν*'s. For comparison we display the *B*-yield with (*YB*) or without (*Y*<sup>0</sup>

the *Mi<sup>ν</sup>*�*<sup>c</sup>* 's can be approximated by the following expressions [25, 26]

*<sup>ε</sup>*<sup>2</sup> ∼ − <sup>3</sup>*M*2*ν*�*<sup>c</sup>*

which is rather accurate, especially in the case of IO *mν*'s.

8*π*�*Hu*�<sup>2</sup>

  � *m*<sup>2</sup> 1D *m*2*<sup>ν</sup> s*<sup>2</sup> 12 , <sup>2</sup>*m*<sup>2</sup> 2D *<sup>m</sup>*3*<sup>ν</sup>* , *<sup>m</sup>*<sup>2</sup> 3D*s*<sup>2</sup> 12 2*m*1*<sup>ν</sup>* �

� *<sup>m</sup>*<sup>2</sup> √ 1D |∆*m*31|

are consistent with Eq. (65) for *mG*� 11 TeV.

(*M*1*ν*�*<sup>c</sup>* , *<sup>M</sup>*2*ν*�*<sup>c</sup>* , *<sup>M</sup>*3*ν*�*<sup>c</sup>* ) <sup>∼</sup>

[25, 26]

**Figure 2.** Contours on the *λ* − *m*2D plane, yielding the central *YB* in Eq. (64), consistently with the inflationary requirements, for *<sup>λ</sup><sup>H</sup>* = *<sup>λ</sup>H*¯ = 0.5, *kS* = <sup>1</sup> and *<sup>y</sup>*<sup>33</sup> = 0.6 and various (*m*ℓ*ν*, *<sup>m</sup>*1D, *<sup>ϕ</sup>*1, *<sup>ϕ</sup>*2)'s indicated next to the graph (c) and NO [IO] *miν*'s (black [gray] lines). The corresponding ranges of *Mi<sup>ν</sup><sup>c</sup>* 's are also shown in the table included.

variation of the drawn lines is negligible. The resulting *Mj<sup>ν</sup><sup>c</sup>* 's are displayed in the table included. The conventions adopted for the types and the color of the various lines are also described next to the graph (c) of Fig. 2. In particular, we use black [gray] lines for NO [IO] *miν*'s. The black dashed and the solid gray line terminate at the values of *m*2D beyond which Eq. (64) is non fulfilled due to the violation of Eq. (59 ).

In all cases, two disconnected allowed domains arise according to which of the two contributions in Eq. (36) dominates. The critical point (*λ*c, *<sup>c</sup>*Rc) is extracted from:

$$11 - 12c\_{\mathcal{R}\mathbb{C}}m\_{\mathbb{PS}}^2 = 0 \Rightarrow c\_{\mathcal{R}\mathbb{C}} = c\_{\mathcal{R}}^{\max}/2 \simeq 453 \text{ or } \lambda\_{\mathbb{C}} \simeq 10^{-4}c\_{\mathcal{R}}^{\max}/4.2 \simeq 0.021 \tag{73}$$

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model 27

http://dx.doi.org/10.5772/51888

267

Leptogenesis and Neutrino Masses in an Inflationary SUSY Pati-Salam Model

We would like to thank A.B. Lahanas, G. Lazarides and V.C. Spanos for valuable discussions.

[2] Buchmuller W. Peccei D.R and Yanagida T. *Ann. Rev. Nucl. Part. Sci.*, 55:311, 2005 [hep-

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[11] Ferrara S. Kallosh R. Linde D.A. Marrani A. and Proeyen Van A. *Phys. Rev. D* , 83:025008,

[12] Kallosh R. and Linde D.A. *J. Cosmol. Astropart. Phys.*, 11:011, 2010 [arXiv:1008.3375].

[16] Komatsu E. *et al*. [WMAP Collaboration] *Astrophys. J. Suppl.*, 192:18, 2011

[18] Bolz M. Brandenburg A. and Buchmüller W. *Nucl. Phys. B*, 606:518, 2001 [hep-ph/00

[10] Lee M.H. *J. Cosmol. Astropart. Phys.*, 08:003, 2010 [arXiv:1005.2735].

[13] Antoniadis I. and Leontaris K.G. *Phys. Lett. B* , 216:333, 1989.

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[17] Khlopov Yu. M. and Linde D.A. *Phys. Lett. B* , 138:265, 1984.

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Department of Physics, University of Cyprus, Nicosia, Cyprus

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12052].

[1] Hamaguchi K. *Phd Thesis,* hep-ph/0212305.

**Acknowledgements**

C. Pallis and N. Toumbas

ph/0502169].

9405337].

**Author details**

**References**

where we make use of Eq. (57) and Eq. (31) in the intermediate and the last step respectively. From Eqs. (40), (41) and (42) one can deduce that for *λ* < *λc*, *T*rh remains almost constant; <sup>Γ</sup>I2*<sup>ν</sup><sup>c</sup>*/Γ<sup>I</sup> decreases as *<sup>c</sup>*<sup>R</sup> increases and so the *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* 's, which satisfy Eq. (64), increase. On the contrary, for *<sup>λ</sup>* <sup>&</sup>gt; *<sup>λ</sup>c*, <sup>Γ</sup>I2*<sup>ν</sup><sup>c</sup>*/Γ<sup>I</sup> is independent of *<sup>c</sup>*<sup>R</sup> but *<sup>T</sup>*rh increases with *<sup>c</sup>*<sup>R</sup> and so the fulfilling Eq. (64) *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* 's decrease.

Summarizing, we conclude that our scenario prefers the following ranges for the *Mi<sup>ν</sup><sup>c</sup>* 's:

$$1 \lesssim M\_{1\tilde{\nu}^{\tilde{c}}}/10^{10}\text{ GeV} \lesssim 6, \ 0.6 \lesssim M\_{2\tilde{\nu}^{\tilde{c}}}/10^{12}\text{ GeV} \lesssim 20, \ 0.3 \lesssim M\_{3\tilde{\nu}^{\tilde{c}}}/10^{14}\text{GeV} \lesssim 30,\tag{74a}$$

while the *m*1D and *m*2D are restricted in the ranges:

$$10.3 \lesssim m\_{\rm ID}/\text{GeV} \lesssim 1 \text{ } 1.5 \lesssim m\_{\rm 2D}/\text{GeV} \lesssim 20.\tag{74b}$$

## **6. Conclusions**

We investigated the implementation of nTL within a realistic GUT, based on the PS gauge group. Leptogenesis follows a stage of nMHI driven by the radial component of the Higgs field, which leads to the spontaneous breaking of the PS gauge group to the SM one with the GUT breaking v.e.v identified with the SUSY GUT scale and without overproduction of monopoles. The model possesses also a resolution to the strong CP and the *µ* problems of the MSSM via a PQ symmetry which is broken during nMHI and afterwards. As a consequence the axion cannot be the dominant component of CDM, due to the present bounds on the axion isocurvature fluctuation. Moreover, we briefly discussed scenaria in which the potential axino and saxion overproduction problems can be avoided.

Inflation is followed by a reheating phase, during which the inflaton can decay into the lightest, *<sup>ν</sup><sup>c</sup>* <sup>1</sup>, and the next-to-lightest, *<sup>ν</sup><sup>c</sup>* <sup>2</sup>, RH neutrinos allowing, thereby for nTL to occur via the subsequent decay of *<sup>ν</sup><sup>c</sup>* <sup>1</sup> and *<sup>ν</sup><sup>c</sup>* <sup>2</sup>. Although other decay channels to the SM particles via non-renormalizable interactions are also activated, we showed that the production of the required by the observations BAU can be reconciled with the observational constraints on the inflationary observables and the *<sup>G</sup>* abundance, provided that the (unstable) *<sup>G</sup>* masses are greater than 11 TeV. The required by the observations BAU can become consistent with the present low energy neutrino data, the restriction on *m*3D due to the PS gauge group and the imposed mild hierarchy between *m*1D and *m*2D. To this end, *m*1D and *m*2D turn out to be heavier than the ones of the corresponding quarks and lie in the ranges (0.1 − 1) GeV and (<sup>2</sup> <sup>−</sup> <sup>20</sup>) GeV while the obtained *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* , *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* and *<sup>M</sup>*3*<sup>ν</sup><sup>c</sup>* are restricted to the values 10<sup>10</sup> GeV, 1011 − 1012 GeV and 10<sup>13</sup> − 10<sup>15</sup> GeV respectively.

## **Acknowledgements**

We would like to thank A.B. Lahanas, G. Lazarides and V.C. Spanos for valuable discussions.

## **Author details**

26 Open Questions in Cosmology

266 Open Questions in Cosmology

<sup>1</sup> − <sup>12</sup>*c*Rc*m*<sup>2</sup>

fulfilling Eq. (64) *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* 's decrease.

**6. Conclusions**

lightest, *<sup>ν</sup><sup>c</sup>*

described next to the graph (c) of Fig. 2. In particular, we use black [gray] lines for NO [IO] *miν*'s. The black dashed and the solid gray line terminate at the values of *m*2D beyond which

In all cases, two disconnected allowed domains arise according to which of the two

where we make use of Eq. (57) and Eq. (31) in the intermediate and the last step respectively. From Eqs. (40), (41) and (42) one can deduce that for *λ* < *λc*, *T*rh remains almost constant; <sup>Γ</sup>I2*<sup>ν</sup><sup>c</sup>*/Γ<sup>I</sup> decreases as *<sup>c</sup>*<sup>R</sup> increases and so the *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* 's, which satisfy Eq. (64), increase. On the contrary, for *<sup>λ</sup>* <sup>&</sup>gt; *<sup>λ</sup>c*, <sup>Γ</sup>I2*<sup>ν</sup><sup>c</sup>*/Γ<sup>I</sup> is independent of *<sup>c</sup>*<sup>R</sup> but *<sup>T</sup>*rh increases with *<sup>c</sup>*<sup>R</sup> and so the

Summarizing, we conclude that our scenario prefers the following ranges for the *Mi<sup>ν</sup><sup>c</sup>* 's:

<sup>1</sup> *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>*/10<sup>10</sup> GeV 6, 0.6 *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>*/10<sup>12</sup> GeV 20, 0.3 *<sup>M</sup>*3*<sup>ν</sup><sup>c</sup>*/1014GeV 30, (74a)

We investigated the implementation of nTL within a realistic GUT, based on the PS gauge group. Leptogenesis follows a stage of nMHI driven by the radial component of the Higgs field, which leads to the spontaneous breaking of the PS gauge group to the SM one with the GUT breaking v.e.v identified with the SUSY GUT scale and without overproduction of monopoles. The model possesses also a resolution to the strong CP and the *µ* problems of the MSSM via a PQ symmetry which is broken during nMHI and afterwards. As a consequence the axion cannot be the dominant component of CDM, due to the present bounds on the axion isocurvature fluctuation. Moreover, we briefly discussed scenaria in which the potential

Inflation is followed by a reheating phase, during which the inflaton can decay into the

via non-renormalizable interactions are also activated, we showed that the production of the required by the observations BAU can be reconciled with the observational constraints on the inflationary observables and the *<sup>G</sup>* abundance, provided that the (unstable) *<sup>G</sup>* masses are greater than 11 TeV. The required by the observations BAU can become consistent with the present low energy neutrino data, the restriction on *m*3D due to the PS gauge group and the imposed mild hierarchy between *m*1D and *m*2D. To this end, *m*1D and *m*2D turn out to be heavier than the ones of the corresponding quarks and lie in the ranges (0.1 − 1) GeV and (<sup>2</sup> <sup>−</sup> <sup>20</sup>) GeV while the obtained *<sup>M</sup>*1*<sup>ν</sup><sup>c</sup>* , *<sup>M</sup>*2*<sup>ν</sup><sup>c</sup>* and *<sup>M</sup>*3*<sup>ν</sup><sup>c</sup>* are restricted to the values 10<sup>10</sup> GeV,

<sup>R</sup> /2 <sup>≃</sup> 453 or *<sup>λ</sup>*<sup>c</sup> <sup>≃</sup> <sup>10</sup>−4*c*max

0.3 *m*1D/GeV 1, 1.5 *m*2D/GeV 20. (74b)

<sup>2</sup>, RH neutrinos allowing, thereby for nTL to occur

<sup>2</sup>. Although other decay channels to the SM particles

<sup>R</sup> /4.2 ≃ 0.021 (73)

contributions in Eq. (36) dominates. The critical point (*λ*c, *<sup>c</sup>*Rc) is extracted from:

Eq. (64) is non fulfilled due to the violation of Eq. (59 ).

while the *m*1D and *m*2D are restricted in the ranges:

axino and saxion overproduction problems can be avoided.

<sup>1</sup> and *<sup>ν</sup><sup>c</sup>*

10<sup>13</sup> − 10<sup>15</sup> GeV respectively.

<sup>1</sup>, and the next-to-lightest, *<sup>ν</sup><sup>c</sup>*

via the subsequent decay of *<sup>ν</sup><sup>c</sup>*

1011 − 1012 GeV and

PS <sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>c</sup>*R<sup>c</sup> <sup>=</sup> *<sup>c</sup>*max

C. Pallis and N. Toumbas

Department of Physics, University of Cyprus, Nicosia, Cyprus

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**Chapter 11**

**Provisional chapter**

**Light Cold Dark Matter in a Two-Singlet Model**

**Light Cold Dark Matter in a Two-Singlet Model**

Understanding the nature of dark matter (DM) as well as the origin of baryon asymmetry are two of the most important questions in both Cosmology and Particle Physics. The failure of the Standard Model (SM) of the electroweak and strong interactions in accommodating such questions motivates the search for an explanation in the realm of new physics beyond it. The most attractive strategy so far has been to look into extensions of the SM that incorporate electrically neutral and colorless weakly interacting massive particles (WIMPs), with masses from one to a few hundred GeV, coupling constants in the milli-weak scale and lifetimes

The most popular extension of the SM is the minimal supersymmetric standard model (MSSM) in which the neutral lightest supersymmetric particle (LSP) is seen as a candidate for dark matter. Indeed, neutralinos are odd under *R* -parity and are only produced or destroyed in pairs, thus making the LSP stable [1]. They can annihilate through a t-channel sfermion exchange into Standard Model fermions, or via a t-channel chargino-mediated process into *<sup>W</sup>*+*W*−, or through an s-channel pseudoscalar Higgs exchange into fermion pairs. They can also undergo elastic scattering with nuclei through mainly a scalar Higgs exchange [2].

However, most particularly in light of the recent signal reported by CoGeNT [3], which favors a light dark matter (LDM) with a mass in the range 7 − 9GeV and nucleon scattering cross section *<sup>σ</sup>*det <sup>∼</sup> <sup>10</sup>−<sup>4</sup> pb, having a neutralino as a LDM candidate can be challenging. Indeed, systematic studies show that an LSP with a mass around 10GeV and an elastic scattering cross-section off a nucleus larger than <sup>∼</sup> <sup>10</sup>−<sup>5</sup> pb requires a very large tan *<sup>β</sup>* and a relatively light CP-odd Higgs [4]. This choice of parameters leads to a sizable contribution to the branching ratios of some rare decays, which then disfavors the scenario of light neutralinos [5]. Also, in the next-to-minimal supersymmetric standard model (NMSSM) with 12 input parameters [6], realizing a LDM with an elastic scattering cross section capable of generating the CoGeNT signal is possible only in a finely-tuned region of the parameters where the neutralino is mostly singlino and the light CP-even Higgs is singlet-like with a mass below a few GeV. In such a situation, it is very difficult to detect such a light Higgs at the collider. It

and reproduction in any medium, provided the original work is properly cited.

©2012 Abada and Nasri, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Abada and Nasri; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

is clear then that other alternative scenarios for LDM are needed [7].

Abdessamad Abada and Salah Nasri

Abdessamad Abada and Salah Nasri

http://dx.doi.org/10.5772/52243

longer than the age of the Universe.

**1. Introduction**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Provisional chapter**
