The Early Universe as a Source of Gravitational Waves

*Vladimir Gladyshev and Igor Fomin*

#### **Abstract**

In this chapter we consider the issues of the origin and evolution of relic gravitational waves (GW), which appear as a result of quantum fluctuations of the scalar field and the corresponding perturbations of the space-time metric at the early inflationary stage of the evolution of the universe. The main provisions of the inflationary paradigm and the methods of the construction of current cosmological models on its basis are considered. The influence of relic gravitational waves on the anisotropy and polarization of the relic radiation and the importance of estimating such an effect on the verification of cosmological models are discussed as well.

**Keywords:** universe, inflation, scalar field, cosmological perturbations, gravitational waves

### **1. Introduction**

The general relativity (GR) theory proposed by A. Einstein more than a 100 years ago currently finds new confirmations. The possible existence of gravitational waves was predicted by A. Einstein on the basis of solving the equations of general relativity when calculating the power of gravitational radiation [1, 2].

Gravitational waves (GW) are space-time curvature disturbances, which propagate at the speed of light. They occur at any movements of material bodies, leading to inhomogeneous gravity force variation in the environment. Gravitational radiation was predicted by A. Einstein in the general relativity (GR) theory, but so far not detected by direct measurements.

According to general relativity, space-time is curved around the bodies due to the action of gravity and is represented by a symmetric tensor *gμν* with 10 independent components. However, far from the masses (the case of weak gravitational fields), the tensor can be divided into two terms *gμν* ¼ *ημν* þ *hμν* where the first term, i.e., tensor *ημν*, corresponds to the flat space-time of the special theory of relativity and has only four components. The second tensor *hμν* contains information about the curvature caused by the gravitational field and makes small corrections. In the case of gravitational disturbances propagating far from their sources, the components of the tensor *hμν* can be calculated by the method proposed by Einstein [1], similar to that used in electrodynamics for delayed potentials.

The first evidence was received by experimental studies of Joseph H. Taylor and Joel M. Weisberg et al., who studied the effect of slowing down the period of the binary star system PSR 1913 + 16 due to energy losses on gravitational radiation [3]. Until recently, however, there has remained the main task: the direct recording of gravitational waves from space radiation sources by means of ground-based or space gravitational antennas.

increasingly convincing as a necessary step modifying the standard Big Bang theory, which is based on solutions of Einstein's equations for the universe filled with ordinary baryonic matter with a positive energy density obtained by Friedmann. However, the extrapolation of Friedmann solutions to early times leads to many insoluble problems when constructing on their basis the evolution scenarios of the

The exponential (de Sitter) expansion, suggesting *p* ¼ �*ρ*, or a close expansion of the early universe based on the evolution of a certain substance with the equation of state *p*≈ � *ρ*, i.e., with a negative pressure, is a feature of inflation models which allow to solve the problems of the standard model of the Big Bang theory, namely, the problems of the horizon, flatness, homogeneity, isotropy, low concentration of exotic states of matter (domain walls, monopoles, etc.), anisotropy of the back-

Thus, the cosmological models containing a combination of Friedmann solutions and (quasi) de Sitter solutions provide the basis for a current description of the evolution of the universe. In the context of the inflationary paradigm, the early universe expands for some time accelerated and, further, goes into a power-law expansion mode without acceleration corresponding to Friedmann solutions.

In most cosmological models, the geometric description of the universe is based on the Friedmann-Robertson-Walker (FRW) homogeneous isotropic space (spacetime) model, which is associated with a high degree of isotropy of space, measured on the basis of the cosmic microwave background (CMB) radiation research. This identification also relies on a formal result known as the Ehlers-Geren-Sachs theorem, which refers to the universe filled with any ideal barotropic fluid [12]. The metric of Friedmann-Robertson-Walker (FRW) space-time is written

ground radiation, the initial singularity, and some other problems [9].

*ds*<sup>2</sup> ¼ �*dt*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*<sup>2</sup>

*The Early Universe as a Source of Gravitational Waves DOI: http://dx.doi.org/10.5772/intechopen.87946*

a spatially flat, closed, and open model of the universe.

ð Þ*t*

*dr*<sup>2</sup> <sup>1</sup> � *kr*<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*

where *a t*ð Þ is a scale factor which characterizes the size of the universe, f g *r; θ; φ* are the spherical coordinates, and the values *k* ¼ 0, *k* ¼ 1, and *k* ¼ �1 correspond to

The source of the accelerated expansion of the early universe with the equation of state *p* ¼ �*ρ* is a vacuum; the equation of the state *p*≈ � *ρ* corresponds to a scalar (bosonic) field. The Bose-Einstein statistics for an ensemble of bosons, in contrast to an ensemble of fermions obeying the Pauli exclusion principle, implies that there can be several particles in one quantum state, which leads to the formation of boson condensate in which the increase in the concentration of massless bosons is associated with a decrease in the effective pressure corresponding to the equation of state *p*≈ � *ρ*. The initial (quasi) exponential expansion associated with a negative pressure, due to the exotic equation of state, is unstable, which leads to a phase transition, the termination of accelerated expansion, and the fragmentation of the original volume into many areas in which further evolution corresponds to the

Also, the presence of a scalar field violates the symmetry of the system, which leads to the appearance of a mass of initially massless particles, for example, in the

Thus, the inclusion of the scalar field into cosmological models makes it possible

To prevent the rapid decay of the state of *p*≈ � *ρ*, it is necessary to assume the existence of some potential barrier, that is, the minimum potential energy of a

to move from (quasi) de Sitter solutions to the Friedmann ones.

<sup>2</sup> *<sup>d</sup>θ*<sup>2</sup> <sup>þ</sup> sin <sup>2</sup>

*<sup>θ</sup>dφ*<sup>2</sup> � � ! (1)

universe [9].

as follows:

Friedmann solutions.

Higgs field [9].

**223**

Over the years, several methods have been proposed for recording gravitational radiation. Experimental work began in the 1960s of the twentieth century, but before the beginning of the twenty-first century, there was no reliable experimental proof of the ground-based recording of gravitational radiation [4].

This is due to the fact that gravitational waves have small amplitude; in addition, the proposed detection methods have insufficient sensitivity and are rather complicated in technical implementation.

These are broadband gravitational antennas, which offer a lot of opportunities as to the methods of recording gravitational waves and extracting signals, as well as the use of quantum non-perturbative measurements and the inclusion of gravitational antennas in the network.

The creation of new-generation gravitational antennas designed to reliably receive gravitational waves from remote space sources involves the use of highpower lasers, complex computer systems for processing large data arrays, the use of complex seismic protection systems, and the solution of other complex engineering and physical problems.

At present large international experience has been gained in the field of creating laser gravitational antennas, which ensured the ground-based recording of gravitational waves from black hole collision [5, 6] and neutron star merger [7]. Furthermore, the gravitational wave propagation velocity was estimated, which appeared to be equal to the speed of light in vacuum with an accuracy of 10<sup>15</sup> based on almost simultaneous recording of gravitational waves and a short gamma-ray burst from neutron star merger [8].

The modern theory of the early universe is based on the inclusion of the inflationary stage which precedes the stage of the hot universe. The theory of cosmological inflation [9] explains the origin of a large-scale structure and corresponds to observational data [10]. Inflationary expansion of the universe during very early times, once the universe emerged from the quantum gravity (Planck) era, has been proposed in the late 1970s and mainly in the beginning of the 1980s and is becoming more accepted as a necessary stage which modifies the standard Big Bang theory model. According to the theory of inflation, the primordial perturbations appear from quantum fluctuations. These fluctuations had essential amplitudes in scales of Planck length, and during inflation they generate the primordial perturbations which then lead nearer to scales of galaxies with almost the same amplitudes [11].

Thus, the theory of cosmological inflation connects large-scale structure of the universe with microscopic scales. The resultant range of inhomogeneities practically doesn't depend on scenarios of inflation and has a universal form. It leads to unambiguous predictions for a range of anisotropy of the background radiation.

The small quantum perturbations of the scalar field and the corresponding perturbations of the metric generate the relic gravitational waves. This type of gravitational waves was not directly observed; however, the possibility of such observations plays a key role to understand the physical processes in the early universe.

#### **2. Inflationary stage of the early universe**

The models of inflationary (accelerated) expansion of the universe at the early stage of its evolution, that is, at times close to the Planck time, are becoming

Until recently, however, there has remained the main task: the direct recording of gravitational waves from space radiation sources by means of ground-based or

radiation. Experimental work began in the 1960s of the twentieth century, but before the beginning of the twenty-first century, there was no reliable experimental

proof of the ground-based recording of gravitational radiation [4].

the use of quantum non-perturbative measurements and the inclusion of

of complex seismic protection systems, and the solution of other complex

Over the years, several methods have been proposed for recording gravitational

This is due to the fact that gravitational waves have small amplitude; in addition, the proposed detection methods have insufficient sensitivity and are rather compli-

These are broadband gravitational antennas, which offer a lot of opportunities as to the methods of recording gravitational waves and extracting signals, as well as

At present large international experience has been gained in the field of creating laser gravitational antennas, which ensured the ground-based recording of gravitational waves from black hole collision [5, 6] and neutron star merger [7]. Furthermore, the gravitational wave propagation velocity was estimated, which appeared to be equal to the speed of light in vacuum with an accuracy of 10<sup>15</sup> based on almost simultaneous recording of gravitational waves and a short gamma-ray burst

The modern theory of the early universe is based on the inclusion of the inflationary stage which precedes the stage of the hot universe. The theory of cosmological inflation [9] explains the origin of a large-scale structure and corresponds to observational data [10]. Inflationary expansion of the universe during very early times, once the universe emerged from the quantum gravity (Planck) era, has been proposed in the late 1970s and mainly in the beginning of the 1980s and is becoming more accepted as a necessary stage which modifies the standard Big Bang theory model. According to the theory of inflation, the primordial perturbations appear from quantum fluctuations. These fluctuations had essential amplitudes in scales of Planck length, and during inflation they generate the primordial perturbations which then lead nearer to scales of galaxies with almost the same amplitudes [11]. Thus, the theory of cosmological inflation connects large-scale structure of the universe with microscopic scales. The resultant range of inhomogeneities practically doesn't depend on scenarios of inflation and has a universal form. It leads to unambiguous predictions for a range of anisotropy of the background radiation. The small quantum perturbations of the scalar field and the corresponding perturbations of the metric generate the relic gravitational waves. This type of gravitational waves was not directly observed; however, the possibility of such observations plays a key role to understand the physical processes in the early

The models of inflationary (accelerated) expansion of the universe at the early

stage of its evolution, that is, at times close to the Planck time, are becoming

The creation of new-generation gravitational antennas designed to reliably receive gravitational waves from remote space sources involves the use of highpower lasers, complex computer systems for processing large data arrays, the use

space gravitational antennas.

*Progress in Relativity*

cated in technical implementation.

gravitational antennas in the network.

engineering and physical problems.

from neutron star merger [8].

universe.

**222**

**2. Inflationary stage of the early universe**

increasingly convincing as a necessary step modifying the standard Big Bang theory, which is based on solutions of Einstein's equations for the universe filled with ordinary baryonic matter with a positive energy density obtained by Friedmann. However, the extrapolation of Friedmann solutions to early times leads to many insoluble problems when constructing on their basis the evolution scenarios of the universe [9].

The exponential (de Sitter) expansion, suggesting *p* ¼ �*ρ*, or a close expansion of the early universe based on the evolution of a certain substance with the equation of state *p*≈ � *ρ*, i.e., with a negative pressure, is a feature of inflation models which allow to solve the problems of the standard model of the Big Bang theory, namely, the problems of the horizon, flatness, homogeneity, isotropy, low concentration of exotic states of matter (domain walls, monopoles, etc.), anisotropy of the background radiation, the initial singularity, and some other problems [9].

Thus, the cosmological models containing a combination of Friedmann solutions and (quasi) de Sitter solutions provide the basis for a current description of the evolution of the universe. In the context of the inflationary paradigm, the early universe expands for some time accelerated and, further, goes into a power-law expansion mode without acceleration corresponding to Friedmann solutions.

In most cosmological models, the geometric description of the universe is based on the Friedmann-Robertson-Walker (FRW) homogeneous isotropic space (spacetime) model, which is associated with a high degree of isotropy of space, measured on the basis of the cosmic microwave background (CMB) radiation research. This identification also relies on a formal result known as the Ehlers-Geren-Sachs theorem, which refers to the universe filled with any ideal barotropic fluid [12].

The metric of Friedmann-Robertson-Walker (FRW) space-time is written as follows:

$$ds^2 = -dt^2 + a^2(t) \left( \frac{dr^2}{1 - kr^2} + r^2(d\theta^2 + \sin^2\theta d\varphi^2) \right) \tag{1}$$

where *a t*ð Þ is a scale factor which characterizes the size of the universe, f g *r; θ; φ* are the spherical coordinates, and the values *k* ¼ 0, *k* ¼ 1, and *k* ¼ �1 correspond to a spatially flat, closed, and open model of the universe.

The source of the accelerated expansion of the early universe with the equation of state *p* ¼ �*ρ* is a vacuum; the equation of the state *p*≈ � *ρ* corresponds to a scalar (bosonic) field. The Bose-Einstein statistics for an ensemble of bosons, in contrast to an ensemble of fermions obeying the Pauli exclusion principle, implies that there can be several particles in one quantum state, which leads to the formation of boson condensate in which the increase in the concentration of massless bosons is associated with a decrease in the effective pressure corresponding to the equation of state *p*≈ � *ρ*. The initial (quasi) exponential expansion associated with a negative pressure, due to the exotic equation of state, is unstable, which leads to a phase transition, the termination of accelerated expansion, and the fragmentation of the original volume into many areas in which further evolution corresponds to the Friedmann solutions.

Also, the presence of a scalar field violates the symmetry of the system, which leads to the appearance of a mass of initially massless particles, for example, in the Higgs field [9].

Thus, the inclusion of the scalar field into cosmological models makes it possible to move from (quasi) de Sitter solutions to the Friedmann ones.

To prevent the rapid decay of the state of *p*≈ � *ρ*, it is necessary to assume the existence of some potential barrier, that is, the minimum potential energy of a

scalar field. Consequently, in realistic inflation models, the scalar field evolves from a state of "false vacuum" with a non-zero potential energy to a state of "true vacuum," corresponding to the minimum of a potential *V*ð Þ *ϕ* . In other words, the scalar field rolls down (or tunnels) from some initial state to the minimum of *V*ð Þ *ϕ* , and the nature of this process is determined by the shape of the potential.

At the moment, there are many models of cosmological inflation with different potentials of a scalar field and different specifics of its evolution. A large number of current models of the early universe on the basis of the inflationary paradigm are considered in the review [13].

The physical justification for the inclusion of scalar fields in cosmological models is based on the experimental detection of the Higgs boson in the experiment at the Large Hadron Collider [14]. Thus, the scalar field corresponding to the Higgs bosons can be considered as the source of the gravitational field of the early universe. Moreover, the Higgs field can be considered as "inflation," leading to early accelerated expansion of the universe.

Now, in the system of units 8*πG* ¼ *c* ¼ 1, we write the action that determines the dynamics of a scalar field *ϕ* based on Einstein's theory of gravity:

$$S\_E = \int d^4 \mathbf{x} \sqrt{-\mathbf{g}} \left[ \frac{\mathbf{1}}{2} R - \mathbf{g}^{\mu \nu} \frac{\mathbf{1}}{2} \partial\_{\mu} \phi \, \partial\_{\nu} \phi - V(\phi) \right] \tag{2}$$

where *R* is the Ricci scalar and *V*ð Þ *ϕ* is the potential of a scalar field.

From the variation of this action with respect to the metric (1) and a field *ϕ*, for the case of the spatially flat universe, we obtain the equations defining the dynamics of a scalar field [9]:

$$\Im \mathcal{H}^2 = \frac{1}{2}\dot{\phi}^2 + V(\phi) \tag{3}$$

conditions under which an inflationary stage occurs: namely, a scalar field can be located at one of its potential minima, or accelerated expansion occurs for any conditions permitting the onset of inflation for scalar field energy density values

The form of the scalar field's potential is determined from the physics of elementary particles and theories of the unification of fundamental interactions, such as supersymmetric theories and string theories in the context of the inflationary paradigm. Physical mechanisms corresponding to a large number of inflationary potentials were discussed in the review [14]. Due to the fact that the potential of the scalar field has a great importance for determining the physical processes at the stage of cosmological inflation, the potential *V*ð Þ *ϕ* is given to build models of the

However, the finding of exact solutions to the system of Eqs. (3)–(5) for a given potential is impossible in most cases due to their nonlinearity. For this reason, a convenient tool for analyzing inflationary models based on a given scalar field potential is the "slow-roll approximation" which implies that *<sup>V</sup>*ð Þ *<sup>ϕ</sup>* >>*<sup>X</sup>* and *<sup>ϕ</sup>*€≈<sup>0</sup>

The dynamics of the expansion of the universe which determined by the scale factor *a t*ð Þ is no less important when analyzing cosmological models. By setting the expansion law *a t*ð Þ, it is often possible to find the exact solutions of the system of Eqs. (3)–(5) and restore the evolution of the scalar field *ϕ*ð Þ*t* and the potential *V*ð Þ *ϕ* . The different methods for constructing exact and approximate solutions of the equations of cosmological dynamics (3)–(5) can be found, for example, in the papers [15, 16]. We also note that the system of Eqs. (3)–(5) has many solutions that satisfy all the conditions for the inflationary stage that were outlined earlier. Now, we consider the parameters that are necessary for the analysis of infla-

and, therefore, simplifies the initial dynamic equations [9].

tionary stage, namely, the e-fold number and the slow-roll parameters.

*N t*ðÞ¼ ln *a t*ð Þ *end*

ϵ � 2

*H*″ *ϕ <sup>H</sup>* <sup>¼</sup> <sup>ϵ</sup> � <sup>ϵ</sup>\_

> *H*0 *ϕH*‴ *ϕ <sup>H</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

*ξ* � 4

*δ* � 2

and these parameters are defined as follows [13]:

determined by the condition ϵ ¼ 1.

The e-fold number is usually noted as the natural logarithm of the ratio of the scale factor at the end of inflation to the scale factor at the beginning of

*a t*ð Þ*<sup>i</sup>*

where *ti* and *tend* are the times of the beginning and ending of the inflation. The value of the number of e-folds at the end of the inflationary stage is estimated as

When analyzing inflationary models, the slow-roll parameters are important,

*H*0 *ϕ H* � �<sup>2</sup> ¼ *t* ð*end*

*ti*

¼ � *<sup>H</sup>*\_

<sup>2</sup>*H*<sup>ϵ</sup> ¼ � *<sup>H</sup>*€

*<sup>H</sup>* <sup>ϵ</sup>\_ � \_

Based on the relations (8)–(10), one can consider the slow-roll parameters as a function of time or field. During the inflationary stage, ϵ < 1 and its completion are

*Hdt* (7)

*<sup>H</sup>*<sup>2</sup> (8)

<sup>2</sup>*HH*\_ (9)

*δ* � � (10)

comparable to the Planck mass [9].

*The Early Universe as a Source of Gravitational Waves DOI: http://dx.doi.org/10.5772/intechopen.87946*

early universe.

inflation [9]:

*N* ¼ 50 � 60 [9].

**225**

$$-2\dot{H} - 3H^2 = \frac{1}{2}\dot{\phi}^2 - V(\phi) \tag{4}$$

$$
\ddot{\phi} + 3H\dot{\phi} + \frac{dV(\phi)}{d\phi} = 0\tag{5}
$$

where *<sup>H</sup>* <sup>¼</sup> *<sup>a</sup>*\_*=<sup>a</sup>* is the Hubble parameter, *<sup>X</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> *<sup>ϕ</sup>*\_ <sup>2</sup> is the kinetic energy of a canonical scalar field *ϕ*, and the dot denotes the derivative with respect to cosmic time *a*\_ ¼ *da=dt*.

Also, the state parameter *w* of a scalar field can be calculated as

$$w = \frac{p}{\rho} = \frac{X - V}{X + V} = -1 - \frac{2}{3} \frac{\dot{H}}{H^2} \tag{6}$$

To build a consistent model of cosmological inflation, the following conditions must be met:


Currently, along with other models, several types of cosmological inflationary models are considered, which differ in both by the type of potential and the initial

#### *The Early Universe as a Source of Gravitational Waves DOI: http://dx.doi.org/10.5772/intechopen.87946*

scalar field. Consequently, in realistic inflation models, the scalar field evolves from a state of "false vacuum" with a non-zero potential energy to a state of "true vacuum," corresponding to the minimum of a potential *V*ð Þ *ϕ* . In other words, the scalar field rolls down (or tunnels) from some initial state to the minimum of *V*ð Þ *ϕ* ,

At the moment, there are many models of cosmological inflation with different potentials of a scalar field and different specifics of its evolution. A large number of current models of the early universe on the basis of the inflationary paradigm are

The physical justification for the inclusion of scalar fields in cosmological models is based on the experimental detection of the Higgs boson in the experiment at the Large Hadron Collider [14]. Thus, the scalar field corresponding to the Higgs bosons can be considered as the source of the gravitational field of the early universe. Moreover, the Higgs field can be considered as "inflation," leading to early

Now, in the system of units 8*πG* ¼ *c* ¼ 1, we write the action that determines the

*<sup>R</sup>* � *<sup>g</sup>μν* <sup>1</sup> 2

From the variation of this action with respect to the metric (1) and a field *ϕ*, for the case of the spatially flat universe, we obtain the equations defining the dynamics

2

canonical scalar field *ϕ*, and the dot denotes the derivative with respect to cosmic

*dV*ð Þ *ϕ*

*<sup>X</sup>* <sup>þ</sup> *<sup>V</sup>* ¼ �<sup>1</sup> � <sup>2</sup>

To build a consistent model of cosmological inflation, the following conditions

a. The presence of the stage of accelerated expansion, which implies ‐<sup>1</sup> <sup>&</sup>lt; <sup>w</sup> <sup>&</sup>lt; ‐1*=*<sup>3</sup>

c. The reheating of the scalar field with the subsequent formation of photons, i.e.,

Currently, along with other models, several types of cosmological inflationary models are considered, which differ in both by the type of potential and the initial

the transition to the stage of predominance of radiation with w ¼ 1*=*3

3 *H*\_

where *R* is the Ricci scalar and *V*ð Þ *ϕ* is the potential of a scalar field.

<sup>3</sup>*H*<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2

�2*H*\_ � <sup>3</sup>*H*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

*<sup>ϕ</sup>*€ <sup>þ</sup> <sup>3</sup>*Hϕ*\_ <sup>þ</sup>

Also, the state parameter *w* of a scalar field can be calculated as

b.The completion of the stage of accelerated expansion w <sup>¼</sup> ‐1*=*<sup>3</sup>

*<sup>ρ</sup>* <sup>¼</sup> *<sup>X</sup>* � *<sup>V</sup>*

*<sup>w</sup>* <sup>¼</sup> *<sup>p</sup>*

where *<sup>H</sup>* <sup>¼</sup> *<sup>a</sup>*\_*=<sup>a</sup>* is the Hubble parameter, *<sup>X</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>∂</sup>μϕ∂νϕ* � *<sup>V</sup>*ð Þ *<sup>ϕ</sup>*

*<sup>ϕ</sup>*\_ <sup>2</sup> <sup>þ</sup> *<sup>V</sup>*ð Þ *<sup>ϕ</sup>* (3)

*<sup>ϕ</sup>*\_ <sup>2</sup> � *<sup>V</sup>*ð Þ *<sup>ϕ</sup>* (4)

*<sup>d</sup><sup>ϕ</sup>* <sup>¼</sup> <sup>0</sup> (5)

<sup>2</sup> *<sup>ϕ</sup>*\_ <sup>2</sup> is the kinetic energy of a

*<sup>H</sup>*<sup>2</sup> (6)

(2)

� �

and the nature of this process is determined by the shape of the potential.

dynamics of a scalar field *ϕ* based on Einstein's theory of gravity:

considered in the review [13].

*Progress in Relativity*

of a scalar field [9]:

time *a*\_ ¼ *da=dt*.

must be met:

**224**

accelerated expansion of the universe.

*SE* ¼ ð *d*4 *x* ffiffiffiffiffiffi �*<sup>g</sup>* <sup>p</sup> <sup>1</sup> 2 conditions under which an inflationary stage occurs: namely, a scalar field can be located at one of its potential minima, or accelerated expansion occurs for any conditions permitting the onset of inflation for scalar field energy density values comparable to the Planck mass [9].

The form of the scalar field's potential is determined from the physics of elementary particles and theories of the unification of fundamental interactions, such as supersymmetric theories and string theories in the context of the inflationary paradigm. Physical mechanisms corresponding to a large number of inflationary potentials were discussed in the review [14]. Due to the fact that the potential of the scalar field has a great importance for determining the physical processes at the stage of cosmological inflation, the potential *V*ð Þ *ϕ* is given to build models of the early universe.

However, the finding of exact solutions to the system of Eqs. (3)–(5) for a given potential is impossible in most cases due to their nonlinearity. For this reason, a convenient tool for analyzing inflationary models based on a given scalar field potential is the "slow-roll approximation" which implies that *<sup>V</sup>*ð Þ *<sup>ϕ</sup>* >>*<sup>X</sup>* and *<sup>ϕ</sup>*€≈<sup>0</sup> and, therefore, simplifies the initial dynamic equations [9].

The dynamics of the expansion of the universe which determined by the scale factor *a t*ð Þ is no less important when analyzing cosmological models. By setting the expansion law *a t*ð Þ, it is often possible to find the exact solutions of the system of Eqs. (3)–(5) and restore the evolution of the scalar field *ϕ*ð Þ*t* and the potential *V*ð Þ *ϕ* . The different methods for constructing exact and approximate solutions of the equations of cosmological dynamics (3)–(5) can be found, for example, in the papers [15, 16]. We also note that the system of Eqs. (3)–(5) has many solutions that satisfy all the conditions for the inflationary stage that were outlined earlier.

Now, we consider the parameters that are necessary for the analysis of inflationary stage, namely, the e-fold number and the slow-roll parameters.

The e-fold number is usually noted as the natural logarithm of the ratio of the scale factor at the end of inflation to the scale factor at the beginning of inflation [9]:

$$N(t) = \ln \frac{a(t\_{end})}{a(t\_i)} = \int\_{t\_i}^{t\_{end}} H dt \tag{7}$$

where *ti* and *tend* are the times of the beginning and ending of the inflation. The value of the number of e-folds at the end of the inflationary stage is estimated as *N* ¼ 50 � 60 [9].

When analyzing inflationary models, the slow-roll parameters are important, and these parameters are defined as follows [13]:

$$\epsilon \equiv 2 \left( \frac{H\_{\phi}^{\prime}}{H} \right)^{2} = -\frac{\dot{H}}{H^{2}} \tag{8}$$

$$\delta \equiv 2\frac{H\_{\phi}^{\prime}}{H} = \epsilon - \frac{\dot{\epsilon}}{2H\epsilon} = -\frac{\ddot{H}}{2H\dot{H}}\tag{9}$$

$$\xi \equiv 4 \frac{H\_{\phi}^{\prime} H\_{\phi}^{\cdots}}{H^2} = \frac{1}{H} \left( \dot{\epsilon} - \dot{\delta} \right) \tag{10}$$

Based on the relations (8)–(10), one can consider the slow-roll parameters as a function of time or field. During the inflationary stage, ϵ < 1 and its completion are determined by the condition ϵ ¼ 1.

### **3. Cosmological perturbations**

Cosmological perturbations are the source of the evolution of large-scale structure of the universe. An explanation of the distribution of galaxies and clusters of galaxies at large distances in the observable part of the universe on the basis of cosmological perturbations was originally proposed in the works of Harrison [17] and Zeldovich [18]. In the context of inflationary paradigm, the source of cosmological perturbations is quantum fluctuations of a scalar field and the corresponding fluctuations of the metric, which, in a linear order, correspond to three modes that evolve independently.

b.Silk effect due to adiabatic compression of radiation and baryon acoustic oscillations prior to the recombination epoch in high- and low-density zones

In the zero order of the cosmological perturbation theory, the universe is described by a single function of time, namely, by the scale factor *a t*ð Þ. In the first (linear) order, the perturbations of the metric are the sum of three independent modes—scalar, vector, and tensor (relic gravitational waves), each of which is

For the inflationary stage in the linear approximation, one can write the Mukhanov-Sasaki equations for Fourier modes of the scalar *vk* and tensor *uk*

> *z d*2 *z dη*<sup>2</sup>

*a d*2 *a dη*<sup>2</sup>

Eqs. (11) and (12) allow finding the power spectra P*<sup>S</sup>* and P*<sup>T</sup>* and spectral indices *nS* and *nT* of the scalar and tensor perturbations. The formulas for calculating the main cosmological parameters at crossing the Hubble radius

!

*vk* ¼ 0 (11)

*uk* ¼ 0 (12)

� � (15)

¼ 4ϵ (17)

<sup>10</sup><sup>9</sup>P*<sup>S</sup>* <sup>¼</sup> <sup>2</sup>*:*<sup>142</sup> � <sup>0</sup>*:*<sup>049</sup> (18) *nS* ¼ 0*:*9667 � 0*:*0040 (19)

*r* < 0*:*065 (20)

(13)

(14)

(16)

!

*<sup>d</sup>η*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> � <sup>1</sup>

*<sup>d</sup>η*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> � <sup>1</sup>

where *<sup>z</sup>* <sup>¼</sup> *<sup>a</sup>ϕ*\_ *<sup>=</sup>H*, *<sup>k</sup>* is a wave number, and *<sup>η</sup>* is the conformal time.

<sup>P</sup>*S*ð Þ¼ *<sup>k</sup>* <sup>1</sup>

2ϵ

<sup>P</sup>*T*ð Þ¼ *<sup>k</sup>* <sup>2</sup> *<sup>H</sup>*

*nS* � <sup>1</sup> <sup>¼</sup> <sup>2</sup> *<sup>δ</sup>* � <sup>2</sup><sup>ϵ</sup>

*nT* ¼ � <sup>2</sup><sup>ϵ</sup>

*<sup>r</sup>* <sup>¼</sup> <sup>P</sup>*<sup>T</sup>* P*<sup>S</sup>*

The data on the effects of scalar and tensor modes can be obtained from observations of the anisotropy and polarization of the cosmic microwave background (CMB) radiation, which arose as a result of the joint effect on the photon distribution of the perturbation modes. Observational restrictions on the values of

In the context of such verification of cosmological models, let us pay attention to the tendency for the upper limit to decrease by the value of the tensor-scalar ratio

the parameters of cosmological perturbations according to the data of the

*H* 2*π* � �<sup>2</sup>

2*π* � �<sup>2</sup>

1 � ϵ

1 � ϵ

characterized by the spectral function of the wave number *k* [11].

*d*2 *vk*

*The Early Universe as a Source of Gravitational Waves DOI: http://dx.doi.org/10.5772/intechopen.87946*

> *d*2 *uk*

perturbations [11]:

ð Þ *k* ¼ *aH* [19] are

PLANCK are [10]

**227**

for updated observational data [10].

It is known from the classical theory of cosmological perturbations that the analysis of metric inhomogeneities can be simplified to the study of one perturbed quantity [11]. Thus, the quantum theory of cosmological perturbations can be reduced to the quantum description of the fluctuations of a certain scalar field.

Since the background in which the scalar field evolves depends on time, the field mass will also depend on time. This dependence of the field's mass on time will lead to the appearance of particles if the evolution begins with a certain vacuum state. Quantum particle production corresponds to the development and growth of cosmological perturbations.

In inflation models with one scalar field, at the crossing of the Hubble radius, cosmological perturbations "freeze," and their quantum state begins to change in such a way that the condition of constant amplitude is satisfied. The freezing of the vacuum state leads to the appearance of the classical properties [11]. Thus, the theory of cosmological perturbations provides a consistent approach for considering the generation and evolution of cosmological perturbations.

The influence of cosmological perturbations on the anisotropy and polarization of the background radiation is determined on the basis of spectral parameters and observational restrictions on the values of which form the basis of the experimental verification of theoretical models of the early universe. Also, within the framework of the cosmological perturbation theory, it is possible to calculate the spectra of initial density perturbations and relic gravitational waves depending on the values of the parameters of theoretical models [11].

After the end of the inflationary stage, the scalar field reheating and the formation of the first light particles of baryon matter begin. In the hot dense plasma, due to scattering on electrons, photons propagate much slower than the speed of light. When the universe expands so much that the plasma cools down to the recombination temperature, the electrons begin to connect with the protons, forming neutral hydrogen, and the photons begin to spread freely.

The points from which the photons reach the observer form the last scattering surface, whose temperature at the time of recombination is 3000 K and rapidly decreases with the expansion of the universe. The background radiation temperature is isotropic with an accuracy of 10<sup>5</sup> . The low anisotropy manifests itself as the temperature difference in different directions and its value is approximately equal to 3 mK [10].

The kinetic component of the anisotropy of the cosmic background radiation is due to the movement of the observer relative to the background radiation, which corresponds to the dipole harmonic.

In addition to the kinetic component in the anisotropy of the CMB, there are potential terms associated with effects in gravitational fields of very large scales that are comparable to the distance to the last scattering surface, namely:

a. Sachs-Wolfe effect, which corresponds to a change in the photon energy in a variable gravitational field of the universe

**3. Cosmological perturbations**

evolve independently.

*Progress in Relativity*

mological perturbations.

Cosmological perturbations are the source of the evolution of large-scale structure of the universe. An explanation of the distribution of galaxies and clusters of galaxies at large distances in the observable part of the universe on the basis of cosmological perturbations was originally proposed in the works of Harrison [17] and Zeldovich [18]. In the context of inflationary paradigm, the source of cosmological perturbations is quantum fluctuations of a scalar field and the corresponding fluctuations of the metric, which, in a linear order, correspond to three modes that

It is known from the classical theory of cosmological perturbations that the analysis of metric inhomogeneities can be simplified to the study of one perturbed quantity [11]. Thus, the quantum theory of cosmological perturbations can be reduced to the quantum description of the fluctuations of a certain scalar field.

Since the background in which the scalar field evolves depends on time, the field mass will also depend on time. This dependence of the field's mass on time will lead to the appearance of particles if the evolution begins with a certain vacuum state. Quantum particle production corresponds to the development and growth of cos-

In inflation models with one scalar field, at the crossing of the Hubble radius, cosmological perturbations "freeze," and their quantum state begins to change in such a way that the condition of constant amplitude is satisfied. The freezing of the vacuum state leads to the appearance of the classical properties [11]. Thus, the theory of cosmological perturbations provides a consistent approach for considering

The influence of cosmological perturbations on the anisotropy and polarization of the background radiation is determined on the basis of spectral parameters and observational restrictions on the values of which form the basis of the experimental verification of theoretical models of the early universe. Also, within the framework of the cosmological perturbation theory, it is possible to calculate the spectra of initial density perturbations and relic gravitational waves depending on the values

After the end of the inflationary stage, the scalar field reheating and the formation of the first light particles of baryon matter begin. In the hot dense plasma, due to scattering on electrons, photons propagate much slower than the speed of light. When the universe expands so much that the plasma cools down to the recombination temperature, the electrons begin to connect with the protons, forming neutral

The points from which the photons reach the observer form the last scattering surface, whose temperature at the time of recombination is 3000 K and rapidly decreases with the expansion of the universe. The background radiation tempera-

temperature difference in different directions and its value is approximately equal

The kinetic component of the anisotropy of the cosmic background radiation is due to the movement of the observer relative to the background radiation, which

In addition to the kinetic component in the anisotropy of the CMB, there are potential terms associated with effects in gravitational fields of very large scales that

a. Sachs-Wolfe effect, which corresponds to a change in the photon energy in a

are comparable to the distance to the last scattering surface, namely:

. The low anisotropy manifests itself as the

the generation and evolution of cosmological perturbations.

of the parameters of theoretical models [11].

hydrogen, and the photons begin to spread freely.

variable gravitational field of the universe

ture is isotropic with an accuracy of 10<sup>5</sup>

corresponds to the dipole harmonic.

to 3 mK [10].

**226**

b.Silk effect due to adiabatic compression of radiation and baryon acoustic oscillations prior to the recombination epoch in high- and low-density zones

In the zero order of the cosmological perturbation theory, the universe is described by a single function of time, namely, by the scale factor *a t*ð Þ. In the first (linear) order, the perturbations of the metric are the sum of three independent modes—scalar, vector, and tensor (relic gravitational waves), each of which is characterized by the spectral function of the wave number *k* [11].

For the inflationary stage in the linear approximation, one can write the Mukhanov-Sasaki equations for Fourier modes of the scalar *vk* and tensor *uk* perturbations [11]:

$$\frac{d^2 v\_k}{d\eta^2} + \left(k^2 - \frac{1}{z}\frac{d^2 z}{d\eta^2}\right) v\_k = 0 \tag{11}$$

$$\frac{d^2 u\_k}{d\eta^2} + \left(k^2 - \frac{1}{a} \frac{d^2 a}{d\eta^2}\right) u\_k = 0 \tag{12}$$

where *<sup>z</sup>* <sup>¼</sup> *<sup>a</sup>ϕ*\_ *<sup>=</sup>H*, *<sup>k</sup>* is a wave number, and *<sup>η</sup>* is the conformal time.

Eqs. (11) and (12) allow finding the power spectra P*<sup>S</sup>* and P*<sup>T</sup>* and spectral indices *nS* and *nT* of the scalar and tensor perturbations. The formulas for calculating the main cosmological parameters at crossing the Hubble radius ð Þ *k* ¼ *aH* [19] are

$$\mathcal{P}\_S(k) = \frac{1}{2\epsilon} \left(\frac{H}{2\pi}\right)^2 \tag{13}$$

$$\mathcal{P}\_T(k) = 2\left(\frac{H}{2\pi}\right)^2\tag{14}$$

$$n\_S - 1 = 2\left(\frac{\delta - 2\epsilon}{1 - \epsilon}\right) \tag{15}$$

$$m\_T = -\frac{2\epsilon}{1-\epsilon} \tag{16}$$

$$r = \frac{\mathcal{P}\_T}{\mathcal{P}\_S} = 4\mathfrak{e} \tag{17}$$

The data on the effects of scalar and tensor modes can be obtained from observations of the anisotropy and polarization of the cosmic microwave background (CMB) radiation, which arose as a result of the joint effect on the photon distribution of the perturbation modes. Observational restrictions on the values of the parameters of cosmological perturbations according to the data of the PLANCK are [10]

$$10^9 \mathcal{P}\_{\mathbb{S}} = 2.142 \pm 0.049 \tag{18}$$

$$n\_{\mathbb{S}} = 0.9667 \pm 0.0040\tag{19}$$

$$r \le 0.065 \tag{20}$$

In the context of such verification of cosmological models, let us pay attention to the tendency for the upper limit to decrease by the value of the tensor-scalar ratio for updated observational data [10].

Also, we note that the relic gravitational waves were not directly observed, which leads to a large number of theoretical models of cosmological inflation, which provide an explanation of the origin and evolution of the large-scale structure of the universe and correspond to the observational constraints.

b.The potential *V*ð Þ *φ* corresponding to the scale factor (20) implies the evolution

Transformations (21)–(24) define a class of models with the generalized exponential power-law dynamics, and the original scale factor *a t*ð Þ may not correspond to the condition of accelerated expansion *a*€>0; however, the resulting scale factor *a t*ð Þ implies a combination of the de Sitter solution (for *n* ¼ 0) and the power-law expansion (for *λ* ¼ 0), which corresponds to the basic feature of the inflationary paradigm implying a graceful exit from the stage of accelerated expansion to the

As an additional verification tool for cosmological models, we consider the possibility of direct detection of the relic gravitational waves. The detection of relic gravitational waves is extremely important for determining the parameters of the models of early universe. Additionally, such a detection enhances the position of the inflationary paradigm compared to alternative scenarios, for example, the models with a rebound from singularity in which cosmological gravitational waves are

As the main observational characteristic of relic gravitational waves, we consider the energy density, which is usually determined by the dimensionless quantity [20]:

*ρc*

value of the Hubble parameter in the modern era, and *ρGW* is the energy density of

Also, the energy density of relic gravitational waves can be represented in terms

12*H*<sup>2</sup> 0

The frequency and energy density of relic gravitational waves are limited by the

*dρGW*

*<sup>d</sup>*ln *<sup>f</sup>* (27)

<sup>0</sup> is the critical energy density, *H*<sup>0</sup> is the

*PT*ð Þ*k* (28)

*GeV* (30)

<sup>Ω</sup>*GWd*ln *<sup>f</sup>* <sup>&</sup>lt; <sup>1</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>5</sup> (29)

<sup>Ω</sup>*GW*ð Þ¼ *<sup>f</sup>* <sup>1</sup>

<sup>Ω</sup>*GW*ð Þ¼ *<sup>k</sup> <sup>k</sup>*<sup>2</sup>

a. The energy density of relic gravitational waves should not exceed

a. The temperature of the scalar field *T*<sup>∗</sup> and the frequency of gravitational

*Hz* � � *<sup>g</sup>* <sup>∗</sup>

<sup>106</sup>*:*<sup>75</sup> � �<sup>1</sup>*=*<sup>6</sup>

∞ð

*f* 0

*<sup>T</sup>* <sup>∗</sup> <sup>¼</sup> <sup>5</sup>*:*<sup>85</sup> � <sup>10</sup><sup>6</sup> *<sup>f</sup>*

waves *f* at the end of the inflation stage are

of the scalar field *φ*, according to the inflationary paradigm.

power-law non-accelerated expansion.

*The Early Universe as a Source of Gravitational Waves DOI: http://dx.doi.org/10.5772/intechopen.87946*

where *<sup>f</sup>* is the linear frequency, *<sup>ρ</sup><sup>c</sup>* <sup>¼</sup> <sup>3</sup>*H*<sup>2</sup>

**5. Relic gravitational waves**

absent [19].

gravitational waves.

of the power spectrum:

following conditions [20]:

where *f* <sup>0</sup>≈10�<sup>9</sup> Hz.

**229**
