**Light Cold Dark Matter in a Two-Singlet Model**

**Light Cold Dark Matter in a Two-Singlet Model**

Abdessamad Abada and Salah Nasri

Abdessamad Abada and Salah Nasri

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52243

## **1. Introduction**

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 longer than the age of the Universe.

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 is clear then that other alternative scenarios for LDM are needed [7].

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, 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

The simplest of scenarios is then to extend the Standard Model by a real **Z**<sup>2</sup> symmetric scalar field, the dark matter, which has to be a SM gauge singlet that interacts with visible particles via the Higgs field only. Such an extension was first proposed in [8] and further studied in [9] where the **Z**<sup>2</sup> symmetry is extended to a global U(1) symmetry, more extensively in [10]. Specific implications on Higgs detection and LHC physics were discussed in [11] and one-loop vacuum stability and perturbativity bounds discussed in [12]. However, the work of [13] uses constraints from the experiments XENON10 [14] and CDMSII [15] to exclude DM masses smaller than 50, 70 and 75GeV for Higgs masses equal to 120, 200 and 350 GeV respectively. Also, the Fermi-LAT data on the isotropic diffuse gamma-ray emission can potentially exclude this one-singlet dark-matter model for masses as low as 6GeV, assuming a NFW profile for the dark-matter distribution [16].

*<sup>U</sup>* <sup>=</sup> *<sup>m</sup>*˜ <sup>2</sup> 0 <sup>2</sup> *<sup>S</sup>*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>µ</sup>*<sup>2</sup>

and *<sup>χ</sup>*<sup>1</sup> = *<sup>v</sup>*<sup>1</sup> + *<sup>S</sup>*˜

following relations:

<sup>2</sup> *<sup>h</sup>*′<sup>2</sup> <sup>−</sup> *<sup>µ</sup>*<sup>2</sup> 1 2 *χ*2 <sup>1</sup> <sup>+</sup> *<sup>η</sup>*<sup>0</sup> <sup>24</sup> *<sup>S</sup>*<sup>4</sup> 0 + *λ* <sup>24</sup> *<sup>h</sup>*′<sup>4</sup> <sup>+</sup> *<sup>η</sup>*<sup>1</sup>

The mass parameters squared *m*˜ <sup>2</sup>

where the quadratic terms are given by:

*m*2 <sup>0</sup> <sup>=</sup> *<sup>m</sup>*˜ <sup>2</sup> 0 + *λ*0

*M*<sup>2</sup>

masses in (5) by the two relations:

*m*2 *<sup>h</sup>* <sup>=</sup> <sup>1</sup> 2 *M*<sup>2</sup> *<sup>h</sup>* <sup>+</sup> *<sup>M</sup>*<sup>2</sup>

*m*2 <sup>1</sup> <sup>=</sup> <sup>1</sup> 2 *M*<sup>2</sup> *<sup>h</sup>* <sup>+</sup> *<sup>M</sup>*<sup>2</sup>

where *ε* is the sign function.

<sup>1</sup> <sup>=</sup> <sup>−</sup>*µ*<sup>2</sup>

1 + *λ*1 <sup>2</sup> *<sup>v</sup>*<sup>2</sup> <sup>+</sup> *<sup>η</sup>*<sup>1</sup>

mass-squared field eigenmodes by *h* and *S*1, we rewrite:

*<sup>U</sup>*quad <sup>=</sup> <sup>1</sup>

 *h S*1 =

Here *θ* is the mixing angle, given by the relation tan 2*θ* = 2*M*<sup>2</sup>

2 *m*2 0*S*2 0 + 1 2 *m*2 *<sup>h</sup>h*<sup>2</sup> <sup>+</sup> 1 2 *m*2 1*S*2

where the physical fields are related to the mixed ones by a 2 × 2 rotation:

<sup>1</sup> <sup>+</sup> *<sup>ε</sup> M*<sup>2</sup> *<sup>h</sup>* <sup>−</sup> *<sup>M</sup>*<sup>2</sup> 1 *M*<sup>2</sup> *<sup>h</sup>* <sup>−</sup> *<sup>M</sup>*<sup>2</sup> 1

<sup>1</sup> <sup>−</sup> *<sup>ε</sup> M*<sup>2</sup> *<sup>h</sup>* <sup>−</sup> *<sup>M</sup>*<sup>2</sup> 1 *M*<sup>2</sup> *<sup>h</sup>* <sup>−</sup> *<sup>M</sup>*<sup>2</sup> 1

*<sup>U</sup>*quad <sup>=</sup> <sup>1</sup>

2 *m*2 0*S*2 0 + 1 <sup>2</sup> *<sup>M</sup>*<sup>2</sup> *h* ˜ *h*<sup>2</sup> + 1 <sup>2</sup> *<sup>M</sup>*<sup>2</sup> 1*S*˜2 <sup>1</sup> <sup>+</sup> *<sup>M</sup>*<sup>2</sup> 1*h* ˜ *hS*˜

<sup>2</sup> *<sup>v</sup>*<sup>2</sup> <sup>+</sup> *<sup>η</sup>*<sup>01</sup>

<sup>2</sup> *<sup>v</sup>*<sup>2</sup>

2 *v*2

24*χ*<sup>4</sup> 1 + *λ*0 4 *S*2

1, the potential function becomes, up to an irrelevant zero-field energy:

numbers. Electroweak spontaneous symmetry breaking occurs for the Higgs field with the vacuum expectation value *<sup>v</sup>* = 246GeV. The field *<sup>χ</sup>*<sup>1</sup> will oscillate around the vacuum expectation value *<sup>v</sup>*<sup>1</sup> which we take at the electroweak scale 100GeV. Writing *<sup>h</sup>*′ <sup>=</sup> *<sup>v</sup>* <sup>+</sup> ˜

with the mass-squared coefficients related to the original parameters of the theory by the

<sup>1</sup>; *<sup>M</sup>*<sup>2</sup>

<sup>1</sup>; *<sup>M</sup>*<sup>2</sup>

Clearly, we need to diagonalize the mass-squared matrix. Denoting the physical

 cos *θ* sin *θ* − sin *θ* cos *θ*

*<sup>h</sup>* <sup>=</sup> <sup>−</sup>*µ*<sup>2</sup> <sup>+</sup>

 ˜ *h S*˜ 1 

> <sup>1</sup>*h*/ *M*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>M</sup>*<sup>2</sup> *h*

> > <sup>2</sup> + <sup>4</sup>*M*<sup>4</sup>

<sup>2</sup> + <sup>4</sup>*M*<sup>4</sup>

1*h* ;

1*h* 

<sup>0</sup>, *<sup>µ</sup>*2and *<sup>µ</sup>*<sup>2</sup>

<sup>0</sup>*h*′<sup>2</sup> <sup>+</sup> *<sup>η</sup>*<sup>01</sup>

*<sup>U</sup>* = *<sup>U</sup>*quad + *<sup>U</sup>*cub + *<sup>U</sup>*quar, (2)

*λ* <sup>2</sup> *<sup>v</sup>*<sup>2</sup> <sup>+</sup> *λ*1 2 *v*2 1;

<sup>1</sup>*<sup>h</sup>* <sup>=</sup> *<sup>λ</sup>*1*v v*1. (4)

<sup>1</sup>, (5)

. (6)

,and the physical

, (7)

<sup>1</sup> and all the coupling constants are real positive

<sup>4</sup> *<sup>S</sup>*<sup>2</sup> 0*χ*2 1 + *λ*1 <sup>4</sup> *<sup>h</sup>*′2*χ*<sup>2</sup>

Light Cold Dark Matter in a Two-Singlet Model

http://dx.doi.org/10.5772/52243

<sup>1</sup>. (1)

273

*h*

1, (3)

Therefore, a two-singlet extension of the Standard Model was proposed in [17] as a simple model for light cold dark matter. Both scalar fields are **Z**<sup>2</sup> symmetric with one undergoing spontaneous symmetry breaking. The behavior of the model under the DM relic-density constraint and the restrictions from experimental direct detection was studied. It was concluded that the model was capable of bearing a light dark-matter WIMP.

The present chapter describes how we can further the study of the two-singlet scalar model (2SSM) by discussing some of its phenomenological implications. We limit ourselves to small DM masses, from 0.1GeV to 10GeV. We discuss the implications of the model on the meson factories and the Higgs search at the LHC. In fact, it is pertinent at this stage to mention that there are more than one motivation for scalar-singlet extensions of the SM. Indeed, besides providing a possible account for the dark matter in the Universe consistent with the CoGeNT signal, they also provide a solution to the mu problem in the supersymmetric standard model. They can explain the matter-anti matter asymmetry in the Universe [18], and account for the possible occurrence of a light Higgs with a mass less or equal to 100GeV while still in agreement with the electroweak precision tests [19] and potential signatures at *B*-factories [20].

This chapter is organized as follows. The next section introduces briefly the model and summarizes the results of [17] regarding relic-density constraint and direct detection. The two following sections investigate the rare decays of Υ (section three) and *B* (section four) mesons, most particularly their invisible channels. Section five looks into the decay channels of the Higgs particle. In each of these situations, we try, when possible, to deduce preferred regions of the parameter space and excluded ones. The last section is devoted to concluding remarks. Results presented here are found in [20].

## **2. The 2SSM Model**

The Standard Model is extended with two real, spinless and **Z**2-symmetric fields. One is the dark-matter field *S*<sup>0</sup> with unbroken symmetry to ensure the stability of the dark matter, the other is an auxiliary field *χ*<sup>1</sup> for which the symmetry is spontaneously broken. Both fields are Standard-Model gauge singlets and can interact with SM particles only via the Higgs doublet *<sup>H</sup>*. This latter is taken in the unitary gauge such that *<sup>H</sup>*† <sup>=</sup> 1/√<sup>2</sup> (<sup>0</sup> *<sup>h</sup>*′ ), where *<sup>h</sup>*′ is a real scalar. We assume all processes calculable in perturbation theory. The potential function that incorporates *<sup>S</sup>*0, *<sup>h</sup>*′ and *<sup>χ</sup>*<sup>1</sup> is:

$$\mathcal{U} = \frac{\tilde{m}\_0^2}{2} S\_0^2 - \frac{\mu^2}{2} h'^2 - \frac{\mu\_1^2}{2} \chi\_1^2 + \frac{\eta\_0}{24} S\_0^4 + \frac{\lambda}{24} h'^4 + \frac{\eta\_1}{24} \chi\_1^4 + \frac{\lambda\_0}{4} S\_0^2 h'^2 + \frac{\eta\_{01}}{4} S\_0^2 \chi\_1^2 + \frac{\lambda\_1}{4} h'^2 \chi\_1^2. \tag{1}$$

The mass parameters squared *m*˜ <sup>2</sup> <sup>0</sup>, *<sup>µ</sup>*2and *<sup>µ</sup>*<sup>2</sup> <sup>1</sup> and all the coupling constants are real positive numbers. Electroweak spontaneous symmetry breaking occurs for the Higgs field with the vacuum expectation value *<sup>v</sup>* = 246GeV. The field *<sup>χ</sup>*<sup>1</sup> will oscillate around the vacuum expectation value *<sup>v</sup>*<sup>1</sup> which we take at the electroweak scale 100GeV. Writing *<sup>h</sup>*′ <sup>=</sup> *<sup>v</sup>* <sup>+</sup> ˜ *h* and *<sup>χ</sup>*<sup>1</sup> = *<sup>v</sup>*<sup>1</sup> + *<sup>S</sup>*˜ 1, the potential function becomes, up to an irrelevant zero-field energy:

$$
\mathcal{U} = \mathcal{U}\_{\text{quad}} + \mathcal{U}\_{\text{cub}} + \mathcal{U}\_{\text{quad}} \,\prime \tag{2}
$$

where the quadratic terms are given by:

2 Open Questions in Cosmology

*B*-factories [20].

**2. The 2SSM Model**

a NFW profile for the dark-matter distribution [16].

remarks. Results presented here are found in [20].

function that incorporates *<sup>S</sup>*0, *<sup>h</sup>*′ and *<sup>χ</sup>*<sup>1</sup> is:

The simplest of scenarios is then to extend the Standard Model by a real **Z**<sup>2</sup> symmetric scalar field, the dark matter, which has to be a SM gauge singlet that interacts with visible particles via the Higgs field only. Such an extension was first proposed in [8] and further studied in [9] where the **Z**<sup>2</sup> symmetry is extended to a global U(1) symmetry, more extensively in [10]. Specific implications on Higgs detection and LHC physics were discussed in [11] and one-loop vacuum stability and perturbativity bounds discussed in [12]. However, the work of [13] uses constraints from the experiments XENON10 [14] and CDMSII [15] to exclude DM masses smaller than 50, 70 and 75GeV for Higgs masses equal to 120, 200 and 350 GeV respectively. Also, the Fermi-LAT data on the isotropic diffuse gamma-ray emission can potentially exclude this one-singlet dark-matter model for masses as low as 6GeV, assuming

Therefore, a two-singlet extension of the Standard Model was proposed in [17] as a simple model for light cold dark matter. Both scalar fields are **Z**<sup>2</sup> symmetric with one undergoing spontaneous symmetry breaking. The behavior of the model under the DM relic-density constraint and the restrictions from experimental direct detection was studied. It was

The present chapter describes how we can further the study of the two-singlet scalar model (2SSM) by discussing some of its phenomenological implications. We limit ourselves to small DM masses, from 0.1GeV to 10GeV. We discuss the implications of the model on the meson factories and the Higgs search at the LHC. In fact, it is pertinent at this stage to mention that there are more than one motivation for scalar-singlet extensions of the SM. Indeed, besides providing a possible account for the dark matter in the Universe consistent with the CoGeNT signal, they also provide a solution to the mu problem in the supersymmetric standard model. They can explain the matter-anti matter asymmetry in the Universe [18], and account for the possible occurrence of a light Higgs with a mass less or equal to 100GeV while still in agreement with the electroweak precision tests [19] and potential signatures at

This chapter is organized as follows. The next section introduces briefly the model and summarizes the results of [17] regarding relic-density constraint and direct detection. The two following sections investigate the rare decays of Υ (section three) and *B* (section four) mesons, most particularly their invisible channels. Section five looks into the decay channels of the Higgs particle. In each of these situations, we try, when possible, to deduce preferred regions of the parameter space and excluded ones. The last section is devoted to concluding

The Standard Model is extended with two real, spinless and **Z**2-symmetric fields. One is the dark-matter field *S*<sup>0</sup> with unbroken symmetry to ensure the stability of the dark matter, the other is an auxiliary field *χ*<sup>1</sup> for which the symmetry is spontaneously broken. Both fields are Standard-Model gauge singlets and can interact with SM particles only via the Higgs

*<sup>h</sup>*′ is a real scalar. We assume all processes calculable in perturbation theory. The potential

), where

doublet *<sup>H</sup>*. This latter is taken in the unitary gauge such that *<sup>H</sup>*† <sup>=</sup> 1/√<sup>2</sup> (<sup>0</sup> *<sup>h</sup>*′

concluded that the model was capable of bearing a light dark-matter WIMP.

$$\mathcal{U}\_{\text{quad}} = \frac{1}{2} m\_0^2 S\_0^2 + \frac{1}{2} M\_h^2 \tilde{h}^2 + \frac{1}{2} M\_1^2 \tilde{S}\_1^2 + M\_{1h}^2 \tilde{h} \tilde{S}\_1 \tag{3}$$

with the mass-squared coefficients related to the original parameters of the theory by the following relations:

$$m\_0^2 = \dot{m}\_0^2 + \frac{\lambda\_0}{2}v^2 + \frac{\eta\_{01}}{2}v\_1^2; \quad M\_h^2 = -\mu^2 + \frac{\lambda}{2}v^2 + \frac{\lambda\_1}{2}v\_1^2;$$

$$M\_1^2 = -\mu\_1^2 + \frac{\lambda\_1}{2}v^2 + \frac{\eta\_1}{2}v\_1^2; \quad M\_{1h}^2 = \lambda\_1 v \, v\_1. \tag{4}$$

Clearly, we need to diagonalize the mass-squared matrix. Denoting the physical mass-squared field eigenmodes by *h* and *S*1, we rewrite:

$$\mathcal{U}\_{\text{quad}} = \frac{1}{2}m\_0^2 S\_0^2 + \frac{1}{2}m\_h^2 h^2 + \frac{1}{2}m\_1^2 S\_{1'}^2 \tag{5}$$

where the physical fields are related to the mixed ones by a 2 × 2 rotation:

$$
\begin{pmatrix} h \\ \mathcal{S}\_1 \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta \cos \theta \end{pmatrix} \begin{pmatrix} \tilde{h} \\ \tilde{\mathcal{S}}\_1 \end{pmatrix}. \tag{6}
$$

Here *θ* is the mixing angle, given by the relation tan 2*θ* = 2*M*<sup>2</sup> <sup>1</sup>*h*/ *M*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>M</sup>*<sup>2</sup> *h* ,and the physical masses in (5) by the two relations:

$$\begin{aligned} m\_h^2 &= \frac{1}{2} \left[ M\_h^2 + M\_1^2 + \varepsilon \left( M\_h^2 - M\_1^2 \right) \sqrt{\left( M\_h^2 - M\_1^2 \right)^2 + 4 M\_{1h}^4} \right];\\ m\_1^2 &= \frac{1}{2} \left[ M\_h^2 + M\_1^2 - \varepsilon \left( M\_h^2 - M\_1^2 \right) \sqrt{\left( M\_h^2 - M\_1^2 \right)^2 + 4 M\_{1h}^4} \right],\end{aligned} \tag{7}$$

where *ε* is the sign function.

Written now directly in terms of the physical fields, the cubic interaction terms are expressed as follows:

$$\mathcal{U}\_{\rm cub} = \frac{\lambda\_0^{(3)}}{2} \mathcal{S}\_0^2 h + \frac{\eta\_{01}^{(3)}}{2} \mathcal{S}\_0^2 \mathcal{S}\_1 + \frac{\lambda^{(3)}}{6} h^3 + \frac{\eta\_1^{(3)}}{6} \mathcal{S}\_1^3 + \frac{\lambda\_1^{(3)}}{2} h^2 \mathcal{S}\_1 + \frac{\lambda\_2^{(3)}}{2} h \mathcal{S}\_{1\prime}^2 \tag{8}$$

Finally, after spontaneous breaking of the electroweak and **Z**<sup>2</sup> symmetries, the part of the Standard Model Lagrangian that is relevant to dark matter annihilation writes, in terms of

> <sup>+</sup> *<sup>λ</sup>*(3) *hw hW*<sup>−</sup>

*hz <sup>h</sup>*<sup>2</sup> *Zµ*

<sup>2</sup> <sup>+</sup> *<sup>λ</sup>*(4)

The quantities *mf* , *mw* and *mz* are the masses of the fermion *f* , the *W* and the *Z* gauge bosons respectively, and the above coupling constants are given by the following relations:

*v*

*z v*

<sup>1</sup>*<sup>w</sup>* <sup>=</sup> <sup>−</sup><sup>2</sup>

<sup>1</sup>*<sup>z</sup>* <sup>=</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

<sup>1</sup>*<sup>w</sup>* <sup>=</sup> *<sup>m</sup>*<sup>2</sup> *w*

<sup>1</sup>*<sup>z</sup>* <sup>=</sup> *<sup>m</sup>*<sup>2</sup> *z*

The field *S*<sup>0</sup> is odd under the unbroken **Z**<sup>2</sup> symmetry, and so is a stable relic and can therefore constitute the dark matter of the Universe. Its relic density can be obtained using the standard

*<sup>h</sup>*<sup>2</sup> <sup>=</sup> 1.07 <sup>×</sup> <sup>10</sup><sup>9</sup>*xf*

into *S*1*S*<sup>1</sup> for *m*<sup>1</sup> < *m*0/2. The annihilation cross-section into fermions proceeds via s-channel

*<sup>h</sup>* is the normalized Hubble constant,*M*Pl = 1.22 × 1019GeV is the Planck mass, *<sup>g</sup>*<sup>∗</sup> the number of relativistic degrees of freedom at the freeze-out temperature *Tf* , and *xf* = *<sup>m</sup>*0/*Tf* which, for *<sup>m</sup>*<sup>0</sup> in the range 1 − 20GeV, is in the range 18.2 − 19.4. The quantity �*υ*12*σ*ann� is the

sin *θ*;

sin *θ*;

sin *θ*;

*<sup>v</sup>*<sup>2</sup> sin2 *<sup>θ</sup>*; *<sup>λ</sup>h*1*<sup>w</sup>* <sup>=</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

<sup>2</sup>*v*<sup>2</sup> sin2 *<sup>θ</sup>*; *<sup>λ</sup>h*1*<sup>z</sup>* <sup>=</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

*m*2 *w v*

*<sup>µ</sup> <sup>W</sup>*+*<sup>µ</sup>* <sup>+</sup> *<sup>λ</sup>*(3)

*hw <sup>h</sup>*2*W*<sup>−</sup>

<sup>2</sup> <sup>+</sup> *<sup>λ</sup>*(4) <sup>1</sup>*<sup>z</sup> <sup>S</sup>*<sup>2</sup> 1 *Zµ*

<sup>1</sup>*<sup>w</sup> <sup>S</sup>*1*W*<sup>−</sup>

*<sup>µ</sup> <sup>W</sup>*+*<sup>µ</sup>* <sup>+</sup> *<sup>λ</sup>*(4)

*<sup>µ</sup> <sup>W</sup>*+*<sup>µ</sup>*

Light Cold Dark Matter in a Two-Singlet Model

http://dx.doi.org/10.5772/52243

<sup>1</sup>*<sup>w</sup> <sup>S</sup>*<sup>2</sup> 1*W*<sup>−</sup> *<sup>µ</sup> <sup>W</sup>*+*<sup>µ</sup>*

<sup>2</sup> <sup>+</sup> *<sup>λ</sup>h*1*zhS*<sup>1</sup>

*w <sup>v</sup>*<sup>2</sup> sin 2*θ*;

*z*

<sup>√</sup>*g*∗*M*Pl �*υ*12*σ*ann� GeV, (14)

<sup>2</sup>*v*<sup>2</sup> sin 2*θ*. (13)

*f* for *mf* < *m*0/2, and

 *Zµ* <sup>2</sup> . (12)

275

the physical fields *h* and *S*1, as follows:

*f f* + *<sup>λ</sup>*<sup>1</sup> *<sup>f</sup> <sup>S</sup>*<sup>1</sup> ¯

<sup>2</sup> <sup>+</sup> *<sup>λ</sup>*(3) <sup>1</sup>*<sup>z</sup> S*<sup>1</sup> *Zµ*

*<sup>µ</sup> <sup>W</sup>*+*<sup>µ</sup>* <sup>+</sup> *<sup>λ</sup>*(4)

*f f*

cos *<sup>θ</sup>*; *<sup>λ</sup>*<sup>1</sup> *<sup>f</sup>* <sup>=</sup> *mf*

cos *<sup>θ</sup>*; *<sup>λ</sup>*(3)

cos *<sup>θ</sup>*; *<sup>λ</sup>*(3)

*<sup>v</sup>*<sup>2</sup> cos2 *<sup>θ</sup>*; *<sup>λ</sup>*(4)

<sup>2</sup>*v*<sup>2</sup> cos2 *<sup>θ</sup>*; *<sup>λ</sup>*(4)

*f λh f h* ¯

<sup>+</sup>*λ*(3) *hz h Zµ*

<sup>+</sup>*λh*1*whS*1*W*<sup>−</sup>

*<sup>λ</sup>h f* <sup>=</sup> <sup>−</sup> *mf*

*λ*(3) *hw* <sup>=</sup> <sup>2</sup>

*λ*(3) *hz* <sup>=</sup> *<sup>m</sup>*<sup>2</sup> *z v*

*λ*(4) *hw* <sup>=</sup> *<sup>m</sup>*<sup>2</sup> *w*

*λ*(4) *hz* <sup>=</sup> *<sup>m</sup>*<sup>2</sup> *z*

where ¯

**2.1. Dark Matter relic density constraint**

exchange of *h* and *S*<sup>1</sup> and is given by:

approximate solutions to the Boltzmann equations [21]:

Ω*<sup>D</sup>* ¯

thermally averaged annihilation cross section of *S*<sup>0</sup> into fermion pairs *f* ¯

*v*

*m*2 *w v*

*<sup>U</sup>*SM <sup>=</sup> ∑

where the cubic physical coupling constants are related to the original parameters via the following relations:

$$\begin{aligned} \lambda\_0^{(3)} &= \lambda\_0 v \cos \theta + \eta\_{01} v\_1 \sin \theta, \\ \eta\_{01}^{(3)} &= \eta\_{01} v\_1 \cos \theta - \lambda\_0 v \sin \theta; \\ \lambda\_1^{(3)} &= \lambda v \cos^3 \theta + \frac{3}{2} \lambda\_1 \sin 2\theta \left(v\_1 \cos \theta + v \sin \theta\right) + \eta\_1 v\_1 \sin^3 \theta; \\ \eta\_1^{(3)} &= \eta\_1 v\_1 \cos^3 \theta - \frac{3}{2} \lambda\_1 \sin 2\theta \left(v \cos \theta - v\_1 \sin \theta\right) - \lambda v \sin^3 \theta; \\ \lambda\_1^{(3)} &= \lambda\_1 v\_1 \cos^3 \theta + \frac{1}{2} \sin 2\theta \left[\left(2\lambda\_1 - \lambda\right) v \cos \theta - \left(2\lambda\_1 - \eta\_1\right) v\_1 \sin \theta\right] - \lambda\_1 v \sin^3 \theta; \\ \lambda\_2^{(3)} &= \lambda\_1 v \cos^3 \theta - \frac{1}{2} \sin 2\theta \left[\left(2\lambda\_1 - \eta\_1\right) v\_1 \cos \theta + \left(2\lambda\_1 - \lambda\right) v \sin \theta\right] + \lambda\_1 v\_1 \sin^3 \theta. \end{aligned} \tag{9}$$

In the same way, in terms of the physical fields too, the quartic interactions are given by:

$$\begin{split} \mathcal{U}\_{\text{quar}} &= \frac{\eta\_0}{24} \mathcal{S}\_0^4 + \frac{\lambda^{(4)}}{24} h^4 + \frac{\eta\_1^{(4)}}{24} \mathcal{S}\_1^4 + \frac{\lambda\_0^{(4)}}{4} \mathcal{S}\_0^2 h^2 + \frac{\eta\_{01}^{(4)}}{4} \mathcal{S}\_0^2 \mathcal{S}\_1^2 + \frac{\lambda\_{01}^{(4)}}{2} \mathcal{S}\_0^2 h \mathcal{S}\_1 \\ &+ \frac{\lambda\_1^{(4)}}{6} h^3 \mathcal{S}\_1 + \frac{\lambda\_2^{(4)}}{4} h^2 \mathcal{S}\_1^2 + \frac{\lambda\_3^{(4)}}{6} h \mathcal{S}\_1^3 \end{split} \tag{10}$$

with the physical quartic coupling constants written as:

$$\begin{aligned} \lambda^{(4)} &= \lambda\cos^4\theta + \frac{3}{2}\lambda\_1\sin^2 2\theta + \eta\_1\sin^4\theta, \\ \eta\_1^{(4)} &= \eta\_1\cos^4\theta + \frac{3}{2}\lambda\_1\sin^2 2\theta + \lambda\sin^4\theta; \\ \lambda\_0^{(4)} &= \lambda\_0\cos^2\theta + \eta\_{01}\sin^2\theta, \\ \eta\_{01}^{(4)} &= \eta\_{01}\cos^2\theta + \lambda\_0\sin^2\theta; \\ \lambda\_{01}^{(4)} &= \frac{1}{2}\left(\eta\_{01} - \lambda\_0\right)\sin 2\theta, \\ \lambda\_1^{(4)} &= \frac{1}{2}\left[\left(3\lambda\_1 - \lambda\right)\cos^2\theta - \left(3\lambda\_1 - \eta\_1\right)\sin^2\theta\right]\sin 2\theta; \\ \lambda\_2^{(4)} &= \lambda\_1\cos^22\theta - \frac{1}{4}\left(2\lambda\_1 - \eta\_1 - \lambda\right)\sin^22\theta; \\ \lambda\_3^{(4)} &= \frac{1}{2}\left[\left(\eta\_1 - 3\lambda\_1\right)\cos^2\theta - \left(\lambda - 3\lambda\_1\right)\sin^2\theta\right]\sin 2\theta. \end{aligned} \tag{11}$$

Finally, after spontaneous breaking of the electroweak and **Z**<sup>2</sup> symmetries, the part of the Standard Model Lagrangian that is relevant to dark matter annihilation writes, in terms of the physical fields *h* and *S*1, as follows:

$$\mathcal{U}\_{\rm SM} = \sum\_{f} \left(\lambda\_{hf} h \tilde{f} f + \lambda\_{1f} \mathcal{S}\_{1} \tilde{f} f\right) + \lambda\_{hw}^{(3)} h \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} + \lambda\_{1w}^{(3)} \mathcal{S}\_{1} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu}$$

$$\begin{split} &+ \lambda\_{hz}^{(3)} h \left(\mathcal{Z}\_{\mu}\right)^{2} + \lambda\_{1z}^{(3)} \mathcal{S}\_{1} \left(\mathcal{Z}\_{\mu}\right)^{2} + \lambda\_{hw}^{(4)} h^{2} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} + \lambda\_{1w}^{(4)} \mathcal{S}\_{1}^{2} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} \\ &+ \lambda\_{h1w} h \mathcal{S}\_{1} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} + \lambda\_{h2}^{(4)} h^{2} \left(\mathcal{Z}\_{\mu}\right)^{2} + \lambda\_{1z}^{(4)} \mathcal{S}\_{1}^{2} \left(\mathcal{Z}\_{\mu}\right)^{2} + \lambda\_{h1z} h \mathcal{S}\_{1} \left(\mathcal{Z}\_{\mu}\right)^{2}. \end{split}$$

The quantities *mf* , *mw* and *mz* are the masses of the fermion *f* , the *W* and the *Z* gauge bosons respectively, and the above coupling constants are given by the following relations:

$$\begin{aligned} \lambda\_{hf} &= -\frac{m\_f}{v}\cos\theta; & \lambda\_{1f} &= \frac{m\_f}{v}\sin\theta; \\ \lambda\_{hw}^{(3)} &= 2\frac{m\_w^2}{v}\cos\theta; & \lambda\_{1w}^{(3)} &= -2\frac{m\_w^2}{v}\sin\theta; \\ \lambda\_{hz}^{(3)} &= \frac{m\_z^2}{v}\cos\theta; & \lambda\_{1z}^{(3)} &= -\frac{m\_z^2}{v}\sin\theta; \\ \lambda\_{hw}^{(4)} &= \frac{m\_w^2}{v^2}\cos^2\theta; & \lambda\_{1w}^{(4)} &= \frac{m\_w^2}{v^2}\sin^2\theta; & \lambda\_{h1w} &= -\frac{m\_w^2}{v^2}\sin2\theta; \\ \lambda\_{hz}^{(4)} &= \frac{m\_z^2}{2v^2}\cos^2\theta; & \lambda\_{1z}^{(4)} &= \frac{m\_z^2}{2v^2}\sin^2\theta; & \lambda\_{h1z} &= -\frac{m\_z^2}{2v^2}\sin2\theta. \end{aligned} \tag{13}$$

#### **2.1. Dark Matter relic density constraint**

4 Open Questions in Cosmology

following relations:

*λ*(3)

*η*(3)

*η*(3)

*λ*(3)

*λ*(3)

*<sup>U</sup>*cub <sup>=</sup> *<sup>λ</sup>*(3) 0 <sup>2</sup> *<sup>S</sup>*<sup>2</sup>

<sup>0</sup> <sup>=</sup> *<sup>λ</sup>*0*<sup>v</sup>* cos *<sup>θ</sup>* <sup>+</sup> *<sup>η</sup>*01*v*<sup>1</sup> sin *<sup>θ</sup>*,

<sup>01</sup> <sup>=</sup> *<sup>η</sup>*01*v*<sup>1</sup> cos *<sup>θ</sup>* <sup>−</sup> *<sup>λ</sup>*0*<sup>v</sup>* sin *<sup>θ</sup>*;

3 2

2

1

with the physical quartic coupling constants written as:

*<sup>λ</sup>*(4) <sup>=</sup> *<sup>λ</sup>* cos<sup>4</sup> *<sup>θ</sup>* <sup>+</sup>

<sup>1</sup> <sup>=</sup> *<sup>η</sup>*<sup>1</sup> cos<sup>4</sup> *<sup>θ</sup>* <sup>+</sup>

*<sup>λ</sup>*(3) <sup>=</sup> *<sup>λ</sup><sup>v</sup>* cos<sup>3</sup> *<sup>θ</sup>* <sup>+</sup>

<sup>1</sup> <sup>=</sup> *<sup>η</sup>*1*v*<sup>1</sup> cos3 *<sup>θ</sup>* <sup>−</sup> <sup>3</sup>

<sup>1</sup> <sup>=</sup> *<sup>λ</sup>*1*v*<sup>1</sup> cos3 *<sup>θ</sup>* <sup>+</sup>

<sup>2</sup> <sup>=</sup> *<sup>λ</sup>*1*<sup>v</sup>* cos<sup>3</sup> *<sup>θ</sup>* <sup>−</sup> <sup>1</sup>

*<sup>U</sup>*quar <sup>=</sup> *<sup>η</sup>*<sup>0</sup>

<sup>24</sup> *<sup>S</sup>*<sup>4</sup> 0 + *<sup>λ</sup>*(4)

+ *λ*(4) 1 <sup>6</sup> *<sup>h</sup>*3*S*<sup>1</sup> <sup>+</sup>

*η*(4)

*λ*(4)

*η*(4)

*λ*(4) <sup>01</sup> <sup>=</sup> <sup>1</sup>

*λ*(4) <sup>1</sup> <sup>=</sup> <sup>1</sup> 2 

*λ*(4)

*λ*(4) <sup>3</sup> <sup>=</sup> <sup>1</sup> 2 

<sup>0</sup>*<sup>h</sup>* <sup>+</sup> *<sup>η</sup>*(3) 01 <sup>2</sup> *<sup>S</sup>*<sup>2</sup> <sup>0</sup>*S*<sup>1</sup> <sup>+</sup>

as follows:

Written now directly in terms of the physical fields, the cubic interaction terms are expressed

where the cubic physical coupling constants are related to the original parameters via the

*<sup>λ</sup>*<sup>1</sup> sin 2*<sup>θ</sup>* (*v*<sup>1</sup> cos *<sup>θ</sup>* + *<sup>v</sup>* sin *<sup>θ</sup>*) + *<sup>η</sup>*1*v*<sup>1</sup> sin<sup>3</sup> *<sup>θ</sup>*;

In the same way, in terms of the physical fields too, the quartic interactions are given by:

*λ*(4) 0 <sup>4</sup> *<sup>S</sup>*<sup>2</sup>

*λ*(4) 3 <sup>6</sup> *hS*<sup>3</sup>

*<sup>λ</sup>*<sup>1</sup> sin<sup>2</sup> <sup>2</sup>*<sup>θ</sup>* + *<sup>η</sup>*<sup>1</sup> sin4 *<sup>θ</sup>*,

*<sup>λ</sup>*<sup>1</sup> sin<sup>2</sup> <sup>2</sup>*<sup>θ</sup>* + *<sup>λ</sup>* sin<sup>4</sup> *<sup>θ</sup>*;

(3*λ*<sup>1</sup> − *<sup>λ</sup>*) cos2 *<sup>θ</sup>* − (3*λ*<sup>1</sup> − *<sup>η</sup>*1) sin2 *<sup>θ</sup>*

(*η*<sup>1</sup> − <sup>3</sup>*λ*1) cos2 *<sup>θ</sup>* − (*<sup>λ</sup>* − <sup>3</sup>*λ*1) sin2 *<sup>θ</sup>*

<sup>4</sup> (2*λ*<sup>1</sup> <sup>−</sup> *<sup>η</sup>*<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*) sin<sup>2</sup> <sup>2</sup>*θ*;

<sup>24</sup> *<sup>h</sup>*<sup>4</sup> <sup>+</sup> *<sup>η</sup>*(4) 1 <sup>24</sup> *<sup>S</sup>*<sup>4</sup> 1 +

> *λ*(4) 2 <sup>4</sup> *<sup>h</sup>*2*S*<sup>2</sup> 1 +

> > 3 2

> > > 3 2

<sup>0</sup> <sup>=</sup> *<sup>λ</sup>*<sup>0</sup> cos2 *<sup>θ</sup>* <sup>+</sup> *<sup>η</sup>*<sup>01</sup> sin<sup>2</sup> *<sup>θ</sup>*,

<sup>01</sup> <sup>=</sup> *<sup>η</sup>*<sup>01</sup> cos<sup>2</sup> *<sup>θ</sup>* <sup>+</sup> *<sup>λ</sup>*<sup>0</sup> sin<sup>2</sup> *<sup>θ</sup>*;

<sup>2</sup> <sup>=</sup> *<sup>λ</sup>*<sup>1</sup> cos2 <sup>2</sup>*<sup>θ</sup>* <sup>−</sup> <sup>1</sup>

<sup>2</sup> (*η*<sup>01</sup> <sup>−</sup> *<sup>λ</sup>*0) sin 2*θ*,

<sup>6</sup> *<sup>h</sup>*<sup>3</sup> <sup>+</sup> *<sup>η</sup>*(3) 1 <sup>6</sup> *<sup>S</sup>*<sup>3</sup> 1 + *λ*(3) 1 <sup>2</sup> *<sup>h</sup>*2*S*<sup>1</sup> <sup>+</sup>

*<sup>λ</sup>*<sup>1</sup> sin 2*<sup>θ</sup>* (*<sup>v</sup>* cos *<sup>θ</sup>* − *<sup>v</sup>*<sup>1</sup> sin *<sup>θ</sup>*) − *<sup>λ</sup><sup>v</sup>* sin<sup>3</sup> *<sup>θ</sup>*; (9)

<sup>2</sup> sin 2*<sup>θ</sup>* [(2*λ*<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*) *<sup>v</sup>* cos *<sup>θ</sup>* <sup>−</sup> (2*λ*<sup>1</sup> <sup>−</sup> *<sup>η</sup>*1) *<sup>v</sup>*<sup>1</sup> sin *<sup>θ</sup>*] <sup>−</sup> *<sup>λ</sup>*1*<sup>v</sup>* sin<sup>3</sup> *<sup>θ</sup>*;

<sup>2</sup> sin 2*<sup>θ</sup>* [(2*λ*<sup>1</sup> <sup>−</sup> *<sup>η</sup>*1) *<sup>v</sup>*<sup>1</sup> cos *<sup>θ</sup>* <sup>+</sup> (2*λ*<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*) *<sup>v</sup>* sin *<sup>θ</sup>*] <sup>+</sup> *<sup>λ</sup>*1*v*<sup>1</sup> sin<sup>3</sup> *<sup>θ</sup>*.

<sup>0</sup>*h*<sup>2</sup> <sup>+</sup> *<sup>η</sup>*(4) 01 <sup>4</sup> *<sup>S</sup>*<sup>2</sup> 0*S*2 1 + *λ*(3) 2 <sup>2</sup> *hS*<sup>2</sup>

*λ*(4) 01 <sup>2</sup> *<sup>S</sup>*<sup>2</sup> <sup>0</sup>*hS*<sup>1</sup>

<sup>1</sup>, (10)

 sin 2*θ*;

sin 2*θ*. (11)

<sup>1</sup>, (8)

*<sup>λ</sup>*(3)

The field *S*<sup>0</sup> is odd under the unbroken **Z**<sup>2</sup> symmetry, and so is a stable relic and can therefore constitute the dark matter of the Universe. Its relic density can be obtained using the standard approximate solutions to the Boltzmann equations [21]:

$$
\Omega\_D \bar{h}^2 = \frac{1.07 \times 10^9 \text{x}\_f}{\sqrt{\mathcal{g}\_\*} M\_{\text{Pl}} \left< v\_{12} \sigma\_{\text{ann}} \right> \text{GeV}'} \tag{14}
$$

where ¯ *<sup>h</sup>* is the normalized Hubble constant,*M*Pl = 1.22 × 1019GeV is the Planck mass, *<sup>g</sup>*<sup>∗</sup> the number of relativistic degrees of freedom at the freeze-out temperature *Tf* , and *xf* = *<sup>m</sup>*0/*Tf* which, for *<sup>m</sup>*<sup>0</sup> in the range 1 − 20GeV, is in the range 18.2 − 19.4. The quantity �*υ*12*σ*ann� is the thermally averaged annihilation cross section of *S*<sup>0</sup> into fermion pairs *f* ¯ *f* for *mf* < *m*0/2, and into *S*1*S*<sup>1</sup> for *m*<sup>1</sup> < *m*0/2. The annihilation cross-section into fermions proceeds via s-channel exchange of *h* and *S*<sup>1</sup> and is given by:

$$\begin{split} \sigma\_{12}\sigma\_{\text{S}\_{0}\text{S}\_{0}\to ff} &= \frac{\sqrt{\left(m\_{0}^{2}-m\_{f}^{2}\right)^{3}}}{4\pi m\_{0}^{3}}\Theta\left(m\_{0}-m\_{f}\right) \left[\frac{\left(\lambda\_{0}^{(3)}\lambda\_{hf}\right)^{2}}{\left(4m\_{0}^{2}-m\_{h}^{2}\right)^{2}+\epsilon\_{h}^{2}} + \frac{\left(\eta\_{01}^{(3)}\lambda\_{1f}\right)^{2}}{\left(4m\_{0}^{2}-m\_{1}^{2}\right)^{2}+\epsilon\_{1}^{2}}\right] \\ &+ \frac{2\lambda\_{0}^{(3)}\eta\_{01}^{(3)}\lambda\_{hf}\lambda\_{1f}\left(4m\_{0}^{2}-m\_{h}^{2}\right)\left(4m\_{0}^{2}-m\_{1}^{2}\right)}{\left[\left(4m\_{0}^{2}-m\_{h}^{2}\right)^{2}+\epsilon\_{h}^{2}\right]\left[\left(4m\_{0}^{2}-m\_{1}^{2}\right)^{2}+\epsilon\_{1}^{2}\right]}\right) . \end{split} \tag{15}$$

theory, the requirement *<sup>η</sup>*(4)

<sup>0</sup> .

*<sup>σ</sup>*det <sup>≡</sup> *<sup>σ</sup>S*0*N*→*S*0*<sup>N</sup>* <sup>=</sup> *<sup>m</sup>*<sup>2</sup>

the behavior of the physical mutual coupling constant *<sup>η</sup>*(4)

parameters whereas perturbativity is lost mainly for larger values of *m*1.

*N* � *mN* − <sup>7</sup>

<sup>4</sup>*<sup>π</sup>* (*mN* + *<sup>m</sup>*0)

rather small values for *<sup>λ</sup>*(4)

**2.2. Direct detection**

given by the relation [17]:

**3. Upsilon decays**

process is given by the relation:

the branching ratio Br(*µ*) <sup>≡</sup> Br �

In this expression, *xn* ≡

Br(Υ*nS* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> *<sup>S</sup>*1) <sup>=</sup> *GFm*<sup>2</sup>

� 1 − *m*<sup>2</sup>

1GeV.

<sup>01</sup> <sup>&</sup>lt; 1 will also be imposed throughout, as well as a choice of

<sup>01</sup> between *S*<sup>0</sup> and *S*<sup>1</sup> as a function of

Light Cold Dark Matter in a Two-Singlet Model

http://dx.doi.org/10.5772/52243

277

Studying the effects of the relic-density constraint for large ranges of the parameters through

the DM mass *m*<sup>0</sup> shows that, apart from forbidden regions and others where perturbativity is lost, viable solutions in the small-moderate mass ranges of the DM sector exist for most values of the parameters [17]. Forbidden regions are found for most of the ranges of the

On the other hand, experiments like CDMS II [15], XENON 10/100 [14, 25], DAMA/LIBRA [26] and CoGeNT [3] search directly for a dark matter signal, which would typically come from the elastic scattering of a dark matter WIMP off a non-relativistic nucleon target. However, throughout the years, such experiments have not yet detected an unambiguous signal, but rather yielded increasingly stringent exclusion bounds on the dark matter – nucleon elastic-scattering total cross-section *σ*det in terms of the DM mass *m*0. Any viable theoretical dark-matter model has to satisfy these bounds. In the 2SSM, *σ*det is found to be

> <sup>9</sup>*mB* �2

in which *mN* is the nucleon mass and *mB* the baryon mass in the chiral limit [13, 27, 28]. This relation was compared against the experimental bounds from CDMSII and XENON100. We found that strong constraints were imposed on *m*<sup>0</sup> in the range between 10 to 20GeV. We found also that for small values of *m*1, very light dark matter is viable, with *m*<sup>0</sup> as small as

We now further the analysis of the two-singlet model and start by looking at the constraints on the parameter space of the model coming from the decay of the meson Υ in the state *nS* (*<sup>n</sup>* = 1, 3) into one photon *<sup>γ</sup>* and one particle *<sup>S</sup>*1. For *<sup>m</sup>*<sup>1</sup> 8GeV, the branching ratio for this

> *xn* � <sup>1</sup> <sup>−</sup> <sup>4</sup>*α<sup>s</sup>*

> > �

coupling constant, *<sup>α</sup><sup>s</sup>* <sup>=</sup> 0.184 the QCD coupling constant at the scale *<sup>m</sup>*Υ*nS* , the quantity *GF*

<sup>Υ</sup>1(3)*<sup>S</sup>* <sup>→</sup> *<sup>µ</sup>*+*µ*<sup>−</sup>

<sup>3</sup>*<sup>π</sup> <sup>f</sup>*(*xn*)

�

Br(*µ*) <sup>Θ</sup> �

with *<sup>m</sup>*Υ1(3)*<sup>S</sup>* <sup>=</sup> 9.46(10.355)GeV the mass of <sup>Υ</sup>1(3)*S*,

<sup>=</sup> 2.48(2.18) <sup>×</sup> <sup>10</sup>−<sup>2</sup> [29], *<sup>α</sup>* is the QED

*<sup>m</sup>*Υ*nS* <sup>−</sup> *<sup>m</sup>*<sup>1</sup>

�

. (18)

*<sup>b</sup>* sin<sup>2</sup> *<sup>θ</sup>* √2*πα*

1/*m*<sup>2</sup> <sup>Υ</sup>*ns*� <sup>2</sup> *v*<sup>2</sup>

 *λ*(3) <sup>0</sup> cos *θ m*2 *h*

<sup>−</sup> *<sup>η</sup>*(3) <sup>01</sup> sin *θ m*2 1

 

2

, (17)

and the annihilation into *S*<sup>1</sup> pairs given by:

*<sup>v</sup>*12*σS*0*S*0→*S*1*S*<sup>1</sup> <sup>=</sup> � *m*2 <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1 64*πm*<sup>3</sup> 0 <sup>Θ</sup>(*m*<sup>0</sup> − *<sup>m</sup>*1) � *η*(4) <sup>01</sup> �<sup>2</sup> + <sup>4</sup>*η*(4) <sup>01</sup> � *η*(3) <sup>01</sup> �<sup>2</sup> *m*2 <sup>1</sup> <sup>−</sup> <sup>2</sup>*m*<sup>2</sup> 0 + <sup>2</sup>*η*(4) <sup>01</sup> *<sup>η</sup>*(3) <sup>01</sup> *<sup>η</sup>*(3) 1 4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1 + <sup>2</sup>*η*(4) <sup>01</sup> *<sup>λ</sup>*(3) <sup>0</sup> *<sup>λ</sup>*(3) <sup>2</sup> (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *h*) (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *<sup>h</sup>*)<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> *h* + 4 � *η*(3) <sup>01</sup> �<sup>4</sup> (*m*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>2</sup>*m*<sup>2</sup> 0)<sup>2</sup> <sup>+</sup> 4 � *η*(3) <sup>01</sup> �<sup>3</sup> *η*(3) 1 (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1)(*m*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>2</sup>*m*<sup>2</sup> 0) + 4 � *η*(3) <sup>01</sup> �<sup>2</sup> *λ*(3) <sup>0</sup> *<sup>λ</sup>*(3) <sup>2</sup> (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *h*) � (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *<sup>h</sup>*)<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> *h* � (*m*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>2</sup>*m*<sup>2</sup> 0) + � *η*(3) <sup>01</sup> �<sup>2</sup> � *η*(3) 1 �2 (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1)2 + � *λ*(3) 0 �<sup>2</sup> � *λ*(3) 2 �2 (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *<sup>h</sup>*)<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> *h* + <sup>2</sup>*η*(3) <sup>01</sup> *<sup>η</sup>*(3) <sup>1</sup> *<sup>λ</sup>*(3) <sup>0</sup> *<sup>λ</sup>*(3) <sup>2</sup> (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *h*) � (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *<sup>h</sup>*)<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> *h* � (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1) . (16)

Solving (14) with the current value for the dark matter relic density Ω*<sup>D</sup>* ¯ *h*<sup>2</sup> = 0.1123 ± 0.0035 [22] translates into a relation between the parameters of a given theory entering the calculated expression of �*v*12*σ*ann�, hence imposing a constraint on these parameters which will limit the intervals of possible dark matter masses. This constraint is exploited to examine aspects of the theory like perturbativity, while at the same time reducing the number of parameters by one.

Indeed, the model starts with eight parameters. The spontaneous breaking of the electroweak and **Z**<sup>2</sup> symmetries introduces the two vacuum expectation values *v* and *v*<sup>1</sup> respectively, which means we are left with six. Four of the parameters are the three physical masses *m*<sup>0</sup> (dark-matter singlet *S*0), *m*<sup>1</sup> (the second singlet *S*1) and *mh* (Higgs *h*), plus the mixing angle *θ* between *<sup>h</sup>* and *<sup>S</sup>*1. We will fix the Higgs mass to *mh* = 125GeV [23, 24], except in the section discussing Higgs decays where we let *mh* vary in the interval 100GeV − 200GeV. We will take both *<sup>m</sup>*<sup>0</sup> and *<sup>m</sup>*<sup>1</sup> in the interval 0.1GeV − 10GeV. For the purpose of our discussions, it is sufficient to let *θ* vary in the interval 1<sup>o</sup> − 40o. The last parameters are the two physical mutual coupling constants *<sup>λ</sup>*(4) <sup>0</sup> (dark matter – Higgs) and *<sup>η</sup>*(4) <sup>01</sup> (dark matter – *S*<sup>1</sup> particle). But *η*(4) <sup>01</sup> is not free as it is the smallest real and positive solution to the dark-matter relic density constraint (14), which will be implemented systematically throughout [17]. Thus we are left with four parameters, namely, *<sup>m</sup>*0, *<sup>m</sup>*1, *<sup>θ</sup>* and *<sup>λ</sup>*(4) <sup>0</sup> . To ensure applicability of perturbation theory, the requirement *<sup>η</sup>*(4) <sup>01</sup> <sup>&</sup>lt; 1 will also be imposed throughout, as well as a choice of rather small values for *<sup>λ</sup>*(4) <sup>0</sup> .

Studying the effects of the relic-density constraint for large ranges of the parameters through the behavior of the physical mutual coupling constant *<sup>η</sup>*(4) <sup>01</sup> between *S*<sup>0</sup> and *S*<sup>1</sup> as a function of the DM mass *m*<sup>0</sup> shows that, apart from forbidden regions and others where perturbativity is lost, viable solutions in the small-moderate mass ranges of the DM sector exist for most values of the parameters [17]. Forbidden regions are found for most of the ranges of the parameters whereas perturbativity is lost mainly for larger values of *m*1.

#### **2.2. Direct detection**

6 Open Questions in Cosmology

*v*12*σS*0*S*0→*<sup>f</sup>* ¯

*<sup>v</sup>*12*σS*0*S*0→*S*1*S*<sup>1</sup> <sup>=</sup>

one.

*η*(4)

*f* =

�� *m*2 <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *f* �3

+

4*πm*<sup>3</sup> 0

<sup>2</sup>*λ*(3) <sup>0</sup> *<sup>η</sup>*(3)

�� 4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *h* �<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> *h* � �� 4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1 �<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> 1 �

and the annihilation into *S*<sup>1</sup> pairs given by:

� *m*2 <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1

+ <sup>2</sup>*η*(4) <sup>01</sup> *<sup>λ</sup>*(3) <sup>0</sup> *<sup>λ</sup>*(3) <sup>2</sup> (4*m*<sup>2</sup>

+ 4 � *η*(3) 01 �2 *λ*(3) <sup>0</sup> *<sup>λ</sup>*(3) <sup>2</sup> (4*m*<sup>2</sup>

+ � *λ*(3) 0 �<sup>2</sup> � *λ*(3) 2 �2

mutual coupling constants *<sup>λ</sup>*(4)

� (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

(4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

with four parameters, namely, *<sup>m</sup>*0, *<sup>m</sup>*1, *<sup>θ</sup>* and *<sup>λ</sup>*(4)

64*πm*<sup>3</sup> 0

> (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

Θ �

<sup>01</sup> *λh f λ*<sup>1</sup> *<sup>f</sup>*

*<sup>m</sup>*<sup>0</sup> − *mf*

� 4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *h* � �4*m*<sup>2</sup>

<sup>Θ</sup>(*m*<sup>0</sup> − *<sup>m</sup>*1)

� 

 � *η*(4) 01 �2 + <sup>4</sup>*η*(4) 01 � *η*(3) 01 �2

<sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *h*) + 4 � *η*(3) 01 �4

<sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *h*)

<sup>2</sup>*η*(3) <sup>01</sup> *<sup>η</sup>*(3) <sup>1</sup> *<sup>λ</sup>*(3) <sup>0</sup> *<sup>λ</sup>*(3) <sup>2</sup> (4*m*<sup>2</sup>

� (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

[22] translates into a relation between the parameters of a given theory entering the calculated expression of �*v*12*σ*ann�, hence imposing a constraint on these parameters which will limit the intervals of possible dark matter masses. This constraint is exploited to examine aspects of the theory like perturbativity, while at the same time reducing the number of parameters by

Indeed, the model starts with eight parameters. The spontaneous breaking of the electroweak and **Z**<sup>2</sup> symmetries introduces the two vacuum expectation values *v* and *v*<sup>1</sup> respectively, which means we are left with six. Four of the parameters are the three physical masses *m*<sup>0</sup> (dark-matter singlet *S*0), *m*<sup>1</sup> (the second singlet *S*1) and *mh* (Higgs *h*), plus the mixing angle *θ* between *<sup>h</sup>* and *<sup>S</sup>*1. We will fix the Higgs mass to *mh* = 125GeV [23, 24], except in the section discussing Higgs decays where we let *mh* vary in the interval 100GeV − 200GeV. We will take both *<sup>m</sup>*<sup>0</sup> and *<sup>m</sup>*<sup>1</sup> in the interval 0.1GeV − 10GeV. For the purpose of our discussions, it is sufficient to let *θ* vary in the interval 1<sup>o</sup> − 40o. The last parameters are the two physical

<sup>0</sup> (dark matter – Higgs) and *<sup>η</sup>*(4)

<sup>01</sup> is not free as it is the smallest real and positive solution to the dark-matter relic density constraint (14), which will be implemented systematically throughout [17]. Thus we are left

(*m*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>2</sup>*m*<sup>2</sup>

*<sup>h</sup>*)<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> *h*

*<sup>h</sup>*)<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> *h* � (*m*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>2</sup>*m*<sup>2</sup> 0) + � *η*(3) 01 �<sup>2</sup> � *η*(3) 1 �2

*<sup>h</sup>*)<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> *h* +

Solving (14) with the current value for the dark matter relic density Ω*<sup>D</sup>* ¯

� 4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *h* �<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> *h* +

� *λ*(3) <sup>0</sup> *λh f* �2

<sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1 �

> *m*2 <sup>1</sup> <sup>−</sup> <sup>2</sup>*m*<sup>2</sup> 0

> > 0)<sup>2</sup> <sup>+</sup>

(4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1)2

*<sup>h</sup>*)<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> *h* � (4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1)

� *η*(3) <sup>01</sup> *λ*<sup>1</sup> *<sup>f</sup>* �2

, (15)

4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1

1)(*m*<sup>2</sup>

 

<sup>01</sup> (dark matter – *S*<sup>1</sup> particle). But

<sup>0</sup> . To ensure applicability of perturbation

<sup>1</sup> <sup>−</sup> <sup>2</sup>*m*<sup>2</sup> 0)

. (16)

*h*<sup>2</sup> = 0.1123 ± 0.0035

� 4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1 �<sup>2</sup> <sup>+</sup> *<sup>ǫ</sup>*<sup>2</sup> 1

+ <sup>2</sup>*η*(4) <sup>01</sup> *<sup>η</sup>*(3) <sup>01</sup> *<sup>η</sup>*(3) 1

4 � *η*(3) 01 �3 *η*(3) 1

(4*m*<sup>2</sup> <sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

<sup>0</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *h*) On the other hand, experiments like CDMS II [15], XENON 10/100 [14, 25], DAMA/LIBRA [26] and CoGeNT [3] search directly for a dark matter signal, which would typically come from the elastic scattering of a dark matter WIMP off a non-relativistic nucleon target. However, throughout the years, such experiments have not yet detected an unambiguous signal, but rather yielded increasingly stringent exclusion bounds on the dark matter – nucleon elastic-scattering total cross-section *σ*det in terms of the DM mass *m*0. Any viable theoretical dark-matter model has to satisfy these bounds. In the 2SSM, *σ*det is found to be given by the relation [17]:

$$\sigma\_{\text{det}} \equiv \sigma\_{\text{S\!}N \to \text{S\!}nN} = \frac{m\_N^2 \left(m\_N - \frac{7}{5}m\_B\right)^2}{4\pi \left(m\_N + m\_0\right)^2 v^2} \left[\frac{\lambda\_0^{(3)} \cos\theta}{m\_h^2} - \frac{\eta\_{01}^{(3)} \sin\theta}{m\_1^2}\right]^2,\tag{17}$$

in which *mN* is the nucleon mass and *mB* the baryon mass in the chiral limit [13, 27, 28]. This relation was compared against the experimental bounds from CDMSII and XENON100. We found that strong constraints were imposed on *m*<sup>0</sup> in the range between 10 to 20GeV. We found also that for small values of *m*1, very light dark matter is viable, with *m*<sup>0</sup> as small as 1GeV.

## **3. Upsilon decays**

We now further the analysis of the two-singlet model and start by looking at the constraints on the parameter space of the model coming from the decay of the meson Υ in the state *nS* (*<sup>n</sup>* = 1, 3) into one photon *<sup>γ</sup>* and one particle *<sup>S</sup>*1. For *<sup>m</sup>*<sup>1</sup> 8GeV, the branching ratio for this process is given by the relation:

$$\operatorname{Br}\left(\mathbf{Y}\_{n\mathbf{S}}\rightarrow\gamma+\mathbf{S}\_{1}\right)=\frac{G\_{F}m\_{b}^{2}\sin^{2}\theta}{\sqrt{2}\pi\alpha}\,\operatorname{x}\_{n}\left(1-\frac{4\alpha\_{\mathrm{s}}}{3\pi}f(\mathbf{x}\_{n})\right)\operatorname{Br}^{(\mu)}\,\Theta\left(m\_{\mathbf{Y}\_{n\mathbf{S}}}-m\_{1}\right).\tag{18}$$

In this expression, *xn* ≡ � 1 − *m*<sup>2</sup> 1/*m*<sup>2</sup> <sup>Υ</sup>*ns*� with *<sup>m</sup>*Υ1(3)*<sup>S</sup>* <sup>=</sup> 9.46(10.355)GeV the mass of <sup>Υ</sup>1(3)*S*, the branching ratio Br(*µ*) <sup>≡</sup> Br � <sup>Υ</sup>1(3)*<sup>S</sup>* <sup>→</sup> *<sup>µ</sup>*+*µ*<sup>−</sup> � <sup>=</sup> 2.48(2.18) <sup>×</sup> <sup>10</sup>−<sup>2</sup> [29], *<sup>α</sup>* is the QED coupling constant, *<sup>α</sup><sup>s</sup>* <sup>=</sup> 0.184 the QCD coupling constant at the scale *<sup>m</sup>*Υ*nS* , the quantity *GF*

is the Fermi coupling constant and *mb* the *<sup>b</sup>* quark mass [22]. The function *<sup>f</sup>*(*x*) incorporates the effect of QCD radiative corrections given in [30].

But the above expression is not sufficient because a rough estimate of the lifetime of *S*<sup>1</sup> indicates that this latter is likely to decay inside a typical particle detector, which means we ought to take into account its most dominant decay products. We first have a process by which *S*<sup>1</sup> decays into a pair of pions, with a decay rate given by:

$$\Gamma\left(S\_1 \to \pi\pi\right) \simeq \frac{G\_F m\_1}{4\sqrt{2}\pi} \sin^2\theta \left[\frac{m\_1^2}{27} \left(1 + \frac{11m\_\pi^2}{2m\_1^2}\right)^2\right.$$

$$\times \left(1 - \frac{4m\_\pi^2}{m\_1^2}\right)^{\frac{1}{2}} \Theta[\left(m\_1 - 2m\_\pi\right)\left(2m\_K - m\_1\right)]$$

$$+ 3\left(M\_\mu^2 + M\_d^2\right) \left(1 - \frac{4m\_\pi^2}{m\_1^2}\right)^{\frac{3}{2}} \Theta\left(m\_1 - 2m\_K\right)\right].\tag{19}$$

Here, *<sup>α</sup>*′

a decay rate:

10�<sup>12</sup>

10�<sup>10</sup>

Br�1 �Γ� *s*

X

10�<sup>8</sup>

10�<sup>6</sup>

roughly 1GeV and 3GeV.

Γ  *<sup>S</sup>*<sup>1</sup> <sup>→</sup> <sup>ℓ</sup>+ℓ<sup>−</sup>

kinematically allowed final state, will be:

corresponding experimental upper bounds are shown.

<sup>Γ</sup> (*S*<sup>1</sup> → *<sup>S</sup>*0*S*0) =

*<sup>s</sup>* <sup>=</sup> 0.47 is the QCD coupling constant at the spectator-quark-model scale, between

<sup>1</sup> <sup>−</sup> <sup>4</sup>*m*<sup>2</sup> ℓ *m*2 1

3 2

<sup>Θ</sup> (*m*<sup>1</sup> − <sup>2</sup>*m*ℓ), (23)

Light Cold Dark Matter in a Two-Singlet Model

http://dx.doi.org/10.5772/52243

279

<sup>Θ</sup> (*m*<sup>1</sup> − <sup>2</sup>*m*0). (24)

Exp KK

<sup>Υ</sup>1*<sup>S</sup>* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> *<sup>π</sup>*+*π*<sup>−</sup>

<

Exp ΠΠ

Exp ΤΤ

X � KK

X � ΠΠ

X � ΤΤ

We then have the decay of *S*<sup>1</sup> into leptons, the corresponding rate given by:

ℓ*m*<sup>1</sup>

 *η*(3) <sup>01</sup> <sup>2</sup>

32*πm*<sup>1</sup>

<sup>4</sup> � 0.04, *m*<sup>0</sup> � 2GeV, Θ � 15°

1 2 3 4 5 6 7 8

*m*<sup>1</sup> GeV

Br(Υ1*<sup>S</sup>* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> *ττ*) <sup>&</sup>lt; <sup>5</sup> <sup>×</sup> <sup>10</sup>−<sup>5</sup> for 3.5GeV <sup>&</sup>lt; *<sup>m</sup>*<sup>1</sup> <sup>&</sup>lt; 9.2GeV [36]; (ii) Br

**Figure 1.** Typical branching ratios of Υ1*<sup>S</sup>* decaying into *τ*'s, charged pions and charged kaons as functions of *m*1. The

The best available experimental upper bounds on 1*S*–state branching ratios are: (i)

sin2 *θ* 

where *m*ℓ is the lepton mass. Finally, *S*<sup>1</sup> can decay into a pair of dark matter particles, with

The branching ratio for Υ*nS* decaying via *S*<sup>1</sup> into a photon plus *X*, where *X* represents any

<sup>1</sup> <sup>−</sup> <sup>4</sup>*m*<sup>2</sup> 0 *m*2 1

Br(Υ*nS* → *<sup>γ</sup>* + *<sup>X</sup>*) = Br(Υ*nS* → *<sup>γ</sup>* + *<sup>S</sup>*1) × Br(*S*<sup>1</sup> → *<sup>X</sup>*). (25)

<sup>=</sup> *GFm*<sup>2</sup>

4 √2*π*

In particular, *<sup>X</sup>* ≡ *<sup>S</sup>*0*S*<sup>0</sup> corresponds to a decay into invisible particles.

Λ0

In the above decay rate, *mπ*(*K*) is the pion (kaon) mass. Also, chiral perturbation theory is used below the kaon pair production threshold [31, 32], and the spectator quark model above up to roughly 3GeV, with the dressed *<sup>u</sup>* and *<sup>d</sup>* quark masses *Mu* = *Md* ≃ 0.05GeV. Note that this rate includes all pions, charged and neutral. Above the 2*mK* threshold, both pairs of kaons and *η* particles are produced. The decay rate for *K* production is:

$$\Gamma\left(\mathcal{S}\_1 \to KK\right) \simeq \frac{9}{13} \frac{3G\_F M\_s^2 m\_1}{4\sqrt{2}\pi} \sin^2\theta \left(1 - \frac{4m\_K^2}{m\_1^2}\right)^{\frac{3}{2}} \Theta\left(m\_1 - 2m\_K\right). \tag{20}$$

In the above rate, *Ms* ≃ 0.45GeV is the *s* quark-mass in the spectator quark model [33, 34]. For *η* production, replace *mK* by *m<sup>η</sup>* and <sup>9</sup> <sup>13</sup> by <sup>4</sup> 13 .

The particle *S*<sup>1</sup> can also decay into *c* and *b* quarks (mainly *c*). Including the radiative QCD corrections, the corresponding decay rates are given by:

$$\Gamma(\mathbb{S}\_1 \to q\overline{q}) \simeq \frac{3G\_F \overline{m}\_q^2 m\_1}{4\sqrt{2}\pi} \sin^2 \theta \left(1 - \frac{4\overline{m}\_q^2}{m\_h^2}\right)^{\frac{3}{2}} \left(1 + 5.67 \frac{\overline{\mathfrak{a}}\_s}{\pi}\right) \Theta\left(m\_1 - 2\overline{m}\_q\right). \tag{21}$$

The dressed quark mass *<sup>m</sup>*¯ *<sup>q</sup>* ≡ *mq*(*m*1) and the running strong coupling constant *<sup>α</sup>*¯*<sup>s</sup>* ≡ *<sup>α</sup>s*(*m*1) are defined at the energy scale *m*<sup>1</sup> [35]. Gluons can also be produced, with a corresponding decay rate given by the relation:

$$\Gamma\left(\mathbb{S}\_1 \to \mathrm{gg}\right) \simeq \frac{G\_F m\_1^3 \sin^2 \theta}{12\sqrt{2}\pi} \left(\frac{a\_s'}{\pi}\right)^2 \left[6 - 2\left(1 - \frac{4m\_\pi^2}{m\_1^2}\right)^{\frac{3}{2}} - \left(1 - \frac{4m\_\mathcal{K}^2}{m\_1^2}\right)^{\frac{3}{2}}\right] \Theta\left(m\_1 - 2m\_\mathcal{K}\right). \tag{22}$$

Here, *<sup>α</sup>*′ *<sup>s</sup>* <sup>=</sup> 0.47 is the QCD coupling constant at the spectator-quark-model scale, between roughly 1GeV and 3GeV.

We then have the decay of *S*<sup>1</sup> into leptons, the corresponding rate given by:

8 Open Questions in Cosmology

the effect of QCD radiative corrections given in [30].

<sup>Γ</sup> (*S*<sup>1</sup> <sup>→</sup> *ππ*) <sup>≃</sup> *GFm*<sup>1</sup>

<sup>Γ</sup> (*S*<sup>1</sup> <sup>→</sup> *KK*) <sup>≃</sup> <sup>9</sup>

For *η* production, replace *mK* by *m<sup>η</sup>* and <sup>9</sup>

<sup>Γ</sup>(*S*<sup>1</sup> <sup>→</sup> *qq*¯) <sup>≃</sup> <sup>3</sup>*GFm*¯ <sup>2</sup>

decay rate given by the relation:

<sup>Γ</sup> (*S*<sup>1</sup> <sup>→</sup> *gg*) <sup>≃</sup> *GFm*<sup>3</sup>

which *S*<sup>1</sup> decays into a pair of pions, with a decay rate given by:

4 √2*π*

× �

+3 � *M*<sup>2</sup> *<sup>u</sup>* <sup>+</sup> *<sup>M</sup>*<sup>2</sup> *d* � �

kaons and *η* particles are produced. The decay rate for *K* production is:

3*GFM*<sup>2</sup> *<sup>s</sup> m*<sup>1</sup>

4 √2*π*

13

corrections, the corresponding decay rates are given by:

4 √2*π*

<sup>1</sup> sin2 *<sup>θ</sup>* <sup>12</sup>√2*<sup>π</sup>*

*qm*<sup>1</sup>

� *<sup>α</sup>*′ *s π* �2 <sup>6</sup> <sup>−</sup> <sup>2</sup>

sin<sup>2</sup> *θ* �

sin<sup>2</sup> *θ*

<sup>1</sup> <sup>−</sup> <sup>4</sup>*m*<sup>2</sup> *π m*2 1

 *m*2 1 27

�1 2

In the above decay rate, *mπ*(*K*) is the pion (kaon) mass. Also, chiral perturbation theory is used below the kaon pair production threshold [31, 32], and the spectator quark model above up to roughly 3GeV, with the dressed *<sup>u</sup>* and *<sup>d</sup>* quark masses *Mu* = *Md* ≃ 0.05GeV. Note that this rate includes all pions, charged and neutral. Above the 2*mK* threshold, both pairs of

> sin2 *θ* �

In the above rate, *Ms* ≃ 0.45GeV is the *s* quark-mass in the spectator quark model [33, 34].

The particle *S*<sup>1</sup> can also decay into *c* and *b* quarks (mainly *c*). Including the radiative QCD

<sup>1</sup> <sup>−</sup> <sup>4</sup>*m*¯ <sup>2</sup> *q m*2 *h*

The dressed quark mass *<sup>m</sup>*¯ *<sup>q</sup>* ≡ *mq*(*m*1) and the running strong coupling constant *<sup>α</sup>*¯*<sup>s</sup>* ≡ *<sup>α</sup>s*(*m*1) are defined at the energy scale *m*<sup>1</sup> [35]. Gluons can also be produced, with a corresponding

�

<sup>1</sup> <sup>−</sup> <sup>4</sup>*m*<sup>2</sup> *π m*2 1

�3 2 − �

�3 <sup>2</sup> �

<sup>13</sup> by <sup>4</sup> 13 .

� 1 +

<sup>1</sup> <sup>−</sup> <sup>4</sup>*m*<sup>2</sup> *π m*2 1

> <sup>1</sup> <sup>−</sup> <sup>4</sup>*m*<sup>2</sup> *K m*2 1

�3 2

<sup>1</sup> <sup>+</sup> 5.67 *<sup>α</sup>*¯*<sup>s</sup>*

*π* � Θ �

<sup>1</sup> <sup>−</sup> <sup>4</sup>*m*<sup>2</sup> *K m*2 1

�3 2 

11*m*<sup>2</sup> *π* 2*m*<sup>2</sup> 1

�2

<sup>Θ</sup>[(*m*<sup>1</sup> − <sup>2</sup>*mπ*) (2*mK* − *<sup>m</sup>*1)]

<sup>Θ</sup> (*m*<sup>1</sup> − <sup>2</sup>*mK*)

<sup>Θ</sup> (*m*<sup>1</sup> − <sup>2</sup>*mK*). (20)

*<sup>m</sup>*<sup>1</sup> − <sup>2</sup>*m*¯ *<sup>q</sup>*

�

<sup>Θ</sup> (*m*<sup>1</sup> <sup>−</sup> <sup>2</sup>*mK*).

. (21)

(22)

. (19)

�3 2

is the Fermi coupling constant and *mb* the *<sup>b</sup>* quark mass [22]. The function *<sup>f</sup>*(*x*) incorporates

But the above expression is not sufficient because a rough estimate of the lifetime of *S*<sup>1</sup> indicates that this latter is likely to decay inside a typical particle detector, which means we ought to take into account its most dominant decay products. We first have a process by

$$\Gamma\left(\mathbb{S}\_1 \to \ell^+ \ell^-\right) = \frac{G\_F m\_\ell^2 m\_1}{4\sqrt{2}\pi} \sin^2 \theta \left(1 - \frac{4m\_\ell^2}{m\_1^2}\right)^{\frac{3}{2}} \Theta\left(m\_1 - 2m\_\ell\right),\tag{23}$$

where *m*ℓ is the lepton mass. Finally, *S*<sup>1</sup> can decay into a pair of dark matter particles, with a decay rate:

$$
\Gamma\left(\mathcal{S}\_1 \to \mathcal{S}\_0 \mathcal{S}\_0\right) = \frac{\left(\eta\_{01}^{(3)}\right)^2}{32\pi m\_1} \sqrt{1 - \frac{4m\_0^2}{m\_1^2}} \Theta\left(m\_1 - 2m\_0\right). \tag{24}
$$

The branching ratio for Υ*nS* decaying via *S*<sup>1</sup> into a photon plus *X*, where *X* represents any kinematically allowed final state, will be:

$$\operatorname{Br}\left(\mathbf{Y}\_{n\mathbf{S}}\rightarrow\gamma+X\right)=\operatorname{Br}\left(\mathbf{Y}\_{n\mathbf{S}}\rightarrow\gamma+\mathbf{S}\_{1}\right)\times\operatorname{Br}\left(\mathbf{S}\_{1}\rightarrow\mathbf{X}\right).\tag{25}$$

In particular, *<sup>X</sup>* ≡ *<sup>S</sup>*0*S*<sup>0</sup> corresponds to a decay into invisible particles.

**Figure 1.** Typical branching ratios of Υ1*<sup>S</sup>* decaying into *τ*'s, charged pions and charged kaons as functions of *m*1. The corresponding experimental upper bounds are shown.

The best available experimental upper bounds on 1*S*–state branching ratios are: (i) Br(Υ1*<sup>S</sup>* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> *ττ*) <sup>&</sup>lt; <sup>5</sup> <sup>×</sup> <sup>10</sup>−<sup>5</sup> for 3.5GeV <sup>&</sup>lt; *<sup>m</sup>*<sup>1</sup> <sup>&</sup>lt; 9.2GeV [36]; (ii) Br <sup>Υ</sup>1*<sup>S</sup>* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> *<sup>π</sup>*+*π*<sup>−</sup> <

6.3 <sup>×</sup> <sup>10</sup>−<sup>5</sup> for 1GeV <sup>&</sup>lt; *<sup>m</sup>*<sup>1</sup> [37]; (iii) Br <sup>Υ</sup>1*<sup>S</sup>* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> *<sup>K</sup>*+*K*<sup>−</sup> <sup>&</sup>lt; 1.14 <sup>×</sup> <sup>10</sup>−<sup>5</sup> for 2GeV <sup>&</sup>lt; *m*<sup>1</sup> < 3GeV [38]. Figure 1 displays the corresponding branching ratios of Υ1*<sup>S</sup>* decays via *S*<sup>1</sup> as functions of *m*1, together with these upper bounds. Also, the best available experimental upper bounds on <sup>Υ</sup>3*<sup>S</sup>* branching ratios are: (i) Br(Υ3*<sup>S</sup>* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> *µµ*) <sup>&</sup>lt; <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>6</sup> for 1GeV <sup>&</sup>lt; *<sup>m</sup>*<sup>1</sup> <sup>&</sup>lt; 10GeV; (ii) Br(Υ3*<sup>S</sup>* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> Invisible) <sup>&</sup>lt; <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>6</sup> for 1GeV <sup>&</sup>lt; *<sup>m</sup>*<sup>1</sup> <sup>&</sup>lt; 7.8GeV [39]. Typical corresponding branching ratios are shown in figure 2.

invisible decay is into *<sup>S</sup>*0*S*<sup>0</sup> via *<sup>S</sup>*1. The process *<sup>B</sup>*<sup>+</sup> <sup>→</sup> *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>S</sup>*<sup>1</sup> has the following branching

In the above relation, *<sup>m</sup>*<sup>±</sup> <sup>=</sup> *mB* <sup>±</sup> *mK* where *mB* is the *<sup>B</sup>*<sup>+</sup> mass, *<sup>τ</sup><sup>B</sup>* its lifetime, and*Vtb* and

The different *<sup>S</sup>*<sup>1</sup> decay modes are given in (19) - ( 24) above. The branching ratio of *<sup>B</sup>*<sup>+</sup> decaying into *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>S</sup>*0*S*<sup>0</sup> via the production and propagation of an intermediary *<sup>S</sup>*<sup>1</sup> will be:

<sup>4</sup> � 0.02, *m*<sup>0</sup> � 0.4GeV, Θ � 15°

*Vts* are flavor changing CKM coefficients. The function *<sup>f</sup>*<sup>0</sup> (*s*) is given by the relation:

0.63*s m*2 *B*

0 1 2 3 4 5

*m*<sup>1</sup>GeV

**Figure 3.** Typical branching ratio of *<sup>B</sup>*<sup>+</sup> decaying into dark matter via *<sup>S</sup>*<sup>1</sup> as a function of *<sup>m</sup>*1. The SM and experimental bounds

see, the branching ratio is well above the experimental upper bound, and a mixing angle *θ*

*<sup>B</sup>*<sup>+</sup> <sup>→</sup> *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>S</sup>*0*S*<sup>0</sup>

as a function of *m*1. As we

*<sup>B</sup>* (*mb* <sup>−</sup> *ms*)

<sup>−</sup> 0.095*s*<sup>2</sup> *m*4 *B*

*<sup>B</sup>*<sup>+</sup> <sup>→</sup> *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>S</sup>*<sup>1</sup>

<sup>2</sup> <sup>|</sup>*VtbV*<sup>∗</sup> *ts*| <sup>2</sup> *f* <sup>2</sup> 0 *m*2 1 

+

0.591*s*<sup>3</sup> *m*6 *B*

sin2 *<sup>θ</sup>* <sup>Θ</sup> (*m*<sup>−</sup> − *<sup>m</sup>*1). (26)

Light Cold Dark Matter in a Two-Singlet Model

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281

. (27)

SM Exp Inv *S*0*S*<sup>0</sup>

× Br(*S*<sup>1</sup> → *<sup>S</sup>*0*S*0). (28)

<sup>−</sup> − *m*<sup>2</sup> 1

ratio:

Br

Br(*S*1)

10�<sup>5</sup>

Figure 3 displays a typical behavior of Br(*S*1)

10�<sup>4</sup>

Br*B* ��

are shown.

*K*

��*S*0*S*0

0.001

0.01

0.1

*<sup>B</sup>*<sup>+</sup> <sup>→</sup> *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>S</sup>*<sup>1</sup>

<sup>=</sup> <sup>9</sup>

*<sup>f</sup>*<sup>0</sup> (*s*) <sup>=</sup> 0.33 exp

*<sup>B</sup>*<sup>+</sup> <sup>→</sup> *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>S</sup>*0*S*<sup>0</sup>

Λ0

 = Br

√

× *m*2 <sup>+</sup> − *m*<sup>2</sup> 1 *m*<sup>2</sup>

2*τBG*<sup>3</sup> *Fm*<sup>4</sup> *<sup>t</sup> <sup>m</sup>*<sup>2</sup> *bm*<sup>2</sup> <sup>+</sup>*m*<sup>2</sup> −

1024*π*5*m*<sup>3</sup>

**Figure 2.** Typical branching ratios of Υ3*<sup>S</sup>* decaying into muons and dark matter as functions of *m*1. The corresponding experimental upper bounds are shown.

When scanning the parameter space, we see that the Higgs-dark-matter coupling constant *λ*(4) <sup>0</sup> and the dark-matter mass *m*<sup>0</sup> have little effect on the shapes of the branching ratios, apart from excluding, via the relic density and perturbativity constraints, regions of applicability of the model. This shows in figures 1 and 2 where the region *m*<sup>1</sup> 1.9GeV is excluded. Also, the onset of the *<sup>S</sup>*0*S*<sup>0</sup> channel for *<sup>m</sup>*<sup>1</sup> ≥ <sup>2</sup>*m*<sup>0</sup> abates sharply the other channels as this one becomes dominant by far. The effect of the mixing angle *θ* is to enhance all branching ratios as it increases, due to the factor sin2 *θ*. Furthermore, we notice that the dark matter decay channel reaches the invisible upper bound already for *<sup>θ</sup>* ≃ <sup>15</sup>o, for fairly small *<sup>m</sup>*0, say 0.5GeV, whereas the other channels find it hard to get to their respective experimental upper bounds, even for large values of *θ*.

## **4.** *B* **meson decays**

Next we look at the flavor changing process in which the meson *<sup>B</sup>*<sup>+</sup> decays into a *<sup>K</sup>*<sup>+</sup> plus invisible. The corresponding Standard-Model mode is a decay into *<sup>K</sup>*<sup>+</sup> and a pair of neutrinos, with a branching ratio BrSM *<sup>B</sup>*<sup>+</sup> <sup>→</sup> *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *νν*¯ ≃ 4.7 × <sup>10</sup>−<sup>6</sup> [40]. The experimental upper bound is BrExp *<sup>B</sup>*<sup>+</sup> → *<sup>K</sup>*<sup>+</sup> + Inv ≃ <sup>14</sup> × <sup>10</sup>−<sup>6</sup> [41]. Here too, the most prominent *<sup>B</sup>*

invisible decay is into *<sup>S</sup>*0*S*<sup>0</sup> via *<sup>S</sup>*1. The process *<sup>B</sup>*<sup>+</sup> <sup>→</sup> *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>S</sup>*<sup>1</sup> has the following branching ratio:

10 Open Questions in Cosmology

10�<sup>13</sup>

experimental upper bounds are shown.

bounds, even for large values of *θ*.

neutrinos, with a branching ratio BrSM

**4.** *B* **meson decays**

upper bound is BrExp

10�<sup>11</sup>

Br�3 �Γ� *s*

*λ*(4)

X

10�<sup>9</sup>

10�<sup>7</sup>

6.3 <sup>×</sup> <sup>10</sup>−<sup>5</sup> for 1GeV <sup>&</sup>lt; *<sup>m</sup>*<sup>1</sup> [37]; (iii) Br

[39]. Typical corresponding branching ratios are shown in figure 2.

Λ0

<sup>Υ</sup>1*<sup>S</sup>* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> *<sup>K</sup>*+*K*<sup>−</sup> <sup>&</sup>lt; 1.14 <sup>×</sup> <sup>10</sup>−<sup>5</sup> for 2GeV <sup>&</sup>lt;

Exp Inv

Exp ΜΜ

X � *S*0*S*<sup>0</sup>

X � ΜΜ

*m*<sup>1</sup> < 3GeV [38]. Figure 1 displays the corresponding branching ratios of Υ1*<sup>S</sup>* decays via *S*<sup>1</sup> as functions of *m*1, together with these upper bounds. Also, the best available experimental upper bounds on <sup>Υ</sup>3*<sup>S</sup>* branching ratios are: (i) Br(Υ3*<sup>S</sup>* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> *µµ*) <sup>&</sup>lt; <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>6</sup> for 1GeV <sup>&</sup>lt; *<sup>m</sup>*<sup>1</sup> <sup>&</sup>lt; 10GeV; (ii) Br(Υ3*<sup>S</sup>* <sup>→</sup> *<sup>γ</sup>* <sup>+</sup> Invisible) <sup>&</sup>lt; <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>6</sup> for 1GeV <sup>&</sup>lt; *<sup>m</sup>*<sup>1</sup> <sup>&</sup>lt; 7.8GeV

<sup>4</sup> � 0.04, *m*<sup>0</sup> � 2GeV, Θ � 15°

1 2 3 4 5 6 7 8

*m*<sup>1</sup> GeV

**Figure 2.** Typical branching ratios of Υ3*<sup>S</sup>* decaying into muons and dark matter as functions of *m*1. The corresponding

When scanning the parameter space, we see that the Higgs-dark-matter coupling constant

<sup>0</sup> and the dark-matter mass *m*<sup>0</sup> have little effect on the shapes of the branching ratios, apart from excluding, via the relic density and perturbativity constraints, regions of applicability of the model. This shows in figures 1 and 2 where the region *m*<sup>1</sup> 1.9GeV is excluded. Also, the onset of the *<sup>S</sup>*0*S*<sup>0</sup> channel for *<sup>m</sup>*<sup>1</sup> ≥ <sup>2</sup>*m*<sup>0</sup> abates sharply the other channels as this one becomes dominant by far. The effect of the mixing angle *θ* is to enhance all branching ratios as it increases, due to the factor sin2 *θ*. Furthermore, we notice that the dark matter decay channel reaches the invisible upper bound already for *<sup>θ</sup>* ≃ <sup>15</sup>o, for fairly small *<sup>m</sup>*0, say 0.5GeV, whereas the other channels find it hard to get to their respective experimental upper

Next we look at the flavor changing process in which the meson *<sup>B</sup>*<sup>+</sup> decays into a *<sup>K</sup>*<sup>+</sup> plus invisible. The corresponding Standard-Model mode is a decay into *<sup>K</sup>*<sup>+</sup> and a pair of

*<sup>B</sup>*<sup>+</sup> <sup>→</sup> *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *νν*¯

*<sup>B</sup>*<sup>+</sup> → *<sup>K</sup>*<sup>+</sup> + Inv ≃ <sup>14</sup> × <sup>10</sup>−<sup>6</sup> [41]. Here too, the most prominent *<sup>B</sup>*

≃ 4.7 × <sup>10</sup>−<sup>6</sup> [40]. The experimental

$$\mathrm{Br}\left(\mathcal{B}^{+} \to \mathcal{K}^{+} + \mathbb{S}\_{1}\right) = \frac{9\sqrt{2}\tau\_{\mathrm{B}}G\_{F}^{3}m\_{l}^{4}m\_{b}^{2}m\_{+}^{2}m\_{-}^{2}}{1024\pi^{5}m\_{\mathrm{B}}^{3}\left(m\_{b} - m\_{\mathrm{s}}\right)^{2}}\left|V\_{tb}V\_{\mathrm{ts}}^{\*}\right|^{2}f\_{0}^{2}\left(m\_{1}^{2}\right)$$

$$\times\sqrt{\left(m\_{+}^{2} - m\_{1}^{2}\right)\left(m\_{-}^{2} - m\_{1}^{2}\right)}\sin^{2}\theta\,\Theta\left(m\_{-} - m\_{1}\right). \tag{26}$$

In the above relation, *<sup>m</sup>*<sup>±</sup> <sup>=</sup> *mB* <sup>±</sup> *mK* where *mB* is the *<sup>B</sup>*<sup>+</sup> mass, *<sup>τ</sup><sup>B</sup>* its lifetime, and*Vtb* and *Vts* are flavor changing CKM coefficients. The function *<sup>f</sup>*<sup>0</sup> (*s*) is given by the relation:

$$f\_0(\mathbf{s}) = 0.33 \exp\left[\frac{0.63 \mathbf{s}}{m\_B^2} - \frac{0.095 \mathbf{s}^2}{m\_B^4} + \frac{0.591 \mathbf{s}^3}{m\_B^6}\right]. \tag{27}$$

The different *<sup>S</sup>*<sup>1</sup> decay modes are given in (19) - ( 24) above. The branching ratio of *<sup>B</sup>*<sup>+</sup> decaying into *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>S</sup>*0*S*<sup>0</sup> via the production and propagation of an intermediary *<sup>S</sup>*<sup>1</sup> will be:

$$\operatorname{Br}^{(\mathcal{S}\_{\mathcal{I}})} \left( \mathcal{B}^{+} \to K^{+} + \mathcal{S}\_{0}\mathcal{S}\_{0} \right) = \operatorname{Br} \left( \mathcal{B}^{+} \to K^{+} + \mathcal{S}\_{1} \right) \times \operatorname{Br} \left( \mathcal{S}\_{1} \to \mathcal{S}\_{0}\mathcal{S}\_{0} \right). \tag{28}$$

**Figure 3.** Typical branching ratio of *<sup>B</sup>*<sup>+</sup> decaying into dark matter via *<sup>S</sup>*<sup>1</sup> as a function of *<sup>m</sup>*1. The SM and experimental bounds are shown.

Figure 3 displays a typical behavior of Br(*S*1) *<sup>B</sup>*<sup>+</sup> <sup>→</sup> *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>S</sup>*0*S*<sup>0</sup> as a function of *m*1. As we see, the branching ratio is well above the experimental upper bound, and a mixing angle *θ*

as small as 1<sup>o</sup> will not help with this, no matter what the values for *<sup>λ</sup>*(4) <sup>0</sup> and *m*<sup>0</sup> are. So, we conclude that for *m*<sup>1</sup> 4.8GeV, this process excludes the two-singlet model for *m*<sup>0</sup> < *m*1/2. For *<sup>m</sup>*<sup>1</sup> 4.8GeV or *<sup>m</sup>*<sup>0</sup> ≥ *<sup>m</sup>*1/2, the decay does not occur, so no constraints on the model from this process.

This works until about 2.7GeV, beyond which *m*<sup>0</sup> ceases to have any significant direct effect. Increasing *θ* enhances the values of the branching ratio without affecting the width. Also, for all the range of *<sup>m</sup>*1, all of Br(*S*1) <sup>+</sup> BrSM stays below BrExp as long as *<sup>θ</sup>* <sup>&</sup>lt; 10o. As *<sup>θ</sup>* increases beyond this value, the peak region pushes up increasingly above BrExp, like in figure 4, and

We finally examine the implications of the model on the Higgs different decay modes. In this section, we allow the Higgs mass *mh* to vary in the interval 100GeV − 200GeV. First, *h* can decay into a pair of leptons ℓ, predominantly *τ*'s. The corresponding decay rate

can also decay into a pair of quarks *q*, mainly into *b*'s and, to a lesser degree, into *c*'s. Here too the decay rate Γ(*h* → *qq*¯) is given in (21) with similar replacements. Then the Higgs can decay into a pair of gluons. The corresponding decay rate that includes the next-to-next-to-leading

> *dx* <sup>1</sup>−*<sup>x</sup>* 0

where the sum is over all quark flavors *q*. A systematic study of the double integral above shows that, with *mh* in the range 100GeV – 200GeV, the *t* quark dominates in the sum over *q*,

For *mh* smaller than the *W* or *Z* pair-production thresholds, the Higgs can decay into a pair

In this expression, *mV* is the mass of the gauge boson *<sup>V</sup>*, the factor *AV* = 1 for *<sup>W</sup>* and

 *m*2 *V m*2 *h*

arccos 3*<sup>x</sup>* <sup>−</sup> <sup>1</sup>

 − 3 2 

<sup>2</sup>*x*3/2

for *Z* with *θw* the Weinberg angle, and we have the definition:

1 − 6*x* + 4*x*<sup>2</sup>

log *x*. (32)

is given by the relation (23) where we replace *m*<sup>1</sup> by *mh* and sin *θ* by cos *θ*. It

*dy* <sup>1</sup> <sup>−</sup> <sup>4</sup>*xy m*<sup>2</sup> *q m*<sup>2</sup> *h* − *xy*

  2

156.8 <sup>−</sup> 5.7 log *<sup>m</sup>*<sup>2</sup>

*t m*2 *h* Light Cold Dark Matter in a Two-Singlet Model

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283

<sup>Θ</sup> [(*mh* − *mV*) (2*mV* − *mh*)] . (31)

cos<sup>2</sup> *θ*, (30)

hence gets excluded, but all the rest is allowed.

QCD radiative corrections is given by:

<sup>Γ</sup> (*<sup>h</sup>* <sup>→</sup> *gg*) <sup>=</sup> *GFm*<sup>3</sup>

<sup>Γ</sup> (*<sup>h</sup>* <sup>→</sup> *VV*∗) <sup>=</sup> <sup>3</sup>*G*<sup>2</sup>

<sup>9</sup> sin<sup>4</sup> *<sup>θ</sup><sup>w</sup>*

<sup>9</sup> sin2 *<sup>θ</sup><sup>w</sup>* <sup>+</sup> <sup>40</sup>

*h*

 

∑ *q*

<sup>2</sup> 1 + 215 12 *α*¯*s π* + *α*¯ 2 *s π*2

but with non-negligible contributions from the *c* and *b* quarks.

of one real and one virtual gauge bosons, with rates given by:

*Fm*<sup>4</sup> *<sup>V</sup>mh*

*<sup>R</sup>*(*x*) = <sup>3</sup>(<sup>1</sup> <sup>−</sup> <sup>8</sup>*<sup>x</sup>* <sup>+</sup> <sup>20</sup>*x*2) √4*x* − 1

> <sup>−</sup><sup>1</sup> <sup>−</sup> *<sup>x</sup>* 2*x*

<sup>16</sup>*π*<sup>3</sup> cos<sup>2</sup> *<sup>θ</sup> AV <sup>R</sup>*

2 − 13*x* + 47*x*<sup>2</sup>

*m*2 *q m*2 *h*

 <sup>1</sup> 0

4 √2*π*

× *α*¯*<sup>s</sup> π*

**5. Higgs decays**

*<sup>h</sup>* → ℓ+ℓ<sup>−</sup>

Γ 

 7 <sup>12</sup> <sup>−</sup> <sup>10</sup>

Another process involving *B* mesons is the decay of *Bs* into predominately a pair of muons. The Standard Model branching ratio for this process is BrSM *Bs* <sup>→</sup> *<sup>µ</sup>*+*µ*<sup>−</sup> = (3.2 <sup>±</sup> 0.2) <sup>×</sup> <sup>10</sup>−<sup>9</sup> [42], and the experimental upper bound is BrExp *Bs* <sup>→</sup> *<sup>µ</sup>*+*µ*<sup>−</sup> <sup>&</sup>lt; 1.08 <sup>×</sup> <sup>10</sup>−<sup>8</sup> [43]. In the present model, two additional decay diagrams occur, both via intermediary *S*1, yielding together the branching ratio:

$$\mathrm{Br}^{(\mathcal{S}\_{\mathrm{I}})}(\mathcal{B}\_{\mathrm{s}}\rightarrow\boldsymbol{\mu}^{+}\boldsymbol{\mu}^{-}) = \frac{9\pi\_{\mathrm{B}\_{\mathrm{s}}}G\_{\mathrm{F}}^{4}f\_{\mathrm{B}\_{\mathrm{s}}}^{2}m\_{\mathrm{B}\_{\mathrm{s}}}^{5}}{2048\pi^{5}}m\_{\mu}^{2}m\_{\mathrm{I}}^{4}\left|V\_{\mathrm{tb}}V\_{\mathrm{ts}}^{\*}\right|^{2}\frac{\left(1-4m\_{\mu}^{2}/m\_{\mathrm{B}\_{\mathrm{s}}}^{2}\right)^{3/2}}{\left(m\_{\mathrm{B}\_{\mathrm{s}}}^{2}-m\_{\mathrm{I}}^{2}\right)^{2}+m\_{\mathrm{I}}^{2}\Gamma\_{\mathrm{I}}^{2}}\sin^{4}\theta.\tag{29}$$

In this relation, *<sup>τ</sup>Bs* is the *Bs* life-time, *mBs* <sup>=</sup> 5.37GeV its mass, and *fBs* a form factor that we take equal to 0.21GeV. The quantity Γ<sup>1</sup> is the total width of the particle *S*<sup>1</sup> [17].

**Figure 4.** Typical behavior of Br(*S*1) (*Bs* → *<sup>µ</sup>*+*µ*−) as a function of *<sup>m</sup>*1, together with the SM and experimental bounds.

A typical behavior of Br(*S*1) (*Bs* <sup>→</sup> *<sup>µ</sup>*+*µ*−) as a function of *<sup>m</sup>*<sup>1</sup> is displayed in figure 4. The peak is at *mBs* . All three parameters *<sup>λ</sup>*(4) <sup>0</sup> , *m*<sup>0</sup> and *θ* combine in the relic density constraint to exclude few regions of applicability of the model. For example, for the values of figure 4, the region *m*<sup>1</sup> < 2.25GeV is excluded. However, a systematic scan of the parameter space shows that outside the relic density constraint, *<sup>λ</sup>*(4) <sup>0</sup> has no significant direct effect on the shape of Br(*S*1) (*Bs* <sup>→</sup> *<sup>µ</sup>*+*µ*−). As *<sup>m</sup>*<sup>0</sup> increases, it sharpens the peak of the curve while pushing it up. This works until about 2.7GeV, beyond which *m*<sup>0</sup> ceases to have any significant direct effect. Increasing *θ* enhances the values of the branching ratio without affecting the width. Also, for all the range of *<sup>m</sup>*1, all of Br(*S*1) <sup>+</sup> BrSM stays below BrExp as long as *<sup>θ</sup>* <sup>&</sup>lt; 10o. As *<sup>θ</sup>* increases beyond this value, the peak region pushes up increasingly above BrExp, like in figure 4, and hence gets excluded, but all the rest is allowed.

## **5. Higgs decays**

12 Open Questions in Cosmology

from this process.

Br(*S*1)

10�<sup>11</sup>

**Figure 4.** Typical behavior of Br(*S*1)

A typical behavior of Br(*S*1)

peak is at *mBs* . All three parameters *<sup>λ</sup>*(4)

that outside the relic density constraint, *<sup>λ</sup>*(4)

10�<sup>10</sup>

Br*B* �Μ *s*

Br(*S*1)

�Μ�

10�<sup>9</sup>

10�<sup>8</sup>

10�<sup>7</sup>

together the branching ratio:

(*Bs* <sup>→</sup> *<sup>µ</sup>*+*µ*−) =

as small as 1<sup>o</sup> will not help with this, no matter what the values for *<sup>λ</sup>*(4)

The Standard Model branching ratio for this process is BrSM

9*τBsG*<sup>4</sup> *F f* 2 *Bs m*5 *Bs* <sup>2048</sup>*π*<sup>5</sup> *<sup>m</sup>*<sup>2</sup>

Λ0

<sup>10</sup>−<sup>9</sup> [42], and the experimental upper bound is BrExp

conclude that for *m*<sup>1</sup> 4.8GeV, this process excludes the two-singlet model for *m*<sup>0</sup> < *m*1/2. For *<sup>m</sup>*<sup>1</sup> 4.8GeV or *<sup>m</sup>*<sup>0</sup> ≥ *<sup>m</sup>*1/2, the decay does not occur, so no constraints on the model

Another process involving *B* mesons is the decay of *Bs* into predominately a pair of muons.

the present model, two additional decay diagrams occur, both via intermediary *S*1, yielding

*µm*<sup>4</sup>

In this relation, *<sup>τ</sup>Bs* is the *Bs* life-time, *mBs* <sup>=</sup> 5.37GeV its mass, and *fBs* a form factor that we

<sup>4</sup> � 0.03, *m*<sup>0</sup> � 4GeV, Θ � 20°

0 2 4 6 8 10

*m*<sup>1</sup>GeV

exclude few regions of applicability of the model. For example, for the values of figure 4, the region *m*<sup>1</sup> < 2.25GeV is excluded. However, a systematic scan of the parameter space shows

(*Bs* <sup>→</sup> *<sup>µ</sup>*+*µ*−). As *<sup>m</sup>*<sup>0</sup> increases, it sharpens the peak of the curve while pushing it up.

(*Bs* → *<sup>µ</sup>*+*µ*−) as a function of *<sup>m</sup>*1, together with the SM and experimental bounds.

(*Bs* <sup>→</sup> *<sup>µ</sup>*+*µ*−) as a function of *<sup>m</sup>*<sup>1</sup> is displayed in figure 4. The

<sup>0</sup> , *m*<sup>0</sup> and *θ* combine in the relic density constraint to

<sup>0</sup> has no significant direct effect on the shape of

take equal to 0.21GeV. The quantity Γ<sup>1</sup> is the total width of the particle *S*<sup>1</sup> [17].

*<sup>t</sup>* <sup>|</sup>*VtbV*<sup>∗</sup> *ts*| 2 1 − 4*m*<sup>2</sup>

 *m*2 *Bs* <sup>−</sup> *<sup>m</sup>*<sup>2</sup> 1 2 + *m*<sup>2</sup> 1Γ2 1

<sup>0</sup> and *m*<sup>0</sup> are. So, we

sin<sup>4</sup> *θ*. (29)

SM Exp Μ�Μ�

*Bs* <sup>→</sup> *<sup>µ</sup>*+*µ*<sup>−</sup> = (3.2 <sup>±</sup> 0.2) <sup>×</sup>

*Bs* <sup>→</sup> *<sup>µ</sup>*+*µ*<sup>−</sup> <sup>&</sup>lt; 1.08 <sup>×</sup> <sup>10</sup>−<sup>8</sup> [43]. In

*µ*/*m*<sup>2</sup> *Bs* 3/2 We finally examine the implications of the model on the Higgs different decay modes. In this section, we allow the Higgs mass *mh* to vary in the interval 100GeV − 200GeV. First, *h* can decay into a pair of leptons ℓ, predominantly *τ*'s. The corresponding decay rate Γ *<sup>h</sup>* → ℓ+ℓ<sup>−</sup> is given by the relation (23) where we replace *m*<sup>1</sup> by *mh* and sin *θ* by cos *θ*. It can also decay into a pair of quarks *q*, mainly into *b*'s and, to a lesser degree, into *c*'s. Here too the decay rate Γ(*h* → *qq*¯) is given in (21) with similar replacements. Then the Higgs can decay into a pair of gluons. The corresponding decay rate that includes the next-to-next-to-leading QCD radiative corrections is given by:

$$\Gamma(h \to \text{gg}) = \frac{\mathbf{G}\_F m\_h^2}{4\sqrt{2}\pi} \left| \sum\_q \frac{m\_q^2}{m\_h^2} \int\_0^1 dx \int\_0^{1-x} dy \frac{1-4xy}{\frac{m\_q^2}{m\_h^2} - xy} \right|^2$$

$$\times \left(\frac{\bar{\mathbf{a}}\_s}{\pi}\right)^2 \left[1 + \frac{215}{12} \frac{\bar{\mathbf{a}}\_s}{\pi} + \frac{\bar{\mathbf{a}}\_s^2}{\pi^2} \left(156.8 - 5.7 \log \frac{m\_t^2}{m\_h^2}\right) \right] \cos^2 \theta,\tag{30}$$

where the sum is over all quark flavors *q*. A systematic study of the double integral above shows that, with *mh* in the range 100GeV – 200GeV, the *t* quark dominates in the sum over *q*, but with non-negligible contributions from the *c* and *b* quarks.

For *mh* smaller than the *W* or *Z* pair-production thresholds, the Higgs can decay into a pair of one real and one virtual gauge bosons, with rates given by:

$$\Gamma\left(h \to VV^\*\right) = \frac{3G\_F^2 m\_V^4 m\_h}{16\pi^3} \cos^2\theta \, A\_V \, R\left(\frac{m\_V^2}{m\_h^2}\right) \Theta\left[\left(m\_h - m\_V\right)\left(2m\_V - m\_h\right)\right].\tag{31}$$

In this expression, *mV* is the mass of the gauge boson *<sup>V</sup>*, the factor *AV* = 1 for *<sup>W</sup>* and 7 <sup>12</sup> <sup>−</sup> <sup>10</sup> <sup>9</sup> sin2 *<sup>θ</sup><sup>w</sup>* <sup>+</sup> <sup>40</sup> <sup>9</sup> sin<sup>4</sup> *<sup>θ</sup><sup>w</sup>* for *Z* with *θw* the Weinberg angle, and we have the definition:

$$\begin{split} R(\mathbf{x}) &= \frac{3(1 - 8\mathbf{x} + 20\mathbf{x}^2)}{\sqrt{4\mathbf{x} - 1}} \arccos\left(\frac{3\mathbf{x} - 1}{2\mathbf{x}^{3/2}}\right) \\ &- \frac{1 - \mathbf{x}}{2\mathbf{x}} \left(2 - 13\mathbf{x} + 47\mathbf{x}^2\right) - \frac{3}{2} \left(1 - 6\mathbf{x} + 4\mathbf{x}^2\right) \log \mathbf{x}. \end{split} \tag{32}$$

For a heavier Higgs particle, the decay rates into a *V* pair is given by:

$$\Gamma\left(h \to VV\right) = \frac{\mathbb{G}\_F m\_V^4 \cos^2 \theta}{\sqrt{2}\pi m\_h} B\_V \left(1 - \frac{4m\_V^2}{m\_h^2}\right)^{\frac{1}{2}} \left[1 + \frac{\left(m\_h^2 - 2m\_V^2\right)^2}{8m\_V^4}\right] \Theta\left(m\_h - 2m\_V\right), \tag{33}$$

with *BV* = 1 for *<sup>W</sup>* and <sup>1</sup> <sup>2</sup> for *Z*.

While all these decay modes are already present within the Standard Model, the two-singlet extension introduces two additional (invisible) modes, namely a decay into a pair of *S*0's and a pair of *S*1's. The corresponding decay rates are:

$$\Gamma\left(\hbar \to S\_{\bar{i}}S\_{\bar{i}}\right) = \frac{\lambda\_{\bar{i}}^2}{32\pi m\_h} \left(1 - \frac{4m\_{\bar{i}}^2}{m\_h^2}\right)^{\frac{1}{2}} \Theta\left(m\_h - 2m\_{\bar{i}}\right),\tag{34}$$

However, the decay distribution between *S*<sup>0</sup> and *S*<sup>1</sup> is not even. The most dramatic effect

is a gradual shift towards a more dominating dark-matter pair production, a shift competed

*WW* and *ZZ* thresholds, and lowers the production of everything except that of a pair of *S*1,

, *m*<sup>0</sup> � 2GeV, *m*<sup>1</sup> � 2.5GeV

100 120 140 160 180 200

*mh*GeV

Like in the Standard Model, the production of a pair of *b* quarks dominates over the production of the other fermions, and all fermions are not favored by increasing *<sup>λ</sup>*(4)

Changes in *m*<sup>0</sup> and *m*<sup>1</sup> have very little direct effects on all the branching ratios except that of *S*0*S*<sup>0</sup> production where, at small *θ*, increasing *m*<sup>1</sup> (*m*0) increases (decreases) the branching ratio, with reversed effects at larger *θ*. Note though that these masses have indirect impact

Understanding light dark matter is one of the challenges facing popular extensions of the Standard Model. In this chapter, we have furthered the study of a two-singlet extension of the SM we proposed as a model for light dark matter by exploring some of its phenomenological aspects. We have looked into the rare decays of Υ and *B* mesons and studied the implications

<sup>0</sup> . When it is very small, the dominant production is

<sup>0</sup> <sup>=</sup> 0.7. In general, increasing *<sup>θ</sup>* smoothens the crossings of the

<sup>0</sup> <sup>=</sup> 0.01. As it increases, there

Light Cold Dark Matter in a Two-Singlet Model

http://dx.doi.org/10.5772/52243

285

<sup>0</sup> <sup>=</sup> 0.1 and figure

h�ZZ h�WW

<sup>0</sup> .

h�gg h�ΤΤ h�c*c* � h�b*b* � h�*S*1*S*<sup>1</sup> h�*S*0*S*<sup>0</sup>

comes from the coupling constant *<sup>λ</sup>*(4)

Λ0

7 for the larger value *<sup>λ</sup>*(4)

which is instead increased.

10�<sup>5</sup>

**6. Concluding remarks**

0.001

Brh

�

X

0.1

that of a pair of *<sup>S</sup>*1. This is exhibited in figure 5 for which *<sup>λ</sup>*(4)

<sup>4</sup> � 0.1, Θ � 10°

**Figure 6.** Branching ratios for Higgs decays. Small dark-matter Higgs coupling.

of the model on the decay channels of the Higgs particle.

through the relic density constraint by excluding certain regions [17].

against by the increase in *<sup>θ</sup>*. Figure 6 displays the branching ratios for *<sup>λ</sup>*(4)

where *<sup>λ</sup><sup>i</sup>* <sup>=</sup> *<sup>λ</sup>*(3) <sup>0</sup>(2) for *<sup>S</sup>*0(1) given in (9). The total decay rate <sup>Γ</sup> (*h*) of the Higgs particle is the sum of these partial rates. The branching ratio corresponding to a particular decay will be Br(*h* → *X*) = Γ (*h* → *X*) /Γ (*h*).

**Figure 5.** Branching ratios for Higgs decays. Very small dark-matter Higgs coupling.

Typical behaviors of the most prominent branching ratios are displayed in figure 5. A systematic study shows that for all ranges of the parameters, the Higgs decays dominantly into invisible. The production of fermions and gluons is comparatively marginal, whereas that of *W* and *Z* pairs takes relative importance towards and above the corresponding thresholds, and more significantly at larger values of the mixing angle *θ*.

However, the decay distribution between *S*<sup>0</sup> and *S*<sup>1</sup> is not even. The most dramatic effect comes from the coupling constant *<sup>λ</sup>*(4) <sup>0</sup> . When it is very small, the dominant production is that of a pair of *<sup>S</sup>*1. This is exhibited in figure 5 for which *<sup>λ</sup>*(4) <sup>0</sup> <sup>=</sup> 0.01. As it increases, there is a gradual shift towards a more dominating dark-matter pair production, a shift competed against by the increase in *<sup>θ</sup>*. Figure 6 displays the branching ratios for *<sup>λ</sup>*(4) <sup>0</sup> <sup>=</sup> 0.1 and figure 7 for the larger value *<sup>λ</sup>*(4) <sup>0</sup> <sup>=</sup> 0.7. In general, increasing *<sup>θ</sup>* smoothens the crossings of the *WW* and *ZZ* thresholds, and lowers the production of everything except that of a pair of *S*1, which is instead increased.

**Figure 6.** Branching ratios for Higgs decays. Small dark-matter Higgs coupling.

Like in the Standard Model, the production of a pair of *b* quarks dominates over the production of the other fermions, and all fermions are not favored by increasing *<sup>λ</sup>*(4) <sup>0</sup> . Changes in *m*<sup>0</sup> and *m*<sup>1</sup> have very little direct effects on all the branching ratios except that of *S*0*S*<sup>0</sup> production where, at small *θ*, increasing *m*<sup>1</sup> (*m*0) increases (decreases) the branching ratio, with reversed effects at larger *θ*. Note though that these masses have indirect impact through the relic density constraint by excluding certain regions [17].

## **6. Concluding remarks**

14 Open Questions in Cosmology

<sup>Γ</sup> (*<sup>h</sup>* <sup>→</sup> *VV*) <sup>=</sup> *GFm*<sup>4</sup>

with *BV* = 1 for *<sup>W</sup>* and <sup>1</sup>

where *<sup>λ</sup><sup>i</sup>* <sup>=</sup> *<sup>λ</sup>*(3)

10�<sup>5</sup>

0.001

Brh

�

X

0.1

Br(*h* → *X*) = Γ (*h* → *X*) /Γ (*h*).

Λ0

For a heavier Higgs particle, the decay rates into a *V* pair is given by:

<sup>1</sup> <sup>−</sup> <sup>4</sup>*m*<sup>2</sup> *V m*2 *h*

*i* 32*πmh* 1 <sup>2</sup> 1 + *m*2 *<sup>h</sup>* <sup>−</sup> <sup>2</sup>*m*<sup>2</sup> *V* 2

While all these decay modes are already present within the Standard Model, the two-singlet extension introduces two additional (invisible) modes, namely a decay into a pair of *S*0's and

sum of these partial rates. The branching ratio corresponding to a particular decay will be

100 120 140 160 180 200

*mh*GeV

Typical behaviors of the most prominent branching ratios are displayed in figure 5. A systematic study shows that for all ranges of the parameters, the Higgs decays dominantly into invisible. The production of fermions and gluons is comparatively marginal, whereas that of *W* and *Z* pairs takes relative importance towards and above the corresponding

<sup>1</sup> <sup>−</sup> <sup>4</sup>*m*<sup>2</sup> *i m*2 *h*

1 2

<sup>0</sup>(2) for *<sup>S</sup>*0(1) given in (9). The total decay rate <sup>Γ</sup> (*h*) of the Higgs particle is the

, *m*<sup>0</sup> � 2GeV, *m*<sup>1</sup> � 2.5GeV

8*m*<sup>4</sup> *V* <sup>Θ</sup> (*mh* − <sup>2</sup>*mV*), (33)

h�ZZ h�WW

h�gg h�ΤΤ h�c*c* � h�b*b* � h�*S*1*S*<sup>1</sup> h�*S*0*S*<sup>0</sup>

<sup>Θ</sup> (*mh* − <sup>2</sup>*mi*), (34)

*BV* 

*<sup>V</sup>* cos<sup>2</sup> *<sup>θ</sup>*

<sup>2</sup> for *Z*.

<sup>Γ</sup> (*<sup>h</sup>* <sup>→</sup> *SiSi*) <sup>=</sup> *<sup>λ</sup>*<sup>2</sup>

<sup>4</sup> � 0.01, Θ � 10°

**Figure 5.** Branching ratios for Higgs decays. Very small dark-matter Higgs coupling.

thresholds, and more significantly at larger values of the mixing angle *θ*.

a pair of *S*1's. The corresponding decay rates are:

√2*πmh*

> Understanding light dark matter is one of the challenges facing popular extensions of the Standard Model. In this chapter, we have furthered the study of a two-singlet extension of the SM we proposed as a model for light dark matter by exploring some of its phenomenological aspects. We have looked into the rare decays of Υ and *B* mesons and studied the implications of the model on the decay channels of the Higgs particle.

**Figure 7.** Branching ratios for Higgs decays. Larger dark-matter Higgs coupling.

In brief, for both <sup>Υ</sup> and *<sup>B</sup>* decays, the Higgs-DM coupling constant *<sup>λ</sup>*(4) <sup>0</sup> and the DM mass *m*<sup>0</sup> have little effect on the shapes of the branching ratios, apart from combining with the other two parameters in the relic-density and perturbativity constraints to exclude regions of applicability of the model. Also, the effect of increasing the *<sup>h</sup>* − *<sup>S</sup>*<sup>1</sup> mixing angle *<sup>θ</sup>* is to enhance all branching ratios. For Υ decays, the DM channel dominates over the other decay modes in regions where kinematically allowed. It reaches the experimental invisible upper bound for already fairly small values of *<sup>θ</sup>* and *<sup>m</sup>*0. From *<sup>B</sup>*<sup>+</sup> decays, we learn that our model is excluded for *<sup>m</sup>*<sup>1</sup> < 4.8GeV (= *mB* − *mK*) and *<sup>m</sup>*<sup>0</sup> < *<sup>m</sup>*1/2. From *Bs* decay into muons, we learn that for the model to contribute a distinct signal to this process, it is best to restrict 4GeV *m*<sup>1</sup> 6.5GeV with no additional constraint on *m*<sup>0</sup> [20]. Also, in general, keeping *λ*(4) <sup>0</sup> 0.1 to avoid systematic exclusion from direct detection for all these processes is safe.

Before closing the chapter, it is useful to comment briefly on how light dark matter in our model affects Higgs searches. Since *mh* ≫ <sup>2</sup>*m*0, the process *<sup>h</sup>* → *<sup>S</sup>*0*S*<sup>0</sup> is kinematically allowed and, for a large range of the parameter space, the ratio

$$\mathcal{R}\_{\text{decay}}^{(b)} = \frac{\text{Br}\left(h \to S\_0 S\_0\right)}{\text{Br}\left(h \to b\bar{b}\right)}\tag{35}$$

seems to suggest that we should have limited our analysis of the Higgs branching ratios to *mh* < 145GeV. However, it is important to note that these experimental constraints apply to the SM Higgs and can not therefore be used as such if the Higgs interactions are modified. In our model, the mixing of *h* with *S*<sup>1</sup> will result in a reduction of the statistical significance of the Higgs discovery at the LHC. Indeed, the relevant quantity that allows one to use the experimental limits on Higgs searches to derive constraints on the parameters of the model

In this expression, *<sup>X</sup>*SM corresponds to all the Standard Model particles, *Xi*nv = *<sup>S</sup>*0*S*<sup>0</sup> and *<sup>S</sup>*1*S*1, *<sup>σ</sup>* is a cross-section, BrSM (*<sup>h</sup>* → *<sup>X</sup>*) the branching fraction of the SM Higgs decaying

Model. To open up the region *mh* <sup>&</sup>gt; 140GeV requires the ratio <sup>R</sup>*X*SM to be smaller than 0.25 [23, 24], a constraint easily fulfilled in our model. By comparison, the minimal extensions of the Standard Model with just one singlet scalar or a Majorana fermion, even under a *Z*<sup>2</sup> symmetry, are highly constrained in this regard [44]. Finally, if the recent data from ATLAS and CMS turn out to be a signal for a SM-like Higgs with mass about 125GeV, then this will put a very strong constraint on the mixing angle *θ*. Indeed, only for *θ* 0.50 will the ratio

dark matter mass, but as long as *m*<sup>1</sup> 50GeV, which we are assuming in this work.

<sup>4</sup> � 0.01, Θ � 0.4°

*b*)/Γ (*h* → inv) 1. For larger values, the model is ruled out, independently of the

, *m*<sup>0</sup> � 3GeV, *mh* � 125GeV

0 1 2 3 4 5 6 7

*b*)/Γ(*h* → inv) is larger than one for *θ* = 0.4<sup>0</sup> and *mh* = 125GeV. Finally, it is important to find the bounds on the mass SM Higgs that can satisfy the triviality and perturbativity constraints on the coupling constants in the scalar sector of this model up to a scale higher then 1TeV. This requires studying the renormalization group equations of

*m*<sup>1</sup>GeV

<sup>=</sup> cos4 *<sup>θ</sup>*

cos2 *<sup>θ</sup>* + <sup>Γ</sup> (*<sup>h</sup>* → *<sup>X</sup>*inv) /ΓSM

*<sup>h</sup>* the total Higgs decay rate in the Standard

*h*

Light Cold Dark Matter in a Two-Singlet Model

http://dx.doi.org/10.5772/52243

. (36)

287

<sup>R</sup>*X*SM <sup>≡</sup> *<sup>σ</sup>* (*gg* <sup>→</sup> *<sup>h</sup>*) Br(*<sup>h</sup>* <sup>→</sup> *<sup>X</sup>*SM)

into any kinematically allowed mode *X*, and ΓSM

*<sup>σ</sup>*(SM) (*gg* <sup>→</sup> *<sup>h</sup>*) BrSM (*<sup>h</sup>* <sup>→</sup> *<sup>X</sup>*SM)

Λ0

is the ratio:

Γ(*h* → *b*¯

0.0

these coupling constants [45].

**Figure 8.** The ratio Γ(*h* → *b*¯

0.2

0.4

�h�b

*b*�h�inv

�

0.6

0.8

1.0

1.2

can be larger than one for *mh* < 120GeV as can be seen in figure 6. In this situation, the LEP bound on the Higgs mass can be weaker. Also, in our model, the Higgs production at LEP via Higgstrahlung can be smaller than the one in the Standard Model, and so the Higgs can be as light as 100GeV. Such a light Higgs would be in good agreement with the electroweak precision tests. As to the Higgs searches at the LHC, the ATLAS and CMS collaborations have reported the exclusion of a Higgs mass in the interval 145 – 460 GeV [23, 24], which seems to suggest that we should have limited our analysis of the Higgs branching ratios to *mh* < 145GeV. However, it is important to note that these experimental constraints apply to the SM Higgs and can not therefore be used as such if the Higgs interactions are modified. In our model, the mixing of *h* with *S*<sup>1</sup> will result in a reduction of the statistical significance of the Higgs discovery at the LHC. Indeed, the relevant quantity that allows one to use the experimental limits on Higgs searches to derive constraints on the parameters of the model is the ratio:

16 Open Questions in Cosmology

Λ0

<sup>4</sup> � 0.7, Θ � 10°

**Figure 7.** Branching ratios for Higgs decays. Larger dark-matter Higgs coupling.

allowed and, for a large range of the parameter space, the ratio

R(*b*)

10�<sup>6</sup>

10�<sup>4</sup>

Brh

*λ*(4)

�

X

0.01

1

100 120 140 160 180 200

, *m*<sup>0</sup> � 2GeV, *m*<sup>1</sup> � 2.5GeV

h�ZZ

<sup>0</sup> and the DM mass

*<sup>b</sup>*) (35)

h�WW

h�gg

h�ΤΤ h�c*c* � h�b*b* � h�*S*1*S*<sup>1</sup>

h�*S*0*S*<sup>0</sup>

*mh*GeV

*m*<sup>0</sup> have little effect on the shapes of the branching ratios, apart from combining with the other two parameters in the relic-density and perturbativity constraints to exclude regions of applicability of the model. Also, the effect of increasing the *<sup>h</sup>* − *<sup>S</sup>*<sup>1</sup> mixing angle *<sup>θ</sup>* is to enhance all branching ratios. For Υ decays, the DM channel dominates over the other decay modes in regions where kinematically allowed. It reaches the experimental invisible upper bound for already fairly small values of *<sup>θ</sup>* and *<sup>m</sup>*0. From *<sup>B</sup>*<sup>+</sup> decays, we learn that our model is excluded for *<sup>m</sup>*<sup>1</sup> < 4.8GeV (= *mB* − *mK*) and *<sup>m</sup>*<sup>0</sup> < *<sup>m</sup>*1/2. From *Bs* decay into muons, we learn that for the model to contribute a distinct signal to this process, it is best to restrict 4GeV *m*<sup>1</sup> 6.5GeV with no additional constraint on *m*<sup>0</sup> [20]. Also, in general, keeping

<sup>0</sup> 0.1 to avoid systematic exclusion from direct detection for all these processes is safe. Before closing the chapter, it is useful to comment briefly on how light dark matter in our model affects Higgs searches. Since *mh* ≫ <sup>2</sup>*m*0, the process *<sup>h</sup>* → *<sup>S</sup>*0*S*<sup>0</sup> is kinematically

> decay <sup>=</sup> Br(*<sup>h</sup>* <sup>→</sup> *<sup>S</sup>*0*S*0) Br(*h* → *b*¯

can be larger than one for *mh* < 120GeV as can be seen in figure 6. In this situation, the LEP bound on the Higgs mass can be weaker. Also, in our model, the Higgs production at LEP via Higgstrahlung can be smaller than the one in the Standard Model, and so the Higgs can be as light as 100GeV. Such a light Higgs would be in good agreement with the electroweak precision tests. As to the Higgs searches at the LHC, the ATLAS and CMS collaborations have reported the exclusion of a Higgs mass in the interval 145 – 460 GeV [23, 24], which

In brief, for both <sup>Υ</sup> and *<sup>B</sup>* decays, the Higgs-DM coupling constant *<sup>λ</sup>*(4)

$$\mathcal{R}\_{\text{X}\_{\text{SM}}} \equiv \frac{\sigma \left(\text{gg} \to h\right) \text{Br} \left(h \to \text{X}\_{\text{SM}}\right)}{\sigma^{\text{(SM)}} \left(\text{gg} \to h\right) \text{Br}^{\text{SM}} \left(h \to \text{X}\_{\text{SM}}\right)} = \frac{\cos^{4} \theta}{\cos^{2} \theta + \Gamma \left(h \to \text{X}\_{\text{inv}}\right) / \Gamma\_{h}^{\text{SM}}}.\tag{36}$$

In this expression, *<sup>X</sup>*SM corresponds to all the Standard Model particles, *Xi*nv = *<sup>S</sup>*0*S*<sup>0</sup> and *<sup>S</sup>*1*S*1, *<sup>σ</sup>* is a cross-section, BrSM (*<sup>h</sup>* → *<sup>X</sup>*) the branching fraction of the SM Higgs decaying into any kinematically allowed mode *X*, and ΓSM *<sup>h</sup>* the total Higgs decay rate in the Standard Model. To open up the region *mh* <sup>&</sup>gt; 140GeV requires the ratio <sup>R</sup>*X*SM to be smaller than 0.25 [23, 24], a constraint easily fulfilled in our model. By comparison, the minimal extensions of the Standard Model with just one singlet scalar or a Majorana fermion, even under a *Z*<sup>2</sup> symmetry, are highly constrained in this regard [44]. Finally, if the recent data from ATLAS and CMS turn out to be a signal for a SM-like Higgs with mass about 125GeV, then this will put a very strong constraint on the mixing angle *θ*. Indeed, only for *θ* 0.50 will the ratio Γ(*h* → *b*¯ *b*)/Γ (*h* → inv) 1. For larger values, the model is ruled out, independently of the dark matter mass, but as long as *m*<sup>1</sup> 50GeV, which we are assuming in this work.

**Figure 8.** The ratio Γ(*h* → *b*¯ *b*)/Γ(*h* → inv) is larger than one for *θ* = 0.4<sup>0</sup> and *mh* = 125GeV.

Finally, it is important to find the bounds on the mass SM Higgs that can satisfy the triviality and perturbativity constraints on the coupling constants in the scalar sector of this model up to a scale higher then 1TeV. This requires studying the renormalization group equations of these coupling constants [45].

## **Author details**

Abdessamad Abada1,2 and Salah Nasri<sup>2</sup>

<sup>⋆</sup> Address all correspondence to: a.abada@uaeu.ac.ae; snasri@uaeu.ac.ae

1 Laboratoire de Physique des Particules et Physique Statistique, Ecole Normale Supérieure, BP 92 Vieux Kouba, Alger, Algeria

[11] Barger, V., Langacker, P., McCaskey, M., Ramsey-Musolf, M, and Shaughnessy, G.;

Light Cold Dark Matter in a Two-Singlet Model

http://dx.doi.org/10.5772/52243

289

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2 Physics Department, United Arab Emirates University, POB, Al Ain, United Arab Emirates

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18 Open Questions in Cosmology

Abdessamad Abada1,2 and Salah Nasri<sup>2</sup>

BP 92 Vieux Kouba, Alger, Algeria

(arXiv:hep-ph/9506380).

<sup>⋆</sup> Address all correspondence to: a.abada@uaeu.ac.ae; snasri@uaeu.ac.ae

1 Laboratoire de Physique des Particules et Physique Statistique, Ecole Normale Supérieure,

2 Physics Department, United Arab Emirates University, POB, Al Ain, United Arab Emirates

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**Chapter 12**

**Provisional chapter**

**Where Is the PdV in the First Law of Black Hole**

**Where Is the PdV in the First Law of Black Hole**

Ever since Hawking's discovery in 1974, [1–3], that black holes have a temperature associated

*<sup>T</sup>* <sup>=</sup> *<sup>h</sup>*¯

*S* = *α*

squared. In the classical limit the temperature vanishes and the entropy diverges.

where *α* is an undetermined constant, presumed of order one, and ¯*hG* is the Planck length

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

*A*

©2012 Dolan, 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 Dolan; 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.

(we use units in which *c* = 1), the thermodynamics of black holes has been a fascinating area of research. Equation (1) immediately implies that a Schwarzschild black hole in isolation is unstable: it will radiate and in so doing loses energy hence the mass decreases, thus increasing the temperature causing it to radiate with more power leading to a runaway effect. Hawking's result is fundamentally quantum mechanical in nature and came after a number of important developments in the classical thermodynamics of black holes. Penrose [4] realised that the mass of a rotating black hole can decrease, when rotational energy is extracted, and this was followed by the observation that the area never decreases in any classical process. Nevertheless there is still a minimum, irreducible, mass below which one cannot go classically [5, 6]. This lead Bekenstein's to propose that an entropy should be associated with a black hole that is proportional to the area, *A*, of the event horizon, [7, 8].

<sup>8</sup>*πGM* (1)

*hG*¯ , (2)

to them, in the simplest case a temperature inversely proportional to their mass,

**Thermodynamics?**

**Thermodynamics?**

http://dx.doi.org/10.5772/52455

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Brian P. Dolan

**1. Introduction**

In natural units,

Brian P. Dolan


**Provisional chapter**

## **Where Is the PdV in the First Law of Black Hole Thermodynamics? Thermodynamics?**

**Where Is the PdV in the First Law of Black Hole**

Brian P. Dolan Additional information is available at the end of the chapter

Brian P. Dolan

20 Open Questions in Cosmology

290 Open Questions in Cosmology

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[45] Abada, A., and Nasri, S., work in progress.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52455

## **1. Introduction**

Ever since Hawking's discovery in 1974, [1–3], that black holes have a temperature associated to them, in the simplest case a temperature inversely proportional to their mass,

$$T = \frac{\hbar}{8\pi GM} \tag{1}$$

(we use units in which *c* = 1), the thermodynamics of black holes has been a fascinating area of research. Equation (1) immediately implies that a Schwarzschild black hole in isolation is unstable: it will radiate and in so doing loses energy hence the mass decreases, thus increasing the temperature causing it to radiate with more power leading to a runaway effect.

Hawking's result is fundamentally quantum mechanical in nature and came after a number of important developments in the classical thermodynamics of black holes. Penrose [4] realised that the mass of a rotating black hole can decrease, when rotational energy is extracted, and this was followed by the observation that the area never decreases in any classical process. Nevertheless there is still a minimum, irreducible, mass below which one cannot go classically [5, 6]. This lead Bekenstein's to propose that an entropy should be associated with a black hole that is proportional to the area, *A*, of the event horizon, [7, 8]. In natural units,

*S* = *α A hG*¯ , (2)

where *α* is an undetermined constant, presumed of order one, and ¯*hG* is the Planck length squared. In the classical limit the temperature vanishes and the entropy diverges.

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 Dolan; 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 Dolan, licensee InTech. This is an open access chapter distributed under the terms of the Creative

The first law of black hole thermodynamics, in its simplest form, associates the internal energy of a black hole with the mass, *U*(*S*) = *M*, (more precisely the ADM mass, as defined with reference to the time-like Killing vector at infinity [9]) and reads

$$d\mathcal{U} = TdS.\tag{3}$$

as a specific function of *S* is liable to lead to inconsistencies. We shall see below how this

Where Is the PdV in the First Law of Black Hole Thermodynamics?

http://dx.doi.org/10.5772/52455

293

From the point of view of Eintein's equations a pressure is associated with a cosmological constant. There is now very strong evidence that the cosmological constant in our Universe is positive [10, 11]. This poses a problem for the study of black hole thermodynamics for two reasons: firstly there is no asymptotic regime in de Sitter space which allows the unambiguous identification of the ADM mass of a black hole embedded in a space with a positive Λ; secondly positive Λ corresponds to negative pressure, implying thermodynamic instability. The first problem is related to the fact that there are two event horizons for a de Sitter black hole, a black hole horizon and a cosmological horizon, and the radial co-ordinate is time-like for large enough values of *r*, outside the cosmological horizon. The second problem is not necessarily too serious as one can still glean some information from negative pressure systems which are thermodynamically unstable [12] (instability is not an insurmountable barrier to obtaining physical information from a thermodynamic system, after all, as described above, Hawking's formula (1) shows that black holes can have negative heat capacity but it is still a central formula in the understanding of black hole thermodynamics). In contrast for negative Λ there is no cosmological horizon and the pressure is positive, the thermodynamics is perfectly well defined, so we shall restrict our considerations here to negative <sup>Λ</sup> and identify the thermodynamic pressure *<sup>P</sup>* = − <sup>Λ</sup>

The notion that the cosmological constant should be thought of as a thermodynamic variable is not new, and its thermodynamic conjugate is often denoted Θ in the literature, [13–21], but

It may seem a little surprising to elevate Λ to the status of a thermodynamic variable. Λ is usually thought of as a coupling constant in the Einstein action, on the same footing as Newton's constant, and it would seem bizarre to think of Newton's constant as a thermodynamic variable. However the nature of Λ has long been mysterious [22] and we should keep an open mind as to its physical interpretation. Indeed in [23] it was argued that Λ must be included in the pantheon of thermodynamic variables for consistency with the Smarr relation [24], which is essentially dimensional analysis applied to thermodynamic functions. Furthermore [23] suggested that, for a black hole embedded in anti-de Sitter (AdS) space-time, the black hole mass is more correctly interpreted as the enthalpy, *H* beloved of

The *PV* term in this equation can be though of as the contribution to the mass-energy of the black hole due the negative energy density of the vacuum, *ǫ* = −*P*, associated with a positive cosmological constant. If the black hole has volume *V* then it contains energy *ǫV* = −*PV*

This interpretation forces us to face up to the definition of the black hole volume. In [23] *V* is defined as the volume relative to that of empty AdS space-time: the black hole volume is the volume excluded from empty AdS when the black hole is introduced. We shall refer to

*M* = *H*(*S*, *P*) = *U*(*S*, *V*) + *PV*. (9)

the fluid dynamical pressure appearing in Einstein's equations.

Θ was not given a physical interpretation in these works.

chemists, rather than the more traditional internal energy,

and so the total energy is *U* = *M* − *PV*.

<sup>8</sup>*π<sup>G</sup>* with

potential problem is avoided.

**2. Pressure and enthalpy**

In particular, for a Schwarzschild black hole, the event horizon radius is *rh* = <sup>2</sup>*GM* and the event horizon area is

$$A = 4\pi r\_h^2 = 16\pi G^2 M^2 \qquad \Rightarrow \qquad M = \frac{1}{4G} \sqrt{\frac{A}{\pi}}.\tag{4}$$

Hence

$$
\mathcal{U} = \frac{1}{4} \sqrt{\frac{\hbar \mathcal{S}}{\pi a \mathcal{G}}} \tag{5}
$$

and

$$T = \frac{\partial U}{\partial S} = \frac{1}{8} \sqrt{\frac{\hbar}{\pi \alpha G S}} = \frac{\hbar}{32 \pi \alpha G M}. \tag{6}$$

Hawking's result (1) then fixes the constant of proportionality in (2) to be one quarter.

The black hole instability referred to above is reflected in the thermodynamic potentials by the fact that the heat capacity of a Schwarzschild black hole,

$$\mathcal{C} = T \frac{\partial \mathcal{S}}{\partial T} = -\frac{\hbar \mathcal{G}}{8\pi T^2} < 0,\tag{7}$$

is negative.

The first law generalises to electrically charged, rotating black holes as

$$d\mathcal{U} = TdS + \Omega d\mathcal{J} + \Phi d\mathcal{Q} \tag{8}$$

where *J* is the angular momentum of the black hole, Ω its angular velocity, and *Q* the electric charge and the electrostatic potential (see e.g. [9]).

In contrast to elementary treatments of the first law of black hole thermodynamics it is noteworthy that (8) lacks the familiar *PdV* term, but a little thought shows that it is by no means obvious how to define the volume of a black hole. For a Schwarzschild black hole the radial co-ordinate, r, is time-like inside the event horizon, where *r* < *rh*, so it would seem non-sensical to associate a volume *<sup>V</sup>* = <sup>4</sup>*<sup>π</sup> rh* <sup>0</sup> *<sup>r</sup>*2*dr* <sup>=</sup> <sup>4</sup>*<sup>π</sup>* <sup>3</sup> *<sup>r</sup>*<sup>3</sup> *<sup>h</sup>* with the black hole. In fact identifying any function of *rh* alone with a volume, *<sup>V</sup>*(*rh*), will lead to inconsistencies in a thermodynamic description since the area, and hence the entropy, is already a function of *rh*, *<sup>S</sup>* = *<sup>π</sup>r*<sup>2</sup> *<sup>h</sup>*, so any volume *<sup>V</sup>*(*rh*) would be determined purely in terms of the entropy. The internal energy, *U*(*S*, *V*), should be a function of two variables, so giving *V*(*S*) uniquely as a specific function of *S* is liable to lead to inconsistencies. We shall see below how this potential problem is avoided.

## **2. Pressure and enthalpy**

2 Open Questions in Cosmology

event horizon area is

Hence

and

is negative.

of *rh*, *<sup>S</sup>* = *<sup>π</sup>r*<sup>2</sup>

The first law of black hole thermodynamics, in its simplest form, associates the internal energy of a black hole with the mass, *U*(*S*) = *M*, (more precisely the ADM mass, as defined

In particular, for a Schwarzschild black hole, the event horizon radius is *rh* = <sup>2</sup>*GM* and the

*<sup>h</sup>* <sup>=</sup> <sup>16</sup>*πG*2*M*<sup>2</sup> <sup>⇒</sup> *<sup>M</sup>* <sup>=</sup> <sup>1</sup>

*hS*¯

*παGS* <sup>=</sup> *<sup>h</sup>*¯

*<sup>U</sup>* <sup>=</sup> <sup>1</sup> 4

*h*¯

Hawking's result (1) then fixes the constant of proportionality in (2) to be one quarter.

The black hole instability referred to above is reflected in the thermodynamic potentials by

*<sup>∂</sup><sup>T</sup>* <sup>=</sup> <sup>−</sup> *hG*¯

where *J* is the angular momentum of the black hole, Ω its angular velocity, and *Q* the electric

In contrast to elementary treatments of the first law of black hole thermodynamics it is noteworthy that (8) lacks the familiar *PdV* term, but a little thought shows that it is by no means obvious how to define the volume of a black hole. For a Schwarzschild black hole the radial co-ordinate, r, is time-like inside the event horizon, where *r* < *rh*, so it would

fact identifying any function of *rh* alone with a volume, *<sup>V</sup>*(*rh*), will lead to inconsistencies in a thermodynamic description since the area, and hence the entropy, is already a function

The internal energy, *U*(*S*, *V*), should be a function of two variables, so giving *V*(*S*) uniquely

*dU* = *TdS*. (3)

*A*

*πα<sup>G</sup>* (5)

<sup>8</sup>*πT*<sup>2</sup> <sup>&</sup>lt; 0, (7)

*dU* = *TdS* + Ω*d J* + Φ*dQ* (8)

<sup>0</sup> *<sup>r</sup>*2*dr* <sup>=</sup> <sup>4</sup>*<sup>π</sup>*

*<sup>h</sup>*, so any volume *<sup>V</sup>*(*rh*) would be determined purely in terms of the entropy.

<sup>3</sup> *<sup>r</sup>*<sup>3</sup>

*<sup>h</sup>* with the black hole. In

<sup>32</sup>*παGM*. (6)

*<sup>π</sup>* . (4)

4*G*

with reference to the time-like Killing vector at infinity [9]) and reads

*A* = 4*πr* 2

*<sup>T</sup>* <sup>=</sup> *<sup>∂</sup><sup>U</sup>*

the fact that the heat capacity of a Schwarzschild black hole,

charge and the electrostatic potential (see e.g. [9]).

seem non-sensical to associate a volume *<sup>V</sup>* = <sup>4</sup>*<sup>π</sup> rh*

*<sup>∂</sup><sup>S</sup>* <sup>=</sup> <sup>1</sup> 8

*<sup>C</sup>* <sup>=</sup> *<sup>T</sup> <sup>∂</sup><sup>S</sup>*

The first law generalises to electrically charged, rotating black holes as

From the point of view of Eintein's equations a pressure is associated with a cosmological constant. There is now very strong evidence that the cosmological constant in our Universe is positive [10, 11]. This poses a problem for the study of black hole thermodynamics for two reasons: firstly there is no asymptotic regime in de Sitter space which allows the unambiguous identification of the ADM mass of a black hole embedded in a space with a positive Λ; secondly positive Λ corresponds to negative pressure, implying thermodynamic instability. The first problem is related to the fact that there are two event horizons for a de Sitter black hole, a black hole horizon and a cosmological horizon, and the radial co-ordinate is time-like for large enough values of *r*, outside the cosmological horizon. The second problem is not necessarily too serious as one can still glean some information from negative pressure systems which are thermodynamically unstable [12] (instability is not an insurmountable barrier to obtaining physical information from a thermodynamic system, after all, as described above, Hawking's formula (1) shows that black holes can have negative heat capacity but it is still a central formula in the understanding of black hole thermodynamics). In contrast for negative Λ there is no cosmological horizon and the pressure is positive, the thermodynamics is perfectly well defined, so we shall restrict our considerations here to negative <sup>Λ</sup> and identify the thermodynamic pressure *<sup>P</sup>* = − <sup>Λ</sup> <sup>8</sup>*π<sup>G</sup>* with the fluid dynamical pressure appearing in Einstein's equations.

The notion that the cosmological constant should be thought of as a thermodynamic variable is not new, and its thermodynamic conjugate is often denoted Θ in the literature, [13–21], but Θ was not given a physical interpretation in these works.

It may seem a little surprising to elevate Λ to the status of a thermodynamic variable. Λ is usually thought of as a coupling constant in the Einstein action, on the same footing as Newton's constant, and it would seem bizarre to think of Newton's constant as a thermodynamic variable. However the nature of Λ has long been mysterious [22] and we should keep an open mind as to its physical interpretation. Indeed in [23] it was argued that Λ must be included in the pantheon of thermodynamic variables for consistency with the Smarr relation [24], which is essentially dimensional analysis applied to thermodynamic functions. Furthermore [23] suggested that, for a black hole embedded in anti-de Sitter (AdS) space-time, the black hole mass is more correctly interpreted as the enthalpy, *H* beloved of chemists, rather than the more traditional internal energy,

$$M = H(S, P) = \mathcal{U}(S, V) + PV. \tag{9}$$

The *PV* term in this equation can be though of as the contribution to the mass-energy of the black hole due the negative energy density of the vacuum, *ǫ* = −*P*, associated with a positive cosmological constant. If the black hole has volume *V* then it contains energy *ǫV* = −*PV* and so the total energy is *U* = *M* − *PV*.

This interpretation forces us to face up to the definition of the black hole volume. In [23] *V* is defined as the volume relative to that of empty AdS space-time: the black hole volume is the volume excluded from empty AdS when the black hole is introduced. We shall refer to this as the 'geometric volume' below. Other suggestions for the volume of a black hole have been made in [25, 26]

An alternative definition of the black-volume is that it is the thermodynamic conjugate of the pressure, under the Legendre transform (9),

$$V \coloneqq \frac{\partial H}{\partial P}' \tag{10}$$

and, for negative Λ, the ADM mass is *M* = *m* [15], which, following the philosophy of [23],

, (17)

http://dx.doi.org/10.5772/52455

Where Is the PdV in the First Law of Black Hole Thermodynamics?

. (18)

*h*)

(19)

295

4*πrh*

<sup>3</sup> . (20)

we identify with the enthalpy, *H*(*S*, *P*). Solving (15) for *m* immediately yields

*<sup>m</sup>* <sup>=</sup> *rh* 2 <sup>1</sup> <sup>−</sup> <sup>Λ</sup> 3 *r* 2 *h* 

> 2 *S π* 1 <sup>2</sup> 1 + 8*SP* 3

The usual thermodynamic relations can now be used to determine the temperature and the

1 2

*<sup>π</sup>* <sup>=</sup> <sup>4</sup>*πr*<sup>3</sup> *h*

(<sup>1</sup> <sup>+</sup> <sup>8</sup>*PS*) = (<sup>1</sup> <sup>−</sup> <sup>Λ</sup>*r*<sup>2</sup>

4 1 *πS*

3 *S* 3 2 √

That the resulting thermodynamic volume (for a non-rotating black hole) is identical to the geometric volume is quite remarkable, but appears co-incidental as this equality no longer holds for rotating (Kerr-AdS) black holes, as we shall see. It does however hold

As mentioned in the introduction, equation (20) has a potential problem associated with it, in that it implies that the volume and the entropy cannot be considered to be independent thermodynamic variables, *S* determines *V* uniquely – they cannot be varied independently and so *V* seems redundant. Indeed this may the reason why *V* was never considered in the early literature on black hole thermodynamics. But this is an artifact of the non-rotating approximation, *V* and *S* can, and should, be considered to be independent variables for a

The line element for a charged rotating black hole in 4-dimensional AdS space is [29]

<sup>∆</sup> *dr*<sup>2</sup> <sup>+</sup> *<sup>ρ</sup>*<sup>2</sup> ∆*θ*

*dθ*<sup>2</sup> +

*<sup>L</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>*mr* <sup>+</sup> *<sup>q</sup>*2, <sup>∆</sup>*<sup>θ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

*<sup>L</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>8*πP*.

<sup>∆</sup>*<sup>θ</sup>* sin2 *<sup>θ</sup> ρ*2

*adt* <sup>−</sup> *<sup>r</sup>*<sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup>

*<sup>L</sup>*<sup>2</sup> cos<sup>2</sup> *<sup>θ</sup>*,

*<sup>L</sup>*<sup>2</sup> , (22)

<sup>Ξ</sup> *<sup>d</sup><sup>φ</sup>*

2

, (21)

from which *H*(*S*, *P*) = *M* = *m*, with (16), identifies the enthalpy as

volume,

rotating black hole.

*ds*<sup>2</sup> <sup>=</sup> <sup>−</sup> <sup>∆</sup>

where

*ρ*2

*dt* <sup>−</sup> *<sup>a</sup>* sin<sup>2</sup> *<sup>θ</sup>*

*ρ*<sup>2</sup> = *r*

and the cosmological constant is Λ = − <sup>3</sup>

<sup>Ξ</sup> *<sup>d</sup><sup>φ</sup>*

<sup>∆</sup> <sup>=</sup> (*r*<sup>2</sup> <sup>+</sup> *<sup>a</sup>*2)(*L*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*2)

2 <sup>+</sup> *<sup>ρ</sup>*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> cos2 *<sup>θ</sup>*, <sup>Ξ</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

*T* =

*V* =

 *∂H ∂S P*

 *∂H ∂P S*

for non-rotating black holes in all dimensions [28].

*<sup>H</sup>*(*S*, *<sup>P</sup>*) = <sup>1</sup>

<sup>⇒</sup> *<sup>T</sup>* <sup>=</sup> <sup>1</sup>

<sup>⇒</sup> *<sup>V</sup>* <sup>=</sup> <sup>4</sup>

which we shall call the 'thermodynamic volume'.

With the definition of the thermodynamic volume (10) we are in a position to state the definitive version of the first law of black hole thermodynamics,

$$\begin{array}{c}\hline\hline d\mathcal{U} = TdS + \Omega d\mathcal{J} + \Phi d\mathcal{Q} - Pd\mathcal{V} \\\\ \hline \end{array} \tag{11}$$

which follows from the Legendre transform of

$$dM = dH = TdS + \Omega d\!\!/ + \Phi d\!\!/ Q + VdP.\tag{12}$$

Equation (12), in Θ *d*Λ notation, appeared in [27].

### **3. Thermodynamic volume**

The suggested definition of the thermodynamic volume (10) must be tested for consistency. For example, for a non-rotating black hole in four-dimensional space-time, the line element is given, in Schwarzschild co-ordinates, by,

$$ds^2s = -f(r)dt^2 + f^{-1}(r)dr^2 + r^2d\Omega^2,\tag{13}$$

with

$$f(r) = 1 - \frac{2m}{r} - \frac{\Lambda}{3}r^2,\tag{14}$$

and *d*Ω<sup>2</sup> = *dθ*<sup>2</sup> + sin2 *θdφ*<sup>2</sup> the solid angle area element.<sup>1</sup> The event horizon is defined by *<sup>f</sup>*(*rh*) = 0,

$$\frac{\Lambda}{3}r\_{\text{fl}}^3 - r\_{\text{fl}} + 2m = 0,\tag{15}$$

but we do not need to solve this equation explicitly in order to analyse (10). We already know that

$$S = \pi r\_{h'}^2 \qquad P = -\frac{\Lambda}{8\pi} \tag{16}$$

<sup>1</sup> From now on we set *G* = *h*¯ = 1 to avoid cluttering formulae.

and, for negative Λ, the ADM mass is *M* = *m* [15], which, following the philosophy of [23], we identify with the enthalpy, *H*(*S*, *P*). Solving (15) for *m* immediately yields

$$m = \frac{r\_h}{2} \left( 1 - \frac{\Lambda}{3} r\_h^2 \right),$$

from which *H*(*S*, *P*) = *M* = *m*, with (16), identifies the enthalpy as

$$H(\mathcal{S}, P) = \frac{1}{2} \left(\frac{\mathcal{S}}{\pi}\right)^{\frac{1}{2}} \left(1 + \frac{8SP}{3}\right). \tag{18}$$

The usual thermodynamic relations can now be used to determine the temperature and the volume,

$$T = \left(\frac{\partial H}{\partial S}\right)\_P \quad \Rightarrow \quad T = \frac{1}{4} \left(\frac{1}{\pi S}\right)^{\frac{1}{2}} (1 + 8PS) = \frac{(1 - \Lambda r\_h^2)}{4\pi r\_h} \tag{19}$$

$$V = \left(\frac{\partial H}{\partial P}\right)\_S \quad \Rightarrow \quad V = \frac{4}{3} \frac{S^{\frac{3}{2}}}{\sqrt{\pi}} = \frac{4\pi r\_h^{\frac{3}{2}}}{3}. \tag{20}$$

That the resulting thermodynamic volume (for a non-rotating black hole) is identical to the geometric volume is quite remarkable, but appears co-incidental as this equality no longer holds for rotating (Kerr-AdS) black holes, as we shall see. It does however hold for non-rotating black holes in all dimensions [28].

As mentioned in the introduction, equation (20) has a potential problem associated with it, in that it implies that the volume and the entropy cannot be considered to be independent thermodynamic variables, *S* determines *V* uniquely – they cannot be varied independently and so *V* seems redundant. Indeed this may the reason why *V* was never considered in the early literature on black hole thermodynamics. But this is an artifact of the non-rotating approximation, *V* and *S* can, and should, be considered to be independent variables for a rotating black hole.

The line element for a charged rotating black hole in 4-dimensional AdS space is [29]

$$ds^2 = -\frac{\Delta}{\rho^2} \left( dt - \frac{a \sin^2 \theta}{\Sigma} d\phi \right)^2 + \frac{\rho^2}{\Delta} dr^2 + \frac{\rho^2}{\Delta\_\theta} d\theta^2 + \frac{\Delta\_\theta \sin^2 \theta}{\rho^2} \left( a dt - \frac{r^2 + a^2}{\Sigma} d\phi \right)^2,\tag{21}$$

where

4 Open Questions in Cosmology

been made in [25, 26]

pressure, under the Legendre transform (9),

which we shall call the 'thermodynamic volume'.

which follows from the Legendre transform of

Equation (12), in Θ *d*Λ notation, appeared in [27].

**3. Thermodynamic volume**

with

*<sup>f</sup>*(*rh*) = 0,

know that

is given, in Schwarzschild co-ordinates, by,

definitive version of the first law of black hole thermodynamics,

this as the 'geometric volume' below. Other suggestions for the volume of a black hole have

An alternative definition of the black-volume is that it is the thermodynamic conjugate of the

*<sup>V</sup>* :<sup>=</sup> *<sup>∂</sup><sup>H</sup>*

With the definition of the thermodynamic volume (10) we are in a position to state the

The suggested definition of the thermodynamic volume (10) must be tested for consistency. For example, for a non-rotating black hole in four-dimensional space-time, the line element

*<sup>d</sup>*2*<sup>s</sup>* <sup>=</sup> <sup>−</sup>*f*(*r*)*dt*<sup>2</sup> <sup>+</sup> *<sup>f</sup>* <sup>−</sup>1(*r*)*dr*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*

*<sup>f</sup>*(*r*) = <sup>1</sup> <sup>−</sup> <sup>2</sup>*<sup>m</sup>*

Λ 3 *r* 3

*S* = *πr* 2

<sup>1</sup> From now on we set *G* = *h*¯ = 1 to avoid cluttering formulae.

and *d*Ω<sup>2</sup> = *dθ*<sup>2</sup> + sin2 *θdφ*<sup>2</sup> the solid angle area element.1 The event horizon is defined by

but we do not need to solve this equation explicitly in order to analyse (10). We already

*<sup>h</sup>*, *<sup>P</sup>* <sup>=</sup> <sup>−</sup> <sup>Λ</sup>

*<sup>r</sup>* <sup>−</sup> <sup>Λ</sup> 3 *r*

*<sup>∂</sup><sup>P</sup>* , (10)

<sup>2</sup>*d*Ω2, (13)

2, (14)

<sup>8</sup>*<sup>π</sup>* (16)

*<sup>h</sup>* <sup>−</sup> *rh* <sup>+</sup> <sup>2</sup>*<sup>m</sup>* <sup>=</sup> 0, (15)

*dU* = *TdS* + Ω*d J* + Φ*dQ* − *PdV* (11)

*dM* = *dH* = *TdS* + Ω*d J* + Φ*dQ* + *VdP*. (12)

$$
\Delta = \frac{(r^2 + a^2)(L^2 + r^2)}{L^2} - 2mr + q^2, \qquad \Delta\_\theta = 1 - \frac{a^2}{L^2}\cos^2\theta,
$$

$$
\rho^2 = r^2 + a^2\cos^2\theta, \qquad \Xi = 1 - \frac{a^2}{L^2}.\tag{22}
$$

and the cosmological constant is Λ = − <sup>3</sup> *<sup>L</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>8*πP*. The physical properties of this space-time are well known [15]. The metric parameters *m* and *q* are related to the ADM mass *M* and the electric charge *Q* by

$$M = \frac{m}{\Xi^2}, \qquad Q = \frac{q}{\Xi}.\tag{23}$$

The temperature follows from

= 1 8*πH*

> *J* <sup>2</sup> <sup>≤</sup> *<sup>S</sup>*<sup>2</sup> 4*π*<sup>2</sup>

<sup>|</sup>*Jmax*<sup>|</sup> <sup>=</sup> *<sup>S</sup>*

From (10) and (29) the thermodynamic volume is [30]

is associated with an extremal black hole.

*<sup>V</sup>* <sup>=</sup> *<sup>∂</sup><sup>H</sup> ∂P S*,*J*,*Q*

which is manifestly positive.

and

[23].

2*π*

 1 +

> = 2 3*πH*

The angular velocity and the electric potential also follow from (29) via

Ω = *∂H ∂J S*,*Q*,*P*

The Smarr relation follows from (29), (30), (33), (34) and (35), namely

*H*

Φ = *∂H ∂Q S*,*J*,*P*

The maximum angular momentum,

 1 +

> 1 +

from which we immediately see that *T* ≥ 0 requires

*πQ*<sup>2</sup> *<sup>S</sup>* <sup>+</sup>

> *πQ*<sup>2</sup> *<sup>S</sup>* <sup>+</sup>

> > *πQ*<sup>2</sup> *<sup>S</sup>* <sup>+</sup>

> > > *S*

8*PS* 3

> 8*PS* 3

> > 8*PS* 3

*S* + *πQ*<sup>2</sup> +

<sup>=</sup> <sup>4</sup>*π*<sup>2</sup> *<sup>J</sup>*

<sup>=</sup> <sup>2</sup>*π<sup>Q</sup>*

<sup>2</sup> <sup>+</sup> *PV* <sup>−</sup> *ST* <sup>−</sup> *<sup>J</sup>*<sup>Ω</sup> <sup>−</sup> *<sup>Q</sup>*<sup>Φ</sup>

from which it is clear that the *PV*-term must be included for consistency, as pointed out in

 1 + <sup>8</sup>*PS* 3 

2*H* √

2*H* √

*<sup>S</sup>* + *<sup>π</sup>Q*<sup>2</sup> + <sup>8</sup>*PS*<sup>2</sup>

3 

<sup>1</sup> <sup>−</sup> *<sup>π</sup>Q*<sup>2</sup>

<sup>1</sup> <sup>−</sup> *<sup>π</sup>Q*<sup>2</sup>

<sup>1</sup> <sup>−</sup> *<sup>π</sup>Q*<sup>2</sup>

8*PS*<sup>2</sup> 3

+ 2*π*<sup>2</sup> *J* 2 

*<sup>π</sup><sup>S</sup>* (34)

*<sup>π</sup><sup>S</sup>* . (35)

<sup>2</sup> <sup>=</sup> 0, (36)

*<sup>S</sup>* <sup>+</sup> <sup>8</sup>*PS*

*<sup>S</sup>* <sup>+</sup> <sup>8</sup>*PS*

*<sup>S</sup>* <sup>+</sup> <sup>8</sup>*PS*

− 4*π*<sup>2</sup>

Where Is the PdV in the First Law of Black Hole Thermodynamics?

 *J S* 2 

http://dx.doi.org/10.5772/52455

. (31)

, (32)

, (33)

, (30)

297

*<sup>T</sup>* <sup>=</sup> *<sup>∂</sup><sup>H</sup> ∂S J*,*Q*,*P*

The event horizon, *r*+, lies at the largest root of ∆(*r*) = 0, so, in terms of geometrical parameters,

$$M = \frac{(r\_+^2 + a^2)(L^2 + r\_+^2) + q^2 L^2}{2r\_+ L^2 \Xi^2} \tag{24}$$

and the area of the event horizon is

$$A = 4\pi \frac{r\_+^2 + a^2}{\Xi},\tag{25}$$

giving

$$S = \pi \frac{r\_+^2 + a^2}{\Xi}.\tag{26}$$

The angular momentum is *J* = *aM* and the relevant thermodynamic angular velocity is

$$
\Omega = \frac{a(L^2 + r\_+^2)}{L^2(r\_+^2 + a^2)}.\tag{27}
$$

As explained in [27], Ω here is the difference between the asymptotic angular velocity and the angular velocity at the black hole outer horizon.

The electrostatic potential, again the difference between the potential at infinity and at the horizon, is

$$
\Phi = \frac{qr\_+}{r\_+^2 + a^2}.\tag{28}
$$

To determine the thermodynamic properties, *M* must be expressed in terms of *S*, *J*, *Q* and *P* (or, equivalently, *L*). This was done in [27] and the result is

$$H(\mathcal{S}, P, f, \mathcal{Q}) := \frac{1}{2} \sqrt{\frac{\left(\mathcal{S} + \pi Q^2 + \frac{8\mathcal{P}\mathcal{S}^2}{3}\right)^2 + 4\pi^2 \left(1 + \frac{8\mathcal{P}\mathcal{S}}{3}\right) f^2}{\pi \mathcal{S}}}.\tag{29}$$

This generalises the Christodoulou-Ruffini formula [5, 6] for the mass of a rotating black hole in terms of its irreducible mass, *Mirr*. (The irreducible mass for a black hole with entropy *S* is the mass of a Schwarzschild black hole with the same entropy, *M*<sup>2</sup> *irr* <sup>=</sup> *<sup>S</sup>* <sup>4</sup>*<sup>π</sup>* ).

The temperature follows from

6 Open Questions in Cosmology

parameters,

giving

horizon, is

and the area of the event horizon is

the angular velocity at the black hole outer horizon.

(or, equivalently, *L*). This was done in [27] and the result is

2 

is the mass of a Schwarzschild black hole with the same entropy, *M*<sup>2</sup>

*<sup>H</sup>*(*S*, *<sup>P</sup>*, *<sup>J</sup>*, *<sup>Q</sup>*) :<sup>=</sup> <sup>1</sup>

The physical properties of this space-time are well known [15]. The metric parameters *m* and

The event horizon, *r*+, lies at the largest root of ∆(*r*) = 0, so, in terms of geometrical

<sup>+</sup> + *a*2)(*L*<sup>2</sup> + *r*<sup>2</sup>

*r*2 <sup>+</sup> + *a*<sup>2</sup>

*A* = 4*π*

*S* = *π r*2 <sup>+</sup> + *a*<sup>2</sup>

The angular momentum is *J* = *aM* and the relevant thermodynamic angular velocity is

<sup>Ω</sup> <sup>=</sup> *<sup>a</sup>*(*L*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*<sup>2</sup>

*L*2(*r*<sup>2</sup>

As explained in [27], Ω here is the difference between the asymptotic angular velocity and

The electrostatic potential, again the difference between the potential at infinity and at the

Φ = *qr*<sup>+</sup> *r*2

To determine the thermodynamic properties, *M* must be expressed in terms of *S*, *J*, *Q* and *P*

*S* + *πQ*<sup>2</sup> + <sup>8</sup>*PS*<sup>2</sup>

This generalises the Christodoulou-Ruffini formula [5, 6] for the mass of a rotating black hole in terms of its irreducible mass, *Mirr*. (The irreducible mass for a black hole with entropy *S*

3 2

+ 4*π*<sup>2</sup> 1 + <sup>8</sup>*PS* 3 *J*2 *<sup>π</sup><sup>S</sup>* . (29)

+)

<sup>+</sup> + *a*2)

<sup>+</sup>) + *q*2*L*<sup>2</sup>

<sup>Ξ</sup><sup>2</sup> , *<sup>Q</sup>* <sup>=</sup> *<sup>q</sup>*

Ξ. (23)

<sup>2</sup>*r*+*L*<sup>2</sup> <sup>Ξ</sup><sup>2</sup> (24)

Ξ , (25)

Ξ . (26)

<sup>+</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> . (28)

*irr* <sup>=</sup> *<sup>S</sup>* <sup>4</sup>*<sup>π</sup>* ).

. (27)

*<sup>M</sup>* <sup>=</sup> *<sup>m</sup>*

*q* are related to the ADM mass *M* and the electric charge *Q* by

*<sup>M</sup>* <sup>=</sup> (*r*<sup>2</sup>

$$T = \left. \frac{\partial H}{\partial S} \right|\_{I,Q,P} = \frac{1}{8\pi H} \left[ \left( 1 + \frac{\pi Q^2}{S} + \frac{8PS}{3} \right) \left( 1 - \frac{\pi Q^2}{S} + 8PS \right) - 4\pi^2 \left( \frac{I}{S} \right)^2 \right],\tag{30}$$

from which we immediately see that *T* ≥ 0 requires

$$J^2 \le \frac{S^2}{4\pi^2} \left( 1 + \frac{\pi Q^2}{S} + \frac{8PS}{3} \right) \left( 1 - \frac{\pi Q^2}{S} + 8PS \right). \tag{31}$$

The maximum angular momentum,

$$|j\_{\max}| = \frac{\mathcal{S}}{2\pi} \sqrt{\left(1 + \frac{\pi Q^2}{\mathcal{S}} + \frac{8PS}{\mathcal{S}}\right) \left(1 - \frac{\pi Q^2}{\mathcal{S}} + 8PS\right)}\tag{32}$$

is associated with an extremal black hole.

From (10) and (29) the thermodynamic volume is [30]

$$V = \left. \frac{\partial H}{\partial P} \right|\_{S,l,Q} = \frac{2}{3\pi H} \left[ S \left( S + \pi Q^2 + \frac{8PS^2}{3} \right) + 2\pi^2 l^2 \right] \tag{33}$$

which is manifestly positive.

The angular velocity and the electric potential also follow from (29) via

$$
\Omega = \left. \frac{\partial H}{\partial f} \right|\_{S,Q,P} = \frac{4\pi^2 f \left( 1 + \frac{8PS}{3} \right)}{2H\sqrt{\pi S}} \tag{34}
$$

and

$$\Phi = \left. \frac{\partial H}{\partial Q} \right|\_{S,I,P} = \frac{2\pi Q \left( S + \pi Q^2 + \frac{8PS^2}{3} \right)}{2H\sqrt{\pi S}}.\tag{35}$$

The Smarr relation follows from (29), (30), (33), (34) and (35), namely

$$\frac{H}{2} + PV - ST - J\Omega - \frac{Q\Phi}{2} = 0,\tag{36}$$

from which it is clear that the *PV*-term must be included for consistency, as pointed out in [23].

It is clear from (33) that, in general, *V* is a function of all the four independent thermodynamical variables, *S*, *P*, *J* and *Q*, but for the limiting case *J* = 0,

$$V = \frac{4}{3} \frac{S^{\frac{3}{2}}}{\sqrt{\pi}}.\tag{37}$$

**4. The First Law**

Note that the discriminant,

is positive.

Now

where

first write the enthalpy (29) in the form

To examine the consequences of the *PdV* term in the first law we need to perform a Legendre transform on the enthalpy to obtain the internal energy *U*(*S*, *V*, *J*, *Q*) from *U* = *H* − *PV*. We

*a* + *bP* + *cP*2, (42)

Where Is the PdV in the First Law of Black Hole Thermodynamics?

http://dx.doi.org/10.5772/52455

(43)

299

, (44)

<sup>2</sup>*<sup>c</sup>* . (45)

*<sup>V</sup>*<sup>2</sup> <sup>−</sup> *<sup>c</sup>* . (46)

, (47)

= 0, (48)

*H* =

 *S π S π* + *Q*<sup>2</sup> + 2*J* 2 

<sup>2</sup> *S π* 3 .

> <sup>9</sup> *<sup>J</sup>* 2 *J* <sup>2</sup> <sup>+</sup> *SQ*<sup>2</sup> *π*

<sup>2</sup>*<sup>H</sup>* <sup>⇒</sup> *<sup>P</sup>* <sup>=</sup> <sup>2</sup>*HV* <sup>−</sup> *<sup>b</sup>*

4*π* 3

*<sup>b</sup>*<sup>2</sup> <sup>−</sup> <sup>4</sup>*ac* <sup>=</sup> <sup>64</sup>*π*<sup>2</sup>

<sup>=</sup> *<sup>b</sup>* <sup>+</sup> <sup>2</sup>*cP*

*<sup>H</sup>* <sup>=</sup> <sup>1</sup> 2

*V*<sup>2</sup> ≥ *c* =

*<sup>b</sup>*<sup>2</sup> <sup>−</sup> <sup>4</sup>*ac* <sup>=</sup> <sup>64</sup>*π*<sup>2</sup>

4*π* 3

<sup>9</sup> *<sup>J</sup>* 2 *J* <sup>2</sup> <sup>+</sup> *SQ*<sup>2</sup> *π* 

*b*<sup>2</sup> − 4*ac*

<sup>2</sup> *S π* 3

*<sup>a</sup>* :<sup>=</sup> *<sup>π</sup> S* 1 4 *S π* + *Q*<sup>2</sup> 2 + *J* 2 

*<sup>b</sup>* :<sup>=</sup> <sup>4</sup>*<sup>π</sup>* 3

*c* :=

*<sup>V</sup>* <sup>=</sup> *<sup>∂</sup><sup>H</sup> ∂P S*,*J*,*Q*

This allows us to re-express *H* as a function of *V*,

We can immediately conclude that

with equality only when

is determined purely in terms of *S* alone, independent of both *P* and *Q*. Thus, as explained in the introduction, *V* and *S* cannot be viewed as thermodynamically independent variables as *J* → 0, rendering the description in terms of the thermodynamic potential *U*(*S*, *J*) impossible in this limit.

Expressing the thermodynamic volume (33) in terms of geometrical variables one gets [30]

$$V = \frac{2\pi}{3} \left\{ \frac{(r\_+^2 + a^2) \left(2r\_+^2 L^2 + a^2 L^2 - r\_+^2 a^2\right) + L^2 q^2 a^2}{L^2 \Sigma^2 r\_+} \right\}. \tag{38}$$

Given that the area of the event horizon is

$$A = 4\pi \frac{r\_+^2 + a^2}{\Xi} \tag{39}$$

then, if we define a naïve volume

$$V\_0 := \frac{r\_+ A}{3} = \frac{4\pi}{3} \frac{r\_+ (r\_+^2 + a^2)}{\Xi},\tag{40}$$

equations (24) and (38) give

$$V = V\_0 + \frac{4\pi a^2 M}{3} = V\_0 + \frac{4\pi}{3} \frac{J^2}{M} \tag{41}$$

a formula first derived in [31]. As pointed out in that reference, equation (40) implies that the surface to volume ratio of a black hole is always less than that of a sphere with radius *r*+ in Euclidean geometry. This is the opposite of our usual intuition that a sphere has the smallest surface to volume ratio of any closed surface — the isoperimetric inequality of Euclidean geometry. Thus the surface to volume ratio of a black hole satisfies the reverse of the usual isoperimetric inequality (a similar result holds in higher dimensions [31]). At least this seems to be the case if quantum gravity effects are not taken into account. In one case where quantum gravity corrections can be calculated using the techniques in [32] , the three-dimensional Bañados–Zanelli–Teitelboim (BTZ) black hole [33], they tend to reduce the black hole volume [28] so it seems possible that quantum gravity effects may affect the reverse isoperimetric inequality.

#### **4. The First Law**

To examine the consequences of the *PdV* term in the first law we need to perform a Legendre transform on the enthalpy to obtain the internal energy *U*(*S*, *V*, *J*, *Q*) from *U* = *H* − *PV*. We first write the enthalpy (29) in the form

$$H = \sqrt{a + bP + cP^2} \,\text{.}\tag{42}$$

where

8 Open Questions in Cosmology

in this limit.

*<sup>V</sup>* <sup>=</sup> <sup>2</sup>*<sup>π</sup>* 3

Given that the area of the event horizon is

then, if we define a naïve volume

equations (24) and (38) give

reverse isoperimetric inequality.

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

<sup>+</sup> + *a*2) 2*r*<sup>2</sup>

*<sup>V</sup>*<sup>0</sup> :<sup>=</sup> *<sup>r</sup>*<sup>+</sup> *<sup>A</sup>*

*<sup>V</sup>* = *<sup>V</sup>*<sup>0</sup> +

It is clear from (33) that, in general, *V* is a function of all the four independent

*<sup>V</sup>* <sup>=</sup> <sup>4</sup> 3 *S* 3 2 √

is determined purely in terms of *S* alone, independent of both *P* and *Q*. Thus, as explained in the introduction, *V* and *S* cannot be viewed as thermodynamically independent variables as *J* → 0, rendering the description in terms of the thermodynamic potential *U*(*S*, *J*) impossible

Expressing the thermodynamic volume (33) in terms of geometrical variables one gets [30]

*A* = 4*π*

<sup>3</sup> <sup>=</sup> <sup>4</sup>*<sup>π</sup>* 3

4*πa*2*M*

a formula first derived in [31]. As pointed out in that reference, equation (40) implies that the surface to volume ratio of a black hole is always less than that of a sphere with radius *r*+ in Euclidean geometry. This is the opposite of our usual intuition that a sphere has the smallest surface to volume ratio of any closed surface — the isoperimetric inequality of Euclidean geometry. Thus the surface to volume ratio of a black hole satisfies the reverse of the usual isoperimetric inequality (a similar result holds in higher dimensions [31]). At least this seems to be the case if quantum gravity effects are not taken into account. In one case where quantum gravity corrections can be calculated using the techniques in [32] , the three-dimensional Bañados–Zanelli–Teitelboim (BTZ) black hole [33], they tend to reduce the black hole volume [28] so it seems possible that quantum gravity effects may affect the

<sup>3</sup> <sup>=</sup> *<sup>V</sup>*<sup>0</sup> <sup>+</sup>

*r*2 <sup>+</sup> + *a*<sup>2</sup>

*r*+(*r*<sup>2</sup>

<sup>+</sup> + *a*2)

4*π* 3 *J*2

<sup>+</sup>*L*<sup>2</sup> + *a*2*L*<sup>2</sup> − *r*<sup>2</sup>

*L*2Ξ<sup>2</sup> *r*<sup>+</sup>

+*a*<sup>2</sup>

+ *L*2*q*2*a*<sup>2</sup>

Ξ (39)

Ξ , (40)

*<sup>M</sup>*, (41)

. (38)

*<sup>π</sup>* , (37)

thermodynamical variables, *S*, *P*, *J* and *Q*, but for the limiting case *J* = 0,

$$\begin{aligned} a &:= \frac{\pi}{S} \left\{ \frac{1}{4} \left( \frac{S}{\pi} + \mathbb{Q}^2 \right)^2 + l^2 \right\} \\ b &:= \frac{4\pi}{3} \left\{ \frac{S}{\pi} \left( \frac{S}{\pi} + \mathbb{Q}^2 \right) + 2l^2 \right\} \\ c &:= \left( \frac{4\pi}{3} \right)^2 \left( \frac{S}{\pi} \right)^3. \end{aligned} \tag{43}$$

Note that the discriminant,

$$b^2 - 4ac = \frac{64\pi^2}{9}J^2\left(J^2 + \frac{SQ^2}{\pi}\right),\tag{44}$$

is positive.

Now

$$V = \left. \frac{\partial H}{\partial P} \right|\_{S,l,Q} = \frac{b + 2cP}{2H} \qquad \Rightarrow \qquad P = \frac{2HV - b}{2c}. \tag{45}$$

This allows us to re-express *H* as a function of *V*,

$$H = \frac{1}{2} \sqrt{\frac{b^2 - 4ac}{V^2 - c}}.\tag{46}$$

We can immediately conclude that

$$V^2 \ge c = \left(\frac{4\pi}{3}\right)^2 \left(\frac{S}{\pi}\right)^3,\tag{47}$$

with equality only when

$$b^2 - 4ac = \frac{64\pi^2}{9}J^2\left(f^2 + \frac{SQ^2}{\pi}\right) = 0,\tag{48}$$

*i.e.* when *J* = 0.

It is now straightforward to determine

$$M = H - PV = H - \left(\frac{HV^2}{c} - \frac{bV}{2c}\right) = \frac{bV}{2c} - \frac{\sqrt{\left(V^2 - c\right)\left(b^2 - 4ac\right)}}{2c},\tag{49}$$

which immediately gives

$$M(S, V, I, \mathbb{Q}) = \left(\frac{\pi}{S}\right)^3 \left[ \left(\frac{3V}{4\pi}\right) \left\{ \left(\frac{S}{2\pi}\right) \left(\frac{S}{\pi} + \mathbb{Q}^2\right) + I^2 \right\} \tag{50}$$

$$-|I| \left\{ \left(\frac{3V}{4\pi}\right)^2 - \left(\frac{S}{\pi}\right)^3 \right\}^{\frac{1}{2}} \left(\frac{SQ^2}{\pi} + I^2\right)^{\frac{1}{2}} \right].$$

Note the subtlety in the *J* → 0 limit, (50) is not differentiable at *J* = 0 unless

$$\left(\frac{3V}{4\pi}\right)^2 = \left(\frac{\mathcal{S}}{\pi}\right)^3\tag{51}$$

The work extracted at any infinitesimal step is

*dW* = −*dU* = −*TdS* − Ω*d J* + *PdV* (55)

Where Is the PdV in the First Law of Black Hole Thermodynamics?

3 (3 + 8*PS*)

*<sup>H</sup>*(*Jmax*) (56)

<sup>2</sup> <sup>≈</sup> 0.2929. (58)

(1 + 8*PS*) (59)

, (60)

. (57)

http://dx.doi.org/10.5772/52455

301

and, since *dS* ≥ 0, this is maximised in an isentropic process *dS* = 0. Now with *Q* = 0 and *S* and *P* held constant, the internal energy in equation (53) can be thought of as a function of *J* only, *U*(*J*). The greatest efficiency is then obtained by starting with an extremal black hole

*<sup>η</sup>ext* <sup>=</sup> *<sup>U</sup>*(*Jmax*) <sup>−</sup> *<sup>U</sup>*(0)

2 + 8*PS*

<sup>1</sup> <sup>+</sup> <sup>4</sup>*PS* <sup>−</sup> <sup>1</sup> √

*<sup>η</sup>ext* <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>

In asymptotically flat space, Λ = 0, we set *P* = 0 in *ηext* and obtain the famous result [9]

√

More generally, *ηext* is a maximum for *SP* = 1.837 . . . (obtained by solving a quartic equation) and attains there the value 0.5184 . . . . Thus turning on a negative cosmological constant

What is happening here is that, as |*J*| decreases (giving a positive contribution to *dW*) the volume decreases, which actually tends to *decrease* the work done because of the *PdV* term in (55). But when *P* > 0, the extremal value |*Jmax*| in (32) is increased, which more than

For a charged black hole the internal energy is a function of *J* and *Q* for an isobaric isentropic

 *S π* 

on the charge. The greatest efficiency is achieved starting from an extremal black hole with

*<sup>H</sup>*(*Jmax*, *Qmax*) <sup>=</sup> <sup>3</sup>

with *H*(*Jmax*, *Qmax*) the initial extremal mass, *Mext*. For large *S* efficiencies of up to 75% are

*max* <sup>≥</sup> 0 in (32) imposes the constraint

2

 1 + 8*PS* <sup>3</sup> <sup>+</sup> <sup>16</sup>*PS*

and reducing the angular momentum from *Jmax* to zero, it is given by

where *H*(*Jmax*) = *Mext* is the initial extremal mass. One finds

*<sup>η</sup>ext* <sup>=</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup>*PS*

increases the efficiency of a Penrose process, as first observed in [30].

*Q*<sup>2</sup> ≤ *Q*<sup>2</sup>

*max* and reducing both *J* and *Q* to zero in the final state,

*<sup>η</sup>ext* <sup>=</sup> *<sup>U</sup>*(*Jmax*, *Qmax*) <sup>−</sup> *<sup>J</sup>*(0, 0)

possible [30], which should be compared to 50% in the Λ = 0 case, [9].

*max* <sup>=</sup>

compensates, and overall *ηext* is increased.

process, *U*(*J*, *Q*). The requirement *J*<sup>2</sup>

*Q*<sup>2</sup> = *Q*<sup>2</sup>

there.

Equation (50) can now be used to study the efficiency of a Penrose process. If a black hole has initial mass *Mi*, with internal energy *Ui*, and is taken through a quasi-static series of thermodynamic steps to a state with final internal energy *Uf* , then energy can be extracted if *Uf* < *Ui*. This is the thermodynamic description of a Penrose process [4] and the efficiency is

$$
\eta = \frac{\mathcal{U}\_i - \mathcal{U}\_f}{M\_i}.\tag{52}
$$

We can determine the maximum efficiency for a process at constant *P* by first expressing *U* in (50) in terms of *S*, *P*, *J* and *Q*:

$$\mathcal{U} = \frac{\left(\mathcal{S} + \pi Q^2\right)\left(\mathcal{S} + \pi Q^2 + \frac{8PS^2}{3}\right) + 4\pi^2 \left(1 + \frac{4PS}{3}\right)I^2}{2\sqrt{\pi \mathcal{S} \left[\left(\mathcal{S} + \pi Q^2 + \frac{8PS^2}{3}\right)^2 + 4\pi^2 \left(1 + \frac{8PS}{3}\right)I^2\right]}},\tag{53}$$

which is manifestly positive.

For simplicity consider first the *Q* = 0 case, for which

$$d\mathcal{U} = \mathcal{T}d\mathcal{S} + \Omega d\mathcal{J} - \mathcal{P}dV.\tag{54}$$

The work extracted at any infinitesimal step is

10 Open Questions in Cosmology

which immediately gives

*U*(*S*, *V*, *J*, *Q*) =

in (50) in terms of *S*, *P*, *J* and *Q*:

which is manifestly positive.

*U* =

�

2 � *πS* ��

For simplicity consider first the *Q* = 0 case, for which

*S* + *πQ*<sup>2</sup>

� �

It is now straightforward to determine

*U* = *H* − *PV* = *H* −

�*π S* �3 

� *HV*<sup>2</sup>

�3*V* 4*π*

*<sup>c</sup>* <sup>−</sup> *bV* 2*c* � = *bV* <sup>2</sup>*<sup>c</sup>* <sup>−</sup>

� �� *S* 2*π*

> ��3*V* 4*π*

Equation (50) can now be used to study the efficiency of a Penrose process. If a black hole has initial mass *Mi*, with internal energy *Ui*, and is taken through a quasi-static series of thermodynamic steps to a state with final internal energy *Uf* , then energy can be extracted if *Uf* < *Ui*. This is the thermodynamic description of a Penrose process [4] and the efficiency

> *<sup>η</sup>* <sup>=</sup> *Ui* <sup>−</sup> *Uf Mi*

We can determine the maximum efficiency for a process at constant *P* by first expressing *U*

*<sup>S</sup>* + *<sup>π</sup>Q*<sup>2</sup> + <sup>8</sup>*PS*<sup>2</sup>

*S* + *πQ*<sup>2</sup> + <sup>8</sup>*PS*<sup>2</sup>

3 � + 4*π*<sup>2</sup> � 1 + <sup>4</sup>*PS* 3 � *J*2

> + 4*π*<sup>2</sup> � 1 + <sup>8</sup>*PS* 3 � *J*2

*dU* = *TdS* + Ω*d J* − *PdV*. (54)

3 �2

−|*J*|

Note the subtlety in the *J* → 0 limit, (50) is not differentiable at *J* = 0 unless

�3*V* 4*π* �2 = � *S π* �3

� � *S π* + *Q*<sup>2</sup> � + *J* 2 �

> �2 − � *S π* �3 � 1 <sup>2</sup> � *SQ*<sup>2</sup> *π*

��

*V*<sup>2</sup> − *c*

��*b*<sup>2</sup> − 4*ac*�

+ *J* 2 �1 2 .

. (52)

� , (53)

<sup>2</sup>*<sup>c</sup>* , (49)

(50)

(51)

*i.e.* when *J* = 0.

there.

is

$$dW = -d\mathbf{U} = -T d\mathbf{S} - \Omega d\mathbf{J} + \text{P}dV\tag{55}$$

and, since *dS* ≥ 0, this is maximised in an isentropic process *dS* = 0. Now with *Q* = 0 and *S* and *P* held constant, the internal energy in equation (53) can be thought of as a function of *J* only, *U*(*J*). The greatest efficiency is then obtained by starting with an extremal black hole and reducing the angular momentum from *Jmax* to zero, it is given by

$$\eta\_{\rm ext} = \frac{\mathcal{U}(f\_{\rm max}) - \mathcal{U}(0)}{H(f\_{\rm max})} \tag{56}$$

where *H*(*Jmax*) = *Mext* is the initial extremal mass. One finds

$$\eta\_{\rm ext} = \frac{1 + 2PS}{1 + 4PS} - \frac{1}{\sqrt{2 + 8PS}} \frac{3}{(3 + 8PS)}.\tag{57}$$

In asymptotically flat space, Λ = 0, we set *P* = 0 in *ηext* and obtain the famous result [9]

$$
\eta\_{\rm ext} = 1 - \frac{1}{\sqrt{2}} \approx 0.2929.\tag{58}
$$

More generally, *ηext* is a maximum for *SP* = 1.837 . . . (obtained by solving a quartic equation) and attains there the value 0.5184 . . . . Thus turning on a negative cosmological constant increases the efficiency of a Penrose process, as first observed in [30].

What is happening here is that, as |*J*| decreases (giving a positive contribution to *dW*) the volume decreases, which actually tends to *decrease* the work done because of the *PdV* term in (55). But when *P* > 0, the extremal value |*Jmax*| in (32) is increased, which more than compensates, and overall *ηext* is increased.

For a charged black hole the internal energy is a function of *J* and *Q* for an isobaric isentropic process, *U*(*J*, *Q*). The requirement *J*<sup>2</sup> *max* <sup>≥</sup> 0 in (32) imposes the constraint

$$Q^2 \le Q\_{\text{max}}^2 = \left(\frac{S}{\pi}\right) (1 + 8PS) \tag{59}$$

on the charge. The greatest efficiency is achieved starting from an extremal black hole with *Q*<sup>2</sup> = *Q*<sup>2</sup> *max* and reducing both *J* and *Q* to zero in the final state,

$$\eta\_{\rm ext} = \frac{\mathcal{U}(f\_{\rm max}, Q\_{\rm max}) - f(0, 0)}{H(f\_{\rm max}, Q\_{\rm max})} = \frac{3}{2} \left( \frac{1 + 8PS}{3 + 16PS} \right) \,\tag{60}$$

with *H*(*Jmax*, *Qmax*) the initial extremal mass, *Mext*. For large *S* efficiencies of up to 75% are possible [30], which should be compared to 50% in the Λ = 0 case, [9].

## **5. Critical behaviour**

With knowledge of both *H* and *U* general questions concerning the heat capacity of black holes can be addressed. The heat capacity at constant volume, *CV* = *T*/ *∂T ∂S V*,*J*,*Q* , tends to zero when *J* = 0, though *CV* can be non-zero for *J* �= 0 it does not diverge. For comparison the heat capacity at constant pressure, *CP* = *T*/ *∂T ∂S P*,*J*,*Q* , *CP* vanishes when *T* = 0 and diverges when *<sup>∂</sup><sup>T</sup> <sup>∂</sup><sup>S</sup>* =0.

can be expressed as a function of *PS* and *P J* and the critical point can be found analytically, by solving a cubic equation, but the explicit form is not very illuminating. Numerically it lies

The equation of state cannot be obtained analytically, but its properties near the critical point can be explored by a series expansion and critical exponents extracted. Define the reduced

> *<sup>v</sup>* <sup>=</sup> *<sup>V</sup>* <sup>−</sup> *Vc Vc*

*P J* <sup>−</sup> (*P J*)*crit*

*PS* <sup>−</sup> (*PS*)*crit*

For a given fixed *J* > 0, *p* is the deviation from critical pressure in units of 1/(16*πJ*), but one must be aware that this interpretation precludes taking the *J* → 0 limit in this formulation. Bearing this in mind, (68) and (69) give the *J* > 0, *Q* = 0 equation of state parametrically in

*p* = 0.3472 *t* − 0.1161 *tv* − 0.02883 *v*<sup>3</sup> + *o*(*t*

*t* = 2.881 *p* + 2.201 *pq* + 0.3436 *q*<sup>3</sup> + *o*(*p*2, *pq*2, *q*4). (68)

*v* = −10.44 *p* + 2.284 *q* + *o*(*p*2, *pq*, *q*2). (69)

The critical temperature is obtained from (30), with *Q* = 0,

and the critical volume likewise from (33)

(the authors of [27] fix *P* = <sup>3</sup>

temperature and volume as

*J* at *Jc* ≈ 0.0236).

and

 *T* √*P crit*

 *VP*3/2

*<sup>t</sup>* <sup>=</sup> *<sup>T</sup>* <sup>−</sup> *Tc Tc*

*p* := 16*π*

*q* := 8 

while similar expansion of the thermodynamic volume (33) yields

terms of *q*. Eliminating *q* one arrives at

Expanding the temperature (30) around the critical point, with *Q* set to zero, gives

It is convenient to expand around the critical point using

(*PS*)*crit* ≈ 0.08204, (*P J*)*crit* ≈ 0.002857. (62)

<sup>8</sup>*<sup>π</sup>* <sup>≈</sup> 0.1194, corresponding to *<sup>L</sup>* <sup>=</sup> 1, and find a critical value of

≈ 0.7811 (63)

Where Is the PdV in the First Law of Black Hole Thermodynamics?

http://dx.doi.org/10.5772/52455

303

*crit* <sup>≈</sup> 0.01768 (64)

. (65)

. (67)

2, *tv*2, *v*4). (70)

(66)

at

A full stability analysis was given in [27] and there are both local and global phase transitions. Local stability can be explored visually, by plotting thermodynamic functions, or analytically, examining the curvature of the derivatives of thermodynamics functions.

## **5.1.** *Q* = 0

Let us first focus on the *Q* = 0 case. The blue (lower) curve in the figure below shows the locus of points where *CP* diverges in the *J* − *S* plane, it is given by setting the denominator of *CP*,

$$2 \cdot 144 \left(\pi I P\right)^4 (9 + 32 \cdot SP) + 24 \left(\pi P\right)^2 (PS)^2 (3 + 16 \cdot SP)(3 + 8 \cdot SP)^2 - (PS)^4 (1 - 8 \cdot SP)(3 + 8 \cdot SP)^3 \tag{61}$$

to zero. The red (upper) curve is the *T* = 0 locus, all points above and left of this curve are unphysical as *T* < 0 in this region.

**Figure 1.** *T* = 0 and *CP* → ∞ curves in *J* − *S* plane.

There is also a global phase transition, not shown in the figure, when the free energy of pure AdS is lower than that of a black hole in asymptotically AdS space-time, the famous Hawking-Page phase transition [34]. We shall focus on the second order local phase transition here and examine its critical properties.

In general, at fixed *P* and *J*, there are two values of *S* at which *CP* diverges, and there is a critical point where these two values coalesce into one, the maximum of the lower curve in figure 1. This critical point was first identified in [27]. On purely dimensional grounds *PCP*

can be expressed as a function of *PS* and *P J* and the critical point can be found analytically, by solving a cubic equation, but the explicit form is not very illuminating. Numerically it lies at

$$(PS)\_{crit} \approx 0.08204, \qquad (PI)\_{crit} \approx 0.002857. \tag{62}$$

The critical temperature is obtained from (30), with *Q* = 0,

$$\left(\frac{T}{\sqrt{P}}\right)\_{\text{crit}} \approx 0.7811\tag{63}$$

and the critical volume likewise from (33)

$$
\left(VP^{3/2}\right)\_{crit} \approx 0.01768 \tag{64}
$$

(the authors of [27] fix *P* = <sup>3</sup> <sup>8</sup>*<sup>π</sup>* <sup>≈</sup> 0.1194, corresponding to *<sup>L</sup>* <sup>=</sup> 1, and find a critical value of *J* at *Jc* ≈ 0.0236).

The equation of state cannot be obtained analytically, but its properties near the critical point can be explored by a series expansion and critical exponents extracted. Define the reduced temperature and volume as

$$t = \frac{T - T\_{\mathcal{L}}}{T\_{\mathcal{L}}} \qquad v = \frac{V - V\_{\mathcal{L}}}{V\_{\mathcal{L}}}.\tag{65}$$

It is convenient to expand around the critical point using

$$p \coloneqq 16\pi \left(Pf - (Pf)\_{crit}\right) \tag{66}$$

and

12 Open Questions in Cosmology

diverges when *<sup>∂</sup><sup>T</sup>*

**5.1.** *Q* = 0

of *CP*,

**5. Critical behaviour**

With knowledge of both *H* and *U* general questions concerning the heat capacity of black

zero when *J* = 0, though *CV* can be non-zero for *J* �= 0 it does not diverge. For comparison

A full stability analysis was given in [27] and there are both local and global phase transitions. Local stability can be explored visually, by plotting thermodynamic functions, or analytically,

Let us first focus on the *Q* = 0 case. The blue (lower) curve in the figure below shows the locus of points where *CP* diverges in the *J* − *S* plane, it is given by setting the denominator

to zero. The red (upper) curve is the *T* = 0 locus, all points above and left of this curve are

There is also a global phase transition, not shown in the figure, when the free energy of pure AdS is lower than that of a black hole in asymptotically AdS space-time, the famous Hawking-Page phase transition [34]. We shall focus on the second order local phase

In general, at fixed *P* and *J*, there are two values of *S* at which *CP* diverges, and there is a critical point where these two values coalesce into one, the maximum of the lower curve in figure 1. This critical point was first identified in [27]. On purely dimensional grounds *PCP*

 *∂T ∂S P*,*J*,*Q*

2

−(*PS*)4(1−8 *SP*)(3+8 *SP*)

 *∂T ∂S V*,*J*,*Q*

, *CP* vanishes when *T* = 0 and

, tends to

<sup>3</sup> (61)

holes can be addressed. The heat capacity at constant volume, *CV* = *T*/

examining the curvature of the derivatives of thermodynamics functions.

144 (*πJP*)4(9+32 *SP*)+24 (*πP J*)2(*PS*)2(3+16 *SP*)(3+8 *SP*)

the heat capacity at constant pressure, *CP* = *T*/

*<sup>∂</sup><sup>S</sup>* =0.

unphysical as *T* < 0 in this region.

**Figure 1.** *T* = 0 and *CP* → ∞ curves in *J* − *S* plane.

transition here and examine its critical properties.

$$\eta := 8\left(PS - (PS)\_{crit}\right). \tag{67}$$

Expanding the temperature (30) around the critical point, with *Q* set to zero, gives

$$t = 2.881 \, p + 2.201 \, pq + 0.3436 \, q^3 + o(p^2, pq^2, q^4). \tag{68}$$

while similar expansion of the thermodynamic volume (33) yields

$$v = -10.44 \, p + 2.284 \, q + o(p^2, pq, q^2). \tag{69}$$

For a given fixed *J* > 0, *p* is the deviation from critical pressure in units of 1/(16*πJ*), but one must be aware that this interpretation precludes taking the *J* → 0 limit in this formulation. Bearing this in mind, (68) and (69) give the *J* > 0, *Q* = 0 equation of state parametrically in terms of *q*. Eliminating *q* one arrives at

$$p = 0.3472\,t - 0.1161\,tv - 0.02883\,v^3 + o(t^2, tv^2, v^4). \tag{70}$$

The critical exponent *α* is defined by

$$
\mathbb{C}\_V \propto t^{-\kappa} \tag{71}
$$

and, since as already stated, *CV* does not diverge at *<sup>t</sup>* = 0, *<sup>α</sup>* = 0. To see this explicitly note that *CV* = *<sup>T</sup>*/ *<sup>∂</sup><sup>T</sup> ∂S <sup>V</sup>* and

$$
\left.\frac{\partial T}{\partial \mathbf{S}}\right|\_V = \left.\frac{\partial T}{\partial \mathbf{S}}\right|\_P + \left.\frac{\partial T}{\partial P}\right|\_S \left.\frac{\partial P}{\partial \mathbf{S}}\right|\_V = T\_\mathbf{f} \left(\left.\frac{\partial t}{\partial \mathbf{S}}\right|\_P + \left.\frac{\partial t}{\partial P}\right|\_S \left.\frac{\partial P}{\partial \mathbf{S}}\right|\_V\right). \tag{72}
$$

Now, near the critical point, (68) gives

$$\left. \frac{\partial t}{\partial S} \right|\_P = 8P \left. \frac{\partial t}{\partial q} \right|\_P = o(p\_\prime q^2)\_\prime \tag{73}$$

$$\left. \frac{\partial t}{\partial P} \right|\_S = 8S \left. \frac{\partial t}{\partial q} \right|\_p + 16\pi f \left. \frac{\partial t}{\partial p} \right|\_q = 2.881(16\pi f) + o(p, q), \tag{74}$$

while (69) implies *dp* = 0.2188 *d q* for constant *v*, from which is follows that *<sup>∂</sup><sup>P</sup> ∂S V* is non-zero at the critical point, hence *<sup>∂</sup><sup>T</sup> ∂S <sup>V</sup>* does not vanish at the critical point and so *<sup>α</sup>* <sup>=</sup> 0.

The exponent *β* is defined by

$$v\_{\succ} - v\_{<} = |t|^{\beta} \tag{75}$$

**Figure 2.** Construction associated with Maxwell's equal area law.

*<sup>κ</sup><sup>T</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> *V*

isothermal compressibility near the critical point,

The critical exponent *γ* is related to the isothermal compressibility,

= − 1 *V*  *∂V ∂P S*,*J* − *∂V ∂S P*,*J*

which diverges along the same curve as *CP* does (the adiabatic compression, *<sup>κ</sup><sup>S</sup>* =

*κ<sup>T</sup>* ∝ *t*

*∂p ∂v t*

*γ* can be found by expanding the denominator of *CP* in (61) around the critical point, but a quicker method, since we have the equation of state, is to differentiate (70) with respect to *v*,

, is everywhere finite — see equation (92)). *γ* gives the divergence of the

 *∂V ∂P T*,*J*

and *β* = <sup>1</sup>

− 1 *V ∂V ∂P S*,*J*

2 .

keeping *t* constant, giving


Where Is the PdV in the First Law of Black Hole Thermodynamics?

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305

 *∂T ∂P S*,*J*

 *∂T ∂S P*,*J*

<sup>−</sup>*γ*. (81)

∝ −*t*, (82)

(80)

where *v*> is the greater volume and *v*< the lesser volume across the phase transition, at constant pressure, when *t* < 0 (*v*<sup>&</sup>lt; is negative, since *v* = 0 at the critical point).

Keeping *p* and *t* constant in (70) implies that

$$p\int\_{\upsilon\_{\leq}}^{\upsilon\_{>}}d\upsilon = 0.3742\,\mathrm{t}\int\_{\upsilon\_{\leq}}^{\upsilon\_{>}}d\upsilon - \int\_{\upsilon\_{\leq}}^{\upsilon\_{>}}\left(0.1161\,\mathrm{t}\upsilon + 0.02883\,\mathrm{v}^{3}\right)d\upsilon.\tag{76}$$

Allowing for the area of the rectangle in figure 3, namely 0.3742 |*t*|(*v*<sup>&</sup>gt; − *v*<), Maxwell's equal area law then requires

$$\int\_{v\_{<}}^{v\_{>}} \left( 0.1161 \, t v + 0.02883 \, v^{3} \right) dv = 0 \qquad \Rightarrow \qquad |t| \propto (v\_{>}^{2} + v\_{<}^{2}).\tag{77}$$

It is clear from the figure that *v*<sup>&</sup>gt; − *v*<sup>&</sup>lt; ≫ *v*<sup>&</sup>gt; + *v*<sup>&</sup>lt; so

$$(v\_{>}^{2} + v\_{<}^{2}) = \frac{1}{2}((v\_{>} - v\_{<})^{2} + (v\_{>} + v\_{<})^{2}) \approx \frac{1}{2}(v\_{>} - v\_{<})^{2} \tag{78}$$

giving

**Figure 2.** Construction associated with Maxwell's equal area law.

$$|t| \propto (v\_{>} - v\_{<})^2 \tag{79}$$

and *β* = <sup>1</sup> 2 .

14 Open Questions in Cosmology

that *CV* = *<sup>T</sup>*/ *<sup>∂</sup><sup>T</sup>*

The critical exponent *α* is defined by

*∂T ∂S V*

Now, near the critical point, (68) gives

*∂t ∂S P*

*∂t ∂P S*

Keeping *p* and *t* constant in (70) implies that

= *∂T ∂S P* + *∂T ∂P S*

> <sup>=</sup> <sup>8</sup>*<sup>P</sup> <sup>∂</sup><sup>t</sup> ∂q p*

<sup>=</sup> <sup>8</sup>*<sup>S</sup> <sup>∂</sup><sup>t</sup> ∂q p*

> *∂S*

*dv* = 0.3742 *t*

0.1161 *tv* + 0.02883 *v*<sup>3</sup>

It is clear from the figure that *v*<sup>&</sup>gt; − *v*<sup>&</sup>lt; ≫ *v*<sup>&</sup>gt; + *v*<sup>&</sup>lt; so

<sup>&</sup>lt;) = <sup>1</sup> 2 

*∂S <sup>V</sup>* and

at the critical point, hence *<sup>∂</sup><sup>T</sup>*

The exponent *β* is defined by

*p <sup>v</sup>*<sup>&</sup>gt; *v*<

 *<sup>v</sup>*<sup>&</sup>gt; *v*< 

> (*v*<sup>2</sup> <sup>&</sup>gt; + *v*<sup>2</sup>

giving

equal area law then requires

*CV* ∝ *t*

*∂P ∂S V* = *Tc*

<sup>+</sup> <sup>16</sup>*π<sup>J</sup> <sup>∂</sup><sup>t</sup>*

while (69) implies *dp* = 0.2188 *d q* for constant *v*, from which is follows that *<sup>∂</sup><sup>P</sup>*

constant pressure, when *t* < 0 (*v*<sup>&</sup>lt; is negative, since *v* = 0 at the critical point).

 *<sup>v</sup>*<sup>&</sup>gt; *v*<

*dv* −

*∂p q*

*v*<sup>&</sup>gt; − *v*<sup>&</sup>lt; = |*t*|

where *v*> is the greater volume and *v*< the lesser volume across the phase transition, at

 *<sup>v</sup>*<sup>&</sup>gt; *v*< 

Allowing for the area of the rectangle in figure 3, namely 0.3742 |*t*|(*v*<sup>&</sup>gt; − *v*<), Maxwell's

(*v*<sup>&</sup>gt; − *v*<)<sup>2</sup> + (*v*<sup>&</sup>gt; + *v*<)2

and, since as already stated, *CV* does not diverge at *<sup>t</sup>* = 0, *<sup>α</sup>* = 0. To see this explicitly note

 *∂t ∂S P* + *∂t ∂P S*

*<sup>V</sup>* does not vanish at the critical point and so *<sup>α</sup>* <sup>=</sup> 0.

0.1161 *tv* + 0.02883 *v*<sup>3</sup>

≈ 1 2

*dv* = 0 ⇒ |*t*| ∝ (*v*<sup>2</sup>

<sup>−</sup>*<sup>α</sup>* (71)

*∂P ∂S V* 

= 2.881(16*πJ*) + *o*(*p*, *q*), (74)

*<sup>β</sup>* (75)

<sup>&</sup>gt; + *v*<sup>2</sup>

*dv*. (76)

<sup>&</sup>lt;). (77)

(*v*<sup>&</sup>gt; − *v*<)<sup>2</sup> (78)

*∂S V*

= *o*(*p*, *q*2), (73)

. (72)

is non-zero

The critical exponent *γ* is related to the isothermal compressibility,

$$\kappa\_T = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)\_{T,l} = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)\_{S,l} - \left( \frac{\partial V}{\partial S} \right)\_{P,l} \frac{\left( \frac{\partial T}{\partial P} \right)\_{S,l}}{\left( \frac{\partial T}{\partial S} \right)\_{P,l}} \tag{80}$$

which diverges along the same curve as *CP* does (the adiabatic compression, *<sup>κ</sup><sup>S</sup>* = − 1 *V ∂V ∂P S*,*J* , is everywhere finite — see equation (92)). *γ* gives the divergence of the isothermal compressibility near the critical point,

$$
\kappa\_T \propto t^{-\gamma}.\tag{81}
$$

*γ* can be found by expanding the denominator of *CP* in (61) around the critical point, but a quicker method, since we have the equation of state, is to differentiate (70) with respect to *v*, keeping *t* constant, giving

$$\left.\frac{\partial p}{\partial v}\right|\_{t} \propto -t,\tag{82}$$

hence

$$\left. \kappa\_T \propto -\frac{\partial v}{\partial p} \right|\_t \propto \frac{1}{t} \tag{83}$$

and *γ* = 1.

Lastly setting *t* = 0 in (70) we see that

$$|p| \propto |v|^{\delta} \tag{84}$$

**Figure 3.** Gibbs free energy as a function of pressure and temperature, at fixed angular momentum.

where *a*, *b* and *c* need not be positive and *L* → <sup>1</sup>

and *t*.

Using this in *L* leads to

For notational convenience equation (87) can be written, for fixed *p* and *t*, as

constant. We are to think of *b* and *c* are control parameters that can be varied by varying *p*

Equations (89) and (90) together give *L*(*a*, *b*, *c*) implicitly: a parametric plot of *L*(*b*, *c*), for any fixed *a*, reveals a characteristic "swallow-tail catastrophe" structure. With hindsight the swallow-tail structure is clear: in the *A* − *D* − *E* classification of critical points of functions, [41], (88) has three control parameters and is derived from type *A*<sup>4</sup> in Arnold's classification.

Extremising (88) with respect to *v* determines the value of *v* in terms of *b* and *c* through

*L* = *a* + *bv* + *cv*<sup>2</sup> + *v*<sup>4</sup> (88)

Where Is the PdV in the First Law of Black Hole Thermodynamics?

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307

*b* = −2*cv* − 4*v*3. (89)

*L* = *a* − *cv*<sup>2</sup> − 3*v*4. (90)

*AD L* has been rescaled by a trivial positive

with *δ* = 3, again the mean field result.

To summarise, the critical exponents are

$$\alpha = 0, \quad \beta = \frac{1}{2}, \quad \gamma = 1, \quad \delta = 3. \tag{85}$$

These are the same critical exponents as the Van der Waals fluid and, more importantly, are mean field exponents. The same critical exponents have been found using a virial expansion in [50].

It was first pointed in [35, 36] that a non-rotating, charged black hole has a critical point of the same nature as that of of a Van der Waals fluid, and the critical exponents for the black hole phase transition in this case were calculated in [37] and verified to be mean field exponents, which are indeed the those of a Van der Waals fluid. A similarity between the neutral rotating black hole and the Van der Waals phase transition was first pointed out in [27] and further explored in [30].

The critical point can be visualised by plotting the Gibbs free energy

$$G(T, P\_\prime I) = H(S, P\_\prime I) - TS\_\prime \tag{86}$$

for *J* = 1 and *Q* = 0, as a function of *P* and *T* as in figure 3. We see the "swallow-tail catastrophe" that is typical of the Van der Waals phase transition [38].

This structure is a straightforward consequence of Landau theory, [39]. Near the critical point the Landau free energy is

$$L(T, P, v) = G(T, P) + A\left\{(p - Bt)v + Ctv^2 + Dv^4\right\} + \dots \, \, \, \, \, \tag{87}$$

where *G*(*T*, *P*) is the Gibbs free energy and *A*, *B*, *C* and *D* are positive constants (for simplicity the constant *J* is not made explicit). As stressed in [40] *L* is not strictly speaking a thermodynamic function as it depends on three variables, *p*, *t* and *v* instead of two: *v* is to be determined in terms of *p* and *t* by extremising *L* to obtain the equation of state.

**Figure 3.** Gibbs free energy as a function of pressure and temperature, at fixed angular momentum.

For notational convenience equation (87) can be written, for fixed *p* and *t*, as

$$L = a + bv + cv^2 + v^4\tag{88}$$

where *a*, *b* and *c* need not be positive and *L* → <sup>1</sup> *AD L* has been rescaled by a trivial positive constant. We are to think of *b* and *c* are control parameters that can be varied by varying *p* and *t*.

Extremising (88) with respect to *v* determines the value of *v* in terms of *b* and *c* through

$$b = -2cv - 4v^3.\tag{89}$$

Using this in *L* leads to

16 Open Questions in Cosmology

Lastly setting *t* = 0 in (70) we see that

with *δ* = 3, again the mean field result. To summarise, the critical exponents are

[27] and further explored in [30].

the Landau free energy is

*<sup>κ</sup><sup>T</sup>* <sup>∝</sup> <sup>−</sup> *<sup>∂</sup><sup>v</sup> ∂p t* ∝ 1


2

These are the same critical exponents as the Van der Waals fluid and, more importantly, are mean field exponents. The same critical exponents have been found using a virial expansion

It was first pointed in [35, 36] that a non-rotating, charged black hole has a critical point of the same nature as that of of a Van der Waals fluid, and the critical exponents for the black hole phase transition in this case were calculated in [37] and verified to be mean field exponents, which are indeed the those of a Van der Waals fluid. A similarity between the neutral rotating black hole and the Van der Waals phase transition was first pointed out in

for *J* = 1 and *Q* = 0, as a function of *P* and *T* as in figure 3. We see the "swallow-tail

This structure is a straightforward consequence of Landau theory, [39]. Near the critical point

where *G*(*T*, *P*) is the Gibbs free energy and *A*, *B*, *C* and *D* are positive constants (for simplicity the constant *J* is not made explicit). As stressed in [40] *L* is not strictly speaking a thermodynamic function as it depends on three variables, *p*, *t* and *v* instead of two: *v* is to

(*p* − *Bt*)*v* + *Ctv*<sup>2</sup> + *Dv*<sup>4</sup>

be determined in terms of *p* and *t* by extremising *L* to obtain the equation of state.

*<sup>α</sup>* <sup>=</sup> 0, *<sup>β</sup>* <sup>=</sup> <sup>1</sup>

The critical point can be visualised by plotting the Gibbs free energy

catastrophe" that is typical of the Van der Waals phase transition [38].

*L*(*T*, *P*, *v*) = *G*(*T*, *P*) + *A*

*<sup>t</sup>* (83)

*<sup>δ</sup>* (84)

, *γ* = 1, *δ* = 3. (85)

*G*(*T*, *P*, *J*) = *H*(*S*, *P*, *J*) − *TS*, (86)

+ ... , (87)

hence

and *γ* = 1.

in [50].

$$L = a - cv^2 - 3v^4.\tag{90}$$

Equations (89) and (90) together give *L*(*a*, *b*, *c*) implicitly: a parametric plot of *L*(*b*, *c*), for any fixed *a*, reveals a characteristic "swallow-tail catastrophe" structure. With hindsight the swallow-tail structure is clear: in the *A* − *D* − *E* classification of critical points of functions, [41], (88) has three control parameters and is derived from type *A*<sup>4</sup> in Arnold's classification.

## **5.2.** *Q* �= 0

The above structure was first found in AdS black hole thermodynamics in the charged *J* = 0 case [35, 36], where the equation of state can be found exactly and the critical exponents can be determined [37]. When both *J* and *Q* are non-zero an analytic analysis is much more difficult, for example finding the zero locus of the denominator of *CP* requires solving a quintic equation. However numerical studies show that for a charged rotating black hole, as long as the charge is below the extremal value, the picture is qualitatively the same: the critical exponents are the same, the Landau free energy is still related to type *A*<sup>4</sup> and the Gibbs free energy still takes on a characteristic swallow-tail shape. For fixed values of *J* and *Q*, not both zero, all that changes is the numerical value of the co-efficients in equations (68), (69) and (70) or, equivalently the numerical values of the constants *A*, *B*, *C* and *D* in (87). As long as none of these constants actually changes sign the nature of the critical point does not change and the critical exponents are the same.

## **6. Compressibility and the speed of sound**

In the previous section, the nature of the singularity in the isothermal compressibility near the critical point was discussed, but the adiabatic compressibility

$$\kappa\_{\mathbb{S}} = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)\_{T, I, \mathbb{Q}} \tag{91}$$

where *ρ* = *<sup>M</sup>*

giving *c*<sup>2</sup>

minimum value of 1/2.

For a neutron star *<sup>N</sup>*

lowest for extremal black holes in which case

temperature follows from the degeneracy pressure

extremum value of 2*πM*. Putting in the numbers

compressibility: for an extremal black hole

*<sup>T</sup>*=0<sup>=</sup> <sup>2</sup>*<sup>S</sup>*

  (1 + 4*PS*)

*κT* 

some 33 times larger than *κ<sup>S</sup>*

solar-mass neutron star.

*Pdeg* = (3*π*2)

 *V N*

the black hole adiabatic compressibility at zero temperature is given by (93) with *P* = 0,

where the relevant factors of *c* and *G* are included, and the entropy has been set to the

which is four orders of magnitude less than that of a solar-mass neutron star. We conclude that the zero temperature black hole equation of state, although "softer" than that of a

The "softest" compressibility for a neutral black hole however is the isothermal

11 + 80*PS* + 128(*PS*)2

3 + 48*PS* + 128(*PS*)<sup>2</sup>

 −→ *P*→0

*<sup>T</sup>*=*P*=<sup>0</sup> in (97), but still much larger than degenerate matter in a

*<sup>κ</sup>S*|*T*=*P*=<sup>0</sup> <sup>=</sup> <sup>2</sup>*<sup>S</sup>*

*<sup>κ</sup>S*|*T*=<sup>0</sup> <sup>=</sup> 2.6 <sup>×</sup> <sup>10</sup>−<sup>38</sup> *<sup>M</sup>*

non-rotating black hole, is still very much stiffer than that of a neutron star.

*c*−2 *S* 

*<sup>T</sup>*=<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup>

*<sup>S</sup>* <sup>=</sup> 0.9 (in units with *<sup>c</sup>* <sup>=</sup> 1) when *<sup>P</sup>* <sup>=</sup> 0. In the limiting case *PS* <sup>→</sup> <sup>∞</sup>, *<sup>c</sup>*<sup>2</sup>

These results show that the equation of state is very stiff for adiabatic variations of non-rotating black holes and gets softer as *J* increases. For comparison, the adiabatic compressibility of a degenerate gas of *N* relativistic neutrons in a volume *V* at zero

> <sup>−</sup> <sup>4</sup> 3

<sup>⇒</sup> *<sup>κ</sup><sup>S</sup>* <sup>=</sup> <sup>3</sup>

*<sup>V</sup>* <sup>≈</sup> <sup>10</sup><sup>45</sup> *<sup>m</sup>*−<sup>3</sup> and *<sup>κ</sup><sup>S</sup>* <sup>≈</sup> <sup>10</sup>−<sup>34</sup> *kg*−<sup>1</sup> *m s*2. With zero cosmological constant

<sup>9</sup> <sup>=</sup> <sup>4</sup>*πM*2*G*<sup>3</sup>

*M*⊙

2

4*Pdeg*

Where Is the PdV in the First Law of Black Hole Thermodynamics?

<sup>9</sup> *<sup>c</sup>*<sup>8</sup> , (97)

*kg*−<sup>1</sup> *m s*2, (98)

22 *S*

<sup>3</sup> , (99)

*<sup>V</sup>* is the density. *cS* is unity for incompressible non-rotating black holes and is

1 + 8*PS* <sup>3</sup> <sup>+</sup> <sup>8</sup>*PS* <sup>2</sup>

. (95)

http://dx.doi.org/10.5772/52455

. (96)

*<sup>S</sup>* achieves a

309

is also of interest, and this was studied in [42] on which most of this section is based. From (10) one finds, setting *Q* = 0 for simplicity, that

$$\chi\_{\mathbb{S}} = \frac{36(2\pi I)^4 \mathbb{S}}{(3 + 8PS) \left\{ (3 + 8PS)S^2 + (2\pi I)^2 \right\} \left\{ 2(3 + 8PS)S^2 + 3(2\pi I)^2 \right\}}. \tag{92}$$

This is finite at the critical point, indeed it never diverges for any finite values of *S*, *P* and *J*, and it vanishes as *J* → 0: non-rotating black holes are completely incompressible. Black holes are maximally compressible in the extremal case *T* = 0, when *J* = *Jmax* in (32),

$$\left.\kappa\_{\mathbb{S}}\right|\_{T=0} = \frac{2\,\mathrm{S}\,\left(1+8PS\right)^{2}}{\left(3+8PS\right)^{2}\left(1+4PS\right)}.\tag{93}$$

A speed of sound, *cS*, can also be associated with the black hole, in the usual thermodynamic sense that

$$\left.c\_S^{-2} = \frac{\partial \rho}{\partial P}\right|\_{S,I} = 1 + \rho \,\kappa\_{\mathbb{S}} = 1 + \frac{9 \,(2\pi I)^4}{\left\{2(3 + 8PS)S^2 + 3 \,(2\pi I)^2\right\}^2} \,\text{\textsuperscript{}{}{}{\prime}}\tag{94}$$

where *ρ* = *<sup>M</sup> <sup>V</sup>* is the density. *cS* is unity for incompressible non-rotating black holes and is lowest for extremal black holes in which case

18 Open Questions in Cosmology

change and the critical exponents are the same.

(10) one finds, setting *Q* = 0 for simplicity, that

(3 + 8*PS*)

*c*−2 *<sup>S</sup>* <sup>=</sup> *∂ρ ∂P S*,*J*

**6. Compressibility and the speed of sound**

the critical point was discussed, but the adiabatic compressibility

*<sup>κ</sup><sup>S</sup>* <sup>=</sup> <sup>36</sup>(2*πJ*)4*<sup>S</sup>*

The above structure was first found in AdS black hole thermodynamics in the charged *J* = 0 case [35, 36], where the equation of state can be found exactly and the critical exponents can be determined [37]. When both *J* and *Q* are non-zero an analytic analysis is much more difficult, for example finding the zero locus of the denominator of *CP* requires solving a quintic equation. However numerical studies show that for a charged rotating black hole, as long as the charge is below the extremal value, the picture is qualitatively the same: the critical exponents are the same, the Landau free energy is still related to type *A*<sup>4</sup> and the Gibbs free energy still takes on a characteristic swallow-tail shape. For fixed values of *J* and *Q*, not both zero, all that changes is the numerical value of the co-efficients in equations (68), (69) and (70) or, equivalently the numerical values of the constants *A*, *B*, *C* and *D* in (87). As long as none of these constants actually changes sign the nature of the critical point does not

In the previous section, the nature of the singularity in the isothermal compressibility near

 *∂V ∂P* 

is also of interest, and this was studied in [42] on which most of this section is based. From

This is finite at the critical point, indeed it never diverges for any finite values of *S*, *P* and *J*, and it vanishes as *J* → 0: non-rotating black holes are completely incompressible. Black

*T*,*J*,*Q*

2(3 + 8*PS*)*S*<sup>2</sup> + 3(2*πJ*)<sup>2</sup>

2

9 (2*πJ*)<sup>4</sup> {2(3 + 8*PS*)*S*<sup>2</sup> + 3 (2*πJ*)2}

<sup>2</sup> (1 + 4*PS*)

(91)

. (92)

<sup>2</sup> , (94)

. (93)

*<sup>κ</sup><sup>S</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> *V*

(3 + 8*PS*)*S*<sup>2</sup> + (2*πJ*)<sup>2</sup>

holes are maximally compressible in the extremal case *T* = 0, when *J* = *Jmax* in (32),

= <sup>1</sup> + *ρ κ<sup>S</sup>* = <sup>1</sup> +

*<sup>κ</sup>S*|*T*=<sup>0</sup> <sup>=</sup> <sup>2</sup> *<sup>S</sup>* (<sup>1</sup> <sup>+</sup> <sup>8</sup>*PS*)

(3 + 8*PS*)

A speed of sound, *cS*, can also be associated with the black hole, in the usual thermodynamic

**5.2.** *Q* �= 0

sense that

$$c\_S^{-2}\Big|\_{T=0} = 1 + \left(\frac{1+8PS}{3+8PS}\right)^2.\tag{95}$$

giving *c*<sup>2</sup> *<sup>S</sup>* <sup>=</sup> 0.9 (in units with *<sup>c</sup>* <sup>=</sup> 1) when *<sup>P</sup>* <sup>=</sup> 0. In the limiting case *PS* <sup>→</sup> <sup>∞</sup>, *<sup>c</sup>*<sup>2</sup> *<sup>S</sup>* achieves a minimum value of 1/2.

These results show that the equation of state is very stiff for adiabatic variations of non-rotating black holes and gets softer as *J* increases. For comparison, the adiabatic compressibility of a degenerate gas of *N* relativistic neutrons in a volume *V* at zero temperature follows from the degeneracy pressure

$$P\_{\rm deg} = (3\pi^2)^{\frac{1}{3}} \frac{c\hbar}{4} \left(\frac{V}{N}\right)^{-\frac{4}{3}} \quad \Rightarrow \quad \kappa\_{\mathcal{S}} = \frac{3}{4P\_{\rm deg}}.\tag{96}$$

For a neutron star *<sup>N</sup> <sup>V</sup>* <sup>≈</sup> <sup>10</sup><sup>45</sup> *<sup>m</sup>*−<sup>3</sup> and *<sup>κ</sup><sup>S</sup>* <sup>≈</sup> <sup>10</sup>−<sup>34</sup> *kg*−<sup>1</sup> *m s*2. With zero cosmological constant the black hole adiabatic compressibility at zero temperature is given by (93) with *P* = 0,

$$\left.\kappa\_S\right|\_{T=P=0} = \frac{2S}{9} = \frac{4\pi M^2 G^3}{9\mathcal{L}^8},\tag{97}$$

where the relevant factors of *c* and *G* are included, and the entropy has been set to the extremum value of 2*πM*. Putting in the numbers

$$\left. \kappa\_{\rm S} \right|\_{T=0} = 2.6 \times 10^{-38} \left( \frac{M}{M\_{\odot}} \right)^2 \, \text{kg}^{-1} \, \text{m} \, \text{s}^2 \, \text{\textdegree} \tag{98}$$

which is four orders of magnitude less than that of a solar-mass neutron star. We conclude that the zero temperature black hole equation of state, although "softer" than that of a non-rotating black hole, is still very much stiffer than that of a neutron star.

The "softest" compressibility for a neutral black hole however is the isothermal compressibility: for an extremal black hole

$$\left. \kappa\_T \right|\_{T=0} = \frac{2S \left( 11 + 80PS + 128(PS)^2 \right)}{\left( 1 + 4PS \right) \left( 3 + 48PS + 128(PS)^2 \right)} \quad \underset{P \to 0}{\longrightarrow} \quad \frac{22 \text{ S}}{3} , \tag{99}$$

some 33 times larger than *κ<sup>S</sup> <sup>T</sup>*=*P*=<sup>0</sup> in (97), but still much larger than degenerate matter in a solar-mass neutron star.

## **7. Open questions**

The obvious open question arising from the ideas presented here is: what about Λ > 0?

The analysis of critical behaviour in §5 is only valid for Λ < 0, this critical point lies deep in the region *P* > 0 and does not appear to be of any relevance to astrophysical situations. It is certainly of interest in the AdS-CFT correspondence [43] but the particular analysis of §5, being in 1 + 3-dimensions could only be relevant to 2 + 1-dimensional conformal field theory, which is of course of interest in its own right [44]. Of course one could perform a similar analysis for 4 + 1-dimensional, or yet higher dimensional black holes, to try and gain insight into higher dimensional conformal field theory, and indeed this seems to have been the motivation in [31, 35, 36], but these ideas are not the focus of this volume and will not be pursued here.

The thermodynamics of black holes in de Sitter space-time is a notoriously difficult problem [17, 20, 45–48] as there are two event horizons and no "asymptotically de Sitter" region inside the cosmological horizon. Even with no black hole, a naïve interpretation of the cosmological horizon implies that the transition from Λ = 0 to any infinitesimally small Λ > 0 appears to involve a discontinuous jump from zero to infinite entropy, at least if one associates the usual Hawking-Bekenstein entropy with the cosmological horizon when Λ > 0.

Nevertheless it is argued in [45] that a consistent strategy is to fix the relevant components of the metric at the cosmological horizon, rather than at spacial infinity as would be done in asymptotically flat or AdS space-time. When that is done the same expression for the ADM mass (24) is obtained, but with *L*<sup>2</sup> → −*L*2, so Ξ > 1 while the angular momentum is still given by *J* = *aM*. In this picture, all of the formulae in §3 are applicable for positive Λ and negative *P*, provided *P* is not too negative. If Λ is too large the black hole horizon and the cosmological horizon coincide and demanding that this does not happen puts a lower bound on *P*, for any fixed *S*, *J* and *Q*: with *Q* = 0, for example, this requirement constrains *P* to

$$P > \frac{\sqrt{S^2 + 12\pi f^2} - 2S}{8S}.\tag{100}$$

like beads in a gas. These would certainly be expected to affect the overall compressibility of the gas as well as the speed of sound through the gas. In a radiation dominated Universe, ignoring the matter density, the speed of sound in the photon gas would be given by

Since the speed of sound associated with the embedded black hole "beads" is *cS* <sup>≥</sup> <sup>√</sup>0.9 *<sup>c</sup>* <sup>=</sup> 0.9487 *c* the presence of a significant density of primordial black holes would expected to

<sup>=</sup> <sup>3</sup>*c*<sup>−</sup>2, (101)

Where Is the PdV in the First Law of Black Hole Thermodynamics?

<sup>3</sup> *<sup>ǫ</sup>*) so *<sup>c</sup><sup>γ</sup>* <sup>=</sup> 0.577 *<sup>c</sup>*.

http://dx.doi.org/10.5772/52455

311

*<sup>S</sup>* <sup>=</sup> 1/2 for *PS* <sup>→</sup> <sup>∞</sup>) and

*c*−2 *<sup>γ</sup>* <sup>=</sup> *∂ǫ ∂P S*

where *ǫ* is the energy density (essentially since the equation of state is *P* = <sup>1</sup>

affect speed of sound in the photon gas and thus affect the dynamics.

**Figure 4.** The speed of sound for an electrically neutral, extremal, black hole (with *c* = 1).

figure 4, for *PS* > −1/8. For comparison the asymptotic value (*c*<sup>2</sup>

the speed of sound in a thermal gas of photons (*c*<sup>2</sup>

**8. Conclusions**

now (8),

The square of the speed of sound for an extremal electrically neutral black hole is plotted in

In conclusion there are strong reasons to believe that the cosmological constant should be included in the laws of black hole thermodynamics as a thermodynamic variable, proportional to the pressure of ordinary thermodynamics. The conjugate variable is a thermodynamic volume (10) and the complete first law of black hole thermodynamics is

*<sup>γ</sup>* <sup>=</sup> 1/3) are also shown.

Provided *P* lies above this lower bound we can analytically continue (29) to negative *P*, with the understanding that *S* is the entropy of the black hole event horizon only and does not include any contribution from the cosmological horizon.

Of course *P* < 0 is thermodynamically unstable, but it can be argued, in some circumstances at least, that positive pressures can be analytically continued to negative pressures [12, 39], and in a cosmological context there can now be little doubt that *P* < 0. Adopting the strategy of [45] the maximal efficiency of a rotating black hole in de Sitter space will be less than in the Λ = 0 case, based on simply changing the sign of Λ in §4, and the zero charge efficiency vanishes when the black hole horizon and the cosmological horizon coincide at *PS* = −<sup>1</sup> <sup>8</sup> . Any such deviation from the <sup>Λ</sup> <sup>=</sup> 0 case will however be completely negligible for astrophysical black holes around one solar mass and the observed value of Λ, but it could be more significant during periods of inflation when Λ was larger.

It has been suggested that primordial black-holes may have formed in the early Universe [49] and, if this is the case and if they formed in sufficient numbers at any stage, then one should model the primordial gas as containing a distribution of highly incompressible black holes, like beads in a gas. These would certainly be expected to affect the overall compressibility of the gas as well as the speed of sound through the gas. In a radiation dominated Universe, ignoring the matter density, the speed of sound in the photon gas would be given by

$$\left.c\_{\gamma}^{-2} = \left.\frac{\partial \epsilon}{\partial P}\right|\_{S} = 3c^{-2}\right. \tag{101}$$

where *ǫ* is the energy density (essentially since the equation of state is *P* = <sup>1</sup> <sup>3</sup> *<sup>ǫ</sup>*) so *<sup>c</sup><sup>γ</sup>* <sup>=</sup> 0.577 *<sup>c</sup>*. Since the speed of sound associated with the embedded black hole "beads" is *cS* <sup>≥</sup> <sup>√</sup>0.9 *<sup>c</sup>* <sup>=</sup> 0.9487 *c* the presence of a significant density of primordial black holes would expected to affect speed of sound in the photon gas and thus affect the dynamics.

**Figure 4.** The speed of sound for an electrically neutral, extremal, black hole (with *c* = 1).

The square of the speed of sound for an extremal electrically neutral black hole is plotted in figure 4, for *PS* > −1/8. For comparison the asymptotic value (*c*<sup>2</sup> *<sup>S</sup>* <sup>=</sup> 1/2 for *PS* <sup>→</sup> <sup>∞</sup>) and the speed of sound in a thermal gas of photons (*c*<sup>2</sup> *<sup>γ</sup>* <sup>=</sup> 1/3) are also shown.

## **8. Conclusions**

20 Open Questions in Cosmology

**7. Open questions**

pursued here.

*PS* = −<sup>1</sup>

The obvious open question arising from the ideas presented here is: what about Λ > 0?

The analysis of critical behaviour in §5 is only valid for Λ < 0, this critical point lies deep in the region *P* > 0 and does not appear to be of any relevance to astrophysical situations. It is certainly of interest in the AdS-CFT correspondence [43] but the particular analysis of §5, being in 1 + 3-dimensions could only be relevant to 2 + 1-dimensional conformal field theory, which is of course of interest in its own right [44]. Of course one could perform a similar analysis for 4 + 1-dimensional, or yet higher dimensional black holes, to try and gain insight into higher dimensional conformal field theory, and indeed this seems to have been the motivation in [31, 35, 36], but these ideas are not the focus of this volume and will not be

The thermodynamics of black holes in de Sitter space-time is a notoriously difficult problem [17, 20, 45–48] as there are two event horizons and no "asymptotically de Sitter" region inside the cosmological horizon. Even with no black hole, a naïve interpretation of the cosmological horizon implies that the transition from Λ = 0 to any infinitesimally small Λ > 0 appears to involve a discontinuous jump from zero to infinite entropy, at least if one associates the

Nevertheless it is argued in [45] that a consistent strategy is to fix the relevant components of the metric at the cosmological horizon, rather than at spacial infinity as would be done in asymptotically flat or AdS space-time. When that is done the same expression for the ADM mass (24) is obtained, but with *L*<sup>2</sup> → −*L*2, so Ξ > 1 while the angular momentum is still given by *J* = *aM*. In this picture, all of the formulae in §3 are applicable for positive Λ and negative *P*, provided *P* is not too negative. If Λ is too large the black hole horizon and the cosmological horizon coincide and demanding that this does not happen puts a lower bound on *P*, for any fixed *S*, *J* and *Q*: with *Q* = 0, for example, this requirement constrains *P* to

*S*<sup>2</sup> + 12*πJ*<sup>2</sup> − 2*S*

Provided *P* lies above this lower bound we can analytically continue (29) to negative *P*, with the understanding that *S* is the entropy of the black hole event horizon only and does not

Of course *P* < 0 is thermodynamically unstable, but it can be argued, in some circumstances at least, that positive pressures can be analytically continued to negative pressures [12, 39], and in a cosmological context there can now be little doubt that *P* < 0. Adopting the strategy of [45] the maximal efficiency of a rotating black hole in de Sitter space will be less than in the Λ = 0 case, based on simply changing the sign of Λ in §4, and the zero charge efficiency vanishes when the black hole horizon and the cosmological horizon coincide at

astrophysical black holes around one solar mass and the observed value of Λ, but it could be

It has been suggested that primordial black-holes may have formed in the early Universe [49] and, if this is the case and if they formed in sufficient numbers at any stage, then one should model the primordial gas as containing a distribution of highly incompressible black holes,

<sup>8</sup> . Any such deviation from the <sup>Λ</sup> <sup>=</sup> 0 case will however be completely negligible for

<sup>8</sup>*<sup>S</sup>* . (100)

usual Hawking-Bekenstein entropy with the cosmological horizon when Λ > 0.

*P* >

include any contribution from the cosmological horizon.

more significant during periods of inflation when Λ was larger.

In conclusion there are strong reasons to believe that the cosmological constant should be included in the laws of black hole thermodynamics as a thermodynamic variable, proportional to the pressure of ordinary thermodynamics. The conjugate variable is a thermodynamic volume (10) and the complete first law of black hole thermodynamics is now (8),

$$d\mathcal{U} = TdS + \Omega d\mathcal{J} + \Phi d\mathcal{Q} - PdV. \tag{102}$$

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With this interpretation the ADM mass of the black hole is identified with the enthalpy

$$M = H(\mathcal{S}, \mathcal{P}, \mathcal{J}, \mathcal{Q}) = \mathcal{U}(\mathcal{S}, V, \mathcal{J}, \mathcal{Q}) + PV \tag{103}$$

rather than the internal energy, *U*, of the system.

The inclusion of this extra term increases the maximal efficiency of a Penrose process: for a neutral black hole in asymptotically anti-de Sitter space the maximal efficiency is increased from 0.2929 in asymptotically flat space to 0.5184 in the asymptotically AdS case. For a charged black hole the efficiency can be as high as 75%. A positive cosmological constant is expected to reduce the efficiency of a Penrose process below the asymptotically flat space value.

This point of view makes the relation between asymptotically AdS black holes and the Van der Waals gas, first found in [35, 36], even closer as there is now a critical volume associated with the critical point. The thermodynamic volume then plays the rôle of an order parameter for this phase transition and the critical exponents take the mean field values,

$$
\alpha = 0, \quad \beta = \frac{1}{2}, \quad \gamma = 1, \quad \delta = 3. \tag{104}
$$

While there is no second order phase transition for a black hole in de Sitter space, there are other possible physical effects of including the *PdV* term in the first law. The adiabatic compressibility can be calculated (93) and the speed of sound for such a black hole (95) is greater even than that of a photon gas and approaches *c* when *PS* = −1/8.

Despite much progress the thermodynamics of black holes in de Sitter space-time is still very poorly understood and no doubt much still remains to be discovered.

## **Author details**

Brian P. Dolan1,2

1 Department of Mathematical Physics, National University of Ireland Maynooth, Maynooth, Co. Kildare, Ireland

2 School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin, Ireland

## **References**


22 Open Questions in Cosmology

value.

**Author details**

Brian P. Dolan1,2

**References**

Co. Kildare, Ireland

ibid. vol. 46, p. 206 (1976).

rather than the internal energy, *U*, of the system.

*dU* = *TdS* + Ω*d J* + Φ*dQ* − *PdV*. (102)

*M* = *H*(*S*, *P*, *J*, *Q*) = *U*(*S*, *V*, *J*, *Q*) + *PV* (103)

, *γ* = 1, *δ* = 3. (104)

With this interpretation the ADM mass of the black hole is identified with the enthalpy

The inclusion of this extra term increases the maximal efficiency of a Penrose process: for a neutral black hole in asymptotically anti-de Sitter space the maximal efficiency is increased from 0.2929 in asymptotically flat space to 0.5184 in the asymptotically AdS case. For a charged black hole the efficiency can be as high as 75%. A positive cosmological constant is expected to reduce the efficiency of a Penrose process below the asymptotically flat space

This point of view makes the relation between asymptotically AdS black holes and the Van der Waals gas, first found in [35, 36], even closer as there is now a critical volume associated with the critical point. The thermodynamic volume then plays the rôle of an order parameter

2

While there is no second order phase transition for a black hole in de Sitter space, there are other possible physical effects of including the *PdV* term in the first law. The adiabatic compressibility can be calculated (93) and the speed of sound for such a black hole (95) is

Despite much progress the thermodynamics of black holes in de Sitter space-time is still very

1 Department of Mathematical Physics, National University of Ireland Maynooth, Maynooth,

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2 School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin, Ireland

for this phase transition and the critical exponents take the mean field values,

*<sup>α</sup>* <sup>=</sup> 0, *<sup>β</sup>* <sup>=</sup> <sup>1</sup>

greater even than that of a photon gas and approaches *c* when *PS* = −1/8.

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[25] M.K. Parikh. Volume of black holes. *Phys. Rev.*, D73:124021, 2006.

surface gravity. 2012. [arXiv:1005.1116 [gr-qc]].

*Quantum Grav.*, 28:125020, 2011. [arXiv:1008.5023].

*Quantum Grav.*, 28:235017, 2011. [arXiv:1106.6260].

*JHEP*, 1002:029, 2010. [arXiv:0712.0155 [hep-th]].

equations. *Comm. Math. Phys.*, 10:280, 1968.

[22] S. Weinberg. The cosmological constant problem. *Rev. Mod. Phys.*, 61:1, 1989.


**Chapter 13**

**Plasma Vortices in Planetary Wakes**

H. Pérez-de-Tejada, Rickard Lundin and

Additional information is available at the end of the chapter

view of this issue was presented by Pérez-de-Tejada, 2012).

Measurements conducted in interplanetary space and in the vicinity of planets of the solar system have shown plasma structures produced by the solar wind that resemble fluid dy‐ namic features; namely, vortex rotations within the earth´s magnetosphere and also along the Venus wake. In both planets the solar wind encounters different obstacles since the earth is protected by its intrinsic magnetic field that is compressed by the dynamic pressure of the solar wind to form a large size cavity (the magnetosphere) that bounds its direct approach to the earth´s vicinity. At Venus the conditions are different since the planet does not have an internal magnetization that would produce an earth-type magnetic obstacle to the solar wind. Instead, the latter reaches directly upon the upper layers of the planet´s atmosphere and interacts with its ionized components (the ionosphere). The outcome of this interaction is a plasma wake of large extent whose geometry is similar to that of the earth´s magneto‐ spheric tail but that arises from conditions that are different in both planets. While there is evidence for the observation of fluid-like vortices as the solar wind streams along the wake of the earth and Venus there is a major issue as to the manner in which they are produced. In fact, since the solar wind is a collisionless plasma; namely its charged particles barely exe‐ cute collisions among them (their mean free path is comparable to one astronomical unit) it should not be expected that it behaves as a continuum when it interacts with planetary ob‐ stacles. The opposite has been verified by a variety of observations with indications that the physical properties of both the solar wind fluxes and the planetary particles that are being eroded through their interaction can be described in terms of fluid dynamic processes (a re‐

The motion of the solar wind particles as they interact with the earth´s magnetic field is de‐ scribed in terms of gyromagnetic trajectories as they move across the magnetic field lines.

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

© 2012 Pérez-de-Tejada 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,

D. S. Intriligator

**1. Introduction**

http://dx.doi.org/10.5772/52998

**Chapter 13**

## **Plasma Vortices in Planetary Wakes**

H. Pérez-de-Tejada, Rickard Lundin and D. S. Intriligator

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52998

## **1. Introduction**

Measurements conducted in interplanetary space and in the vicinity of planets of the solar system have shown plasma structures produced by the solar wind that resemble fluid dy‐ namic features; namely, vortex rotations within the earth´s magnetosphere and also along the Venus wake. In both planets the solar wind encounters different obstacles since the earth is protected by its intrinsic magnetic field that is compressed by the dynamic pressure of the solar wind to form a large size cavity (the magnetosphere) that bounds its direct approach to the earth´s vicinity. At Venus the conditions are different since the planet does not have an internal magnetization that would produce an earth-type magnetic obstacle to the solar wind. Instead, the latter reaches directly upon the upper layers of the planet´s atmosphere and interacts with its ionized components (the ionosphere). The outcome of this interaction is a plasma wake of large extent whose geometry is similar to that of the earth´s magneto‐ spheric tail but that arises from conditions that are different in both planets. While there is evidence for the observation of fluid-like vortices as the solar wind streams along the wake of the earth and Venus there is a major issue as to the manner in which they are produced. In fact, since the solar wind is a collisionless plasma; namely its charged particles barely exe‐ cute collisions among them (their mean free path is comparable to one astronomical unit) it should not be expected that it behaves as a continuum when it interacts with planetary ob‐ stacles. The opposite has been verified by a variety of observations with indications that the physical properties of both the solar wind fluxes and the planetary particles that are being eroded through their interaction can be described in terms of fluid dynamic processes (a re‐ view of this issue was presented by Pérez-de-Tejada, 2012).

The motion of the solar wind particles as they interact with the earth´s magnetic field is de‐ scribed in terms of gyromagnetic trajectories as they move across the magnetic field lines.

© 2012 Pérez-de-Tejada 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.

Their circular (Larmor) motion is influenced by local drift displacements that are produced by space gradients of the magnetic field intensity and that carry them along the boundary of the magnetosphere. Such process also occurs while there are local electric currents produced by the different drift of the positive (mostly proton) and negative (electron) components of the solar wind. Throughout the front part (dayside) of the magnetosphere the motion of those particles is directed by magnetic (Lorentz) forces which guide them along the magnet‐ ic field lines in an environment where the local magnetic energy density is larger than their kinetic energy density (Under such conditions the transport speed of magnetic signals, i.e. the Alfven speed, is larger than the particles´ speed and the flow is labeled subalfvenic). The gyrotropic motion of the solar wind particles as they encounter the earth´s magnetic field provides a mechanism that leads to a continuum transport of their properties despite the fact that they do not collide with each other. Alternate conditions are encountered along the magnetospheric tail where the magnetic field intensity has decreased significantly with the downstream distance from the earth and the speed of the solar wind particles has increased to nearly its freestream values (the local flow is expected to achieve superalfvenic condi‐ tions). Data that will be addressed below show the presence of vortical plasma structures within the magnetospheric tail and that reveal effects that can be related to Kelvin-Helm‐ holtz instabilities (Wolfe et al., 1980: Terada et al:, 2002) associated to a fluid dynamic re‐ sponse in their motion rather than to trajectories directed by the magnetic field.

the solar wind as it streams around and behind Venus does not seem to be only related to gyromagnetic trajectories but to a fluid-like behavior generated by other particle motions. Strong magnetic and plasma turbulence that has been inferred from different spacecraft measurements (Bridge et al., 1967; Vőrős et al., 2008) suggest that the motion of the particles may be dictated by local turbulence which should lead to a stochastic distribution of their trajectories. It is possible that wave-particle interactions may be ultimately responsible for the fluid-like character of the solar wind interaction with planetary ionospheres and the gen‐ eration of plasma and magnetic vortical structures as those reported by Pope et al., (2009)

Plasma Vortices in Planetary Wakes http://dx.doi.org/10.5772/52998 319

Measurements conducted with various spacecraft that have probed the earth´s magneto‐ spheric tail have shown that the plasma flow that streams within that region of space exhib‐ its changes in its direction of motion that are suggestive of vortical structures. From the analysis of plasma data first obtained with the International Sun Earth Explorer (ISEE) satel‐ lites and more recently with the Cluster spacecrafts it has been possible to derive that there are regions along the tail direction of the magnetosphere where the plasma exhibits a sense of rotation (clockwise in the morning sector of the magnetosphere and counter clockwise in the evening sector both viewed from above the ecliptic plane) which suggests a wave mo‐ tion that in some instances leads to vortical structures that move tailward with a speed of 300-400 km/s. The inferred features have a scale size of several earth radii nearly comparable to the width of the magnetospheric tail and it has been inferred that their rotation period is of the order of 5-20 min (Hones et al. 1978; 1981; 1983). The vortex structures form part of a wave which has a several earth radii wavelength and that increases downstream along the tail direction. From observations carried out with the Cluster spacecrafts it has also been possible to examine the plasma composition and the vortex flow dynamics at the time when there are enhanced values of the solar wind dynamic pressure (Tian et al. 2010). As a whole it is believed that vortices may derive from Kelvin-Helmholtz instabilities at the boundary of the magnetosphere (magnetopause) or at the inner edge of a plasma boundary layer with‐

A suitable example of the plasma and magnetic field data obtained with a Cluster spacecraft by the morning flank of the magnetosphere and far downstream from the earth (at x = -11 Re; y = - 15 Re; z = 3 Re; Re being the earth radius) was presented by Tian et al., (2010) and is reproduced in Figure 1. Those measurements lead to the observation of a series of vortices that can be inferred from the ion velocity components marked in the lower panels of that figure. The vortex regions are indicated by the shaded bands when the v<sup>x</sup> component be‐ comes large, the vy component changes sign (red profile), and the vortex rotation is derived from changes in the velocity vector orientation in the frame at the left side of the bottom panel. Comparable changes are also seen in the value of the Bx and By magnetic field compo‐ nents (second panel) which are derived by tracing the z-axis along the mean magnetic field

from the Venus Express measurements.

in the magnetosphere.

direction.

**2. Plasma vortices in the earth´s magnetosphere**

The overall manner in which the solar wind responds as it interacts with the Venus iono‐ sphere is different since in the absence of an intrinsic planetary magnetization its particles reach the ionospheric plasma and, at the same time, also encounter ions that are located in the outer exosphere of the planet. Thus, the solar wind particles do not enter a region domi‐ nated by an intrinsic planetary magnetic field where they would be guided to carry out gy‐ romagnetic trajectories as it is the case in the earth´s magnetosphere. Instead, the processes produced as a result of their interaction with the planetary ions are more complex since the latter ions are accelerated by a (convective) electric field that derives from the relative veloci‐ ty difference that exists between them and the solar wind (the planetary oxygen ions of the Venus exosphere are subject to the effects of that electric field). The result of that process is labeled mass loading and leads the planetary ions to execute gyromagnetic trajectories as they travel through the solar wind and its convected solar magnetic field. The magnetic field intensity of the solar wind becomes enhanced around the upper boundary of the dayside ionosphere (ionopause) in a layer labeled °magnetic barrier° that is produced by the solar wind – ionosphere interaction. Despite the acceleration of the exospheric planetary ions by the solar wind measurements show that the bulk of the solar wind momentum is mostly transferred to the upper Venus ionosphere where it strongly contributes to produce the nightward directed trans-terminator flow inferred from the Pioneer Venus Orbiter (PVO) data that was reported by Knudsen et al., (1980).

Processes associated with this effect are different from those expected in gyrotropic trajecto‐ ries since they lead to the observation of vortical plasma structures in the near Venus wake associated with superalfvenic flow conditions (the kinetic energy density of the plasma be‐ ing larger than the local magnetic energy density). The origin of the continuum response of the solar wind as it streams around and behind Venus does not seem to be only related to gyromagnetic trajectories but to a fluid-like behavior generated by other particle motions. Strong magnetic and plasma turbulence that has been inferred from different spacecraft measurements (Bridge et al., 1967; Vőrős et al., 2008) suggest that the motion of the particles may be dictated by local turbulence which should lead to a stochastic distribution of their trajectories. It is possible that wave-particle interactions may be ultimately responsible for the fluid-like character of the solar wind interaction with planetary ionospheres and the gen‐ eration of plasma and magnetic vortical structures as those reported by Pope et al., (2009) from the Venus Express measurements.

## **2. Plasma vortices in the earth´s magnetosphere**

Their circular (Larmor) motion is influenced by local drift displacements that are produced by space gradients of the magnetic field intensity and that carry them along the boundary of the magnetosphere. Such process also occurs while there are local electric currents produced by the different drift of the positive (mostly proton) and negative (electron) components of the solar wind. Throughout the front part (dayside) of the magnetosphere the motion of those particles is directed by magnetic (Lorentz) forces which guide them along the magnet‐ ic field lines in an environment where the local magnetic energy density is larger than their kinetic energy density (Under such conditions the transport speed of magnetic signals, i.e. the Alfven speed, is larger than the particles´ speed and the flow is labeled subalfvenic). The gyrotropic motion of the solar wind particles as they encounter the earth´s magnetic field provides a mechanism that leads to a continuum transport of their properties despite the fact that they do not collide with each other. Alternate conditions are encountered along the magnetospheric tail where the magnetic field intensity has decreased significantly with the downstream distance from the earth and the speed of the solar wind particles has increased to nearly its freestream values (the local flow is expected to achieve superalfvenic condi‐ tions). Data that will be addressed below show the presence of vortical plasma structures within the magnetospheric tail and that reveal effects that can be related to Kelvin-Helm‐ holtz instabilities (Wolfe et al., 1980: Terada et al:, 2002) associated to a fluid dynamic re‐

318 Open Questions in Cosmology

sponse in their motion rather than to trajectories directed by the magnetic field.

data that was reported by Knudsen et al., (1980).

The overall manner in which the solar wind responds as it interacts with the Venus iono‐ sphere is different since in the absence of an intrinsic planetary magnetization its particles reach the ionospheric plasma and, at the same time, also encounter ions that are located in the outer exosphere of the planet. Thus, the solar wind particles do not enter a region domi‐ nated by an intrinsic planetary magnetic field where they would be guided to carry out gy‐ romagnetic trajectories as it is the case in the earth´s magnetosphere. Instead, the processes produced as a result of their interaction with the planetary ions are more complex since the latter ions are accelerated by a (convective) electric field that derives from the relative veloci‐ ty difference that exists between them and the solar wind (the planetary oxygen ions of the Venus exosphere are subject to the effects of that electric field). The result of that process is labeled mass loading and leads the planetary ions to execute gyromagnetic trajectories as they travel through the solar wind and its convected solar magnetic field. The magnetic field intensity of the solar wind becomes enhanced around the upper boundary of the dayside ionosphere (ionopause) in a layer labeled °magnetic barrier° that is produced by the solar wind – ionosphere interaction. Despite the acceleration of the exospheric planetary ions by the solar wind measurements show that the bulk of the solar wind momentum is mostly transferred to the upper Venus ionosphere where it strongly contributes to produce the nightward directed trans-terminator flow inferred from the Pioneer Venus Orbiter (PVO)

Processes associated with this effect are different from those expected in gyrotropic trajecto‐ ries since they lead to the observation of vortical plasma structures in the near Venus wake associated with superalfvenic flow conditions (the kinetic energy density of the plasma be‐ ing larger than the local magnetic energy density). The origin of the continuum response of Measurements conducted with various spacecraft that have probed the earth´s magneto‐ spheric tail have shown that the plasma flow that streams within that region of space exhib‐ its changes in its direction of motion that are suggestive of vortical structures. From the analysis of plasma data first obtained with the International Sun Earth Explorer (ISEE) satel‐ lites and more recently with the Cluster spacecrafts it has been possible to derive that there are regions along the tail direction of the magnetosphere where the plasma exhibits a sense of rotation (clockwise in the morning sector of the magnetosphere and counter clockwise in the evening sector both viewed from above the ecliptic plane) which suggests a wave mo‐ tion that in some instances leads to vortical structures that move tailward with a speed of 300-400 km/s. The inferred features have a scale size of several earth radii nearly comparable to the width of the magnetospheric tail and it has been inferred that their rotation period is of the order of 5-20 min (Hones et al. 1978; 1981; 1983). The vortex structures form part of a wave which has a several earth radii wavelength and that increases downstream along the tail direction. From observations carried out with the Cluster spacecrafts it has also been possible to examine the plasma composition and the vortex flow dynamics at the time when there are enhanced values of the solar wind dynamic pressure (Tian et al. 2010). As a whole it is believed that vortices may derive from Kelvin-Helmholtz instabilities at the boundary of the magnetosphere (magnetopause) or at the inner edge of a plasma boundary layer with‐ in the magnetosphere.

A suitable example of the plasma and magnetic field data obtained with a Cluster spacecraft by the morning flank of the magnetosphere and far downstream from the earth (at x = -11 Re; y = - 15 Re; z = 3 Re; Re being the earth radius) was presented by Tian et al., (2010) and is reproduced in Figure 1. Those measurements lead to the observation of a series of vortices that can be inferred from the ion velocity components marked in the lower panels of that figure. The vortex regions are indicated by the shaded bands when the v<sup>x</sup> component be‐ comes large, the vy component changes sign (red profile), and the vortex rotation is derived from changes in the velocity vector orientation in the frame at the left side of the bottom panel. Comparable changes are also seen in the value of the Bx and By magnetic field compo‐ nents (second panel) which are derived by tracing the z-axis along the mean magnetic field direction.

The rotation of the velocity vector implied from the lower panels in Figure 1 serves to con‐ struct streamline maps of the flow using a numerical code applied to the data points along the spacecraft trajectory. The results are reproduced in Figure 2 where the observed direc‐ tion of the velocity vector projected on the xy plane is marked by the white arrows in the upper panel and the direction of the magnetic field is indicated by white arrows in the lower panel (after Tian et al., 2010). The gradual and persistent change in the direction of the veloc‐ ity vector with distance in the upper panel is indicative of an anticlockwise rotating vortex structure with an approximated two dimensional 1-2 Re scale size. The reported observa‐ tions occurred at the time when there is an enhanced solar wind dynamic pressure thus im‐ plying an intensified compression of the magnetosphere which could in this case have led to

Plasma Vortices in Planetary Wakes http://dx.doi.org/10.5772/52998 321

Measurements conducted with the ISEE vehicles have shown evidence of tailward directed vortical structures along the sides of the magnetosphere. However, there are indications that in the central plasma sheet of the magnetosphere the motion of the particles is observed to be earthward from the tail. An example of such measurements is reproduced in Figure 3 with evidence in the lower panel of a repeated rotation in the direction of the flow which in most cases leads to large (> 100 km/s) speed values (upper panel) at the time when the solar ecliptic longitude ϕSE becomes small; i.e. when the flow is sunward directed (Hones et al., 1981).

**Figure 3.** – Speed values (upper panel) and flow direction (lower panel) measured with the ISEE 2 spacecraft on March 10, 1979 within the earth´s magnetosphere. A repeated counterclockwise rotation of the solar ecliptic longitude an‐

A general schematic view of the flow distribution along the sides of the magnetosphere that has been derived from the ISEE measurements is reproduced in Figure 4. Tailward directed vortex structures occur by those regions and the diagram illustrates plasma patterns that may arise from Kelvin Helmholtz instabilities at the magnetopause and that lead to sudden

gle ϕSE is detected in the evening sector of the plasma sheet (after Hones et al., 1981).

changes in the flow direction when observed in the earth´s rest frame.

earthward moving directed vortices.

**Figure 1.** – A series of flow vortices in the earth´s magnetosphere observed by a Cluster spacecraft on July 6, 2003. The magnetic field and its components are indicated in the top two panels and the ion velocity and its components in the two lower panels (the velocity components in a reference frame in which the z and the x axes are parallel and perpendicular to the average magnetic field are in the middle panel). The profiles were selected from those presented in Figure 7 of Tian et al., (2010).

**Figure 2.** – Map of flow streamlines projected on the xy plane derived from the velocity rotation measured in the 13:36:45 UT – 13:38:20 UT time interval of July 6, 2003 in Figure 1. The streamlines are traced on the ion density (top panel) and on the ion temperature (lower panel) distributions (the white arrows in the top panel represent the meas‐ ured velocity vector direction, and those in the lower panel the direction of the magnetic field). The streamlines de‐ scribe conditions from the inner plasma sheet across the dawn side of the magnetosphere and were selected from those presented in Figure 11 of Tian et al., (2010).

The rotation of the velocity vector implied from the lower panels in Figure 1 serves to con‐ struct streamline maps of the flow using a numerical code applied to the data points along the spacecraft trajectory. The results are reproduced in Figure 2 where the observed direc‐ tion of the velocity vector projected on the xy plane is marked by the white arrows in the upper panel and the direction of the magnetic field is indicated by white arrows in the lower panel (after Tian et al., 2010). The gradual and persistent change in the direction of the veloc‐ ity vector with distance in the upper panel is indicative of an anticlockwise rotating vortex structure with an approximated two dimensional 1-2 Re scale size. The reported observa‐ tions occurred at the time when there is an enhanced solar wind dynamic pressure thus im‐ plying an intensified compression of the magnetosphere which could in this case have led to earthward moving directed vortices.

Measurements conducted with the ISEE vehicles have shown evidence of tailward directed vortical structures along the sides of the magnetosphere. However, there are indications that in the central plasma sheet of the magnetosphere the motion of the particles is observed to be earthward from the tail. An example of such measurements is reproduced in Figure 3 with evidence in the lower panel of a repeated rotation in the direction of the flow which in most cases leads to large (> 100 km/s) speed values (upper panel) at the time when the solar ecliptic longitude ϕSE becomes small; i.e. when the flow is sunward directed (Hones et al., 1981).

**Figure 1.** – A series of flow vortices in the earth´s magnetosphere observed by a Cluster spacecraft on July 6, 2003. The magnetic field and its components are indicated in the top two panels and the ion velocity and its components in the two lower panels (the velocity components in a reference frame in which the z and the x axes are parallel and perpendicular to the average magnetic field are in the middle panel). The profiles were selected from those presented

**Figure 2.** – Map of flow streamlines projected on the xy plane derived from the velocity rotation measured in the 13:36:45 UT – 13:38:20 UT time interval of July 6, 2003 in Figure 1. The streamlines are traced on the ion density (top panel) and on the ion temperature (lower panel) distributions (the white arrows in the top panel represent the meas‐ ured velocity vector direction, and those in the lower panel the direction of the magnetic field). The streamlines de‐ scribe conditions from the inner plasma sheet across the dawn side of the magnetosphere and were selected from

in Figure 7 of Tian et al., (2010).

320 Open Questions in Cosmology

those presented in Figure 11 of Tian et al., (2010).

**Figure 3.** – Speed values (upper panel) and flow direction (lower panel) measured with the ISEE 2 spacecraft on March 10, 1979 within the earth´s magnetosphere. A repeated counterclockwise rotation of the solar ecliptic longitude an‐ gle ϕSE is detected in the evening sector of the plasma sheet (after Hones et al., 1981).

A general schematic view of the flow distribution along the sides of the magnetosphere that has been derived from the ISEE measurements is reproduced in Figure 4. Tailward directed vortex structures occur by those regions and the diagram illustrates plasma patterns that may arise from Kelvin Helmholtz instabilities at the magnetopause and that lead to sudden changes in the flow direction when observed in the earth´s rest frame.

closer to the center of the tail. While along the path line B there should be a complete clock‐ wise rotation (measured from the bottom to the top) a wavy periodic reversal will occur again along the path line C, and a counterclockwise rotation will prevail along the path line D. These variations serve to describe the structure of the vortices as seen in the earth´s rest frame. On the other hand, the flow pattern in the rest frame of the tailward moving wave is sketched in Figure 5b to show how the shape of the vortex structure is maintained within

Plasma Vortices in Planetary Wakes http://dx.doi.org/10.5772/52998 323

**Figure 5.** – (left panel) Dawn sector of the flow pattern in Figure 4 depicting the sequence of flow vectors observed at four different distances from the magnetopause. (right panel) Flow in the rest frame of the tailward moving wave

that wave (Hones et al., 1981).

(Hones et al., 1981).

**Figure 4.** – Schematic description of plasma vortices within the earth´s magnetosphere inferred from the ISEE meas‐ urements. The flow pattern represented by the solid lines is mostly tailward through the magnetosphere as it is indi‐ cated by the white arrows at the bottom (Hones et al., 1981).

The vortex structures are superimposed on a general circulation flow pattern within the magnetosphere in which the plasma is driven tailward along the sides by the kinetic pres‐ sure of the solar wind and then is forced back up in the sunward direction through the cen‐ tral plasma sheet region (Axford and Hines, 1961). The detail variation due to flow motion within and around those vortices is described in Figure 5a as it would be expected at four different distances from the magnetopause and that are represented by the dash lines A, B, C, and D. For a tailward moving vortex pattern (indicated by changes in the arrows of the flow direction) a waving motion should be apparent along the path line A which is located closer to the center of the tail. While along the path line B there should be a complete clock‐ wise rotation (measured from the bottom to the top) a wavy periodic reversal will occur again along the path line C, and a counterclockwise rotation will prevail along the path line D. These variations serve to describe the structure of the vortices as seen in the earth´s rest frame. On the other hand, the flow pattern in the rest frame of the tailward moving wave is sketched in Figure 5b to show how the shape of the vortex structure is maintained within that wave (Hones et al., 1981).

**Figure 4.** – Schematic description of plasma vortices within the earth´s magnetosphere inferred from the ISEE meas‐ urements. The flow pattern represented by the solid lines is mostly tailward through the magnetosphere as it is indi‐

The vortex structures are superimposed on a general circulation flow pattern within the magnetosphere in which the plasma is driven tailward along the sides by the kinetic pres‐ sure of the solar wind and then is forced back up in the sunward direction through the cen‐ tral plasma sheet region (Axford and Hines, 1961). The detail variation due to flow motion within and around those vortices is described in Figure 5a as it would be expected at four different distances from the magnetopause and that are represented by the dash lines A, B, C, and D. For a tailward moving vortex pattern (indicated by changes in the arrows of the flow direction) a waving motion should be apparent along the path line A which is located

cated by the white arrows at the bottom (Hones et al., 1981).

322 Open Questions in Cosmology

**Figure 5.** – (left panel) Dawn sector of the flow pattern in Figure 4 depicting the sequence of flow vectors observed at four different distances from the magnetopause. (right panel) Flow in the rest frame of the tailward moving wave (Hones et al., 1981).

The wavy pattern in the velocity direction that is implied from the measurements repro‐ duced in Figures 2 and 4 could result from effects associated with Kelvin-Helmholtz insta‐ bilities produced along the boundary of the magnetosphere. Such structures require, however, fluid dynamic processes produced under conditions in which the flow is superalf‐ venic; namely, that the local value of the kinetic energy density of the plasma is larger than the local magnetic energy density. As a whole this peculiarity is not applicable to the plasma that pervades within the inner magnetosphere where the magnetic energy density is far larger than the local kinetic energy density. However, different conditions occur along the tail where the magnetic field intensity gradually decreases with the downstream distance from the earth thus reducing the ratio of the magnetic energy density to the kinetic energy density. A suitable comparative calculation of the conditions that were present at the time when the plasma vortices of the Cluster data of Figure 1 were identified can be conducted by using the corresponding pressure profiles of those data and that are reproduced in Fig‐ ure 6. Notable is that the plasma vortices in Figure 1 were detected after the dynamic pres‐ sure of the solar wind exhibited a sudden increase indicated by the dashed vertical line at 12:55 UT in Figure 6. The dynamic pressure PD ~ 3 nPa of the solar wind after that event shown in the upper panel is over 10 times larger than the ~ 0.16 nPa magnetic pressure that can be derived from the ~ 20 nT magnetic field intensity values measured at the same time and that are shown in the third panel. The implication here is that the local flow is superalf‐ venic thus suggesting conditions that could have led to Kelvin-Helmholtz waves along the magnetopause.

#### **3. Plasma vortices in the Venus wake (PVO measurements)**

Measurements conducted within and along the flanks of the Venus wake have revealed conditions that are also suitable for the onset of fluid dynamic processes that lead to the generation of vortex structures. From observations made with the early Mariner 5 space‐ craft it was noted that the solar wind that streams around Venus is subject to a large de‐ crease of its momentum flux as its speed U and density n in the wake become significantly smaller than the values measured under freestream conditions (Bridge et al., 1967; Shefer et al., 1979). A summary of those measurements is presented in Figure 7 to show the presence of a velocity boundary layer that extends along the flanks of the Ve‐ nus wake (indicated by the black shaded region) where both plasma properties exhibit strong deficient values. Such changes are bounded by a plasma transition labeled with the items 2 and 4 along the trajectory of the spacecraft in the lower panel and also at the top of the upper panel, being different from the Venus bow shock encountered at the items labeled 1 and 5 (in the latter case the plasma density is larger in the downstream side as a result of the compressed density values that the solar wind acquires across the bow shock). Much of the missing amount of momentum flux (nU2 ) of the solar wind in the wake that is implied from the profiles in Figure 7 between items 2 and 4 has been found

to be comparable to the momentum flux of the plasma that streams in the trans-termina‐ tor flow within the Venus upper ionosphere (Perez-de-Tejada, 1986) and that was meas‐ ured with the Pioneer Venus Orbiter (PVO) spacecraft (Knudsen et al: 1980). The implication of this agreement is that there is an efficient transport of solar wind momen‐ tum to the Venus upper ionosphere that could be accounted for through the onset of vis‐ cous forces in a continuum fluid interpretation. As it was indicated before such processes require small scale interactions among the particles of both populations in view that the solar wind is a collisionless plasma. Wave-particle interactions resulting from the turbu‐ lence associated with strong fluctuations in the magnetic field that are measured along the flanks of the wake (Bridge et al., 1967; Vőrős et al., 2008) should be ultimately respon‐ sible for the erosion of the ionospheric particles produced by the solar wind and, at the same time, for the the Kelvin-Helmholtz instabilities that develop under the measured su‐

**Figure 6.** – Pressure variations of the solar wind dynamic pressure (top panel) and its static pressure (second panel), together with the magnetic field intensity (third panel) and the plasma pressure (bottom panel) of the plasma sheet

Plasma Vortices in Planetary Wakes http://dx.doi.org/10.5772/52998 325

that were measured on July 6, 2003 with a Cluster spacecraft (Figure 12 in Tian et al., 2010).

peralfvenic flow conditions (Pèrez-de-Tejada et al., 2011).

The wavy pattern in the velocity direction that is implied from the measurements repro‐ duced in Figures 2 and 4 could result from effects associated with Kelvin-Helmholtz insta‐ bilities produced along the boundary of the magnetosphere. Such structures require, however, fluid dynamic processes produced under conditions in which the flow is superalf‐ venic; namely, that the local value of the kinetic energy density of the plasma is larger than the local magnetic energy density. As a whole this peculiarity is not applicable to the plasma that pervades within the inner magnetosphere where the magnetic energy density is far larger than the local kinetic energy density. However, different conditions occur along the tail where the magnetic field intensity gradually decreases with the downstream distance from the earth thus reducing the ratio of the magnetic energy density to the kinetic energy density. A suitable comparative calculation of the conditions that were present at the time when the plasma vortices of the Cluster data of Figure 1 were identified can be conducted by using the corresponding pressure profiles of those data and that are reproduced in Fig‐ ure 6. Notable is that the plasma vortices in Figure 1 were detected after the dynamic pres‐ sure of the solar wind exhibited a sudden increase indicated by the dashed vertical line at 12:55 UT in Figure 6. The dynamic pressure PD ~ 3 nPa of the solar wind after that event shown in the upper panel is over 10 times larger than the ~ 0.16 nPa magnetic pressure that can be derived from the ~ 20 nT magnetic field intensity values measured at the same time and that are shown in the third panel. The implication here is that the local flow is superalf‐ venic thus suggesting conditions that could have led to Kelvin-Helmholtz waves along the

**3. Plasma vortices in the Venus wake (PVO measurements)**

shock). Much of the missing amount of momentum flux (nU2

Measurements conducted within and along the flanks of the Venus wake have revealed conditions that are also suitable for the onset of fluid dynamic processes that lead to the generation of vortex structures. From observations made with the early Mariner 5 space‐ craft it was noted that the solar wind that streams around Venus is subject to a large de‐ crease of its momentum flux as its speed U and density n in the wake become significantly smaller than the values measured under freestream conditions (Bridge et al., 1967; Shefer et al., 1979). A summary of those measurements is presented in Figure 7 to show the presence of a velocity boundary layer that extends along the flanks of the Ve‐ nus wake (indicated by the black shaded region) where both plasma properties exhibit strong deficient values. Such changes are bounded by a plasma transition labeled with the items 2 and 4 along the trajectory of the spacecraft in the lower panel and also at the top of the upper panel, being different from the Venus bow shock encountered at the items labeled 1 and 5 (in the latter case the plasma density is larger in the downstream side as a result of the compressed density values that the solar wind acquires across the bow

wake that is implied from the profiles in Figure 7 between items 2 and 4 has been found

) of the solar wind in the

magnetopause.

324 Open Questions in Cosmology

**Figure 6.** – Pressure variations of the solar wind dynamic pressure (top panel) and its static pressure (second panel), together with the magnetic field intensity (third panel) and the plasma pressure (bottom panel) of the plasma sheet that were measured on July 6, 2003 with a Cluster spacecraft (Figure 12 in Tian et al., 2010).

to be comparable to the momentum flux of the plasma that streams in the trans-termina‐ tor flow within the Venus upper ionosphere (Perez-de-Tejada, 1986) and that was meas‐ ured with the Pioneer Venus Orbiter (PVO) spacecraft (Knudsen et al: 1980). The implication of this agreement is that there is an efficient transport of solar wind momen‐ tum to the Venus upper ionosphere that could be accounted for through the onset of vis‐ cous forces in a continuum fluid interpretation. As it was indicated before such processes require small scale interactions among the particles of both populations in view that the solar wind is a collisionless plasma. Wave-particle interactions resulting from the turbu‐ lence associated with strong fluctuations in the magnetic field that are measured along the flanks of the wake (Bridge et al., 1967; Vőrős et al., 2008) should be ultimately respon‐ sible for the erosion of the ionospheric particles produced by the solar wind and, at the same time, for the the Kelvin-Helmholtz instabilities that develop under the measured su‐ peralfvenic flow conditions (Pèrez-de-Tejada et al., 2011).

of that spacecraft along its nearly polar oriented trajectory. The measurements conducted in the energy cycle for each orbit correspond to those obtained near the outbound (southern hemisphere) crossing of the upper boundary of the ionosphere (ionopause) and represent the most intense ion fluxes that were observed during that cycle. Each reading in the first and second columns describes the time (in UT) when it was made together with the number of the corresponding energy step and the volt per charge value (in parenthesis). The count number of the flux intensity (which leads to the intensity values given in the right side col‐ umn) and the latitudinal collector of the plasma instrument where it was received (marked by the number 3 or 4) are indicated in the third column. These values together with the cal‐ culated particle speed if they represent either H+ or O+ ions, and the azimuthal sector (and also the azimuthal angle α), are presented in the fourth through the seventh columns. The most notable example in the data for orbit 80 shown at the top are measurements in which there are fluxes with a sunward directed component (negative α values). These are observed between the energy steps number 11 and 16 (first column), and there are also ion fluxes measured along the solar wind direction (positive α values) between the energy steps num‐ ber 17 and 21. Comparable variations are also present in the data set of the other orbits with a distribution of the azimuthal α angle that is persistent in orbit 68 (as in orbit 80), and then

Plasma Vortices in Planetary Wakes http://dx.doi.org/10.5772/52998 327

A schematic representation of the direction of the particle fluxes in orbit 80 is described in the upper panel of Figure 8 for 3 different energy cycles of measurements together with the PVO trajectory projected on one quadrant using cylindrical coordinates (the outbound pass with cycles II and III occurs in the southern hemisphere but has been projected to the same quadrant of the inbound pass of cycle I that occurs in the northern hemisphere). The posi‐ tion of the spacecraft during the energy cycle I initiated at 19:34.27 UT in the inbound pass, and the energy cycles II and III initiated at 19:59.16 UT and at 20:08.43 UT in the outbound pass are shown in that figure (each cycle is marked by a rectangular shape). The arrows in‐ dicate the latitudinal direction of arrival of the particle fluxes corresponding to the orienta‐ tion of the collector of the plasma instrument where they were observed (measurements made in collector labeled 4 in Table I during the outbound pass correspond to particle fluxes reaching the most northbound direction detected by the instrument (22.5º < θ < 69º) and is opposite to those made in collector 1 which would be the most southbound directed collec‐ tor of the instrument [Intriligator et al., 1980]. During the energy cycle I in the inbound pass and also in the energy cycle III in the outbound pass there is a tendency for the particle flux‐ es to be directed away from the wake, implying the observation of northbound fluxes in the northern hemisphere (collectors 3 and 4) and also the observation of southbound fluxes in the southern hemisphere (collectors 1 and 2). Different conditions can be identified in the energy cycle II measured in the vicinity of the outbound crossing of the ionopause that is reported in Table I. In this cycle the (dominant) particle fluxes now converge toward the wake (all are detected in collectors 3 or 4) and exhibit directions either with a sunward com‐ ponent (negative α values) or with an anti-sunward component (positive α values). This lat‐ er variation is indicated by large differences in the azimuthal sector number of the measurements which implies angles that may differ by up to 180º between both cases (α = 0

fluctuating orientations in orbits 65 and 66.

corresponding to the antisolar direction).

**Figure 7.** – Thermal speed, density, and bulk speed of the solar wind measured with the Mariner 5 spacecraft (its tra‐ jectory projected in cylindrical coordinates is shown in the lower panel). The labels 1 through 5 along the trajectory and at the top of the upper panel mark important events in the plasma properties (bow shock, intermediate plasma transition) (after Bridge et al..1967).

The presence of a velocity boundary layer along the flanks of the Venus wake has been con‐ firmed with observations conducted with the PVO and more recently with the Venus Ex‐ press (VEX) spacecraft (Pérez-de-Tejada et al., 2011). In the data obtained with both vehicles it has been possible to identify features within that layer that can be interpreted in terms of fluid dynamic processes and, in particular, vortical structures with properties similar to those detected in the earth´s magnetosphere. The first indication on the existence of plasma vortices in the Venus wake was inferred from the observation of changes in the velocity di‐ rection of plasma particles along the PVO trajectory across the Venus wake (Pérez-de-Tejada et al., 1982). In some instances the ion fluxes are not directed away from the sun but their motion has a sunward directed component. A description of such change serves to identify the manner in which the flow is arranged to form vortical structures and their position with‐ in the wake.

From the collection of plasma ion fluxes measured during the first seasons of observation of the PVO we will first refer to a set of plasma data in which the velocity vectors of the ion fluxes exhibit sudden changes in the direction of the particle motion. Those data are present‐ ed in Table I and apply to PVO measurements conducted in orbits 80, 68, 65 and 66, which were conducted in the vicinity of the midnight plane during the first season of observation of that spacecraft along its nearly polar oriented trajectory. The measurements conducted in the energy cycle for each orbit correspond to those obtained near the outbound (southern hemisphere) crossing of the upper boundary of the ionosphere (ionopause) and represent the most intense ion fluxes that were observed during that cycle. Each reading in the first and second columns describes the time (in UT) when it was made together with the number of the corresponding energy step and the volt per charge value (in parenthesis). The count number of the flux intensity (which leads to the intensity values given in the right side col‐ umn) and the latitudinal collector of the plasma instrument where it was received (marked by the number 3 or 4) are indicated in the third column. These values together with the cal‐ culated particle speed if they represent either H+ or O+ ions, and the azimuthal sector (and also the azimuthal angle α), are presented in the fourth through the seventh columns. The most notable example in the data for orbit 80 shown at the top are measurements in which there are fluxes with a sunward directed component (negative α values). These are observed between the energy steps number 11 and 16 (first column), and there are also ion fluxes measured along the solar wind direction (positive α values) between the energy steps num‐ ber 17 and 21. Comparable variations are also present in the data set of the other orbits with a distribution of the azimuthal α angle that is persistent in orbit 68 (as in orbit 80), and then fluctuating orientations in orbits 65 and 66.

A schematic representation of the direction of the particle fluxes in orbit 80 is described in the upper panel of Figure 8 for 3 different energy cycles of measurements together with the PVO trajectory projected on one quadrant using cylindrical coordinates (the outbound pass with cycles II and III occurs in the southern hemisphere but has been projected to the same quadrant of the inbound pass of cycle I that occurs in the northern hemisphere). The posi‐ tion of the spacecraft during the energy cycle I initiated at 19:34.27 UT in the inbound pass, and the energy cycles II and III initiated at 19:59.16 UT and at 20:08.43 UT in the outbound pass are shown in that figure (each cycle is marked by a rectangular shape). The arrows in‐ dicate the latitudinal direction of arrival of the particle fluxes corresponding to the orienta‐ tion of the collector of the plasma instrument where they were observed (measurements made in collector labeled 4 in Table I during the outbound pass correspond to particle fluxes reaching the most northbound direction detected by the instrument (22.5º < θ < 69º) and is opposite to those made in collector 1 which would be the most southbound directed collec‐ tor of the instrument [Intriligator et al., 1980]. During the energy cycle I in the inbound pass and also in the energy cycle III in the outbound pass there is a tendency for the particle flux‐ es to be directed away from the wake, implying the observation of northbound fluxes in the northern hemisphere (collectors 3 and 4) and also the observation of southbound fluxes in the southern hemisphere (collectors 1 and 2). Different conditions can be identified in the energy cycle II measured in the vicinity of the outbound crossing of the ionopause that is reported in Table I. In this cycle the (dominant) particle fluxes now converge toward the wake (all are detected in collectors 3 or 4) and exhibit directions either with a sunward com‐ ponent (negative α values) or with an anti-sunward component (positive α values). This lat‐ er variation is indicated by large differences in the azimuthal sector number of the measurements which implies angles that may differ by up to 180º between both cases (α = 0 corresponding to the antisolar direction).

**Figure 7.** – Thermal speed, density, and bulk speed of the solar wind measured with the Mariner 5 spacecraft (its tra‐ jectory projected in cylindrical coordinates is shown in the lower panel). The labels 1 through 5 along the trajectory and at the top of the upper panel mark important events in the plasma properties (bow shock, intermediate plasma

The presence of a velocity boundary layer along the flanks of the Venus wake has been con‐ firmed with observations conducted with the PVO and more recently with the Venus Ex‐ press (VEX) spacecraft (Pérez-de-Tejada et al., 2011). In the data obtained with both vehicles it has been possible to identify features within that layer that can be interpreted in terms of fluid dynamic processes and, in particular, vortical structures with properties similar to those detected in the earth´s magnetosphere. The first indication on the existence of plasma vortices in the Venus wake was inferred from the observation of changes in the velocity di‐ rection of plasma particles along the PVO trajectory across the Venus wake (Pérez-de-Tejada et al., 1982). In some instances the ion fluxes are not directed away from the sun but their motion has a sunward directed component. A description of such change serves to identify the manner in which the flow is arranged to form vortical structures and their position with‐

From the collection of plasma ion fluxes measured during the first seasons of observation of the PVO we will first refer to a set of plasma data in which the velocity vectors of the ion fluxes exhibit sudden changes in the direction of the particle motion. Those data are present‐ ed in Table I and apply to PVO measurements conducted in orbits 80, 68, 65 and 66, which were conducted in the vicinity of the midnight plane during the first season of observation

transition) (after Bridge et al..1967).

326 Open Questions in Cosmology

in the wake.


northbound directed diverging particle fluxes are measured in the inbound pass (energy cy‐ cle I initiated at 19:53.22 UT), and southbound directed fluxes that also diverge from the wake are observed in the energy cycle III of the outbound pass initiated at 20:26.12 UT.

Plasma Vortices in Planetary Wakes http://dx.doi.org/10.5772/52998 329

In addition, particle fluxes measured in the inner ionosheath during cycle II in the outbound pass (initiated at 20:16.59 UT) and that were included in Table I, converge toward the wake and like those in orbit 80 also include velocity vectors with a sunward directed component (most notable flux intensities in the 4 – 16 energy steps) but there are also sporadic weak fluxes with an anti-sunward directed component in the lowest energies that were measured. The tendency in the data set of the outbound pass of orbits 80 and 68 is that the solar wind streaming near the ionopause (cycle labeled II in Figure 8) can acquire an orientation with a sunward directed component leading to a vortex structure (~180° change of the azimuthal α

**Figure 8.** - Representative position of energy cycles (rectangular shapes) where measurements were made along the trajectory of the PVO in orbit 80 (upper panel) and in orbit 68 (lower panel) projected on a quadrant in cylindrical coordinates. The arrows show schematically the (latitudinal) velocity direction of ion fluxes detected at different ener‐

angle in that region).

gy steps within each cycle.

**Table 1.** Table I: Values of the ion flux intensity (right side column) measured during an energy cycle in the outbound pass of the VO in orbits 80, 68, 65, and 66, through the Venus near wake. The time when the ion fluxes are measured are given in the second column, and the corresponding energy step number with its volts per charge value (in parenthesis) are presented in the first column. Their count number and the latitudinal collector of the plasma instrument where the fluxes were measured is shown in the third column (fluxes in collectors 3 and 4 are northbounddirected implying that fluxes converge toward the plasma wake in the southern (outbound measured) hemisphere). Their speed for H+ and O+ ions is indicated in the fourth and fifth columns. The azimuthal sector and the corresponding azimuthal angle α where the fluxes were detected are indicated in the sixth and seventh columns (sectors above and below ~347.5 correspond to positive and negative α values where in the later case it implies sunward directed fluxes).

A similar distribution of velocity vectors in the Venus wake is observed in the data of orbit 68 of Table I in energy cycles whose position along the PVO trajectory is reproduced in the lower panel of Figure 8. As it was the case for orbit 80 in the upper panel of that figure the northbound directed diverging particle fluxes are measured in the inbound pass (energy cy‐ cle I initiated at 19:53.22 UT), and southbound directed fluxes that also diverge from the wake are observed in the energy cycle III of the outbound pass initiated at 20:26.12 UT.

In addition, particle fluxes measured in the inner ionosheath during cycle II in the outbound pass (initiated at 20:16.59 UT) and that were included in Table I, converge toward the wake and like those in orbit 80 also include velocity vectors with a sunward directed component (most notable flux intensities in the 4 – 16 energy steps) but there are also sporadic weak fluxes with an anti-sunward directed component in the lowest energies that were measured. The tendency in the data set of the outbound pass of orbits 80 and 68 is that the solar wind streaming near the ionopause (cycle labeled II in Figure 8) can acquire an orientation with a sunward directed component leading to a vortex structure (~180° change of the azimuthal α angle in that region).

**Table 1.** Table I: Values of the ion flux intensity (right side column) measured during an energy cycle in the outbound pass of the VO in orbits 80, 68, 65, and 66, through the Venus near wake. The time when the ion fluxes are measured are given in the second column, and the corresponding energy step number with its volts per charge value (in parenthesis) are presented in the first column. Their count number and the latitudinal collector of the plasma instrument where the fluxes were measured is shown in the third column (fluxes in collectors 3 and 4 are northbounddirected implying that fluxes converge toward the plasma wake in the southern (outbound measured) hemisphere).

A similar distribution of velocity vectors in the Venus wake is observed in the data of orbit 68 of Table I in energy cycles whose position along the PVO trajectory is reproduced in the lower panel of Figure 8. As it was the case for orbit 80 in the upper panel of that figure the

Their speed for H+ and O+ ions is indicated in the fourth and fifth columns. The azimuthal sector and the corresponding azimuthal angle α where the fluxes were detected are indicated in the sixth and seventh columns (sectors above and below ~347.5 correspond to positive and negative α values where in the later case it implies

sunward directed fluxes).

328 Open Questions in Cosmology

**Figure 8.** - Representative position of energy cycles (rectangular shapes) where measurements were made along the trajectory of the PVO in orbit 80 (upper panel) and in orbit 68 (lower panel) projected on a quadrant in cylindrical coordinates. The arrows show schematically the (latitudinal) velocity direction of ion fluxes detected at different ener‐ gy steps within each cycle.

## **4. Plasma vortices in the Venus wake (VEX measurements)**

More conclusive evidence in the observation of sunward directed plasma fluxes in the Ve‐ nus wake has been recently reported from measurements conducted with the ASPERA-4 in‐ strument of the Venus Express spacecraft [Lundin et al., 2011]. From a collection of velocity vectors derived from measurements conducted in 380 orbits it was possible to produce the average pattern of O+ and solar wind (SW) H+ ion velocity vectors presented in Figure 9 which are projected in cylindrical coordinates. Notable is that most of the velocity vectors of the O+ ions are deviated towards the inner wake. On the other hand, for the majority of the solar wind H+ ions the velocity vectors by the central tail at x < -2RV have a sunward direct‐ ed component. Such variation mostly occurs in the central wake but in the solar wind H+ ion panel there is also evidence of that behavior in the velocity vectors located downstream from the polar region. The overall pattern of the velocity vectors is consistent with plasma fluxes that reverse direction and agrees with a similar structure that was inferred from the PVO observations; that is, velocity vectors with a sunward component and some directed away from the wake can be separately identified in Figure 9, thus indicating the presence of divergent and sunward directed ion fluxes as it was inferred from the PVO data.

A more detail description of the distribution of velocity vectors measured with the ASPERA-4 instrument of the VEX spacecraft is presented in Figure 10 with an account of separate observa‐ tions of the solar wind H+ and the O+ ion fluxes with values now averaged along the z-direc‐ tion and subsequently projected to the xy plane. Most notable in the left panel of Figure 10 is a pattern located in the central wake within which the velocity vectors of the H+ ions in the dusk side (negative y-values) show a common tendency to be deflected towards the dawn side (pos‐ itive y-values), and also vectors located at and in the vicinity of the midnight meridian that ex‐ hibit a strong solar oriented component. This pattern is consistent with the onset of reverse flow conditions that seem to occur slightly shifted towards the dawn side. As a whole the ve‐ locity vectors in that region contain cases in which they are directed anti-sunward, some are deflected toward positive values in the y-axis, and in various observations they are mostly sun‐ ward oriented. A scheme with different properties is observed in the distribution of velocity vectors of the planetary O+ ion population shown in the right panel of Figure 10. In this case the velocity vectors across the wake show a more thorough deflection toward the dawn side (posi‐ tive y-values) but there is little or no indication in the xy-projection (values averaged along the

Plasma Vortices in Planetary Wakes http://dx.doi.org/10.5772/52998 331

**Figure 10.** – (left panel) Distribution of velocity vectors of the solar wind H+ ion population in the Venus wake ob‐ tained with the ASPERA-4 instrument of the Venus Express projected on the xy plane (the data show evidence of a region in the central wake where the velocity vectors acquire a sunward directed component). (right panel) Distribu‐ tion of velocity vectors of the O+ ion population indicating that ions in the dusk (upper) side of the retrograde rotat‐

From the sunward directed fluxes reported in the left panel of Figure 10 the reversed flow direction should mostly occur for the H+ population and, as it is the case for obstacles im‐ mersed in directional flows, the solar wind particles in the wake will be forced in the up‐ stream direction leading to a vortex structure. A schematic representation of the distribution of flow streamlines that is compatible with this view is indicated in the left panel of Figure 11 with a circulation pattern in the wake that is consistent with the plasma data. The rota‐ tion of the flow direction in the wake is supported by the diagram presented in the right panel of Figure 11 where a wide velocity boundary layer with a thickness that is larger downstream from the terminator stresses the low local flow speeds that should occur near the ionosphere. Such velocity layer derives from the observations shown in Figure 7 and is

ing Venus ionosphere are deviated toward the positive (dawn side) y-direction (Lundin et al., 2011).

z-axis) that they have a sunward directed component.

**Figure 9.** – Velocity vectors of out-flowing O+ (upper panel) and solar wind H+ (lower panel) ions measured in the Venus wake with the ASPERA-4 instrument of the Venus Express spacecraft projected in cylindrical coordinates. The velocity scales for the O+ and the H+ ions are noted in the upper left corner in each diagram. Notice that the H+ flow velocities in the wake are low and barely discernible. However, applying unit vectors we find that most solar wind H+ velocity vectors are sunward directed near the tail central plane at x < 2 RV.

A more detail description of the distribution of velocity vectors measured with the ASPERA-4 instrument of the VEX spacecraft is presented in Figure 10 with an account of separate observa‐ tions of the solar wind H+ and the O+ ion fluxes with values now averaged along the z-direc‐ tion and subsequently projected to the xy plane. Most notable in the left panel of Figure 10 is a pattern located in the central wake within which the velocity vectors of the H+ ions in the dusk side (negative y-values) show a common tendency to be deflected towards the dawn side (pos‐ itive y-values), and also vectors located at and in the vicinity of the midnight meridian that ex‐ hibit a strong solar oriented component. This pattern is consistent with the onset of reverse flow conditions that seem to occur slightly shifted towards the dawn side. As a whole the ve‐ locity vectors in that region contain cases in which they are directed anti-sunward, some are deflected toward positive values in the y-axis, and in various observations they are mostly sun‐ ward oriented. A scheme with different properties is observed in the distribution of velocity vectors of the planetary O+ ion population shown in the right panel of Figure 10. In this case the velocity vectors across the wake show a more thorough deflection toward the dawn side (posi‐ tive y-values) but there is little or no indication in the xy-projection (values averaged along the z-axis) that they have a sunward directed component.

**4. Plasma vortices in the Venus wake (VEX measurements)**

330 Open Questions in Cosmology

divergent and sunward directed ion fluxes as it was inferred from the PVO data.

**Figure 9.** – Velocity vectors of out-flowing O+ (upper panel) and solar wind H+ (lower panel) ions measured in the Venus wake with the ASPERA-4 instrument of the Venus Express spacecraft projected in cylindrical coordinates. The velocity scales for the O+ and the H+ ions are noted in the upper left corner in each diagram. Notice that the H+ flow velocities in the wake are low and barely discernible. However, applying unit vectors we find that most solar wind H+

velocity vectors are sunward directed near the tail central plane at x < 2 RV.

More conclusive evidence in the observation of sunward directed plasma fluxes in the Ve‐ nus wake has been recently reported from measurements conducted with the ASPERA-4 in‐ strument of the Venus Express spacecraft [Lundin et al., 2011]. From a collection of velocity vectors derived from measurements conducted in 380 orbits it was possible to produce the average pattern of O+ and solar wind (SW) H+ ion velocity vectors presented in Figure 9 which are projected in cylindrical coordinates. Notable is that most of the velocity vectors of the O+ ions are deviated towards the inner wake. On the other hand, for the majority of the solar wind H+ ions the velocity vectors by the central tail at x < -2RV have a sunward direct‐ ed component. Such variation mostly occurs in the central wake but in the solar wind H+ ion panel there is also evidence of that behavior in the velocity vectors located downstream from the polar region. The overall pattern of the velocity vectors is consistent with plasma fluxes that reverse direction and agrees with a similar structure that was inferred from the PVO observations; that is, velocity vectors with a sunward component and some directed away from the wake can be separately identified in Figure 9, thus indicating the presence of

**Figure 10.** – (left panel) Distribution of velocity vectors of the solar wind H+ ion population in the Venus wake ob‐ tained with the ASPERA-4 instrument of the Venus Express projected on the xy plane (the data show evidence of a region in the central wake where the velocity vectors acquire a sunward directed component). (right panel) Distribu‐ tion of velocity vectors of the O+ ion population indicating that ions in the dusk (upper) side of the retrograde rotat‐ ing Venus ionosphere are deviated toward the positive (dawn side) y-direction (Lundin et al., 2011).

From the sunward directed fluxes reported in the left panel of Figure 10 the reversed flow direction should mostly occur for the H+ population and, as it is the case for obstacles im‐ mersed in directional flows, the solar wind particles in the wake will be forced in the up‐ stream direction leading to a vortex structure. A schematic representation of the distribution of flow streamlines that is compatible with this view is indicated in the left panel of Figure 11 with a circulation pattern in the wake that is consistent with the plasma data. The rota‐ tion of the flow direction in the wake is supported by the diagram presented in the right panel of Figure 11 where a wide velocity boundary layer with a thickness that is larger downstream from the terminator stresses the low local flow speeds that should occur near the ionosphere. Such velocity layer derives from the observations shown in Figure 7 and is produced by viscous transport of solar wind momentum to the upper ionospheric plasma. Since low speeds are expected close to the ionosphere the solar wind will reach a position within the wake where the large local flow pressure values will produce a change in its di‐ rection thus leading to a vortex structure as that depicted for a flow past an obstacle indicat‐ ed in the right panel of Figure 11. It is to be noted that the fluid dynamic response of the solar wind to the large pressure values present in the wake is different from processes asso‐ ciated with Kelvin-Helmholtz instabilities along the boundary of the ionosphere and that, as it is the case in the earth´s magnetosphere, also contribute to produce vortex structures.

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**Figure 11.** – (left panel) Schematic representation of a vortex structure generated downstream from a polar region of the Venus ionosphere (the streamlines indicate the direction of motion of the solar wind). (right panel) Flow pattern within a velocity boundary layer that extends past an obstacle (higher pressures in the wake reverse the flow direction near the obstacle leading to a vortex structure, Schlichting, 1968).

The fluid dynamic interpretation of both phenomena serves to account for the processes that produce the vortex structures in the earth´s magnetosphere and in the Venus plasma wake, and should also be applicable to the conditions that are expected in other planets; namely, those with an appreciable or strong intrinsic magnetic field (Jupiter, Saturn, Uranus, Nep‐ tune, and Mercury), or in other un-magnetized planets in which there may only be fossil remnants of an early magnetic field as it is the case in Mars (Acuña et al., 1999).

## **Author details**


## **References**

produced by viscous transport of solar wind momentum to the upper ionospheric plasma. Since low speeds are expected close to the ionosphere the solar wind will reach a position within the wake where the large local flow pressure values will produce a change in its di‐ rection thus leading to a vortex structure as that depicted for a flow past an obstacle indicat‐ ed in the right panel of Figure 11. It is to be noted that the fluid dynamic response of the solar wind to the large pressure values present in the wake is different from processes asso‐ ciated with Kelvin-Helmholtz instabilities along the boundary of the ionosphere and that, as it is the case in the earth´s magnetosphere, also contribute to produce vortex structures.

**Figure 11.** – (left panel) Schematic representation of a vortex structure generated downstream from a polar region of the Venus ionosphere (the streamlines indicate the direction of motion of the solar wind). (right panel) Flow pattern within a velocity boundary layer that extends past an obstacle (higher pressures in the wake reverse the flow direction

The fluid dynamic interpretation of both phenomena serves to account for the processes that produce the vortex structures in the earth´s magnetosphere and in the Venus plasma wake, and should also be applicable to the conditions that are expected in other planets; namely, those with an appreciable or strong intrinsic magnetic field (Jupiter, Saturn, Uranus, Nep‐ tune, and Mercury), or in other un-magnetized planets in which there may only be fossil

and D. S. Intriligator3

remnants of an early magnetic field as it is the case in Mars (Acuña et al., 1999).

near the obstacle leading to a vortex structure, Schlichting, 1968).

**Author details**

332 Open Questions in Cosmology

H. Pérez-de-Tejada1\*, Rickard Lundin2

1 Institute of Geophysics, UNAM, México, México

3 Carmel Research Center, Santa Mónica, California, USA

2 Institute of Space Physics, Umea, Sweden


[16] Shefer, R., Lazarus, A., & Bridge, H. (1979). A re-examination of plasma measure‐

[17] Tian, A., et al. (2010). A series of plasma flow vortices in the tail plasma sheet associ‐ ated with solar wind pressure enhancement. *J. Geophys. Res.*, 115, A09204, doi:

[18] Terada, N., Machida, S., & Shinagawa, H. (2002). Global hybrid simulation of the Kel‐ vin-Helmholtz instability at the Venus ionopause,. *J. Geophys. Res.,*, 107, 14.

[19] Wolfe, R., et al. (1980). The onset and development of Kelvin-Helmholtz instability at

[20] Vőrős, Z., et al. (2008). Intermittent turbulence, noisy fluctuations, and wavy struc‐ tures in the Venusian magnetosheath and wake,. *Journal of. Geophys. Res.-Planets,*, 113,

ments from the Mariner 5 Venus encounter,. *J. Geophys. Res.,*, 84, 2109.

the Venus ionopause,. *Journal of Geophys. Res.,*, 85, 7697-7707.

JA014989.

334 Open Questions in Cosmology

E00B21, doi:10.1029/2008.

## *Edited by Gonzalo J. Olmo*

In the last years we have witnessed how the field of Cosmology has experienced a metamorphosis. From being essentially the search for three numbers (the expansion rate, the deceleration parameter, and the cosmological constant), it has become a precision science. This scientific discipline is determined to unravel the most minute details of the elementary processes that took place during the most primitive stages of the Universe and also of the mechanisms driving the cosmic expansion and the growth of structures at the largest scales. To achieve these goals one needs not only the development of new experimental and observational techniques but also a deep understanding of the underlying theoretical frameworks. This book gathers the work of leading experts in these fields and provides a broad view of some of the most relevant open questions faced by Cosmology at the beginning of the twenty-first century.

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Open Questions in Cosmology

Open Questions in

Cosmology