# Cosmological evolution in exponential gravity

###### Abstract

We explore the cosmological evolution in the exponential gravity (). We summarize various viability conditions and explicitly demonstrate that the late-time cosmic acceleration following the matter-dominated stage can be realized. We also study the equation of state for dark energy and confirm that the crossing of the phantom divide from the phantom phase to the non-phantom (quintessence) one can occur. Furthermore, we illustrate that the cosmological horizon entropy globally increases with time.

###### pacs:

04.50.Kd, 95.36.+x, 98.80.-k## I Introduction

There exist two representative approaches to account for the current accelerated expansion WMAP ; Komatsu:2010fb ; SN1 of the universe Copeland:2006wr ; Nojiri:2006ri ; Sotiriou:2008rp ; DeFelice:2010aj ; Review-Tsujikawa ; Linder:2010qn . One is to introduce “dark energy” in the framework of general relativity. The other is to consider a modified gravitational theory, such as gravity.

Several viable theories of gravity have been constructed; e.g., power-law Amendola:2006we ; Li:2007xn , Nojiri-Odintsov Nojiri-Odintsov , Hu-Sawicki Hu:2007nk , Starobinsky’s Starobinsky:2007hu , Appleby-Battye Appleby:2007vb , and Tsujikawa’s Tsujikawa:2007xu models (for more detailed references, see a recent review on gravity DeFelice:2010aj ). It is known that these models can satisfy the following conditions for the viability: (i) positivity of the effective gravitational coupling, (ii) stability of cosmological perturbations Nojiri:2003ft ; Dolgov:2003px ; Faraoni:2006sy ; Song:2006ej , (iii) asymptotic behavior to the standard -Cold-Dark-Matter () model in the large curvature regime, (iv) stability of the late-time de Sitter point Amendola:2006we ; Muller:1987hp ; Faraoni:2005vk , (v) constraints from the equivalence principle, and (vi) solar-system constraints Solar-System-Constraints .

Recently, an interesting model of , called “exponential gravity”, has been proposed in Refs. Exponential-type-f(R)-gravity ; Cognola:2007zu ; Linder:2009jz with being constants. The important feature of the exponential gravity is that it has only one more parameter than the model. The constraints from the violation of the equivalence principle Tsujikawa:2009ku and cosmological observations Ali:2010zx on the exponential gravity have been examined. The exponential gravity in the framework of gravity has been extended to a gravitational theory in terms of the torsion scalar Linder:2010py (for a related work on torsion gravity, see Bengochea:2008gz ). We note that the cosmological dynamics in the gravitational theory consisting only of the exponential term without the Einstein-Hilbert one has also been studied in Ref. Abdelwahab:2007jp .

In this paper, we explicitly investigate the cosmological evolution in the exponential gravity model given by Cognola et al. Cognola:2007zu and Linder Linder:2009jz in more detail by using the analysis method in Ref. Hu:2007nk . We also check the above six viability conditions for the model. In particular, we demonstrate that after the matter-dominated stage, the current accelerated expansion of the universe and the crossing of the phantom divide from the phantom phase to the non-phantom (quintessence) one can be realized. It is interesting to note that the crossing of the phantom divide is implied by the cosmological observational data observational status , while the exponential gravity is a ghost free theory. In addition, we illustrate that the cosmological horizon entropy globally increases with time. We use units of and denote the gravitational constant by with the Planck mass of GeV.

The paper is organized as follows. In Sec. II, we review the model of the exponential gravity in Refs. Cognola:2007zu ; Linder:2009jz and summarize its viability conditions. In Sec. III, we explore the cosmological evolution of the model. We examine the horizon entropy in Sec. IV. Finally, conclusions are given in Sec. V.

## Ii Exponential gravity

### ii.1 The model

The action of gravity with matter is given by

(1) |

where is the determinant of the metric tensor , is the action of matter which is assumed to be minimally coupled to gravity, i.e., the action is written in the Jordan frame, and denotes matter fields. Here, we use the standard metric formalism.

Taking the variation of the action in Eq. (1) with respect to , one obtains Sotiriou:2008rp

(2) |

where is the Einstein tensor, , is the covariant derivative operator associated with , is the covariant d’Alembertian for a scalar field, and is the contribution to the energy-momentum tensor from all perfect fluids of generic matter.

In this paper, we concentrate on the exponential gravity in Refs. Cognola:2007zu ; Linder:2009jz , given by

(3) |

where and . Note that corresponds to the characteristic curvature modification scale.

### ii.2 Viability conditions on exponential gravity

For the model of the exponential gravity in Eq. (3), it is straightforward to show that the conditions for the viability can be satisfied, which are summarized as follows:

(i) Positivity of the effective gravitational coupling

When , . This is required for the positivity of the effective gravitational coupling to avoid anti-gravity. In the sense of the quantum theory, the graviton is not a ghost.

(ii) Stability of cosmological perturbations

When and , , where the prime denotes differentiation with respect to . This is required for the stability of cosmological perturbations Dolgov:2003px ; Faraoni:2006sy ; Song:2006ej . In the sense of the quantum theory, the scalaron, which is a new scalar degree of freedom in gravity, is not a tachyon Starobinsky:2007hu .

(iii) Asymptotic behavior to the model in the large curvature regime

Since for , this model is reduced to the model in the large curvature regime . Such a behavior is necessary for the presence of the matter-dominated stage.

(iv) Stability of the late-time de Sitter point

When , Tsujikawa:2009ku , where and is the value of the scalar curvature at the de Sitter point. This condition is required for the stability of the late-time de Sitter point Amendola:2006we ; Muller:1987hp ; Faraoni:2005vk . The quantity characterizes the deviation from the model because for the model. In the exponential gravity, by using and , one finds that for . Hence, for .

(v) Constraints from the violation of the equivalence principle

It is known that gravity can satisfy local gravity constraints from the violation of the equivalence principle under the chameleon mechanism Mota:2003tc ; Chameleon mechanism . By making the following conformal transformation Conformal Transformation : , the action of gravity in Eq. (1) can be rewritten in the Einstein frame, where with the scalar field . In what follows, a tilde represents the quantity in the Einstein frame. We consider a spherically symmetric body with radius in the Minkowski space-time. Here, is the distance of the center of the body, and is a conserved matter density in the Einstein frame with the energy density of matter in the Jordan frame. We assume that a spherically symmetric body has constant densities of and inside () and outside (), respectively. In this case, the effective potential has two minima at the field values and satisfying the conditions and with a heavier mass squared and a lighter mass squared , respectively. The thin-shell parameter is defined as Chameleon mechanism , where is the gravitational potential at the surface of the body and .

The tightest experimental bound on obtained from the violation of the equivalence principle for the accelerations of the Earth and the moon toward the Sun is given by Capozziello:2007eu ; Tsujikawa:2008uc . This is the thin-shell parameter for the Earth. By using the value of the gravitational potential for the Earth and , the condition on is reduced to Tamaki:2008mf . The field value can be found by solving with , which gives .

For the exponential gravity, Tsujikawa:2009ku and , where is the current scalar curvature, is the current Hubble parameter, is the current density parameter of non-relativistic matter (cold dark matter and baryon), is the energy density of non-relativistic matter at the present time, and is the critical density. As a consequence, by using and the homogeneous baryon/dark matter density , we find Tsujikawa:2009ku . When , which is the stability condition for the late-time de Sitter point in the exponential gravity, the above constraint on is satisfied very well. For example, if , . In what follows, the superscript denotes the present value.

(vi) Solar-system constraints

The bound on the thin-shell parameter coming from the solar-system constraint DeFelice:2010aj is weaker than that from the violation of the equivalence principle shown above.

## Iii Cosmological evolution

We assume the flat Friedmann-Lemaître-Robertson-Walker (FLRW) space-time with the metric,

(4) |

where is the scale factor. From Eq. (2), we obtain the following gravitational field equations:

(5) | |||||

(6) |

where is the Hubble parameter, the dot denotes the time derivative of , and and are the energy density and pressure of all perfect fluids of generic matter, respectively.

Equation (5) can be rewritten to

(7) |

while the scalar curvature is expressed as

(8) |

To solve Eqs. (7) and (8), we introduce the following variables Hu:2007nk :

(9) | |||||

(10) |

with

(11) | |||||

(12) |

where is the energy density of dark energy and is the energy density of radiation at the present time. In our analysis, the contribution from radiation is also taken into consideration. Equations (7) and (8) are reduced to a coupled set of ordinary differential equations

(13) | |||||

(14) | |||||

The equation of state for dark energy is given by

(15) |

derived by the continuity equation

(16) |

On the other hand, the effective equation of state is defined as

(17) |

where and are the total energy density and pressure of the universe, respectively. Here, , and are the pressure of dark energy, non-relativistic matter and radiation, respectively.

In Figs. 1, 2 and 3, we depict the cosmological evolutions of the density parameters of dark energy , non-relativistic matter and radiation as functions of the redshift for , and , respectively. In the high regime (), the universe is at the matter-dominated stage (, ). As decreases, the dark energy becomes dominant over matter for , where is the crossover point in which . Explicitly, we have , and for , and , respectively. The values of become smaller for the larger values of . At the present time (), , and for , and , respectively. In Fig. 4, we also show the cosmological evolution of for . The qualitative behaviors of for and are similar to that for . Thus, the current accelerated expansion of the universe following the matter-dominated stage can be realized in the exponential gravity.

We note that in solving Eq. (21) numerically, we have taken the initial conditions at as and , where , and for , and , respectively. The values of have been chosen so that , i.e., the exponential gravity at can be very close to the model, in which . Since in the high regime (), . Consequently, the value of the combination is set as . Therefore, we have only one free parameter in the exponential gravity in Eq. (3). Furthermore, from Eq. (13) we see that at and it follows from Eq. (15) that at . All numerical calculations have been executed for Komatsu:2010fb .

The cosmological evolution of the equation of state for dark energy in Eq. (15) is shown in Fig. 5. From the figure, we see that starts at the phase of a cosmological constant and evolves from the phantom phase () to the non-phantom (quintessence) phase (). The crossing of the phantom divide occurs at , where , and for , and , respectively. The values of become smaller for the larger values of . Moreover, the present values of are , and for , and , respectively. Since is a constant, the larger is, the closer the exponential gravity is to the model. The results on are qualitatively the same as the analysis in Refs. Linder:2009jz ; Ali:2010zx . Thus, the crossing of the phantom divide from the phantom phase to the non-phantom one can be realized in the exponential gravity. We remark that the similar behaviors can occur in Hu-Sawicki Hu:2007nk ; Martinelli:2009ek , Appleby-Battye Appleby:2009uf , and Starobinsky’s M-S-Y models as well.

In Fig. 6, we also illustrate the cosmological evolution of the effective equation of state in Eq. (17). The present values of are , and for , and , respectively. We remark that does not cross the line of the phantom divide unlike due to the null energy condition .

Finally, we mention that an gravity model with realizing a crossing of the phantom divide from the non-phantom phase to the phantom one, which is the opposite transition from the above one, has been reconstructed in Ref. BGNO-PC . In addition, the behavior of gravity with realizing multiple crossings of the phantom divide Bamba:2009kc and that of gravity around a crossing of the phantom divide by taking into account the presence of cold dark matter Bamba:2009dk have also been explored.

## Iv Horizon entropy

In Ref. Bamba:2009id , it is shown that it is possible to obtain a picture of equilibrium thermodynamics on the apparent horizon in the FLRW background for gravity as well as that of non-equilibrium thermodynamics due to a suitable redefinition of an energy momentum tensor of the “dark” component that respects a local energy conservation. For a recent review on the Black hole entropy on scalar-tensor and gravity, see Ref. Faraoni:2010yi .

In general relativity, the Bekenstein-Hawking horizon entropy is expressed as , where is the area of the apparent horizon Bekenstein:1973ur ; Bardeen:1973gs ; Hawking:1974sw . The Bekenstein-Hawking entropy

(22) |

is a global geometric quantity which is proportional to the horizon area with a constant coefficient . This quantity is not directly affected by the difference of gravitational theories. We regard the horizon entropy in Eq. (22) as the one in the equilibrium description Bamba:2009id . On the other hand, in the context of modified gravity theories including gravity a horizon entropy associated with a Noether charge has been proposed by Wald Wald entropy . The Wald entropy is a local quantity defined in terms of quantities on the bifurcate Killing horizon. More specifically, it depends on the variation of the Lagrangian density of gravitational theories with respect to the Riemann tensor. This is equivalent to , where is the effective gravitational coupling in gravity Cognola:2005de . Therefore, we use the Wald entropy in the exponential gravity in Eq. (3)

(23) |

In what follows, a hat denotes the quantity in the non-equilibrium description of thermodynamics.

It can be shown that the horizon entropy in the equilibrium description has the following relation with in the non-equilibrium description Bamba:2009id :

(24) |

with

(25) |

where is the new term which can be interpreted as a term of entropy produced in the non-equilibrium thermodynamics. The difference between and appears in gravity due to . Note that is identical to in general relativity due to . From Eq. (24), we see that the change of the horizon entropy in the equilibrium framework involves the information of both and in the non-equilibrium framework.

In Figs. 7, 8 and 9, we show the cosmological evolution of the horizon entropy in Eq. (23) in the non-equilibrium description of thermodynamics and in Eq. (22) in the equilibrium description of thermodynamics for , and , respectively. In these figures, we illustrate the normalized quantities and with being the present value of the horizon entropy . Furthermore, we also depict the evolution of . We note that as , increases with time as long as continues to decreases to the de Sitter point, in which becomes a constant.

In the high regime (), since the deviation of the exponential gravity from the model, i.e., general relativity, is very small, the evolution of is similar to that of . In other words, for the high regime (the higher curvature regime) because . As decreases (and also decreases), the deviation of the exponential gravity from the model emerges, i.e., and decreases. Hence, there appears a difference between the evolution of and that of . Note that . The present values of are , and for , and , respectively. It is clear from Figs. 7, 8 and 9 that both and globally increases with time for any values of . This confirms that the second law of thermodynamics on the apparent horizon always holds. The similar behaviors for both and have been obtained in the Starobinsky’s model Bamba:2009id . Furthermore, we see that the larger is, the closer the evolution of is to that of .

## V Conclusions

In the present paper, we have studied the cosmological evolution in the exponential gravity. We have summarized various viability conditions and explicitly illustrated that the late-time cosmic acceleration can be realized after the matter-dominated stage. We have also shown that the crossing of the phantom divide from the phantom phase to the non-phantom one can occur and the cosmological horizon entropy globally increases with time. Phenomenologically, at least in the light of the background cosmological evolution, the exponential gravity can be regarded as one of the most promising viable modified gravitational theories because (a) it satisfies all conditions for the viability; (b) in substance it has only one model parameter; and (c) both the current cosmic acceleration following the matter-dominated stage and the crossing of the phantom divide can be realized.

## Acknowledgments

We thank Mr. Hayato Motohashi for his important suggestions. We are grateful to Professor Shinji Tsujikawa and Dr. Antonio De Felice for helpful discussions. The work is supported in part by the National Science Council of R.O.C. under: Grant #s: NSC-95-2112-M-007-059-MY3 and NSC-98-2112-M-007-008-MY3 and National Tsing Hua University under the Boost Program and Grant #: 97N2309F1.

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