Thermodynamics in $F(R)$ gravity with phantom crossing

We study thermodynamics of the apparent horizon in $F(R)$ gravity. In particular, we demonstrate that a $F(R)$ gravity model with realizing a crossing of the phantom divide can satisfy the second law of thermodynamics in the effective phantom phase as well as non-phantom one.


I. INTRODUCTION
ent horizon area, Friedmann equations follow from the first law of thermodynamics [43].
The equivalent considerations for the FRW universe with the viscous fluid have also been studied [44].
In addition, it was proposed [45] that in F (R) gravity, a non-equilibrium thermodynamic treatment should be required in order to derive the corresponding gravitational field equation by using the procedure in Ref. [39]. It was reconfirmed in Ref. [46] in F (R) gravity as well as Ref. [47] in scalar-tensor theories. The first [48] and second [49] laws of thermodynamics on the apparent horizon in generalized theories of gravitation have recently been analyzed by taking into account the non-equilibrium thermodynamic treatment. Reinterpretations of the non-equilibrium correction [45] through the introduction of a mass-like function [50] and other approaches [51,52] have also been explored. Incidentally, the horizon entropy in fourdimensional modified gravity [53] and a quantum logarithmic correction to the expression of the horizon entropy in a cosmological context [54,55,56] have been examined. Moreover, the first law of the ordinary equilibrium thermodynamics in F (R) gravity, scalar-tensor theories, the Gauss-Bonnet gravity and more general Lovelock gravity have been discussed in Refs. [57,58,59,60], while the corresponding studies on the second law in the accelerating universe, F (R) gravity, the Gauss-Bonnet gravity and the Lovelock gravity have been done in Refs. [61,62,63], respectively. Studies of thermodynamics in braneworld scenario [64,65,66] as well as its properties of dark energy [67] have also been performed.
It was pointed out in Ref. [68] that thermodynamics in the phantom phase usually leads to a negative entropy. Moreover, it was noted [69] that in the framework of general relativity the horizon entropy decreases in phantom models. However, the conditions that the black hole entropy can be positive in the F (R) gravity models [21,22,23] with the solar-system tests have been analyzed in Ref. [70]. These recent studies have motivated us to explore whether in the framework of F (R) gravity the second law of thermodynamics can be satisfied in the phantom phase. To illustrate the point, in the present paper we consider a F (R) gravity model with a crossing of the phantom divide [15] as it contains an effective phantom phase.
The paper is organized as follows. In Sec. II, we explain the first and second laws of thermodynamics in F (R) gravity. In Sec. III, we demonstrate that the model of F (R) gravity with the phantom crossing [15] can satisfy the generalized second law of the thermodynamics.
Finally, conclusions are given in Sec. IV.

II. THERMODYNAMICS IN F (R) GRAVITY
In this section, we study the first and second laws of thermodynamics of the apparent horizon in F (R) gravity. We consider the four-dimensional flat spacetime.
The action of F (R) gravity with matter is written as where g is the determinant of the metric tensor g µν and L matter is the matter Lagrangian.
From the action in Eq. (2.1), the field equation of modified gravity is given by where the prime denotes differentiation with respect to R, ∇ µ is the covariant derivative operator associated with g µν , ≡ g µν ∇ µ ∇ ν is the covariant d'Alembertian for a scalar field, R µν is the Ricci curvature tensor, and T (matter) µν = diag (ρ, p, p, p) is the contribution to the energy-momentum tensor from all ordinary matters with ρ and p being the energy density and pressure of all ordinary matters, respectively.

5)
where ρ c and p c can be regarded as the energy density and pressure generated due to the difference of F (R) gravity from general relativity, given by respectively, with the scalar curvature of R = 6 Ḣ + 2H 2 . Here, H =ȧ/a is the Hubble parameter and the dot denotes the time derivative of ∂/∂t. We define the effective energy density and pressure of the universe as ρ eff ≡ ρ t /F ′ (R) and p eff ≡ p t /F ′ (R) with ρ t = ρ + ρ c and p t = p + p c , respectively. Hence, from Eqs. (2.5) and (2.6) we see that even in F (R) gravity, the gravitational field equations are expressed as H 2 = κ 2 ρ eff /3 andḢ = −κ 2 (ρ eff + p eff ) /2, which are the same as those in general relativity.
The continuity equation in terms of the effective energy density and pressure of the universe is given byρ Similarly, the (semi-)continuity equation of ordinary matters has the forṁ ρ + 3H (ρ + p) = q . (2.10) One can take q = 0 because the gravity is determined only by ordinary matters. Assuming that the energy fluid, generated from the modification of gravity, behaves as a perfect fluid, we have similar semi-continuity equations aṡ where q c and q t (= q + q c ) are quantities of expressing energy exchange. Using Eqs. (2.5), (2.6) and (2.12), we obtain Clearly, from Eq. (2.13), we find that q t = 0 for general relativity with F (R) = R, whereas q t does not generally vanish in F (R) gravity since there could exist some energy exchange with the horizon.

B. First law of thermodynamics
We now illustrate the first law of thermodynamics in F (R) gravity. By using the spherical symmetry, the metric (2.3) can be written as wherer = a(t)r, x 0 = t and x 1 = r, and h αβ is the two-dimensional metric h αβ = diag (−1, a 2 ). The dynamical apparent horizon is determined by the relation h αβ ∂ αr ∂ βr = 0.
The radius of the apparent horizon for the FRW spacetime is given by [43,49] The associated temperature T of the apparent horizon is determined through the surface gravity of where h is the determinant of the metric h αβ . We note that the recent type Ia Supernovae data suggests that in the accelerating universe the enveloping surface should be the apparent horizon rather than the event one from the thermodynamic point of view [61].

Substituting this equation into Eq. (2.19) and multiplying the resultant equation by the
The left-hand side of Eq. (2.20) can be rewritten as which leads to the Clausius relation where the entropy S and the energy flux δQ are defined by where we have used (∂H 2 /∂ρ) = κ 2 / (3F ′ (R)). By combining Eqs. (2.9), (2.10) and (2.26), we obtain is the total work density [73]. Here, T (t) µν = diag (ρ t , p t , p t , p t ) is the contribution to the energy-momentum tensor from all ordinary matters and energy fluid. This may be regarded as the work generated through the evolution of the apparent horizon [73]. It follows from Eqs. (2.17), (2.18) and (2.30) that the first law of thermodynamics in modified gravity can be constructed as which characterizes the non-equilibrium thermodynamics of the apparent horizon in F (R) gravity.

C. Second law of thermodynamics
Next, we investigate the second law of thermodynamics in F (R) gravity. From Eq. (2.34), the first law of thermodynamics in terms of the horizon entropy S h is expressed as The Gibbs equation in terms of all matter and energy fluid is given by where T t and S t denote the temperature and entropy of total energy inside the horizon, respectively. We assume that where b is a constant with 0 < b < 1. If there is no energy exchange between the outside and inside of the apparent horizon, i.e., q t = 0, thermal equilibrium realizes and therefore b = 1.
The second law of thermodynamics in F (R) gravity can be described by [49]

III. SECOND LAW OF THERMODYNAMICS IN A F (R) GRAVITY MODEL RE-ALIZING A CROSSING OF THE PHANTOM DIVIDE
In this section, we examine whether a F (R) gravity model [15] with the phantom crossing can satisfy the second law of thermodynamics discussed in Sec. II. In the model [15], the Hubble rate H(t) is given by where γ is a positive constant and t s is the time when the Big Rip singularity appears. Here, we only consider the period 0 < t < t s . When t → 0, i.e., t ≪ t s , H(t) behaves as In the FRW background (2.3), the effective EoS w eff is given by [8] w eff = −1 − 2Ḣ/ (3H 2 ).
The form of F (R) is given by where Here, t 0 is the present time andp ± are arbitrary constants. We remark that the stability for the obtained solutions in Eq. (3.3) under a quantum correction coming from conformal anomaly has been examined in Ref. [15]. It has been shown that the quantum correction could be small when the phantom crossing occurs, although it becomes important near the Big Rip singularity.
As the denominator inside the first large braces { } on the right-hand side of Eq.
In this case, we obtain Note that R > 0 for R ∼ 60γ (20γ − 1) /t 2 and γ > 1/20. Using Eqs. (2.41), (3.4) and (3.6) and taking into account the fact thatp ± are arbitrary constants, the necessary condition to where we have assumed γ > 1/20. We remark that for simplicity, we have chosen positive coefficients of R −β + /2 and R −β − /2 in the large braces { } of Eq. (3.6). By taking the values ofp ± so that the relation (3.7) can be met, the second law of thermodynamics, i.e., J ≥ 0, can be satisfied.
It follows from w eff = −1 − 2Ḣ/ (3H 2 ) that w eff = −1 whenḢ = 0. Solving w eff = −1 with respect to t by using Eq. (3.1), we find that w eff crosses the phantom divide at the time t = t c , given by . (3.8) On the other hand, when t → t s , we have In this case, the scale factor is given by a(t) ∼ā (t s − t) −10 with a constant ofā. When t → t s , a → ∞ and therefore the Big Rip singularity appears. In this limit, w eff = −16/15 < −1, corresponding to the phantom phase. The form of F (R) is given by which is reduced to for t 2 s R ≫ 1, whereF In this case, we obtain For a power-low type F (R) gravity described as F (R) = c 1 M 2 (R/M 2 ) −n , where c 1 and n are dimensionless constants and M denotes a mass scale, the scale factor a(t) is given by [15,74]. The form of F (R) in Eq. (3.11) corresponds to the case with n = −7/2. Accordingly, a(t) =ā (t s − t) −10 , which impliesä = 110ā (t s − t) −12 > 0. Thus, a late-time cosmic acceleration can be realized. This is the outcome of the F (R) gravity model in Eq. (3.11). It should be noted that whether the gravity model of F (R) ∼ F R 7/2 in Eq. (3.11) can pass Solar System tests still needs to be examined [75].
From Eq. (3.9), we obtain R = 1260/ (t s − t) 2 , which leads to J = (72 − 71b)F /4536 R 9/2 based on Eq. (3.13). In the thermal equilibrium limit, i.e., b ∼ 1, we find J ∼ F /4536 R 9/2 . On the other hand, for general relativity with F (R) = R, it follows from Eq. (2.41) with b = 1 that J =Ḣ 2 = (1/15876) R 2 by assuming the same behavior of H in Eq. (3.9). The main difference between the expressions of J in the thermal equilibrium limit and that for general relativity is only the power of R. This comes from the difference of the action between the present F (R) gravity F (R) ∼F R 7/2 in Eq. (3.11) and general relativity with F (R) = R.
Finally, we remark that even in the effective phantom era of this F (R) gravity model, the second law of thermodynamics can be satisfied due to the non-equilibrium thermodynamic treatment. Hence, this model is more similar to a phantom model with ordinary thermodynamics suggested in Ref. [76].

IV. CONCLUSION
We have investigated the first and second laws of thermodynamics of the apparent horizon in F (R) gravity [49]. We have shown that in the F (R) gravity model with realizing the crossing of the phantom divide proposed in Ref. [15], the second law of thermodynamics can be satisfied in not only the non-phantom phase but also the effective phantom one. In addition to cosmological constraints and solar system tests on the models of F (R) gravity, such an examination whether the second law of thermodynamics can be met in those models is important. The demonstration in this work can be regarded as a meaningful step to construct a more realistic model of F (R) gravity, which could correctly describe the expansion history of the universe.