Thermodynamics of Quasi-Topological Cosmology

In this paper, we study thermodynamical properties of the apparent horizon in a universe governed by quasi-topological gravity. Our aim is twofold. First, by using the variational method we derive the general form of Friedmann equation in quasi-topological gravity. Then, by applying the first law of thermodynamics on the apparent horizon, after using the entropy expression associated with the black hole horizon in quasi-topological gravity, and replacing the horizon radius, $r_{+}$, with the apparent horizon radius, $\tilde{r}_{A}$, we derive the corresponding Friedmann equation in quasi-topological gravity. We find that these two different approaches yield the same result which show the profound connection between the first law of thermodynamics and the gravitational field equations of quasi-topological gravity. We also study the validity of the generalized second law of thermodynamics in quasi-topological cosmology. We find that, with the assumption of the local equilibrium hypothesis, the generalized second law of thermodynamics is fulfilled for the universe enveloped by the apparent horizon for the late time cosmology.


I. INTRODUCTION
The most general Lagrangian which keeps the field equations of motion for the metric of second order, as the pure Einstein-Hilbert action, is Lovelock Lagrangian [1]. This Lagrangian is constructed from the dimensionally extended Euler densities and can be written as where α p and L p are arbitrary constant and Euler density, respectively. In an (n+1)-dimensional spacetime m = [n/2]. L 0 set to be one, and therefore α 0 plays the role of the cosmological constant. Because of the topological origin of the Lovelock terms, the second order (Gauss-Bonnet) term does not have any dynamical effect in four dimensions. Similarly, the cubic interaction only contributes to the equations of motion when the bulk dimension is seven or greater. In other words, although the equations of motion of pth order Lovelock gravity are second-order differential equations, the pth order Lovelock term has no contribution to the field equations in 2p and lower dimensions. Is it possible to construct a gravitational action with cubic curvature interactions or higher which has contribution in five dimension? The answer is positive and the corresponding theory is called "quasi-topological" gravity which was recently proposed in Refs. [2,3] and [4] with cubic and quartic terms of Riemann tensor, respectively. This new gravitational theory provides a useful toy model to study a broader class of four (and higher) dimensional CFT's, involving three or more independent parameters [5]. Various aspects of pth-order quasi-topological terms which have at most second order derivatives of the metric in the field equations for spherically symmetric spacetimes in five and higher dimensions except 2p dimensions have been investigated [6][7][8][9]. Nowadays, it is a general belief that there is a profound connection between the gravitational field equations and the laws of thermodynamics. It was shown that the gravitational field equation of a static spherically symmetric spacetime in Einstein, Gauss-Bonnet and more general Lovelock gravity can be recast as the first law of thermodynamics [10]. The studies were also extended to other gravity theories such as f (R) gravity [11] and scalar-tensor gravity [12]. In the cosmological setup, it was shown that the differential form of the Friedmann equation of Friedmann-Robertson-Walker (FRW) universe can be transformed to the first law of thermodynamics on the apparent horizon [13,14]. In the context of brane cosmology, it was shown that the Friedmann equations on the brane can be expressed as dE = T dS + W dV on the apparent horizon [15][16][17]. This procedure also leads to extract an expression for the entropy at the apparent horizon on the brane, which is useful in studying the thermodynamical properties of the black hole horizon on the brane [15][16][17].
Is the inverse procedure also possible? That is starting from the first law of thermodynamics to extract the general field equations of gravitational theory. Jacobson [18] was the first who disclosed that the Einstein field equation can be derived from the relation between the horizon area and entropy, together with the Clausius relation δQ = T δS.
Also, in the cosmological setup, it was shown that the corresponding Friedmann equations of Einstein, Gauss-Bonnet and Lovelock gravity can be derived by applying the energy balance relation −dE = T dS to the apparent horizon of a Friedmann-Robertson-Walker universe (FRW) with any spatial curvature [19]. Here, −dE is actually just the heat flux δQ in [18] crossing the apparent horizon within an infinitesimal internal of time dt. In the framework of Horava-Lifshitz gravity, it was shown that the corresponding Friedmann equation cannot be derived by applying the first law of thermodynamics on the apparent horizon and using the entropy expression for static spherically symmetric black holes in this gravity theory [20]. The reason of failure seems to be due to the fact that Horava-Lifshiz gravity is not diffeomorphism invariant [21]. Indeed, the action of Horava-Lifshiz gravity is invariant only under a restricted class of diffeomorphism [22]. This implies that the connection between first law of thermodynamics and gravitational field equations is not a generic feature of any theory of gravity.
In this paper we will address the question on the connection between thermodynamics and gravity by investigating whether and how the relation can be found in quasi-topological cosmology. This is the first study on the quasitopological cosmology and in particular investigating thermodynamical aspects of this gravity theory. For this purpose, we first derive the Friedmann equations in quartic and higher order quasi-topological gravity by varying the action of the quasi-topological gravity. Then, to show the consistency of this theory with thermodynamics, we extract the corresponding Friedmann equations by applying the first law of thermodynamics, dE = T h dS h + W dV , on the apparent horizon of a FRW universe governed by quasi-topological gravity. Our strategy is to pick up the entropy expression associated with the black hole horizon in quasi-topological gravity and replacing the black hole horizon radius r + by the apparent horizon radiusr A . We will also examine the time evolution of the total entropy, including the entropy associated with the apparent horizon in quasi-topological gravity together with the matter field entropy inside the apparent horizon. We find that, in the late time, the generalized second law (GSL) of thermodynamics is fulfilled for the universe governed by quasi-topological gravity.
This paper is outlined as follows. In the next section, we introduce the action of the quasi-topological gravity and derive the general form of the Friedmann equation by using the variational method in this gravity theory. In section III, we extract the Friedmann equation of quartic quasi-topological cosmology by applying the first law of thermodynamics, dE = T h dS h + W dV , on the apparent horizon. We also generalize our study to higher order quasitopological theory in this section. We investigate the validity of GSL of thermodynamics for a universe enveloped by the apparent horizon in quasi-topological gravity in section IV. We finish our paper with conclusions in section V.

II. QUASI-TOPOLOGICAL COSMOLOGY
In this section we derive the field equations governing the evolution of the universe in quasi-topological gravity. The most general gravitational theory which produces second-order equation of motion is the i-order Lovelock gravity with action [1] where Λ is the cosmological constant, and the α i 's are Lovelock coefficients with dimensions (length) 2i−2 , and L i is the ith order Lovelock Lagrangian ( In the action (2), the term proportional to α i contributes to the equations of motion in dimensions with n ≥ 2i. For example, the terms associated to i = 3 or higher do not contribute to the field equations in five dimensions.
Recently, a new gravity theory called quasi-topological gravity has been introduced, which has contribution to the field equations in five dimensions from the i-order (i ≥ 3) term in Riemann tensor. The gravity part of the action of the quartic quasi-topological theory in (n + 1)-dimensions in the absence of cosmological constant is given by [4] where L M is the Lagrangian of the matter and In Eq.(5) L 1 = R is the Einstein-Hilbert Lagrangian, L 2 = R abcd R abcd − 4R ab R ab + R 2 is the second order Lovelock (Gauss-Bonnet) Lagrangian, X 3 is the curvature-cubed Lagrangian [3] and X 4 is the fourth order term of quasi-topological gravity [4] where the coefficients c i are given by  The action (4) not only works in five dimensions, but also yields second-order equations of motion for spherically symmetric spacetimes [4]. Our aim here is to derive the corresponding Friedmann equations of quartic and higher order quasi-topological gravity. We consider a homogeneous and isotropic FRW universe in (n + 1)-dimensions which is described by the line element where k is the spatial curvature constant with values 1, 0 and −1 correspond to closed, flat and open universe, respectively, and dΩ 2 n−1 represents the line elements of an (n − 1)-dimensional unit sphere. We use the variational method for deriving the Friedmann equation from the action principle. Using metric (8), the Lagrangian of the fourth order quasi-topological gravity can be written as where γ is the determinant of the metric of t-constant hypersurface, b i 's are b 1 = a 6 k + a 4 k 2μ 2 l 2 + a 2 k 3μ 3 l 4 + k 4μ 4 l 6 , b 2 = a 6 + 2a 4 kμ 2 l 2 + 3a 2 k 2μ 3 l 4 + 4k 3μ 4 l 6 , b 3 = a 4μ 2 l 2 + 3a 2 kμ 3 l 4 + 6k 2μ 4 l 6 , b 4 = a 2μ 3 l 4 + 4kμ 4 l 6 , b 5 =μ 4 l 6 . and the dimensionless parametersμ j 's arê Varying the Lagrangian (9) with respect to N (t), we arrive at On the other hand the variation of the Lagrangian of matter with respect to g 00 = −N leads to Now, one can absorb N (t) in t coordinate. That is, one can set N (t) = 1. Using (10) and (11), one obtains As in the case of black hole solutions presented in [4], the form of the field equation (12) allows us to generalize this equation to the case of mth-order quasi-topological gravity: Here, we pause to study the field equations under a small perturbation around the Friedmann-Robertson-Walker metric. The authors of [3], examined the linearized equations of motion for a graviton perturbation around the AdS metric in cubic quasi-topological gravity, and showed that the linearized graviton equation in an AdS background is only a second-order equation. Here, we examine the same fact under a small perturbation around the FRW metri and find that the linearized field equation is a second order equation. This fact is different from the small perturbation around FRW metric in the new massive garvity [23]. In the latter case, the linearized field equation contains more than two-derivative and one may have ghosty vacuum at initial times while it becomes free of ghosts at later times in the cosmological scenario [23].

III. FRIEDMAN EQUATION FROM THE FIRST LAW
In this section we would like to derive the Friedmann equation of quasi-topological cosmology by applying the first law of thermodynamics on the apparent horizon of FRW universe. The entropy associated with the event horizon of higher dimensional static spherically symmetric black holes in cubic quasi-topological gravity has the following form [3,6] where n = 5 and G n+1 is the (n + 1) dimensional gravitational constant. Here A = nΩ n r n−1 + is the surface area of the black hole horizon, and We further assume the entropy expression (14) is also valid for the apparent horizon of the FRW universe in quasitopological gravity. Replacing the horizon radius r + with the apparent horizon radiusr A , the entropy expression (29) can be written where we have defined Taking differential form of relation (16), we have We rewrite the line element of the FRW metric as where x 0 = t, x 1 = r,r = a(t)r, and h ab =diag (−1, a 2 /(1 − kr 2 )) represents the two dimensional metric. The dynamical apparent horizon, a marginally trapped surface with vanishing expansion, is determined by the relation h ab ∂ ar ∂ br = 0. It is a matter of calculation to show that the radius of the apparent horizon for the FRW universe becomes [24]r The temperature associated with the apparent horizon is defined as −hh ab ∂ br is the surface gravity. It is easy to show that the surface gravity at the apparent horizon of FRW universe can be written as Since forṙ A < 2Hr A , we have κ < 0, which leads to the negative temperature, thus one may, in general, define the temperature on the apparent horizon as T h = |κ|/2π. In addition, since we associate with the apparent horizon a temperature, thus one may expect that the apparent horizon have a kind of Hawking radiation just like a black hole event horizon. This issue was previously addressed [25], by showing the connection between temperature on the apparent horizon and the Hawking radiation. This study gives more solid physical implication of the temperature associated with the apparent horizon. The energy conservation law ∇ µ T µν = 0 leads to the continuity equation in the forṁ The next quantity we need to have is the work density. In our case it can be calculated as [24] The work density is regarded as the work done when the apparent horizon radius changes fromr A tor A + dr A . Then, we suppose the first law of thermodynamics on the apparent horizon of the universe in quasi-topological gravity holds and has the form where S h is the entropy associated with the apparent horizon in quasi-topological cosmology given in Eq. (16). The term W dV in the first law comes from the fact that we have a volume change for the total system enveloped by the apparent horizon. For a pure de Sitter space, ρ = −p, and the work term reduces to the standard −pdV , thus we obtain exactly the standard first law of thermodynamics, dE = T dS − pdV .
Assuming the total energy content of the universe inside an n-sphere of radiusr A is E = ρV , where V = Ω nr n A is the volume enveloped by an n-dimensional sphere. Taking differential form of the total energy, after using the continuity equation (21), we obtain Substituting Eqs. (18), (22) and (24) in the first law (23) and using the the definition of the temperature associated with the apparent horizon, we get the differential form of the Friedmann equation in cubic quasi-topological gravity as Using the continuity equation (21), we obtain Integrating (26) yields where an integration constant has been absorbed into the energy density ρ. Substitutingr A from Eq. (19) we obtain In this way we derived the (n + 1)-dimensional Friedmann equation governing the evolution of the universe in cubic order quasi-topological gravity by applying the first law of thermodynamics on the apparent horizon. The above analysis can be extended to higher order quasi-topological gravity. The entropy associated with spherically symmetric black hole solutions in quartic order of quasi-topological gravity is given by [4] The extension to higher order quasi-topological black holes are quite straightforward and can be written in the compact form Using the same formalism as we have done in this section, we obtain the general form of the Friedmann equation aŝ or in a compact form as Which is compatible with the field equation derived through the use of the variation of the action.

IV. GSL IN QUASI-TOPOLOGICAL GRAVITY
Next, we examine the time evolution of the total entropy. For this purpose, we take the time derivative of Eq. (31) and get Using the continuity equation (21) and solving forṙ A we obtaiṅ Calculating T hṠh , and substitutingṙ A from Eq. (34) in it, we obtain where we have defined the equation of state w = p/ρ, as usual. Since T ≥ 0, we haveṙ A ≤ 2Hr A , thus the sign of Eq. (36) depends on w. If w ≥ −1, thenṠ h ≥ 0 and the second law of thermodynamics is fulfilled. However, some astrophysical evidences show that our universe is currently accelerating and in particular the equation of state parameter can cross the phantom line, i.e. w < −1, indicating that the second law of thermodynamics,Ṡ h ≥ 0, does not hold. Nevertheless, as we shall see below, the GSL of thermodynamics,Ṡ h +Ṡ m ≥ 0, is still preserved throughout the history of the universe. In order to verify the GSL of thermodynamics, we have to study the time evolution of the total entropy including the entropy S h associated with the apparent horizon together with the matter field entropy S m inside the apparent horizon. The entropy of the universe inside the horizon can be related to its energy and pressure by Gibbs equation [26] T m dS m = d(ρV ) + pdV = V dρ + (ρ + p)dV = Ω nr n A dρ + (ρ + p)nΩ nr where T m is the temperature of the matter field inside the apparent horizon. We assume the temperature of the perfect fluid inside the apparent horizon scales as the temperature of the apparent horizon T h . Thus, we suppose the temperature T m = T h . We limit ourselves to the assumption of the local equilibrium hypothesis, that the energy would not spontaneously flow between the horizon and the fluid, the latter would be at variance with the FRW geometry. Therefore, from the Gibbs equation, we get where we have used the continuity equation (21). Adding equations (36) and (38), after substitutingṙ A from Eq. (34), we obtain Expanding the r.h.s. of the above equation for the late time wherer A ≫ l, we arrive at The expression in the bracket, in the late time cosmology is positive which indicate thatṠ h +Ṡ m ≥ 0. This implies that for the late time cosmology, the GSL of thermodynamics is fulfilled in the universe governed by quasi-topological gravity, regardless of the nature of the energy content of the universe.

V. CONCLUSIONS
In this paper we investigated the thermodynamical properties of the apparent horizon in quasi-topological gravity. We first derived the Friedmann equation governing the evolution of the universe in quartic and higher order quasitopological gravity by varying the corresponding action of quasi-topological gravity. Then, by applying the first law of thermodynamics, dE = T h dS h + W dV on the apparent horizon of FRW universe, we extracted the Friedmann equation of quartic and higher order quasi-topological gravity. Here E = ρV is the total energy inside the apparent horizon and T h and S h are the temperature and entropy associated with the apparent horizon, respectively. The entropy expression depends on the gravity theory and it is generally accepted that the apparent horizon entropy in each gravity theory has the same expression as the entropy of black hole horizon, but replacing the black hole horizon radius r + with the apparent horizon radiusr A . We found that the result obtained by the first law is exactly coincides with the ones derived from the action principle. We also studied the time evolution of the total entropy, including the entropy associated with the apparent horizon together with the matter field entropy enveloped by the apparent horizon. Our study implies that, with the assumption of the local equilibrium hypothesis, the GSL of thermodynamics is preserved for the late time cosmology in the universe governed by quasi-topological gravity.