A general thermodynamical description of the event horizon in the FRW universe

The Friedmann equation in the Friedmann-Robertson-Walker(FRW) universe with any spatial curvature is derived from the first law of thermodynamics on the event horizon. The key idea is to redefine a Hawking temperature on the event horizon. Furthermore, we obtain the evolution equations of the universe including the quantum correction and explore the evolution of the universe in the $f(R)$ gravity. In addition, we also investigate the generalized second law of thermodynamics in Einstein gravity and the $f(R)$ gravity. This perspective also implies that the first law of thermodynamics on the event horizon have a general description in respect of the evolution of the FRW universe.


Introduction
Since the discovery by Bardeen, Bekenstein, Hawking, etc. [1,2,3]in the 1970s, the relationship between black hole physics and thermodynamics have been generally accepted by physicists. In 1995, Jacobson [4] argued that Einstein equation could be derived from the relation of thermodynamics and pointed out that Einstein equation is an equation of state. This is an important discovery that there exists a deep connection between Einstein gravity theory and thermodynamics. Subsequently, the ideas of thermodynamics were generally applied to gravity theory, string theory, cosmology, etc. (see, for example, [5,6,7,8,9,10]) Further, the study of thermodynamics of the apparent horizon has made great progress [11,12,13,14,15,16]. For example, Cai and Kim [11] confirmed the equivalency between the first law of thermodynamics on the apparent horizon and Friedmann equations in the FRW universe. On the other hand, there exists an event horizon since the universe is in accelerated expansion according to astronomical observation. While Wang et al. [17] claimed the event horizon is unphysical from the point of view of the laws of thermodynamics, Chakraborty [18] concluded that the universe bounded by the event horizon may be a Bekenstein system by redefining a Hawking temperature. However, the redefinition of the temperature on the event horizon isn't general in his paper, and the thermodynamical depiction is reasonable just in the flat universe and some models.
In this article, we redefine a Hawking temperature on the event horizon. We prove the equivalency between the first law of thermodynamics on the event horizon and Friedmann equations of the FRW universe with any spatial curvature in Einstein gravity. Further, we obtain the evolution of the quantum universe based on the first law of thermodynamics on the event horizon. These equations of evolution of the universe can't be obtained just by Einstein equation, so the method of thermodynamical depiction is more general. Besides, we also get Friedmann equations of the FRW universe in the f (R) universe from the point of view of the first law of thermodynamics. The equivalency between the first law of thermodynamics on the event horizon and Friedmann equations in the FRW universe with any spatial curvature isn't only suitable for Einstein gravity, but for generalized gravity theories like f (R) gravity. Hence, the thermodynamical depiction on the event horizon has the universality. Besides special explanation, the Greek indices, µ, ν, ..., etc. run over 0, 1, 2, 3, and the Latin indices, i, j, ..., etc. run over 1, 2, 3. Throughout the paper the units are chosen with c = = 1 and the signature of the spacetime is taken as (−, +, +, +).

Redefinition of the Hawking temperature on the event horizon
In the homogenous and isotropic universe, the metric can be expressed as where i, j can take value 0 and 1, R = a(t)r in which a(t) is the scalar factor and the 2-dimensional metric h ab = diag(−1, a 2 /(1 − kr 2 )) in which k is the spatial curvature constant. A scalar quantity is defined as The apparent horizon R A is defined by the scalar quantity χ = 0, which gives . Then the surface gravity on the apparent horizon is defined as and the corresponding Hawking temperature is defined as Analogous to the apparent horizon, we suggest that the surface gravity on the event horizon should be defined as where R E is the event horizon defined as So the temperature on the event horizon turns into This is a general temperature that depicts the event horizon of the FRW universe in Einstein gravity. Now we would like to justify Eq.(5) and Eq. (7). When the scale factor is the form a(t) = t α (α > 1) and the spatial curvature constant k = 0, the surface gravity on the event horizon is reduced to and the Hawking temperature on the event horizon becomes These relations are the same as that of the paper [18]. From the above results, we confirm that the relation (5) has the physical interpretation of the surface gravity on the event horizon. So the relation (7) is also reasonable due to the relation between the surface gravity and the Hawking temperature. The energy flux across the event horizon during an infinitesimal time interval dt can be calculated as [19,20,21] where k µ is a null vector and T µν = (ρ + p)u µ u ν + pg µν is the energy-momentum tensor. Thus, we can get the energy flux Using the Friedmann equationḢ − k a 2 = −4πG(ρ + p), the energy flux turns into On the other hand, we use the Bekenstein entropy-area relation and get From Eq. (12) and Eq.(13), we can see the first law of thermodynamics δQ | RE = T E dS E is kept on the event horizon. Therefore, we can conclude that the event horizon is a Bekenstein system and the first law of thermodynamics is equivalent to Einstein equation in Einstein gravity.

Evolution of the quantum universe based on the first law of thermodynamics on the event horizon
By redefining Hawking temperature(Eq. (7)), we confirm the validity of the first of thermodynamics on the event horizon in the above section. In the following sections, we take the first law of thermodynamics δQ = T dS on the event horizon as the fundamental starting point to derive the dynamic equations of evolution of the universe.
In this section, we will consider the quantum corrections of the entropy of the event horizon and derive these equations of evolution of the quantum universe .
In the de Sitter space with radius l, there exists an event horizon where the Hawking temperature T = 1 2πl and the entropy S = A 4G in which A = 4πl 2 is the horizon area [11,22]. Due to the similarity between the properties of the horizon of spacetime and that of black hole, we take the quantum corrected entropy of black hole as the entropy of the event horizon [23,24] where α is a constant. According to Ref. [23], α ∼ O(1). Thus we obtain and the energy flux is According to the first law of thermodynamics δQ = T dS, we get where β = αG π is a constant. This is the Friedmann equation describing the quantum corrected universe. Now we take the scale factor a(t) = t c (c > 1) in order to see the properties of evolution of the universe clearly. Thus, the radius of the event horizon turns into R E = c c−1 H −1 and Eq.(17) turns into where we redefine the effective energy densityρ and the effective pressurep, respectively. On the other hand, the continuity equation for the effective perfect fliud isρ + 3H(ρ +p) = 0.
This is another Friedmann equation under the quantum corrections. In order to see the properties of the accelerated expansion of the universe clearly, we combine Eq. (19) and Eq. (23), and get the resulẗ Comparing with the equationä a = − 4 3 πG(ρ + 3p) which can be obtained by Einstein equation, we find Eq. (24) has an extra term 4λ 3 πG(ρ + 3p)H 2 . This term represents quantum correction effects. As we have pointed out, the extra term is very small in this era. So Eq.(24) doesn't only show physical consistency with classical limit but also describes quantum effects. Therefore, the thermodynamical depiction based on the event horizon under the redefination of Hawking temperature is more general than Einstein equation in describing the dynamic evolution of the universe.

Evolution of the universe based on the first law of thermodynamics on the event horizon in the f (R) theory
In this section, we will investigate the evolution property of the universe in the theory of f (R) gravity. According to Eq.(11), the energy flux whereρ = ρ + ρ g is the total energy density of the matter energy density ρ and the effective gravity energy density ρ g , andp = p+ p g is the total pressure of the matter pressure p and the effective gravity pressure p g . In this gravity theory the relation of entropy-area [24,25] is Hence According to the first law of thermodynamics, we get the following equation However,ρ andp can't be determined just by the first law of thermodynamics. So the equation of evolution of the universe can't be also determined just by thermodynamics.
In order to determine the total energy densityρ and the total pressure densityp, we use the variational principle. In the f (R) theory, the Einstein-Hilbert action can be written as where κ 2 = 8πG. Using the variational principle δS = 0, we obtain where G µν = R µν − 1 2 g µν R is the Einstein tensor, T (m) µν = (ρ + p)u µ u ν + pg µν is the energy-momentum tensor of the matter, and is the energy-momentum tensor of the gravity. Then we get the effective gravity energy density ρ g and the effective gravity pressure p g and respectively. Thus, substituting Eq.(32) and Eq.(33) into Eq.(28), we finally get the Friedmann equation in the FRW universe On the other hand, the continuity equation for the effective perfect fluid in the f (R) gravity isρ + 3H(ρ +p) = 0.
Combining Eq.(34) and Eq.(35), we obtain another Friedmann equation These Friedmann equations Eq.(34) and Eq.(36) are the same as those of Ref. [12] which describe the evolution of the universe based on the apparent horizon. Note that the first law of thermodynamics is obtained by the standard Friedmann equation in the section 2, so the first law of thermodynamics on the event horizon and the Einstein's equation are equivalent in Einstein gravity. However, in the f (R) gravity, we can also get the correct Friedmann equation. This implies that the thermodynamical depiction is general in describing the evolution of the universe.

Conclusion
In Ref. [18], the redefinition of the Hawking temperature isn't general, and his conclusions are only suitable for the flat spacetime and some models. For example, the first law of thermodynamics is out of work on the holographic DE model under his assumption. In order to solve these difficulties, we redefine the surface gravity and the Hawking temperature on the event horizon. Subsequently, we prove the equivalency between the first law of thermodynamics on the event horizon and Friedmann equations in the FRW universe with any spatial curvature in Einstein gravity. That's to say, the first law of thermodynamics on the event horizon is kept in the FRW universe with any spatial curvature in Einstein gravity. This also indicates that the event horizon is a Bekenstein system. Starting with the first law of thermodynamics on the event horizon, we obtain Friedmann equations of the quantum universe. At present, the quantum corrected effects are very small so that we get quantum corrected Friedmann equations which are consistent with the standard Friedmann equations under classical limit. Furthermore, we obtain Friedmann equations of the FRW universe with any spatial curvature in f (R) gravity based on the first law of thermodynamics. Therefore, we may conclude that the first law of thermodynamics on the event horizon has a general depiction in respect of the evolution of the universe.