Generalized Second Law of Thermodynamics in Parabolic LTB Inhomogeneous Cosmology

Transcription

1 Commun. Theor. Phys. 64 (2015) Vol. 64, No. 5, November 1, 2015 Generalized Second Law of Thermodynamics in Parabolic LTB Inhomogeneous Cosmology A. Sheykhi, 1,2, H. Moradpour, 1,2 K. Rezazadeh Sarab, 3 and B. Wang 4, 1 Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran 2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box , Maragha, Iran 3 Department of Physics, University of Kurdistan, Pasdaran St, Sanandaj, Iran 4 INPAC and Department of Physics, Shanghai Jiao Tong University, Shanghai , China (Received April 9, 2015; revised manuscript received July 27, 2015) Abstract We study thermodynamics of the parabolic Lemaitre Tolman Bondi (LTB) cosmology supported by a perfect fluid source. This model is the natural generalization of the flat Friedmann Robertson Walker (FRW) universe, and describes an inhomogeneous universe with spherical symmetry. After reviewing some basic equations in the parabolic LTB cosmology, we obtain a relation for the deceleration parameter in this model. We also obtain a condition for which the universe undergoes an accelerating phase at the present time. We use the first law of thermodynamics on the apparent horizon together with the Einstein field equations to get a relation for the apparent horizon entropy in LTB cosmology. We find out that in LTB model of cosmology, the apparent horizon s entropy could be feeded by a term, which incorporates the effects of the inhomogeneity. We consider this result and get a relation for the total entropy evolution, which is used to examine the generalized second law of thermodynamics for an accelerating universe. We also verify the validity of the second law and the generalized second law of thermodynamics for a universe filled with some kinds of matters bounded by the event horizon in the framework of the parabolic LTB model. PACS numbers: Sf, k, Jk Key words: parabolic LTB cosmology, thermodynamics, entropy, generalized second law 1 Introduction The deep connection between thermodynamics and gravity has been well established through a lot of investigations in the past four decades. Research on this topic was started with the discovery of black holes thermodynamics in 1970 s by Hawking and Bekenstein. [1 4] In 1995, Jacobson [5] disclosed the direct relation between thermodynamics and field equations describing the spacetime geometry, by showing that the hyperbolic second order partial differential Einstein field equations can be derived from the Clausius relation δq = TδS where δq and T are the energy flux across the horizon of spacetime and the Unruh temperature seen by an accelerating observer just inside the horizon. Jacobson s derivation of the Einstein field equations from thermodynamics opened a new window for understanding the thermodynamical nature of gravity. After Jacobson, a lot of works have been done to disclose the profound connection between gravity and thermodynamics. [6 14] The profound connection provides a thermodynamical interpretation of gravity, which makes it interesting to explore the cosmological properties through thermodynamics. It has been shown that the differential form of the Friedmann equation in the FRW universe can be written in the form of the first law of thermodynamics on the apparent horizon. [15 32] It was also shown that the FRW models need a dark energy component in order to satisfy the generalized second law of thermodynamics. [33 35] On the other hand, according to the law of black hole s mechanics the area of the event horizon of any classical black holes is always a non-decreasing function of time. The similarity between the area of the horizon and the entropy function in thermodynamics may reveal that the area of the horizon is proportional to the black hole entropy. However, the second law of black hole thermodynamics can be violated by considering the quantum effects, such as the Hawking radiation. Thus, Bekenstein [2,4] introduced the so-called generalized second law (GSL) of thermodynamics, which states that the total entropy S tot, which is defined as S tot = S h + S m, (1) in general, is a non-decreasing function, i.e., S tot 0. Here S h and S m are, respectively, the black hole entropy and the entropy of the surrounding matter. The GSL in an accelerating FRW universe enveloped by the apparent horizon has been studied extensively in the Supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM), Iran asheykhi@shirazu.ac.ir b@sjtu.edu.cn c 2015 Chinese Physical Society and IOP Publishing Ltd

2 598 Communications in Theoretical Physics Vol. 64 literatures. [36 40] Recently, it was shown that the requirement of the GSL in a fractal universe implies that our universe is currently undergoing a phase of accelerated expansion. [41] This means that thermodynamics can force the universe to accelerate. The standard model of cosmology is based on the cosmological principle, which states that our universe is homogeneous and isotropic on the large scales. Although isotropy has been confirmed by various observations of the cosmic microwave background radiation (CMBR), homogeneity has been challenged by various observations such as filaments, sheets, super-clusters, and voids. On small scales, the existence of massive objects such as clusters of galaxies, black holes and so on, spoils the homogeneity of space, locally. It therefore looks interesting to seek for metrics, which represent spherically symmetric and inhomogeneous spacetime, which merge smoothly to the cosmological background. In the present paper, we concentrate our attention on the parabolic LTB model, which is the natural generalization of the flat FRW cosmology and describes an inhomogeneous universe. [42] The first law of thermodynamic is available on the apparent horizon of the LTB model [43] and it was also shown that, unlike the FRW models, [33 35] some LTB models do not need a dark energy component to satisfy the GSL. [44] We presume that the background is filled by a perfect fluid. Such fluid can be reinterpreted as mixtures of either baryons and radiation [45 46] or an inhomogeneous dust (dark matter) plus a homogeneous dark energy. [46 47] Here, we do not want to investigate such possible decompositions of the hypothetical prefect fluid. Indeed, we are going to study some thermodynamical properties of the parabolic LTB universes filled by a prefect fluid. Considering some characteristics of the realistic models, such as spherical symmetry, not referring to either initial or final moment, we find that it is appropriate to define the apparent horizon as a boundary hypersurface to study thermodynamics of the parabolic LTB model. From the first law of thermodynamics, we will get a relation for the entropy of the apparent horizon in this model. Also, we will examine the GSL of thermodynamics for an accelerating universe in this model. Throughout this paper we set c = k B = 1 for simplicity. This paper is structured as follows. In the next section we review basic equations in the parabolic LTB cosmology. In Sec. 3 we derive an expression for the entropy associated with the apparent horizon in the parabolic LTB model. In Sec. 4, we check the validity of the GSL of thermodynamics for the parabolic LTB model. We investigate the validity of the second law and the GSL of thermodynamics in the universe filled with dust and radiation in Sec. 5. In Sec. 6 we examine the second law and the GSL of thermodynamics for the universe filled with holographic dark energy. The last section is devoted to some concluding remarks. 2 Cosmology of the Parabolic LTB Model The parabolic LTB model has been proposed to describe an inhomogeneous universe with spherical symmetry. In normalized comoving coordinates the metric of the parabolic LTB model is [43,48 50] ds 2 = dt 2 + R 2 dr 2 + R 2 (dθ 2 + sin 2 θdφ 2 ) = h ab dx a dx b + R 2 (dθ 2 + sin 2 θdφ 2 ), (2) where R = R(t, r) is the radius of the spherical surface and R (t, r) plays the role of the scale factor. In this paper, dot and prime denote derivative with respect to t and r, respectively. In above equation h ab is the metric of the two-dimensional hypersurface normal to the 2-sphere and for it we have h ab = diag ( 1, R 2 ), (a, b = 0, 1 with x 0 = t, x 1 = r). (3) We suppose the universe is filled with perfect fluid with energy-momentum tensor T µν = (ρ + p)u µ u ν + pg µν, (4) where ρ and p are, respectively, the matter density and pressure of the fluid and u µ is the four-velocity of the fluid with normalization u µ u µ = 1. For a comoving observer, this special source of energy implies u α = 0 and p = p(t) while the later condition leads to p = 0. [42,51] In addition, the perfect fluid source puts the G θθ G rr = 0 condition on the Einstein tensor (G αβ ), which leads to R(t, r) = F(r)A(t). (5) We see that the FRW Universe can be restores for F(r) = r. Following Refs. [43, 49 50], we introduce the mass function as F(t, r) = RṘ2, (6) which is related to the mass contained within the comoving radius r. Then, the Einstein equations can be written as 8πGρ = F (t, r) R 2 R = Ṙ2 R 2 + 2ṘṘ RR, (7) F(t, 8πGp = r) = Ṙ2 R2Ṙ R 2 2 R R. (8) One can get the same result as Eq. (5) by imposing the p = 0 condition on the Eq. (8). Therefore, we call Eq. (5) as the prefect fluid condition. The conservation of energy, i.e. ν T µν = 0, gives the continuity equation ρ + 3H(ρ + p) = 0, (9) where H is the Hubble parameter and it is defined as H = 1 3 (H L + 2H T ), (10) where H L = Ṙ /R is the longitudinal Hubble expansion and H T = Ṙ/R is the transversal Hubble expansion.[52] From Eq. (5), it is apparent that the prefect fluid condition yields H = H L = H T = Ȧ/A. The dynamical apparent horizon, which is essentially the marginally trapped

3 No. 5 Communications in Theoretical Physics 599 surface with vanishing expansion is a spherical surface of radius R = R A satisfying [43,48 50] which leads to h ab a R b R = 0, (11) R A = RṘ2, (12) Ṙ 2 A = 1. (13) For an expanding apparent horizon ṘA is positive and thus Eq. (13) turns into Ṙ A = 1. (14) In this case, the apparent horizon coincides with the trapping horizon, which is defined as a hypersurface foliated by marginal sphere. [43,49 50] Bearing the comoving coordinate in mind, since we would like to study thermodynamics of this model in case of an accelerating universe, we have to get a relation for the deceleration parameter q. For a comoving observer at coordinate (r, t, θ, φ), the cosmological redshift (z) satisfies [53 55] d ln(1 + z) dt = H L = Ṙ R. (15) Using condition (5), one can rewrite this equation as d ln(1 + z) dt = Ȧ A, (16) which is in agreement with those of FRW. Following Ref. [54] we define the effective redial deceleration parameter as q eff = 1 + d ln(h L) d ln(1 + z). (17) It is worth noticing that the deceleration parameter, q eff, measures the rate of the universe expansion in the comoving coordinate and this rate should be measured in the radial direction, so we define q eff in terms of the longitudinal component of the Hubble parameter H L, which is in the radial direction. [53 55] Clearly, H T which is in transverse direction is not suitable for measuring the rate of the universe expansion. Using the above equations, we obtain the deceleration parameter for the parabolic LTB model as q eff = R R, (18) Ṙ 2 which is compatible with the fact that R plays the role of the scale factor. [54] The prefect fluid condition (Eq. (5)) implies q eff = AÄ/Ȧ2, which is the same as those we have in the FRW spacetime. Let us verify this result by using the covariant quantities definitions. [56] Decelerating quantity q is defined as q 1 H 2 S S, (19) where H Ṡ/S = θ/3 and S is the average length scale, [56] θ is the expansion and simple calculations yield θ = (Ṙ R 2 + 2RR Ṙ)/R R 2. In addition, the prefect fluid condition (5) implies θ = 3Ȧ/A = 3H and thus we get q = AÄ/Ȧ2 for decelerating quantity, which is equal with q eff evaluated heretofore. For an accelerating universe, q eff < 0 and for a decelerating universe q eff > 0. [54] To get an expression for R in the above equation, we use Eq. (8). We find R = 4πGpR Ṙ2 2R. (20) Differentiating the above equation with respect to the comoving coordinate r, we obtain R ṘṘ = R 4πGpR + Ṙ2 R 2R 2. (21) Inserting the above equation in Eq. (18) we reach [( q eff = R 2 Ṙ )( Ṙ ) ] Ṙ 2 R Ṙ2 R 2R 2 + 4πGp. (22) Substituting Ṙ /R from Eq. (7), we arrive at ( q eff = 2ṘR ) 2 [ ] 4πG(ρ + p) Ṙ2 8πGρR 2 Ṙ2 R 2. (23) The above equation is the deceleration parameter for the parabolic LTB model. If we assume an equation of state as p(t) = ω(r, t)ρ(r, t), (24) where ω is the equation of state parameter, then we can rewrite Eq. (23) as ( q eff = 2ṘR ) 2 [ ] 4πGρ(ω + 1) Ṙ2 8πGρR 2 Ṙ2 R 2. (25) Using ρ 0 Ṙ2 /4πGR 2 = Ȧ2 /4πGA 2, Eq. (25) can be written as: q eff 4ρ 0 = (2ρ ρ 0 ) 2 [ρ(ω + 1) ρ 0]. (26) From Eqs. (8) and (26) we get ρ < 3ρ R 4πGR. (27) This equation signaling us the condition that, an inhomogeneous prefect fluid filed can ultimate the accelerating universe (q eff < 0). As one can see from the above equation, to have an accelerated expansion, i.e. q eff < 0, the equation of state parameter should satisfy the condition ω(r, t) < ρ 0 ρ 1. (28) We will use the above condition in Sec. 4 to examine the GSL for an accelerating universe. This condition is also used in Secs. 5 and 6 for checking the validity of the GSL in a universes filled with matter, radiation and holographic dark energy in the framework of the parabolic LTB model. 3 Apparent Horizon Entropy in the Parabolic LTB Model Now, we want to obtain an expression for the entropy of the apparent horizon in the parabolic LTB model by applying the first law of thermodynamic on the apparent horizon together with the Einstein field equations. To this

4 600 Communications in Theoretical Physics Vol. 64 end, we consider a system with the apparent horizon as its boundary. We denote the entropy of the apparent horizon and its temperature with S h and T h, respectively. The associated temperature with the apparent horizon can be given by the Unruh temperature formula as T h = κ /2π, where κ is the surface gravity. The surface gravity can be defined as [49] κ 1 2 h a( hh ab b R), (29) where h is the determinant of h ab. Then, one can easily show that the temperature of the parabolic LTB model can be written as [49] T h = ( ṘṘ + R R ) 4π R. (30) Now, we assume that the first law of thermodynamics on the apparent horizon is satisfied and has the form de = δq + W dv, (31) where δq = T h ds h is the heat that crosses the apparent horizon inward it. The negative sign is appeared because T h ds h = ( κ /2π)dS h is the heat crosses the apparent horizon outward it and it is the result of the usage of the absolute value of the surface gravity in the temperature relation. In the above equation W is the work density and following, [57 58] we define it as In our case, it becomes W 1 2 T ab h ab. (32) W = 1 (ρ p). (33) 2 The work density term is regarded as the work done by the change of the apparent horizon. One can see that in Eq. (31), the work density is replaced with the negative pressure if we compare with the standard first law of thermodynamics, de = T ds pdv. For a pure de Sitter space, ρ = p, then the work term reduces to the standard pdv and we obtain exactly the standard first law of thermodynamics. [28] We also assume E = ρv is the total energy content of the universe inside a 3-sphere of radius R A, where V = (4π/3)RA 3 is the volume enveloped by 3-dimensional sphere with the area of A = 4πRA 2. Taking differential form of the relation E = ρ((4π/3)ra 3 ) for the total matter and energy inside the apparent horizon, we get de = 4π 3 R3 A ρdt + 4πR2 AṘAdt. (34) Using the continuity equation (9), we obtain de = ( 4πHR 3 A (ρ + p) + 4πR2 AṘA)dt. (35) Using the above equation in Eq. (31), we arrive at T h Ṡ h = 2πR 2 A (ρ + p)(2hr A ṘA). (36) Now, we can examine the second law of thermodynamics, which implies Ṡh 0. According to Eq. (28), in the accelerating universe the dominant energy condition may be violated, ρ + p < 0. Let us note that for an accelerating universe the equation of state parameter satisfies w = p/ρ < 1/3, and even in some cases it can cross the phantom line (w = 1), namely w < 1. This can be achieved for some phantom model of dark energy or in some case such as our model as can be seen from Eq. (28). In this case 1 + w < 0 and hence ρ + ρw < 0, which leads to ρ + p < 0. Therefore in an accelerating universe with w < 1, the dominant energy condition is broken. In order the second law of thermodynamics satisfies in this case, we should have HR A 1/2, where we have used Eq. (14). If we use the Einstein equation (7) and also Eq. (10), we can easily show that this condition requires ρ 3(1 Ṙ3 ), (37) 8πGR2Ṙ and validity of this condition depends on the choice of the mass function F(t, r) satisfying the Einstein equations. However, for our universe, HR A 1 and the required condition cannot be satisfied. Therefore, if the dominant energy condition is violated then the second law of thermodynamics will not be valid. If we assume that the dominant energy condition is valid, (ρ + p) 0, then Ṡh 0, provided HR A 0, which implies ρ 3(1 Ṙ3 ). (38) 8πGR2Ṙ Again, the validity of this condition depends on the mass function F(t, r). In our universe, HR A 1, therefore the required condition is satisfied. We conclude that in this case the second law of thermodynamics is valid for the dominant energy regime. Combining Eqs. (7), (8), (10), and (30) in Eq. (36), we get Ṡ h = 2π G R AṘA + 2π R3 Ṙ [ R/R ṘṘ /R ] R 2 R Ṙ[ R/R + 2ṘṘ /R ] + 3( R 2 Ṙ R)/R [R Ṙ/RṘ + R 1] 3G (ṘṘ + R R). (39) If Ṡh be a well-defined function of t and r, Ṡ h must not diverge, then we can integrate from this equation and reach at S h = A 4G + 2π R3 Ṙ [ R/R ṘṘ /R ] R 2 R Ṙ[ R/R + 2ṘṘ /R ] + 3( R 2 Ṙ R/R )[R Ṙ/RṘ + R 1] 3G (ṘṘ + R R) dt + f(r),(40) where A = 4πR 2 A. This is the entropy associated with the apparent horizon of the parabolic LTB model obtained by using the first law of thermodynamics on the apparent horizon as well as the Einstein equations. f(r) comes from this fact that the entropy is a function of r and t while we only integrate with respect to t. As one can see the entropy

5 No. 5 Communications in Theoretical Physics 601 consists three terms, namely S h = S h + S h + f(r), (41) where S h = A/4G obeys the area law, and S h is defined as S h = Sh dt = 2π R3 Ṙ [ R/R ṘṘ /R ] R 2 R Ṙ[ R/R + 2ṘṘ /R ] + 3( R 2 Ṙ R/R )[R Ṙ/RṘ + R 1] 3G (ṘṘ + R R) dt, (42) S h would be vanished in the FRW limit where, R(t, r) = a(t)r and a(t) is the scale factor of the universe signalling us that the FRW limit implies f(r) = 0. Also, S h can be regarded as the entropy production rate due to inhomogeneity in small scales. Therefore, the total entropy is S h = S h + S h. We will use this result in the next section for examining the GSL of thermodynamics. To get the explicit form of the apparent horizon entropy S h (t, r), we need the explicit form of the mass function F(t, r) satisfying the Einstein equations and it can be obtained by the study the cosmology of the parabolic LTB model. However, this is not our aim here and we leave it for future investigations. 4 GSL of Thermodynamics for LTB Cosmology In this section we turn to investigate the validity of the GSL of thermodynamics in a region enclosed by the apparent horizon for an accelerating inhomogeneous universe. To check the GSL, we have to examine the time evolution of the total entropy S tot = S h + S m. We have obtained a relation for the apparent horizon entropy S h in the pervious section and here we want to get a relation for the evolution of the entropy of the matter fields inside the apparent horizon, S m. To this end, we use the Gibbs equation. We assume that the temperature of the matter fields inside the apparent horizon is T m, therefore, from Gibbs equation we have [59] T m ds m = d(ρv ) + pdv = V dρ + (ρ + p)dv. (43) We can rewrite the above equation as T m Ṡ m = 4πR 3 A H(ρ + p) + 4πR2 A (ρ + p)ṙa. (44) We also suppose that the temperature of the matter fields inside the apparent horizon is proportional to the apparent horizon temperature. Therefore, we take T m = bt h, (45) where b is the proportionality constant and it is of order one, since although the apparent horizon is out of the thermodynamic equilibrium, but physically the temperature of a thermal system should be approximately equal to its boundary temperature. Using relation (45) in Eq. (44), we get T h Ṡ m = 4π b R3 AH(ρ + p) + 4π b R2 A(ρ + p)ṙa. (46) Adding Eqs. (36) and (46) and also using the equation of state (24) as well as the condition of the expanding apparent horizon (14), we arrive at T h Ṡ tot = T h (Ṡh + Ṡm) = 2π b R2 Aρ(ω + 1)[2HR A (b 1) (b 2)]. (47) If T h Ṡ tot 0 then the GSL of thermodynamics is fulfilled. In the following, we will use the above equation to study the GSL for an accelerating universe. In order to check the validity of the GSL, we consider different values for the equation of state parameter ω and the proportionality constant b. First, we examine the case ω(r, t) < 1, which implies that the equation of state parameter crosses the phantom line. In this case, the acceleration condition (28) is satisfied if and only if ρ < 0. The validity of GSL also requires 2HR A (b 1) (b 2) 0, (48) or equivalently, 4πGρR 2 3(b 2) Ṙ 2 2(b 3 1)Ṙ3 2, (49) where we have used Eqs. (7), (10), and (12). If 0 < b < 1 then for b < (Ṙ3 2)/(Ṙ3 1) the required condition (49) is satisfied since 3(b 2)/2(b 1)Ṙ3 3/2 < 0 in this interval while 3(b 2)/2(b 1)Ṙ3 3/2 < 4πGρR 2 /Ṙ2 < 0. Therefore, the GSL holds for b < (Ṙ3 2)/(Ṙ3 1). For b 1 we have (3(b 2))/2(b 1)Ṙ3 3/2 and the required condition (49) is violated. Thus, in the phantom regime, if the temperature of the matter fields inside the apparent horizon is infinitesimally smaller the apparent horizon temperature then the GSL cannot be satisfied. If b = 1 then Eq. (47) turns into T h Ṡ tot = 2πRA 2 ρ(ω + 1). (50) Since in this case (ω + 1) < 0, therefore, Ṡ tot > 0 and the GSL is fulfilled. Therefore, we conclude that for the phantom regime, if the matter fields inside the apparent horizon is in equilibrium with the apparent horizon then the GSL is satisfied. If 1 < b < 2 then the condition (48) is satisfied since from this condition, we should have HR A (b 2)/2(b 1) and (b 2)/2(b 1) < 0 in this interval while HR A > 0. Therefore, in the phantom regime, for 1 < b < 2 the GSL is fulfilled. If b 2 then for b < (2 Ṙ3 )/(1 Ṙ3 ), since Ṙ2 > 0, condition (48) is satisfied and for this interval the GSL is also fulfilled. Finally, we see that if cosmos was filled by a phantom fluid, then the acceleration condition of the universe leads to the ρ < 0 condition. The GSL will be violated, if and only if the temperature of the matter fields

6 602 Communications in Theoretical Physics Vol. 64 inside the apparent horizon is infinitesimally smaller than the apparent horizon temperature (b 1 ). Now, we examine the case ω > 1. In this case, the acceleration condition (28) may be violated, except we have ρ > 0. In order to have the GSL for the case ω > 1, we should have or equivalently, 2HR A (b 1) (b 2) 0, (51) 4πGρR 2 3(b 2) Ṙ 2 2(b 3 1)Ṙ3 2. (52) If 0 < b < 1 then the GSL is fulfilled provided 4πGρR 2 /Ṙ2 3(b 2)/2(b 1)Ṙ3 3/2 and so from Eq. (28), ω < 1 + 2[(b 1)Ṙ2 /(3(b 2) 6(b 1)Ṙ2 )] in this manner, the validity of the GSL signaling to an accelerating universe. But for b 1 we have 3(b 2)/2(b 1)Ṙ3 3/2 and the required condition (52) can not be satisfied. Therefore, for the dominant energy regime, if the temperature of the matter fields inside the apparent horizon is infinitesimally smaller than the apparent horizon temperature then the GSL is violated. If b = 1 then according to Eq. (50) the GSL is valid. This implies that if ω > 1 and the temperature of the matter fields inside the apparent horizon be equal to the apparent horizon temperature then the GSL is satisfied. If 1 < b < 2 then according to Eq. (51) to hold the GSL, we should have HR A (b 2)/2(b 1) that it is satisfied because (b 2)/2(b 1) 0 and HR A > 0. Therefore, the GSL is valid for this interval of b. 5 GSL for Accelerating Universe Filled with Dust and Radiation In this section, we pay our attention to the situations in there the hypothetical perfect fluid, as the background supporter, can behave as either a dust filed or a radiation field. In order to do it, we point to restrictions on the density energy, which lead to allow the state parameter (ω) of the dust and radiation fields and satisfy the accelerating condition (q eff < 0). In agreement with Eq. (27), if the energy density of the universe (ρ) satisfies the ρ < ρ 0, (53) condition then, from Eq. (28), it is apparent that ω = 0 can be considered as a possible solution for the hypothetical prefect fluid source, which fills the background. Inserting ω = 0 into the Eq. (8), we get A(t) = A(t t 0 ) 2/3, which is in agreement with the expansion rate of the flat LTB model supported with a dust. [44,54] Here, t 0 is called the bang singularity time. [54] In this manner, from Eq. (26), the universe expansion undergoes an accelerating phase. It is clear that, the origin of this acceleration comes from inhomogeneities of dust in small scales. Now, Eqs. (36) and (47) be transformed to T h Ṡ h = 2πR 2 Aρ[2HR A 1], (54) T h Ṡ tot = T h (Ṡh + Ṡm) = 2π b R2 A ρ[2hr A(b 1) (b 2)]. (55) In driving (54), we have used Eq. (14). Since for our universe HR A 1 we have S h 0. For satisfying GSL ( S tot 0) we get HR A > b 2 2(b 1). (56) For every b Eq. (56) and so GSL be satisfied except, when b approaches 1 from down (b 1 ). When the energy density meets the ρ 3ρ 0 /4 condition then ω = 1/3 is a possible solution telling us that our hypothetical prefect fluid source can behave as a radiation field. This condition on ρ is compatible with Eq. (27). The validity of S h > 0 and GSL is the same as the dust case T h Ṡ h = 8π 3 R2 Aρ[2HR A 1], (57) T h Ṡ tot = T h (Ṡh + Ṡm) = 8π 3b R2 A ρ[2hr A(b 1) (b 2)]. (58) 6 GSL for Universe Filled with Holographic Dark Energy In this section, we want to examine the second law of thermodynamics and also the GSL for the universe filled with holographic dark energy (HDE) bounded by the event horizon R E in the framework of the parabolic LTB model. As geometrically event horizon can not be evaluated for the parabolic LTB model, so we try to evaluate R E from physical consideration. The energy density of the HDE can be written as [43,60] ρ D = 3c2 RE 2, (59) where c is an arbitrary dimensionless parameter estimated by observational data [61] and we have set Mp 2 = 1 for simplicity. From the continuity equation (9) we obtain H = 2ṘE 3(ω D + 1)R E, (60) where, ω D = p D /ρ D is the equation of state parameter of the holographic dark energy. Motivated by Ref. [43], we repeat the calculations of obtaining the relation of the horizon entropy evolution for the event horizon and in this way, we get T h Ṡ h = 2πR 2 E(ρ + p)(2hr E ṘE). (61) Using Eq. (60) in the above relation, we arrive at ( T h Ṡ h = 2πREṘEρ 2 4 ) D (ω D + 1) 3(ω D + 1) 1. (62) For ω D < 1, the second law of thermodynamics is fulfilled (Ṡh 0), provided ω D 1/3. This condition is in contradiction with our assumption ω D < 1. Therefore, if ω D crosses the phantom line then the second law of thermodynamics does not hold. On the other hand, for ω D > 1 we have Ṡh 0 provided ω D 1/3. Therefore, for 1 < ω D < 1/3 the second law of thermodynamics is

7 No. 5 Communications in Theoretical Physics 603 fulfilled. Also, if we repeat the calculations of the total entropy evolution by using the event horizon instead of the apparent horizon, then we find T h Ṡ tot = 2π b R2 E ρ D(ω D + 1) (2HR E (b 1) ṘE(b 2)). (63) Substituting Eq. (60) in the above equation, we obtain T h Ṡ tot = 2π b R2 EṘEρ D (ω D + 1) ( 4(b 1) ) 3(ω D + 1) (b 2). (64) Now, we use the above equation to examine the GSL. Again we consider two cases. In the first case where ω D < 1 the GSL is satisfied, provided 4(b 1)/3(ω D + 1) (b 2) 0, which can be transformed to ω D 1 + 4(b 1)/3(b 2). Therefore, we should have 1 + 4(b 1)/3(b 2) ω D < 1 and this condition is satisfied when 4(b 1)/3(b 2) 0. This condition is satisfied only for 1 b < 2. Therefore, for the case ω D < 1, the GSL is satisfied when the temperature of matter fields inside the event horizon be equal to or higher than the event horizon temperature until doubled. In the second case where ω D > 1, the GSL is preserved provided ω D 1 + 4(b 1)/3(b 2). Therefore, we should have 1 < ω D 1 + 4(b 1)/3(b 2) and this condition is fulfilled when 4(b 1)/3(b 2) 0. This relation holds for 0 < b 1 and b > 2. Therefore, for the dominant energy regime, the GSL is satisfied provided the temperature of the matter fields inside the event horizon be smaller than or equal to the event horizon temperature and it also holds if the temperature of the matter fields inside the event horizon be larger than twice the apparent horizon temperature. If 4(b 1)/3(b 2) < Ṙ2 /4πGρ D R 2, where the validity of this relation depends on the choice of the mass function F(t, r) satisfying the Einstein equations, then ω D < 1 + Ṙ2 /4πGρ D R 2 = ρ 0 /ρ D 1 and the equation of state parameter of the HDE will satisfy the acceleration condition (28). In this case, the validity of the GSL imposes the acceleration condition. 7 Conclusions In this work, we examine thermodynamics of the parabolic LTB cosmology. This model is the natural generalization of the flat FRW universe and describes an inhomogeneous universe with spherical symmetry. After reviewing some basic equations in the parabolic LTB cosmology, we use the definition of the deceleration parameter and obtaine its expression in the framework of parabolic LTB model. We also obtain a condition for the equation of state parameter, which implies an accelerating universe. We derive some of the thermodynamical quantities of this model such as the temperature and work density. To get the temperature of this model, we use the Unruh temperature formula and for the work density, its general definition is used. Then, we apply the first law of thermodynamics on the apparent horizon as well as the Einstein s equations to get a relation for the evolution of the apparent horizon entropy. Using this relation, we have found out the inhomogeneity s effects on the apparent horizon s entropy. To get the explicit form of the apparent horizon entropy, one needs the explicit form of the mass function satisfying the Einstein field equations and it can be obtain by studying the cosmology of the parabolic LTB model, which we leave it for future investigations. We examine the GSL of thermodynamics for LTB model. The GSL states that the total entropy including the apparent horizon entropy and the entropy of the matter fields inside the apparent horizon is a non-decreasing function. To get a relation for the time evolution of the entropy of the matter fields inside the apparent horizon, the Gibbs equation has been used. We show that for the phantom regime, if the temperature of the matter fields inside the apparent horizon is equal to the apparent horizon temperature, then the GSL is violated. We show that for LTB model the GSL is preserved provided the equation of state parameter satisfies the dominant energy condition and the temperature of the matter fields inside the apparent horizon to be equal or larger than the temperature of the apparent horizon. We consider the event horizon as the boundary of the system. We show that if the equation of state parameter of the holographic dark energy crosses the phantom line, then the second law of thermodynamics is not valid. For the dominant energy regime, the GSL holds provided the temperature of the matter fields inside the event horizon be smaller than or equal to the event horizon temperature and it also holds if the temperature of the matter fields inside the event horizon be larger than twice the apparent horizon temperature. In this case, we have also found a condition to have an accelerating universe. Acknowledgments A. Sheykhi thanks Shiraz University Research Council. References [1] J.M. Bardeen, B. Carter, and S. Hawking, Commum. Math. Phys. 31 (1973) 161. [2] J.D. Bekenstein, Phys. Rev. D 7 (1973) [3] S.W. Hawking, Nature (London) 248 (1974) 30. [4] J.D. Bekenstein, Phys. Rev. D 9 (1974) 12. [5] T. Jacobson, Phys. Rev. Lett. 75 (1995) [6] C. Eling, R. Guedens, and T. Jacobson, Phys. Rev. Lett. 96 (2006)

8 604 Communications in Theoretical Physics Vol. 64 [7] M. Akbar and R.G. Cai, Phys. Lett. B 635 (2006) 7. [8] M. Akbar and R.G. Cai, Phys. Lett. B 648 (2007) 243. [9] T. Padmanabhan, Class. Quant. Gravit. 19 (2002) [10] T. Padmanabhan, Phys. Rep. 406 (2005) 49. [11] T. Padmanabhan, Int. J. Mod. Phys. D 15 (2006) [12] A. Paranjape, S. Sarkar, and T. Padmanabhan, Phys. Rev. D 74 (2006) [13] D. Kothawala, S. Sarkar, and T. Padmanabhan, Phys. Lett. B 652 (2007) 338. [14] T. Padmanabhan and A. Paranjape, Phys. Rev. D 75 (2007) [15] M. Akbar and R.G. Cai, Phys. Rev. D 75 (2007) [16] R.G. Cai and L.M. Cao, Phys. Rev. D 75 (2007) [17] R.G. Cai and S.P. Kim, J. High Eergy Phys (2005) 050. [18] A.V. Frolov and L. Kofman, J. Cosmology Astro. Phys (2003) 009. [19] U.K. Danielsson, Phys. Rev. D 71 (2005) [20] R. Bousso, Phys. Rev. D 71 (2005) [21] G. Calcagni, J. High Energy Phys (2005) 060. [22] U.H. Danielsson, Phys. Rev. D 71 (2005) [23] E. Verlinde, arxiv:hep-th/ [24] B. Wang, E. Abdalla, and R.K. Su, Phys. Lett. B 503 (2001) 394. [25] B. Wang, E. Abdalla, and R.K. Su, Mod. Phys. Lett. A 17 (2002) 23. [26] R.G. Cai and Y.S. Myung, Phys. Rev. D 67 (2003) [27] R.G. Cai, L.M. Cao, and Y.P. Hu, J. High Energy Phys (2008) 090. [28] A. Sheykhi, Eur. Phys. J. C 69 (2010) 265. [29] R.G. Cai and L.M. Cao, Nucl. Phys. B 785 (2007) 135. [30] A. Sheykhi, B. Wang, and R.G. Cai, Nucl. Phys. B 779 (2007) 1. [31] A. Sheykhi, B. Wang, and R.G. Cai, Phys. Rev. D 76 (2007) [32] A. Sheykhi, J. Cosmology Astro. Phys. 05 (2009) 019. [33] N. Radicella and D. Pavón, Phys. Lett. B 704 (2011) 260. [34] N. Radicella and D. Pavón, Gen. Relativ. Gravit. 44 (2012) 685. [35] H. Moradpour, A. Sheykhi, N. Riazi, and B. Wang, Adv. H. Energy Phys (2014) [36] B. Wang, Y. Gong, and E. Abdalla, Phys. Rev. D 74 (2006) [37] J. Zhou, B. Wang, Y. Gong, and E. Abdalla, Phys. Lett. B 652 (2007) 86. [38] A. Sheykhi and B. Wang, Phys. Lett. B 678 (2009) 434. [39] A. Sheykhi, Class. Quant. Gravit. 27 (2010) [40] A. Sheykhi and B. Wang, Mod. Phys. Lett. A 25 (2010) [41] A. Sheykhi, Z. Teimori, and B. Wang, Phys. Lett. B 718 (2013) [42] A. Krasinski, Inhomogeneous Cosmological Models, Cambridge University Press, Cambridge (1997). [43] S. Chakraborty, N. Mazumder, and R. Biswas, Gen. Relativ. Gravit. 43 (2011) [44] P. Mishra and T.P. Singh, Phys. Rev. D 89 (2014) [45] J.A.S. Lima and J. Tiomno, Gen. Relativ. Gravit. 20 (1988) [46] R.A. Sussman, Class. Quant. Gravit. 9 (1992) [47] R.A. Sussman, I. Quiros, and O.M. González, Gen. Relativ. Gravit. 37 (2005) [48] B. Wang, E. Abdalla, and T. Osada, Phys. Rev. Lett. 85 (2000) [49] R. Biswas, N. Mazumder, and S. Chakraborty, Int. J. Theor. Phys. 51 (2012) 101. [50] R. Biswas, N. Mazumder, and S. Chakraborty, arxiv: [51] G. Siemieniec-Oziȩb lo and Z. Klimek, Acta Phys. Polonica. B 9 (1978) 79. [52] M. Roos, arxiv: [53] H. Alnes and M. Amarzguioui, Phys. Rev. D 74 (2006) [54] H. Wang and T. Zhang, Astrophys. J. 748 (2012) 111. [55] K. Enqvist, Gen. Relativ. Gravit. 40 (2008) 451. [56] G.F.R. Ellis and H.V. Elst, arxiv:gr-qc/ [57] S.A. Hayward, S. Mukohyana, and M.C. Ashworth, Phys. Lett. A 256 (1999) 347. [58] S.A. Hayward, Class. Quant. Gravit. 15 (1998) [59] G. Izquierdo and D. Pavon, Phys. Lett. B 633 (2006) 420. [60] A.G. Cohen, D.B. Kaplan, and A.E. Nelson, Phys. Rev. Lett. 82 (1999) [61] Q.G. Huang and M. Li, J. Cosmology Astro. Phys (2004) 013.