Generalized Second Law of Thermodynamics for FRW Cosmology with Power-Law Entropy Correction
Ujjal Debnath, Surajit Chattopadhyay, Ibrar Hussain, Mubasher Jamil, Ratbay Myrzakulov
aa r X i v : . [ g r- q c ] F e b Generalized Second Law of Thermodynamics for FRW Cosmologywith Power-Law Entropy Correction
Ujjal Debnath , ∗ Surajit Chattopadhyay, † IbrarHussain , ‡ Mubasher Jamil , § and Ratbay Myrzakulov ¶ Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711 103, India. Department of Computer Application (Mathematics Section),Pailan College of Management and Technology,Bengal Pailan Park, Kolkata-700 104, India. School of Electrical Engineering and Computer Science (SEECS),National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan. Center for Advanced Mathematics and Physics (CAMP),National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan. Eurasian International Center for Theoretical Physics,Eurasian National University, Astana 010008, Kazakhstan. (Dated: October 8, 2018)
Abstract:
In this work, we have considered the power law correction of entropy on thehorizon. If the flat FRW Universe is filled with the n components fluid with interactions,the GSL of thermodynamics for apparent and event horizons have been investigated forequilibrium and non-equilibrium cases. If we consider a small perturbation around the deSitter space-time, the general conditions of the validity of GSL have been found. Also if aphantom dominated Universe has a polelike type scale factor, the validity of GSL has alsobeen analyzed. Further we have obtained constraints on the power-law parameter α in thephantom and quintessence dominated regimes. Finally we obtain conditions under whichGSL breaks down in a cosmological background. ∗ Electronic address: [email protected] , [email protected] † Electronic address: surajit˙[email protected], [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] , [email protected] ¶ Electronic address: [email protected], [email protected]
I. INTRODUCTION
In Einstein gravity, the evidence of a connection between thermodynamics and Einstein fieldequations was first discovered in [1] by deriving the Einstein equation from the proportionality ofentropy and horizon area together with the first law of thermodynamics δQ = T dS in the Rindlerspacetime. The horizon area of black hole is associated with its entropy, the surface gravity isrelated with its temperature in black hole thermodynamics [2, 3]. The Friedmann equation in aradiation dominated Friedmann-Robertson-Walker (FRW) Universe can be written in an analogousform of the Cardy-Verlinde formula, an entropy formula for a conformal field theory [4]. As is wellknown, event horizons, whether of black holes or cosmological, mimic black bodies and possessa non-vanishing temperature and entropy, the latter obeying the Bekenstein-Hawking entropyformula [5, 6] S = A ( c = G = ~ = 1), where A = 4 πR h is the area of the horizon, R h is the radiusof the horizon and G is the Newton’s gravitational constant. The first law of thermodynamics forthe cosmological horizon is given by − dE = T dS , where T = πR h is the Hawking temperature[7, 8].Recently, it was demonstrated that cosmological apparent horizons are also endowed with ther-modynamical properties, formally identical to those of event horizons [9]. In a spatially flat deSitter spacetime, the event horizon and the apparent horizon of the Universe coincide and there isonly one cosmological horizon. When the apparent horizon and the event horizon of the Universeare different, it was found that the first law and generalized second law (GSL) of thermodynam-ics hold on the apparent horizon, while they break down if one considers the event horizon [10].Recently, it has been demonstrated that if the expansion of the Universe is dominated by phan-tom energy, black holes will decrease their mass and eventually disappear altogether [11, 12]. Thismeans a threat for the GSL as these collapsed objects are the most entropic entities of the Universe[11]. This brief consideration spurs the researchers to explore the thermodynamic consequences ofphantom - dominated Universes. In doing so one must take into account that ever accelerating Uni-verses have a future event horizon (or cosmological horizon) [11]. The thermodynamical propertiesassociated with the apparent horizon have been found in a quasi-de Sitter geometry of inflationaryUniverse [13]. Setare [14] considered the interacting holographic model of dark energy to investi-gate the validity of the generalized second laws of thermodynamics in non-flat (closed) Universeenclosed by the event horizon. In [15], it is shown that GSL is generally valid for a system of darkenergy interacting with dark matter and radiation in FRW Universe. Further in Horava Lifshitzcosmology, it has been shown that under detailed balance the generalized second law is generallyvalid for flat and closed geometry and it is conditionally valid for an open Universe, while beyonddetailed balance it is only conditionally valid for all curvatures [16]. In a comprehensive review, theGSL has been extended in various generalized gravity theories including Lovelock, Gauss-Bonnet,braneworld, scalar-tensor and f ( R ) models [17].In Einstein’s gravity, the entropy of the horizon is proportional to the area of the horizon, S ∝ A . When gravity theory is modified by adding extra curvature terms in the action principle, itmodifies to the entropy-area relation, for instance, in f ( R ) gravity, the relation is S ∝ f ′ ( R ) A [18].On the other hand, quantum corrections to the semi-classical entropy law have been introduced inrecent years, namely logarithmic and power law corrections . Logarithmic corrections, arises fromloop quantum gravity due to thermal equilibrium fluctuations and quantum fluctuations [19–25] S = A α ln A . (1)On its part, power law corrections appear in dealing with the entanglement of quantum fields inand out the horizon [26–28] S = A h − K α A − α i , (2)where, A = 4 πR h ; K α = α (4 π ) α − (4 − α ) r − αc . (3)Here, r c is the crossover scale and α is the dimensionless constant whose value is currently underdebate. The second term in Eq. (2) can be considered as a power-law correction to the entropy-arealaw, arising from entanglement of the wave-function of the scalar field between the ground stateand the exited state [26–28]. The correction term is also more significant for higher excitations. Itis important to note that the correction term decreases faster with A and hence in the semi-classicallimit (large area) the entropy-area law is recovered.The plan of the paper is as follows: In section II, we write down basic equations of cosmologyfor our further use. In section III, we investigate the GSL with power-law entropy correction forboth apparent horizon (subsection - A) and future event horizon (subsection - B). In section IV,we will discuss the GSL at both horizons in the non-equilibrium setting. Finally we conclude thispaper. II. BASIC EQUATIONS
We consider a homogeneous and isotropic spatially flat ( k = 0) FRW Universe which is describedby the line element ds = − dt + a ( t ) (cid:2) dr + r ( dθ + sin θdφ ) (cid:3) . (4)Now assume that the Universe is filled with a perfect fluid of n-components (such as dark energy,dark matter, radiation and so on): ρ = P ni =1 ρ i and p = P ni =1 p i where ρ and p are total energydensity and pressure of the combined fluid. So the Einstein’s field equations are given by H = 8 π ρ, (5)and ˙ H = − π ( ρ + p ) . (6)The energy conservation equation is given by˙ ρ + 3 H ( ρ + p ) = 0 . (7)Now consider there is an interaction between all fluid components [29]. We cannot specify theform of interaction between the components as the nature of dark energy and dark matter are notunderstood yet. These interactions modify the form of the equation of states of the interactingfluids. Interest in these models has been spurred when it was found that the evolution of theuniverse, from early deceleration to late time acceleration can be explained [30]. In additionsuch an interacting dark energy model can accommodate a transition of the dark energy from aquintessence state w D > − w D < − ρ i + 3 H ( ρ i + p i ) = Q i , (8)where Q i is an interaction term which can be an arbitrary function of cosmological parameters likethe Hubble parameter and energy densities [33]. This term allows the energy exchange betweenthe components of the perfect fluid and may alleviate the coincidence problem. From above twoequations (7) and (8), we find P ni =1 Q i = 0. III. GSL OF THERMODYNAMICS WITH POWER LAW ENTROPY CORRECTION
Using Gibb’s law for each component of the fluid, we have T i dS i = d ( ρ i V ) + p i dV, (9)where T i is the temperature and S i is the entropy of the i -th component of the fluid. If R h be theradius of the horizon then the volume can be written as V = πR h (assuming spherical symmetry).From the above equation, after simplification, we obtain˙ S i = 43 πR h Q i T i + 4 πR h ( ˙ R h − HR h ) ρ i + p i T i . (10)Now the total changes of entropy inside the horizon is given by˙ S I = n X i =1 ˙ S i = 43 πR h n X i =1 Q i T i + 4 πR h ( ˙ R h − HR h ) n X i =1 ρ i + p i T i . (11)In cosmological models of accelerated Universe, there are horizons to which we can assign anentropy as a measure of information behind them. The most natural horizon of the Universe is theapparent horizon whose radius is R A = H (also called the Hubble horizon). Another cosmologicalhorizon which conceptually more resembles to the black-hole horizon is the future event horizon,whose radius, R E is defined by [34] R E = a ( t ) Z ∞ t dt ′ a ( t ′ ) < ∞ . (12)It describes the distance that light travels from the present time to an arbitrary time in future.Despite the presence of an infinity, the horizon R E can be finite. For future event horizon, thederivative of the radius of event horizon can be written as˙ R E = HR E − . (13)Only in a de Sitter spacetime we have R A = R E . For generality, in our study we consider boththe choices: R A and R E . Note that cosmological event horizon does not always exist for all FRWuniverses, the apparent horizon and the Hubble horizon always do exist. It is the radius obtained by solving the equation g µν ∂ µ ˜ r∂ ν ˜ r = 0, where ˜ r = a ( t ) r i.e. the LHS vanishes at theapparent horizon R A . For FRW Universe, it gives R A = q H + ka , thus R A = H for k = 0. A. On the apparent horizon R A If we take the natural horizon of the Universe as the apparent horizon i.e., R h = R A = H − ,from the equation (2), the entropy on the apparent horizon can be written as S A = πH (cid:20) − α − α ( r c H ) α − (cid:21) . (14)The rate of change of the entropy on the apparent horizon (using (14)) is obtained as˙ S A = − π ˙ HH h − α r c H ) α − i . (15)From (11), we obtain the rate of change of the entropy inside the apparent horizon as˙ S I = 4 π H n X i =1 Q i T i − π ˙ HH + 1 H ! n X i =1 p i + ρ i T i . (16)Hence adding (15) and (16), we have the rate of change of total entropy as˙ S = ˙ S I + ˙ S A = 4 π H n X i =1 Q i T i − π ˙ HH + 1 H ! n X i =1 p i + ρ i T i − π ˙ HH h − α r c H ) α − i . (17) • GSL in Thermal Equilibrium: In thermal equilibrium , we have ∀ i : T i = T i.e. the horizon temperature matches with thetemperature of n-component fluid. In the present state of the Universe, this general assumption isnot so valid since radiation temperature is higher compared to non-relativistic cold dark matter.However thermal equilibrium did occur in the early radiation dominated Universe when all energy-matter was in thermal equilibrium with the radiation. When the horizon is the apparent horizon,we take the temperature as the Hawking temperature T = H π , after simplification, we get from(17) and (6) that ˙ S = 2 π ˙ HH " α r c H ) α − + ˙ HH . (18)It may be derived that GSL would hold if˙ S ≥ , ⇒ ˙ H " α r c H ) α − + ˙ HH ≥ . (19)From above we see that the GSL is always true for α = 0 (for any sign of ˙ H ). We investigatetwo interesting case here: • For quintessence dominated era, ˙
H <
0, the GSL holds (a) for all α < α >
H < − α ( r c ) α − H α but GSL breaks down for α > > ˙ H > − α ( r c ) α − H α . • For phantom dominated era, ˙
H >
0, the GSL holds (a) for all α > α <
H > − α ( r c ) α − H α but GSL breaks down for α < < ˙ H < − α ( r c ) α − H α .Equation (19) puts some constraint on α .At this juncture it would be investigated whether GSL remains valid in the case of smallperturbations around the de Sitter space (quasi-de-Sitter spacetime). As an illustration we consider, H = H + H ǫt + O ( ǫ ); ǫ = ˙ HH ; | ǫ |≪ . (i) When ˙ H > α ≥ − ǫ ( r c H ) − α ⇒ α ( r c H ) ≥ − ǫ ( r c H ) α . (20)(ii) When ˙ H < α ≤ − ǫ ( r c H ) − α ⇒ α ( r c H ) ≤ − ǫ ( r c H ) α . (21)Two constraints on α are now available in order GSL to valid via (20) and (21) in the case ofsmall perturbations around the de Sitter space. Note that precise value of α can not be determinedvia the inequalities. However observational constraints could be helpful for this purpose but thatis out of scope of this paper. B. On the event horizon R E In this section, the validity of GSL would be investigated on the event horizon. From equation(2), we obtain the entropy on the event horizon as S E = πR E " − α − α (cid:18) R E r c (cid:19) − α . (22)Differentiating (22) w.r.t. time t , we obtain˙ S E = 2 πR E ( HR E − " − α (cid:18) R E r c (cid:19) − α . (23)Adding (11) and (23), the rate of change of total entropy for event horizon is obtained as˙ S = ˙ S I + ˙ S E = 4 π H n X i =1 Q i T i − πR E n X i =1 p i + ρ i T i + 2 πR E ( HR E − " − α (cid:18) R E r c (cid:19) − α . (24)The GSL requires ˙ S ≥
0, i.e. the sum of the entropies of the perfect fluids inside the eventhorizon and the entropy attributed to the horizon is a non-decreasing function of the comov-ing time. In the following we discuss the validity of this law specially in the presence of dark energy. • GSL in Thermal Equilibrium:
In thermal equilibrium, we have ∀ i : T i = T . So equation(24) becomes ˙ S = ˙ HT R E + 2 πR E ( HR E − " − α (cid:18) R E r c (cid:19) − α . (25)For the future event horizon, and in the absence of a well-defined temperature, we assume that T is proportional to the Hawking temperature [35] T = bH π , (26)where b is an arbitrary constant of order unity. Equation (25) yields˙ S = 2 π ˙ HbH R E + 2 πR E ( HR E − " − α (cid:18) R E r c (cid:19) − α . (27)GSL will be satisfied if˙ S ≥ ⇒ r c R E ( HR E − ddt log( R E H b ) ≥ α (cid:18) r c R E (cid:19) α . (28)As the expression (28) appears to be very complicated and it is not possible to draw any definiteconclusion regarding the parameters of the model, we consider a particular choice of scale factorthat pertains to a phantom dominated Universe of polelike type described by [36] a ( t ) = a ( t s − t ) − n a > n > t s > t. (29)For this choice of scale factor, we have H = nt s − t ; R E = t s − tn + 1 ; ˙ H = n ( t s − t ) . (30)So equation (27) reduces to the form˙ S = 2 π ( t s − t )( n + 1) " − b + α (cid:18) t s − tr c ( n + 1) (cid:19) − α . (31)The GSL is satisfied if ˙ S ≥ t s − t ≥ r c ( n + 1) (cid:20) b − bα (cid:21) − α with b > . (32)Combining the expressions (29) and (32), we get an upper limit of the scale factor a ( t ) as a ( t ) ≤ a { r c ( n + 1) } n (cid:20) b − bα (cid:21) nα − . Therefore the GSL would be valid if the scale factor lies below the r.h.s of the above expression.
IV. GSL IN THERMAL NON-EQUILIBRIUM
If we drop the condition of thermal-equilibrium, the problem of investigating the validity ofthe GSL becomes more complicated. This situation arises since dark matter, radiation and darkenergy have different temperatures [37]. In this case, the components of temperatures T i ’s are alldistinct. The equation (6) can be written as˙ H = − π n X i =1 ( ρ i + p i ) . (33)Now n X i =1 (cid:18) ρ i + p i T i (cid:19) = p + ρ T + n X i =2 (cid:18) ρ i + p i T i (cid:19) = − ˙ H πT − T n X i =2 ( ρ i + p i ) + n X i =2 (cid:18) ρ i + p i T i (cid:19) . (34)We assume a particular choice for T = bH π and the other T i ’s are considered arbitrarily. Further-more, considering p i = w i ρ i for i = 2 , , ...., n we obtain n X i =1 (cid:18) ρ i + p i T i (cid:19) = − ˙ H bH − n X i =2 " (1 + w i ) ρ i ( πbH − n X i =2 T i ) . (35)In this model ρ and { ρ , ρ , ..., ρ n } consist of ingredients in thermal equilibrium (in each subset),which even if do not interact with the elements of the other subset, can interact with each other.Using (35) in (17) we get that for GSL to be valid on the apparent horizon R A we require˙ S = πbr c H " Hr c ( ˙ H − ( − b ) H ) + bα ( r c H ) α ˙ H + 4 bH ( ˙ H + H ) r c n X i =2 (1 + w i ) ρ i ( πbH − n X i =2 T i ) ≥ . (36)Using (35) in (24) we get that for GSL to be valid on the event horizon R E we require˙ S = 2 πR E " ( HR E − − α (cid:18) R E r c (cid:19) − α ! + R E bH ( ˙ H + 2 bH n X i =2 (1 + w i ) ρ i πbH − n X i =2 T i !) ≥ . (37)From the above expressions (36) and (37) we get general conditions for the validity of GSL forapparent and event horizons in case of thermal non-equilibrium. However, it is not possible toobtain any specific constraints on the model parameters for the validity of GSL.0 V. CONCLUSION
In this paper, we considered the FRW cosmological spacetime composed of interacting compo-nents. We discussed the generalized second law of thermodynamics by considering the power lawcorrection of entropy on the horizon. We focused on both the apparent and future event horizonsand expressed the time derivatives of the total entropies in terms of the model parameters α and r c . Considering the GSL in thermal equilibrium on the apparent horizon we find from equation(19) the conditions for validity of the GSL on the quintessence and phantom dominated era. Also,considering a small perturbation around the de Sitter space-time we found the conditions on themodel parameters required for the validity of GSL. Considering the GSL in thermal equilibriumon the event horizon we find that the GSL is valid if (cid:16) t s − tr c ( n +1) (cid:17) − α ≥ b − bα (equation (32)). Inequation (37) we expressed the time derivative of total entropy in terms of α and r c . If we con-sider a small perturbation around the de Sitter space-time, the general conditions of the validityof GSL have been found. Also if a phantom dominated Universe has a polelike type scale factor,the validity of GSL has also been analyzed. Acknowledgment
The authors would like to thank anonymous referee for giving useful comments to improve thiswork. [1] T. Jacobson,
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