Irreversible Thermodynamics of the Universe in F (R) Gravity

J. Astrophys. Astr. (2019) 40:38 © Indian Academy of Sciences https://doi.org/10.1007/s12036-019-9609-y

Irreversible thermodynamics of the universe in f (R) gravity

ATREYEE BISWAS

Department of Natural Science, Maulana Abul Kalam Azad University of Technology, Haringhata, Nadia 741249, India. E-mail: [email protected]

MS received 7 May 2019; accepted 15 September 2019

Abstract. In this study, the irreversible thermodynamics of the universe in the framework of f (R) gravity has been studied following standard Eckart theory of non-equilibrium thermodynamics. For a spatially ﬂat universe, validity of GSLT and thermodynamic equilibrium have been examined both for apparent and event horizon as bounding horizons.The general result has been veriﬁed for exponential models of f (R) gravity and it is shown that GSLT holds on both apparent and event horizon, but thermodynamic equilibrium does not hold on event horizon.

Keywords. Irreversible thermodynamics—f (R)gravity—GSLT—Thermodynamic equilibrium.

1. Introduction from cosmic plasma, decoupling of photons from matter during recombination era, nucleosynthesis, etc., need In 1970s Stephen Hawking (Hawking 1975)madearev- non-equilibrium thermodynamics for explanation. olutionary discovery that black hole behaves as a black Now, current observations like supernovae Ia (SNeIa) body and emits thermal radiation (Hawking radiation). (Riess et al. 1998), cosmic microwave background It was also discovered (Bardeen et al. 1973) that the four (CMB) (Bennet et al. 2003), large-scale structure (LSS) laws of thermodynamics are actually analogous to the (Hawkins et al. 2003), etc., reveal that our universe is four laws of black hole mechanics, with temperature going through an accelerating phase. There are two and entropy, respectively, playing the role of surface approaches to explain such acceleration of universe. gravity and area of the surface of bounding horizon. One approach is to assume the existence of an exotic Further Jacobson (1995) derived Einstein’s field equa- matter called dark energy which has negative pres- tions from first law of thermodynamics and Padmanavan sure. Another approach is to consider the theory of (2002) derived first law of thermodynamics from Ein- modified gravity. f (R) gravity is one of the popular stein’s field equations for a general static spherically models among modified gravity models where nega- symmetric space-time. These two deductions strongly tive and positive powers of Ricci curvature scalar R support the idea that black hole thermodynamics can naturally combine the inflation at early times and the be extended to more general space-time. More specif- cosmic acceleration at late times. In Sobouti (2007), ically, it can be said that not only a black hole but the it was also argued that the f (R) gravity can serve as whole universe can be considered to be a thermody- dark matter (DM). The above discussions motivated namical object. Now, universal thermodynamics should us to investigate thermodynamic properties of universe be irreversible in nature, neither reversible nor quasi- in irreversible thermodynamic context in f (R) grav- reversible. It was first realized by Eling et al. (2006), ity model. We therefore structure this paper as follows: when following the work of Jacobson (1995), they tried In Section 2 the basic features of f (R) gravity have to reproduce Einstein’s field equations from first law been described. In Section 3, a general prescription of of thermodynamics in f (R) gravity and found out that irreversible thermodynamics in the frame of f (R) grav- a non-equilibrium thermodynamic treatment is neces- ity model has been given. In Section 4, for a general sary for this successful derivation. Also, it will be worth f (R) gravity model, we investigated the validity of the to mention that processes like decoupling of neutrinos generalized second law of thermodynamics (GSLT) and

0123456789().: V,-vol 38 Page 2 of 6 J. Astrophys. Astr. (2019) 40:38 thermodynamic equilibrium (TE). In Section 5 we dis- and f (R) cuss these results for a particular viable model of R = 6(H˙ + 2H 2) (9) gravity, namely, exponential model both for apparent = a˙ ρ and event horizon as bounding horizon. Finally, in Sec- Here H a is the Hubble parameter. Also D and pD tion 6 we discuss the summary and conclusion of this are the curvature contribution to the energy density and work. pressure, respectively.

3. General prescription of irreversible ( ) 2. f R gravity framework thermodynamics in f (R) gravity

The modified EinsteinÐHilbert action in the Jordan In this work we have followed the theory of irreversible ( ) frame in f R gravity is given by (Nojiri & Odintsov thermodynamics previously discussed by Gong et al. 2011) (2009)andChakraborty & Biswas (2013a,b). The sum- √ f (R) mary of the theory is as follows: S = −gd4x + L (1) J πG matter In non-equilibrium thermodynamics. the Clausius 16 δ relation dS = Q is replaced by the entropy balance where G, g, R and Lmatter are the gravitational con- T equation: stant, the determinant of the metric gμν, the Ricci scalar and the Lagrangian density of the matter inside the uni- δQ dS = + d S = d S + d S (10) verse respectively. f (R) is an arbitrary function of the T T i e i Ricci scalar. One can derive the following equation on where the extra term di S arises due to internal produc- varying the action (1) with respect to gμν: tion process. In particular > , m 1 = 0 for irreversible process; FGμν = 8πGTμν− gμν(RF− f )+∇μ∇ν F−gμνF di S , 2 0 for reversible process. (2) Classically δQ represents the heat transfer between the df = , = − 1 m system and its surrounding, while di S is the change of Here F dR Gμν Rμν 2 Rgμν and Tμν is the energy-momentum tensor of the matter. entropy associated with the uncompensated heat when The gravitational field equation in its standard form the system undergoes an−→ irreversible process. (Starobinsky 1980; Motohashi et al. 2010) is written as Let us denote σ and J to be the internal entropy s production density and the entropy flow density respec- = π m + D Gμν 8 G Tμν Tμν (3) tively. Assuming local equilibrium, one can write: where de S −→ =− Js .d (11) D 1 dt 8πGTμν = (1 − F)Gμν − gμν(RF − f ) 2 di S = σ +∇μ∇ν F − gμνF (4) dV (12) dt V μ(m) If one takes Tν = diag(−ρ, p, p, p) in the per- Here V = volume inside the bounding horizon of the fect fluid form, then for a spatially flat universe, the system, being the surface of the horizon. If heat con- FRW equations read as duction is considered to be responsible for causing the 2 = π (ρ + ρ ) entropy flow, then it can be written that 3H 8 G D (5) −→ ˙ =− π (ρ + ρ + + ) d S A Jq 2H 8 G D p pD (6) e = (13) dt T where di S = σ. 1 ˙ V (14) 8πGρD = (RF − f ) − 3H F dt 2 −→ + 2( − ˙ ) where Jq = heat current, T = temperature of the system 3H 1 F (7) −→ and A = surface area of the horizon. J and σ can be π = −1( − ) + ¨ + ˙− q 8 GpD RF f F 2H F calculated from the following relations: 2 −→ −→ Jq (1 − F)(2H˙ + 3H 2) (8) J = s T J. Astrophys. Astr. (2019) 40:38 Page 3 of 6 38

−→ 1 The above expression shows that total entropy change σ = J .∇ q T of the system depends on the non-equilibrium factor λ. In case of reversible system, the non-equilibrium Let us consider the universe be a thermodynamical sys- factor λ →∞. Infinitely large value of thermal con- tem with R the radius of the bounding horizon. Now ductivity λ means the heat transfer rate is infinitely large if we assume the universe to be a Bekenstein system and therefore the temperature will eventually be same bounded by apparent/event horizon, the entropy of the everywhere and therefore the system will be in equilib- horizon in f (R) gravity is then given by S = AF . 4 rium. We see that when λ →∞, the contributing part Therefore, one can obtain corresponding to irreversibility in Equation (18) will d S AF˙ + AF˙ become zero and the equation then perfectly represents e = (15) dt 4 the reversible system. Now, comparing the above equation with Equation (13) and putting A = 4π R2 (area of the surface of the hori- 4. Generalized second law of thermodynamics and zon), we can derive the following equation: thermodynamic equilibrium ˙ −→ TF F˙ R | Jq |= + (16) In the context of isolated macroscopic systems, the 2 2F R A entropy should never decrease, because such a system where for a flat universe the Hawking temperature T = always evolves towards thermodynamic equilibrium, a ˙ state having maximum entropy. Thus, for a matter filled 1 − R 2π R 1 2 . universe, bounded by a horizon, the generalized second Now, according to EckartÐFourier law: law of thermodynamics (GSLT) and thermodynamical −→ −→ equilibrium (TE) hold if the following conditions hold: Jq =−λ ∇ T dST ≥ I dt 0 (for GSLT) which states that there will be an energy flux if there is 2 d ST < λ> II 2 0 (for Thermodynamic Equilibrium) a temperature gradient ( 0) is the−→ thermal conduc- dt tivity). Now, putting the value of | J | in the equation q From Equation (18), it can be easily conjectured that for σ and using the Fourier law, one can obtain: GSLT is valid if either of the following conditions is −→ 1 hold: σ = J .∇ q T • > −→ 1 X 0, = Jq . − ∇T • X < −6λ. T 2 −→ d2 S 1 −→ Jq Now, from Equation (18), one can calculate T as = − J . dt2 T 2 q −λ 2 ˙ ˙ −→ d ST R 2X dST | J |2 = + − 2π R X˙ (19) = q dt2 R X dt λ 2 T ˙ 2 T 2 F2 F˙ R Now, if we assume GSLT holds, then following cases = + may arise in order to make decision on thermodynamic 4λT 2 2F R equilibrium (TE): 2 2 ˙ R˙ = F F + λ (17) 4 2F R 5. Validity of GSLT and TE in exponential model ( ) Using Equations (14), (15)and(17), the change of total of f R gravity entropy can be obtained as f (R) theory of modified gravity as an alternative of dST X dark energy has been widely discussed in recent past = 2π R X 1 + (18) dt 6λ (Sobouti 2007; Capozziello & Laurentis 2011; Clifton ˙ et al. 2011; Felice & Tsujikawa 2010; Nojiri & Odintsov = ˙ + FR where X R F 2 . 2011) and we do not have enough scope in the present 38 Page 4 of 6 J. Astrophys. Astr. (2019) 40:38 article to discuss all the features of every viable model of f (R) gravity analysed before. In this work our aim was to investigate the thermodynamic properties of the universe on the basis of the theory discussed in the previous section in one viable model of f (R) gravity, namely, exponential model of f (R) gravity. Some recent analysis on exponential gravity model suggests that observational constraints can be well satisfied from the cosmological point of view, in such a way that f (R) gravity and CDM model turn out to be nearly indistin- guishable, as investigated in a previous analysis (Bamba et al. 2010a,b; Yang et al. 2010; Chen et al. 2015). In Figure 1. Graph for dST . addition, the exponential gravity model can be extended dt to cover the inflationary stage as well. The exponential model is given by

− R R f (R) = R − β RS 1 − e S (20) where β and RS are two constant parameters of this model. Following Bamba et al. (2010a,b), we put β = 2 18 m0 H0 1.8andRS = β . From Equation (20), we calculate F, F˙ and F¨ as follows: − R F = 1 − βe RS (21) − R 3β( − − ) RS 2 2 ˙ 6H j q 2 e d ST d ST = F = (22) Figure 2. Graph for 2 and 2 with m0 R . , = . , =−dt. , = dt, =− . S 0 23 H0 74 3 q0 0 588 j0 1 s0 0 238. 4 + 2 + + ¨ − R 6H s q 8q 6 F = βe RS 2 RS dST d ST We plot graphs for dt and 2 against the non- λ dt 36H 6 ( j − q − 2)2 equilibrium factor (thermal conductivity) with the use − (23) of current observed value of matter density parameter R2 S and the cosmographic parameters obtained for standard where q, j, s are respectively the deceleration param- CDM model (Xia et al. 2012) as follows: eter, jerk parameter and state ﬁnder parameter deﬁned ˙ ¨ (3) = 0.23, H = 74.3, by q =−1 − H , j = H − 3q − 2, s = H + 4 j + m0 0 H 2 H 3 H 4 =− . , = , =− . . 3q(q + 4) + 6. q0 0 588 j0 1 s0 0 238 Now, we examine GSLT and TE for the exponential From Figures 1 and 2 it is evident that at present ( ) model of f R gravity on apparent and event horizon as epoch both GSLT and TE holds on apparent horizon. 2 bounding horizon of the universe in two separate cases. dST d ST From analytic expression of dt and dt2 in equation Apparent Horizon: In the case of apparent horizon, (24)and(25) respectively it is not possible to understand Equations (18)and( 19) take the following forms: which particular condition depicted in Table 1 the model dST − X follows for the validity of GSLT and TE. Therefore, = 2π H 1 X 1 + (24) dt 6λ we consider an arbitrary value of λ = 0.1(asλ is a 2 ˙ small positive constant) and then calculate the terms d ST 2X dST ˙ = H(q + 1) + X, X˙ , dST , H(q + ) + X dt2 X dt dt 1 2 X and obtain the following − results: −2π H 1 X˙ (25) ˙ ˙ −1 X = 0.391688, X =−62.8283, = ( + ) + FH ˙ = ( 2 + − Here X q 1 F 2 and X H 2q q ˙ ¨ dST X j)F + F + 3 (q + 1)F˙ = 0.0547463, H(q + 1) + 2 =−290.196 2H 2 dt X J. Astrophys. Astr. (2019) 40:38 Page 5 of 6 38

dS > Table 1. Condition for TE when dt 0.

Cases Conclusion

˙ ˙ π ˙ ˙ R 2X dST 2 R X I X > 0, + > 0 TE hods if < ˙ R X dt R ˙ + 2X R X ˙ ˙ < R + 2X˙ > II X 0, R X 0 TE does not hold ˙ ˙ R 2X˙ III X > 0, + < 0 TE holds R X R˙ ˙ 2π RX˙ ˙ 2X dST IV X < 0, + < 0 TE holds if > ˙ R X dt R ˙ + 2X R X

Based on the above result we can naively comment that this model of f (R) gravity GSLT follows the condition X > 0 in order to satisfy GSLT and for holding TE it 2 follows the last condition of Table 1,i.e. Figure 3. d ST Graph for dt2 . 2X˙ dS X˙ < 0, H(q + 1) + < 0, T X dt 2π R X˙ 6. Conclusion > A . ˙ ( + ) + 2X H q 1 X In this paper we have studied thermodynamic properties of universe in f (R) gravity in the context of irreversible Event Horizon In case of event horizon, Equation (18) thermodynamics. Following standard Eckart theory of and Equation (19) take the following forms: irreversible thermodynamics, we ﬁrst derived condition dST X for validity of GSLT and thermodynamic equilibrium = 2π RE X 1 + (26) dt 6λ both on apparent and event horizon for a general f (R) 2 ˙ d S − 2X dS gravity model. Later we considered the exponential T = H − R 1 + T ( ) dt2 E X dt model of f R gravity as a particular example. Using current observed values of matter density parameter and − π ˙ 2 RE X (27) cosmography parameters, we have shown that though where RE is the radius of event horizon and X = GSLT is satisﬁed both on apparent horizon and event ˙ ( − ) + RE F ˙ =− ( + ) + horizon, but thermodynamic equilibrium does not hold HRE 1 F 2 , X HF 1 qHRE ¨ on event horizon. Since there is a constant transfer 3 (HR − 1) + RE F E 2 of energy (in this case, heat) between the system and Using the current observed value for matter den- surroundings, thermal equilibrium is not possible, i.e. sity parameter and cosmography parameters as before maximum entropy can not be attained. Consequently, = 0.23, H = 74.3, q =−0.588, j = 1, s = m0 0 0 0 0 we can say that when a system undergoes irreversible −0.238, we obtain: thermodynamic process, thermodynamic equilibrium X = 71.5419RE − 0.971414 should not hold. In this context, we can conclude that ˙ X = 3209.71RE − 70.2737 event horizon is more preferable than apparent horizon.

Therefore, X ≶ 0 accordingly RE ≶ 0.0135783 and ˙ X ≶ 0 accordingly RE ≶ 0.0218941. > = 1 = Since it is well known that RE RA H Acknowledgements 0.0134589502, we can conclude that both X and X˙ are positive. Since X > 0, GSLT is satisﬁed on event hori- I would like to express my sincere gratitude to Prof. zon. Subenoy Chakraborty of the Department of Mathemat- In order to check whether TE holds on event horizon, ics, Jadavpur University, for his persistent support and 2 d ST R λ guidance during this project. I am thankful to him for we plot dt2 against E and thermal conductivity . From Figure 3 it is evident that thermodynamic equi- his valuable suggestions and ideas he shared with me librium does not hold on event horizon. for accomplishing this work. 38 Page 6 of 6 J. Astrophys. Astr. (2019) 40:38

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