Thermodynamic implications of the gravitationally induced particle creation scenario
aa r X i v : . [ g r- q c ] M a r Thermodynamic implications of the gravitationally induced particle creationscenario
Subhajit Saha a Department of Physical Sciences,Indian Institute of Science Education and Research Kolkata,Mohanpur 741246, West Bengal, India.
Anindita Mondal b Department of Astrophysics & Cosmology,S. N. Bose National Centre for Basic Sciences,Kolkata 700106, West Bengal, India. (Dated: The 27 th March, 2017)A rigorous thermodynamic analysis has been done at the apparent horizon of a spatially flatFriedmann-Lemaitre-Robertson-Walker universe for the gravitationally induced particle creationscenario with constant specific entropy and an arbitrary particle creation rate Γ. Assuming aperfect fluid equation of state p = ( γ − ρ with ≤ γ ≤
2, the first law, the generalized secondlaw (GSL), and thermodynamic equilibrium have been studied and an expression for the totalentropy (i.e., horizon entropy plus fluid entropy) has been obtained which does not contain Γexplicitly. Moreover, a lower bound for the fluid temperature T f has also been found which isgiven by T f ≥ (cid:18) γ − γ − (cid:19) H . It has been shown that the GSL is satisfied for Γ3 H ≤
1. Further,when Γ is constant, thermodynamic equilibrium is always possible for < Γ3 H <
1, while for Γ3 H ≤ min n , γ − γ − o and Γ3 H ≥
1, equilibrium can never be attained. Thermodynamic argumentsalso lead us to believe that during the radiation phase, Γ ≤ H . When Γ is not a constant,thermodynamic equilibrium holds if ¨ H ≥ γ H (cid:0) − Γ3 H (cid:1) , however, such a condition is by nomeans necessary for the attainment of equilibrium.Keywords: Adiabatic particle creation; Apparent horizon; Laws of thermodynamics; Ther-modynamic equilibriumPACS Numbers: 98.80.-k, 05.70.Ln, 04.40.Nr, 95.36.+x a Electronic Address: [email protected] b Electronic Address: [email protected]
1. INTRODUCTION
There have been several attempts to incorporate the present stage of cosmic acceleration into standardcosmology, the most notably being the introduction of an ”exotic” matter termed dark energy (DE) which isbelieved to have a huge negative pressure, however, its nature and origin is still a mystery despite extensiveresearch over the past one and a half decades. Several DE models have been proposed in the literature butobservational data from various sources such as Supernovae Type Ia (SNe Ia), Cosmic Microwave Background(CMB), and Baryon Accoustic Oscillations (BAO) have established that the cosmological constant Λ is themost viable candidate among them. The cosmic concordance ΛCDM model in which the Universe is believedto contain a cosmological constant Λ associated with DE, and cold (i.e., pressureless) dark matter (abbreviatedCDM) fits rather well the current astronomical data.Nevertheless, there are severe drawbacks corresponding to a finite but incredibly small value of Λ such as thefine-tuning problem which leads to a discrepancy of 50 to 120 orders of magnitude with respect to its observedvalue which is about 3 × − eV . Then there is the coincidence problem which is related to the questionof ”why are the energy densities of pressureless matter and DE of the same order precisely at the presentepoch although they evolve so differently with expansion?” Several models such as decaying vacuum models,interacting scalar field descriptions of DE, and a single fluid model with an antifriction dynamics have beenproposed with a view to alleviate such problems. Moreover, in order to solve the flatness and horizon problems,an inflationary stage for the very early universe was introduced but this again gave rise to several new problems,like the initial conditions, the graceful exit, and multiverse problems.Other attempts to explain the late time accelerating stage are modified gravity models, inhomogeneouscosmological models etc. but each one of them comes with several problems that are yet to be settled. Becauseof these said difficulties in various cosmological models, another well known proposal has been suggested —the gravitationally induced particle creation mechanism. Schrodinger [1] pioneered the microscopic descriptionof such a mechanism which was further developed by Parker and others based on quantum field theory incurved spacetimes [2–6]. Prigogine and collaborators [7] provided a macroscopic description of particle creationmechanism induced by the gravitational field. A covariant description was later proposed [8, 9] and the physicaldifference between particle creation and bulk viscosity was clarified [10]. The process of particle creation isclassically described by introducing a backreaction term in the Einstein field equations whose negative pressuremay provide a self-sustained mechanism of cosmic acceleration. Indeed, many phenomenological particle creationmodels have been proposed in the literature [11–16]. It has also been shown that phenomenological particleproduction [17–20] can not only incorporate the late time cosmic acceleration but also provide a viable alternativeto the concordance ΛCDM model.Despite rigorous investigation of various aspects of particle creation mechanism, its thermodynamic implica-tions have never been explored. Such a study has been undertaken in this paper and the essence of this workis that the particle creation rate has been considered arbitrary, not a phenomenological one. The conclusionsdrawn from the present analysis are valid for any expression of the creation rate, constant or otherwise. Thepaper is organized as follows. Section 2 contains a brief discussion of the gravitationally induced adiabatic par-ticle creation scenario, Section 3 along with Subsections A, B, and C are dedicated to detailed thermodynamicanalysis of the process, while Section 4 provides a short discussion and possible scope for future work.
2. GRAVITATIONALLY INDUCED PARTICLE CREATION MECHANISM: A BRIEF DISCUSSION
Let us consider a spatially flat, homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW)universe with matter content endowed with the mechanism of particle creation. The dynamics of such a modelis governed by the Friedman equations given by H = ρ, (1)˙ H = −
12 ( ρ + p + Π) . (2)In the above equations, ρ and p are the energy density and thermostatic pressure of the cosmic fluid respectivelyand they are related by the equation of state (EoS) p = ( γ − ρ with ≤ γ ≤ H = ˙ a ( t ) a ( t ) is the Hubbleparameter [ a ( t ) is the scale factor of the Universe], and Π is the creation pressure related to the gravitationally In this manuscript, without any loss of generality, we have assumed that the physical constants, namely, c , G , ¯ h , and κ B , as wellas 8 π are unity. induced process of particle creation. The lower bound on γ ensures that the perfect fluid does not becomeexotic, or equivalently, the strong energy condition remains valid. As a consequence, the energy conservationlaw gets reduced to ˙ ρ + Θ( ρ + p + Π) = 0 . (3)Now, the non-conservation of the total number N of particles in an open thermodynamic system producesan equation given by ˙ n + Θ n = n Γ . (4)In Eqs. (3) and (4), Θ is the fluid expansion scalar which turns out to be 3 H in our case, Γ denotes the rateof change of the number of particles ( N = na ) in a comoving volume a , and n is the number density ofparticles. So, a positive Γ implies production of particles while a negative Γ indicates particle annihilation.Further, a non-zero Γ produces an effective bulk viscous pressure [21–27] of the fluid and hence non-equilibriumthermodynamics comes into the picture.Using Eqs. (3) and (4), and the Gibb’s relation T ds = d (cid:16) ρn (cid:17) + pd (cid:18) n (cid:19) , (5)we can obtain an equation relating the creation pressure Π and the creation rate Γ, which can be expressed asΠ = − ΓΘ ( ρ + p ) , (6)under the customary assumption that the specific entropy s (in other words, the entropy per particle) is constant,i.e., the process is adiabatic (or isentropic). Thus, a dissipative fluid is equivalent to a perfect fluid with a non-conserved particle number. Eq. (2) now reduces to˙ HH = − γ (cid:18) − Γ3 H (cid:19) (7)The deceleration parameter q takes the form q = − ˙ HH −
1= 3 γ (cid:18) − Γ3 H (cid:19) − , (8)and the effective EoS parameter for this model (denoted by w eff ) becomes w eff = p + Π ρ = γ (cid:18) − Γ3 H (cid:19) − , (9)which represents quintessence era for Γ < H and phantom era for Γ > H , while Γ = 3 H corresponds to acosmological constant, owing to the fact that w eff = −
3. THERMODYNAMIC ANALYSIS
In the following subsections, we shall study the first law, the generalized second law (GSL) , and thermo-dynamic equilibrium for an arbitrary particle creation rate Γ. We shall consider an apparent horizon as ourthermodynamic boundary, since, unlike the event horizon, a cosmic apparent horizon always exists and it coin-cides with the event horizon in the case of a last de Sitter expansion. Moreover, in a flat FLRW universe, the The idea of incorporating the GSL in cosmology was first developed by Ram Brustein [28]. This second law is based on theconjecture that causal boundaries and not only event horizons have geometric entropies proportional to their area. apparent horizon coincides with the Hubble horizon H − . So, the apparent horizon can be considered to belocated at R A = H and its first order derivative with respect to the cosmic time t can be evaluated as˙ R A = − ˙ HH = 3 γ (cid:18) − Γ3 H (cid:19) . (10)The (Bekenstein) entropy and (Hawking) temperature of the apparent horizon are given by S A = (cid:18) c G ¯ h (cid:19) πR A
4= 18 R A , (11)and T A = (cid:18) ¯ hcκ B (cid:19) πR A = 4 R A (12)respectively. A. First law
The first law of thermodynamics at the horizon is governed by the Clausius relation − dE A = T A dS A . (13)The differential dE A of the amount of energy crossing the apparent horizon can be evaluated as (see Eq. (27)of Ref. [29]) − dE A = 12 R A ( ρ + p ) Hdt = 3 γ dt. (14)Again, using the expressions of T A and S A given in Eqs. (11) and (12), the expression T A dS A becomes T A dS A = 3 γ (cid:18) − Γ3 H (cid:19) dt, (15)where we have used relation (10).From the above analysis, we find that the first law holds at the apparent horizon whenever Γ = 0, or looselyspeaking, whenever Γ ≪ H . B. Generalized second law: An expression for total entropy
According to thermodynamics, the equilibrium configuration of an isolated macroscopic physical systemshould be the maximum entropy state, consistent with the constraints imposed on the system. Thus if S isthe total entropy of the system, the following conditions should hold — (a) dS ≥ d S < d ( S A + S f ) ≥ d ( S A + S f ) < S f is the entropy of the cosmic fluid contained within the horizon. The inequality (i) issometimes called the GSL.The Gibb’s equation can be rewritten in the form T f dS f = dE f + pdV, (17)where T f is the temperature of the cosmic fluid respectively, and E f = ρV is the energy of the fluid.Now, the assumption of a constant specific entropy leads us to an evolution equation for the fluid temperaturegiven by (see the second relation in Eq. (35) of Ref. [30])˙ T f T f = (Γ − Θ) ∂p∂ρ . (18)Noting from Eq. (7) that (Γ − Θ) = γ (cid:16) ˙ HH (cid:17) , the above equation leads to the integralln (cid:18) T f T (cid:19) = 2( γ − γ Z dHH . On integration, we obtain, T f = T H γ − γ , (19)where T is the constant of integration. Note that Γ does not appear explicitly in the equation.From Eq. (17), the differential of the fluid entropy can be obtained in the following form: dS f = 3 γ (cid:18) γ − (cid:19) T − (cid:18) − Γ3 H (cid:19) H − γ ) γ dt. (20)The differential of the total entropy can then be evaluated as d ( S A + S f ) = 3 γ H (cid:18) − Γ3 H (cid:19) (cid:20) (cid:18) γ − (cid:19) T − H γ − (cid:21) dt. (21)It can be easily seen from the previous equation that GSL holds if Γ ≤ H , or equivalently, if Γ3 H ≤
1. Therefore,the GSL is not consistent with the phantom fluid. Furthermore, S T is a constant of motion when Γ = 3 H , i.e.,when w eff = −
1, a cosmological constant.Another remarkable fact is that Eq. (21) gives us an opportunity [by replacing dt by dH ˙ H and using Eq. (7)]to derive an expression for the total entropy in terms of the Hubble parameter H in the form S T = S A + S f = 18 H " − γ − γ − ! T − H γ . (22)The essence of Eq. (22) lies in the fact that the particle creation rate Γ does not occur explicitly in the equation.Requiring that the total entropy be always positive, we can, in principle, obtain a lower bound on T given by T ≥ γ − γ − ! H γ . (23)Eq. (23) implies that we can also impose a lower bound on the fluid temperature T f as T f ≥ γ − γ − ! H . (24)For radiation era (i.e., γ = ) and matter dominated era (i.e., γ = 1), the lower bounds on T f become T f ≥ H and T f ≥ H respectively. Using Maple software, the total entropy S T has been plotted against γ for H = 67and T = 10 , and presented in Figure 1. FIG. 1. The total entropy S T for allowed values of γ , taking H = 67 and T = 10 . C. Thermodynamic equilibrium
Case I:
Γ is constant — If the particle creation rate Γ is assumed to be constant, then the second orderdifferential of the total entropy can be obtained from Eq. (21) as d S T dt = d dt ( S A + S f )= 9 γ (cid:18) − Γ3 H (cid:19) (cid:18) − H (cid:19) " (cid:18) γ − (cid:19) T − H γ − ( − (cid:18) γ − (cid:19) − Γ3 H − H !) . (25)In Table I, we have explored relevant subintervals of Γ in order to test for the validity of thermodynamicequilibrium. From the table, it is evident that thermodynamic equilibrium holds unconditionally for < Γ3 H < Γ3 H ≤ min n , γ − γ − o and Γ3 H ≥
1. Thus, thermodynamic equilibrium in this case isinconsistent with the cosmological constant as well as the phantom fluid.From different observational sources, it has been well established that the radiation phase was followed by amatter dominated era which eventually transited to a second de Sitter phase. Accordingly, it can be expectedthat in the radiation dominated era, the entropy increased but the thermodynamic equilibrium was not achieved[31]. If this were not true, the Universe would have attained a state of maximum entropy and would have stayedin it forever unless acted upon by some ”external agent.” However, it is a well known fact [6] that the productionof particles was suppressed during the radiation phase, so in this model, there would be no external agent toremove the system from thermodynamic equilibrium. Therefore, our present analysis leads us to conclude thatduring the radiation phase, if Γ is constant, then Γ3 H ≤ , or equivalently, Γ ≤ H . Table I : Equilibrium configuration for different subintervals of ΓSubintervals of Γ Sign of (cid:0) − Γ3 H (cid:1) (cid:0) − H (cid:1) Sign of n − (cid:16) γ − (cid:17) (cid:16) − Γ3 H − H (cid:17)o Equilibrium?Γ ≤ H Non-negative Non-negative for Γ3 H < γ − γ − Never for Γ3 H ≤ min n , γ − γ − o H < Γ < H Negative Positive AlwaysΓ ≥ H Non-Negative Positive Never
Case II:
Γ is not constant — For a variable Γ, Eq. (25) can be generalized as d S T dt = 27 γ "((cid:18) − Γ3 H (cid:19) − (cid:18) γ (cid:19) ¨ HH ) (cid:26) (cid:18) γ − (cid:19) T − H γ − (cid:27) − (cid:18) γ − (cid:19) (cid:18) γ − (cid:19) × T − H γ − (cid:18) − Γ3 H (cid:19) , (26)where we have substituted the value of ˙Γ evaluated as˙Γ = (6 H − Γ) ˙ HH + (cid:18) γ (cid:19) ¨ HH .
It is evident from Eq. (26) that it is quite difficult to perform an analysis similar to the one that we have done inthe previous case. The only definite conclusion which can be made here is that the thermodynamic equilibriumholds if ¨ H ≥ γ H (cid:0) − Γ3 H (cid:1) .
4. DISCUSSION AND FUTURE WORK
This paper dealt with a rigorous thermodynamic analysis at the apparent horizon of a spatially flat FLRWuniverse for the gravitationally induced particle creation scenario with constant specific entropy and an arbitraryparticle creation rate Γ. Assuming a perfect fluid EoS p = ( γ − ρ with ≤ γ ≤ • The first law holds at the apparent horizon either for a zero particle creation rate or, loosely speaking,when the creation rate is infinitesimally small as compared to 3 H . • The GSL holds if Γ ≤ H , or equivalently, if Γ3 H ≤
1, which implies that the GSL is not consistent withthe phantom fluid. • For a constant particle creation rate, thermodynamic equilibrium always holds for < Γ3 H <
1, whileit never holds for Γ3 H ≤ min n , γ − γ − o and Γ3 H ≥
1. Thus, thermodynamic equilibrium in this case isinconsistent with the cosmological constant as well as the phantom fluid. • When Γ is not constant, the only definite conclusion which can be made is that the thermodynamicequilibrium holds if ¨ H ≥ γ H (cid:0) − Γ3 H (cid:1) , however, such a condition is by no means necessary for theattainment of equilibrium.An expression for the total entropy with no explicit dependence on Γ has also been found. Such an expressionsuggests that for Γ = 3 H , i.e., a cosmological constant, the total entropy is a constant of motion. Further,imposing the condition that the total entropy is always positive, a lower bound on the fluid temperature T f has been obtained. It is evident that T f ≥ H and T f ≥ H for radiation and matter dominated erasrespectively. Thermodynamic arguments also lead us to believe that if Γ is a constant, then Γ ≤ H during theradiation phase.For future work, thermodynamics of the particle creation scenario at any arbitrary horizon can be investigated.The present thermodynamic analysis can also help to constrain various parameters of phenomenological particlecreation rates that have been considered in recent literature [18, 19, 31–38]. Further, attempts to incorporatematter creation in inhomogeneous cosmological models can be made and its thermodynamic implications canbe studied. ACKNOWLEDGMENTS
Subhajit Saha is supported by SERB, Govt.of India under National Post-doctoral Fellowship Scheme [FileNo. PDF/2015/000906]. Anindita Mondal wishes to thank DST, Govt. of India for providing Senior ResearchFellowship. The authors are thankful to the anonymous reviewer for insightful comments which have helped toimprove the quality of the manuscript significantly. [1] E. Schrodinger, Physica , 899 (1939).[2] L. Parker, Phys. Rev. Lett. , 562 (1968).[3] L. Parker, Phys. Rev. , 1057 (1969).[4] N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space (Cambridge Univ. Press, Cambridge, England,1982).[5] V. Mukhanov and S. Winitzki,
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