Thermodynamic implications of the gravitationally induced particle creation scenario
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Abstract
A rigorous thermodynamic analysis has been done as regards the apparent horizon of a spatially flat Friedmann–Lemaitre–Robertson–Walker universe for the gravitationally induced particle creation scenario with constant specific entropy and an arbitrary particle creation rate \(\Gamma \). Assuming a perfect fluid equation of state \(p=(\gamma 1)\rho \) with \(\frac{2}{3} \le \gamma \le 2\), the first law, the generalized second law (GSL), and thermodynamic equilibrium have been studied, and an expression for the total entropy (i.e., horizon entropy plus fluid entropy) has been obtained which does not contain \(\Gamma \) explicitly. Moreover, a lower bound for the fluid temperature \(T_f\) has also been found which is given by \(T_f \ge 8\left( \frac{\frac{3\gamma }{2}1}{\frac{2}{\gamma }1}\right) H^2\). It has been shown that the GSL is satisfied for \(\frac{\Gamma }{3H} \le 1\). Further, when \(\Gamma \) is constant, thermodynamic equilibrium is always possible for \(\frac{1}{2}<\frac{\Gamma }{3H} < 1\), while for \(\frac{\Gamma }{3H} \le \text {min}\left\{ \frac{1}{2},\frac{2\gamma 2}{3\gamma 2} \right\} \) and \(\frac{\Gamma }{3H} \ge 1\), equilibrium can never be attained. Thermodynamic arguments also lead us to believe that during the radiation phase, \(\Gamma \le H\). When \(\Gamma \) is not a constant, thermodynamic equilibrium holds if \(\ddot{H} \ge \frac{27}{4}\gamma ^2 H^3 \left( 1\frac{\Gamma }{3H}\right) ^2\), however, such a condition is by no means necessary for the attainment of equilibrium.
1 Introduction
There have been several attempts to incorporate the present stage of cosmic acceleration into standard cosmology, the most notably being the introduction of an “exotic” component termed dark energy (DE) which is believed to have a huge negative pressure; however, its nature and origin is still a mystery despite extensive research over the past one and a half decades. Several DE models have been proposed in the literature but observational data from various sources such as supernovae type Ia (SNe Ia), cosmic microwave background (CMB), and baryon acoustic oscillations (BAO) have established that the cosmological constant \(\Lambda \) is the most viable candidate among them. The cosmic concordance \(\Lambda \)CDM model in which the universe is believed to contain a cosmological constant \(\Lambda \) associated with DE, and cold (i.e., pressureless) dark matter (abbreviated CDM) fits rather well the current astronomical data.
Nevertheless, there are severe drawbacks corresponding to a finite but incredibly small value of \(\Lambda \) such as the finetuning problem which leads to a discrepancy of 50–120 orders of magnitude with respect to its observed value which is about \(3\times 10^{11}\,\text {eV}^4\). Then there is the coincidence problem which is related to the question of “why are the energy densities of pressureless matter and DE of the same order precisely at the present epoch 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 been proposed 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, inhomogeneous cosmological models, etc. but each one of them comes with several problems that are yet to be settled. Because of these said difficulties in various cosmological models, another well known proposal has been suggested—the gravitationally induced particle creation mechanism. Schrödinger [1] pioneered the microscopic description of such a mechanism which was further developed by Parker et al. based on quantum field theory in curved spacetimes [2, 3, 4, 5, 6]. Prigogine et al. [7] provided a macroscopic description of particle creation mechanism induced by the gravitational field. A covariant description was later proposed [8, 9] and the physical difference between particle creation and bulk viscosity was clarified [10]. The process of particle creation is classically described by introducing a backreaction term in the Einstein field equations whose negative pressure may provide a selfsustained mechanism of cosmic acceleration. Indeed, many phenomenological particle creation models have been proposed in the literature [11, 12, 13, 14, 15, 16]. It has also been shown that phenomenological particle production [17, 18, 19, 20] cannot only incorporate the late time cosmic acceleration but also provide a viable alternative to the concordance \(\Lambda \)CDM model.
Despite rigorous investigation of various aspects of particle creation mechanism, its thermodynamic implications have never been explored. Such a study has been undertaken in this paper and the essence of this work is that the particle creation rate has been considered arbitrary, not a phenomenological one. The conclusions drawn from the present analysis are valid for any expression of the creation rate, constant or otherwise. The paper is organized as follows. Section 2 contains a brief discussion of the gravitationally induced adiabatic particle creation scenario, Sect. 3 along with subsections A, B, and C is dedicated to a detailed thermodynamic analysis of the process, while Sect. 4 provides a short discussion and possible scope for future work.
2 Gravitationally induced particle creation mechanism: a brief discussion
3 Thermodynamic analysis
3.1 First law
From the above analysis, we find that the first law holds at the apparent horizon whenever \(\Gamma =0\), or loosely speaking, whenever \(\Gamma \ll 3H\).
3.2 Generalized second law: an expression for total entropy
3.3 Thermodynamic equilibrium
Equilibrium configuration for different subintervals of \(\Gamma \)
Subintervals of \(\Gamma \)  Sign of \(\left( 1\frac{\Gamma }{3H}\right) \left( 1\frac{2\Gamma }{3H}\right) \)  Sign of \(\left\{ 1\left( \frac{2}{\gamma }1\right) \left( \frac{1\frac{\Gamma }{3H}}{1\frac{2\Gamma }{3H}}\right) \right\} \)  Equilibrium? 

\(\Gamma \le \frac{3H}{2}\)  Nonnegative  Nonnegative for \(\frac{\Gamma }{3H} < \frac{2\gamma 2}{3\gamma 2}\)  Never for \(\frac{\Gamma }{3H} \le \text {min}\left\{ \frac{1}{2},\frac{2\gamma 2}{3\gamma 2} \right\} \) 
\(\frac{3H}{2}<\Gamma <3H\)  Negative  Positive  Always 
\(\Gamma \ge 3H\)  Nonnegative  Positive  Never 
From different observational sources, it has been well established that the radiation phase was followed by a matter dominated era which eventually transited to a second de Sitter phase. Accordingly, it can be expected that 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 stayed in it forever unless acted upon by some “external agent.” However, it is a wellknown fact [6] that the production of particles was suppressed during the radiation phase, so in this model there would be no external agent to remove the system from thermodynamic equilibrium. Therefore, our present analysis leads us to conclude that during the radiation phase, if \(\Gamma \) is constant, then \(\frac{\Gamma }{3H} \le \frac{1}{3}\), or equivalently, \(\Gamma \le H\).
4 Discussion and future work

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 3H.

The GSL holds if \(\Gamma \le 3H\), or equivalently, if \(\frac{\Gamma }{3H} \le 1\), which implies that the GSL is not consistent with the phantom fluid.

For a constant particle creation rate, thermodynamic equilibrium always holds for \(\frac{1}{2}<\frac{\Gamma }{3H}<1\), while it never holds for \(\frac{\Gamma }{3H} \le \text {min}\left\{ \frac{1}{2},\frac{2\gamma 2}{3\gamma 2} \right\} \) and \(\frac{\Gamma }{3H} \ge 1\). Thus, thermodynamic equilibrium in this case is inconsistent with the cosmological constant as well as the phantom fluid.

When \(\Gamma \) is not constant, the only definite conclusion which can be drawn is that the thermodynamic equilibrium holds if \(\ddot{H} \ge \frac{27}{4}\gamma ^2 H^3 \left( 1\frac{\Gamma }{3H}\right) ^2\); however, such a condition is by no means necessary for the attainment of equilibrium.
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 particle creation rates that have been considered in the recent literature [18, 19, 31, 32, 33, 34, 35, 36, 37, 38]. Further, attempts to incorporate matter creation in inhomogeneous cosmological models can be made and its thermodynamic implications can be studied.
Footnotes
 1.
In this manuscript, without any loss of generality, we have assumed that the physical constants, namely c, G, \(\hbar \), and \(\kappa _B\), as well as \(8\pi \), are unity.
 2.
The idea of incorporating the GSL in cosmology was first developed by Brustein [28]. This second law is based on the conjecture that causal boundaries and not only event horizons have geometric entropies proportional to their area.
Notes
Acknowledgements
Subhajit Saha is supported by SERB, Govt. of India under National Postdoctoral Fellowship Scheme [File No. PDF/2015/000906]. Anindita Mondal wishes to thank DST, Govt. of India for providing Senior Research Fellowship. The authors are thankful to the anonymous reviewer for insightful comments which have helped to improve the quality of the manuscript significantly.
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