# Thawing and freezing quintessence models: a thermodynamic consideration

## Abstract

Thawing and freezing quintessence models are compared thermodynamically. Both of them are found to disobey the generalized second law of thermodynamics. However, for freezing models, there is still a scope as this breakdown occurs in the past, deep inside the radiation dominated era, when a standard scalar field model with a pressureless matter is not a correct description of the matter content. The thawing model has a pathological breakdown in terms of thermodynamics in a finite future.

## 1 Introduction

Convincing observational evidences of the accelerated expansion of the universe [1, 2, 3, 4, 5] for the last twenty years posed the great challenge of modelling the universe or more precisely finding a matter component that can bring such a repulsive gravity effect into being. The natural choice, the cosmological constant \(\Lambda \) fails to be the unique choice for its unassailable discrepancy between the actually required and theoretically predicted values [6]. A scalar field with a potential, called the quintessence field, is one of the favoured options although no potential warrants a compelling theoretical support [7, 8, 9].

In connection with their evolution pattern, some of the quintessence fields are broadly classified into two categories called thawing and freezing models. The thawing model is one for which the effective equation of state parameter (EoS) *w* starts as almost a constant close to \(-\,1\) and “thaws” into an evolving one, whereas a freezing model behaves in a different manner, *w* settles down to a constant value close to \(-\,1\) quite late in the evolution of the universe. Scherrer and Sen [10] provides a brief but elegant description of this classification. Particularly interesting amongst the freezing models are the so called “trackers” [11, 12, 13], in which the energy density of the scalar field evolves almost parallel to the energy density of the dark matter for the most of the history, without dominating over the latter but freezes to a value more than the corresponding dark matter density at a later stage of the evolution. As such a quintessence field does not necessarily have to belong to one of these two categories, but both the thawing and freezing models are important in their own right and attracted a lot of attention, primarily by virtue of their ability to address the coincidence issue [14] – the question why the dark matter and dark energy are of the same order of magnitude now.

A comparison between freezing models with a tracking behaviour and a thawing model, in connection with their compliance with observational data has been given by Thakur et al. [15]. These models are also compared in connection with their stability [16]. It is found that there is hardly any reason to favour any of these two if the complete history of universe in the post radiation-dominated is considered. Thawing and freezing models, in terms of cluster number counts, have been discussed by Devi et al. [18]. The purpose of the present work is to compare the thawing and freezing models considering their thermodynamic behaviour, particularly their viability against the generalized second law (GSL) of thermodynamics. A very general description of thawing and freezing models are adhered to, and tracking behaviour or any such finer details are not considered.

We use a simple definition of thawing and freezing models and plot the rate of change of the total entropy against evolution. This rate should always be positive as the total entropy never decreases according to the GSL. The results show that both forms suffer from a breakdown of GSL. We have tested this for a quintessence along with a cold dark matter (CDM), but also do the same exercise with a pure quintessence both leading to similar results to indicate that this feature of thermodynamic non-compliance is a characteristic of the quintessence field. The freezing models, however, has an advantage over the thawing models as this breakdown of GSL occurs way back in the past (\(z\sim 10^{4}\)), which is not really described by a CDM with a quintessence.

The next section contains a general description of the quintessence models. Section 3 describes a pure quintessence without any dark matter content and describes the thermodynamic behaviour of the thawing and freezing models. Section 4 deals with the thermodynamics of such models along with the dark matter component. The fifth and final chapter contains a discussion of the results obtained.

## 2 Quintessence models: thawing and freezing

*R*is the Ricci scalar and \({\mathscr {L}}_m\) is the Lagrangian density for the fluid distribution. In the consequent Einstein field equations \(G_{\alpha \beta } = -\, T^{(f)} _{\alpha \beta } - T^{(q)} _{\alpha \beta }\), where the superscripts

*f*and

*q*represent the fluid and the quintessence matter respectively, the right hand side is given by,

*a*(

*t*) is the scale factor. In such a spacetime \(\Phi \) is also a function of the cosmic time

*t*alone. Einstein field equations can be written as

*H*is the Hubble parameter, \(\rho _\Phi , p_\Phi \) are the density and pressure of the quintessence field given by

## 3 A pure quintessence

*V*is obtained as (see [17] for the details),

*V*is a real valued function of \(\Phi \). In the limit \(\alpha \rightarrow 0\), the above equation reproduces the exponential potential.

*a*. In the limit \(\alpha \rightarrow 0\), it reduces to a constant, \(w_\Phi = -\,1+\frac{\lambda }{3}\).

In the Fig. 1a, b, \(w_\Phi \) is plotted as a function of *N* for some values of \(\alpha \), where \(N=\ln (\frac{a}{a_0})\). For positive and negative values of \(\alpha \), one has a thawing (Fig. 1a) and a freezing (Fig. 1b) behaviour respectively [18].

One may note that, since \(\lambda \) is positive, \(w_\Phi \geqslant -\,1\) for all values of *a*, no matter whether \(\alpha \) is positive or negative [17].

### 3.1 Compliance of a pure quintessence with GSL

We now investigate whether a pure quintessence model comply with the GSL of thermodynamics which asserts that the change in the combination of the horizon entropy and entropy of the matter inside the horizon does not decrease with time [19, 20].

*A*denotes the area of the apparent horizon and is related to radius of the apparent horizon (\(R_h\)) as, \(A = 4\pi R_h^2\). In a spatially flat FRW space, the horizon radius \(R_h\) is related to Hubble parameter [21, 22, 24] as,

*H*and \({\dot{H}}\) in terms of

*a*and the Eq. (25) looks like

*N*.

From Fig. 2a, it is evident that in case of thawing quintessence (\(\alpha > 0\)), the total entropy increases upto a certain time, after that it does not obey GSL. In fact \(\dot{S_{tot}}\) shoots to an infinitely large value and then drops to an infinitely large negative value. The same feature is obtained for all allowed values of \(\alpha \), only the range of *N* for the onset of this pathological behaviour varies. The freezing quintessence (\(\alpha < 0\)) behaves in an opposite way! It does qualify this inquest for the future, the net entropy increases and settles down to a constant value in future when \(\dot{S_{tot}}\) approaches zero (see Fig. 2b), may mot be as fast as shown in the figure but rather asymptotically as revealed by the zoomed in version in the inset. However,it does not respect GSL in the past. Here also one has a discontinuity in \(\dot{S_{tot}}\) and also a negative value for the same indicating a decrease in \(S_{tot}\).

## 4 Quintessence with cold dark matter

### 4.1 Compliance of CDM plus quintessence distribution with GSL

*N*indicating a decrease in entropy. The freezing models (\(\alpha < 0\)) do respect GSL in the future. Albeit having a decreasing value, \({\dot{S}}_{tot}\) remains positive (as seen in the Fig. 4b), indicating that the entropy is increasing and settles down to a constant value in future, as indicated by Fig. 4b. In past, however, it indeed has a problem. The rate of change of entropy becomes negative at a finite past. However, by the choice of the parameter \(\alpha \), the time of occurrence can be pushed back. For instance, for \(\alpha = -0.3\), the pathology is observed for \(z \sim 10^{4}\) (see Fig. 5a), i.e., before the onset of matter domination over the radiation, where this system of equations will not govern the dynamics of the universe. Similarly, for \(\alpha =-0.1\), this discrepancy is observed at \(z\sim 10^{12}\), which is far beyond the jurisdiction of quintessence along with CDM (see Fig. 5b).

If we carefully notice Eq. (25), it is apparent that the term \(2H^2+{\dot{H}}\) decides the fate the thermodynamic viability of the models. For \({\dot{H}} + 2H^2 < 0\) (i.e., \(q \ge 1\)), the model fails in the thermodynamic inquest. So the violation of GSL is associated with a decelerated universe in future. The deceleration has to be at least as rapid as in the case of a pure radiation dominated universe. However, no alarm is indicated for the good old standard radiation dominated model of the universe which does not contain any dominating scalar field – Eq. (29) clearly indicates that if \(\rho _{\Phi ,0} = 0\), \({\dot{S}}_{tot}\) is a positive semi-definite quantity.

So one can see that the freezing models are stronger against thermodynamic viability, at least in the relevant period when the system under consideration is actually valid.

As in the previous section, here also we have used the figure for \(\lambda = 0.06\), and excluded \(\lambda = 0.01\) and 0.1 as there is no change in the quality. The only difference is that of a minor shift in the epochs, such as that of the onset of the violation of GSL.

## 5 Discussion

Thawing and freezing models are compared in this work in the context of the GSL of thermodynamics. The total entropy (\(S_{tot}\)) is taken as the sum of the horizon entropy and the entropy of the matter inside the horizon. With a simple ansatz [18] for the evolution of the energy density of the quintessence field, one can easily figure out the range of the parameter (\(\alpha \)) values responsible for thawing and freezing behaviour of the field. It is found that both of them are incompatible with GSL, there are situations where *S* decreases, and decreases very fast. For the freezing models, this breakdown can occur at a distant past (\(z \sim 10^{4}\)) where a quintessence model along with a CDM does not account for the evolution, one has to have dominant contribution from a radiation distribution. So this breakdown of GSL may not be real. For the thawing models, however, this pathological breakdown of GSL is in a finite future. Thus the major indication is that the freezing models are favoured compared to the thawing ones on the considerations of thermodynamic viability.

## Notes

### Acknowledgements

Tanima Duary wants to acknowledge CSIR for funding this project. She also wishes to thank her colleagues for lively discussions.

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