# Universe consisting of diffusive dark fluids: thermodynamics and stability analysis

## Abstract

The present work deals with homogeneous and isotropic FLRW model of the Universe having a system of non-interacting diffusive cosmic fluids with barotropic equation of state (constant or variable equation of state parameter). Due to diffusive nature of the cosmic fluids, the divergence of the energy momentum tensor is chosen to be proportional to the diffusive current. The thermodynamic stability analysis of individual fluids is done and the stability conditions are expressed as restrictions on the equation of state parameter.

## 1 Introduction

The greatest challenge of standard cosmology today is to accommodate the recent observational predictions [1, 2, 3, 4, 5, 6, 7, 8, 9]. The modern cosmology is facing the challenging issue of explaining the present accelerated expansion of the universe. In the framework of Einstein gravity, cosmologists are speculating some hypothetical exotic matter (known as dark energy (DE) having large negative pressure) to explain this accelerating phase. It is estimated [10] that about 70% of the cosmic fluid consists of this unknown DE component. The simplest as well as the common candidate for this dark fluid is the cosmological constant (the zero point energy of the quantum fields). Although a large number of available observational data are in support of this cosmological parameter as a DE candidate but there are severe drawback of it at the interface of Cosmology and Particle Physics: the cosmological constant problem [11, 12] and the coincidence problem [13]. As a result there are several alternative proposed dynamical DE models into the picture. These DE models have been studied for the last several years [14], yet the cosmological constant is still the best observationally supported DE candidate. However, these different dynamic DE models cannot be compared from observational view point as these models try to adjust the data seamlessly. As a result, cosmologists have been trying with interacting DE model to have a better understanding of the mechanism of this cosmic acceleration.

From recent past, interacting dark fluid models have been receiving much attention as they can provide small value of the cosmological constant and due to their ability of explaining the cosmic coincidence problem [15, 16, 17]. Moreover, recent observed data [18, 19, 20, 21, 22, 23] favor interacting dark fluid models and it is possible to have an estimate of the coupling parameter in the interaction term by various observations [21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Further,the cosmologists are of the opinion that these interacting dark fluid models may have the solution to the current tensions on \(\sigma _8\) and the local value of the Hubble constant \((H_0)\) [22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Moreover, it is speculated that cosmological perturbation analysis may be affected by the interaction terms and consequently, the lowest multipoles of the CMB spectrum [32, 33] should have an imprint of it.

On the other hand, the unknown nature of DE may have some clue from the thermodynamic laws which are applicable to all types of macroscopic systems and are based on experimental evidence. However, unlike classical mechanics or electromagnetism, thermodynamical analysis can not predict any definite value for observables, it may only give limit on physical processes. So it is reasonable to believe that thermodynamical study of dark cosmic fluid may indicate some unknown character of it. Investigation in this direction has been initiated recently by Barboza et al. [34]. But their stability conditions demand that the DE should have constant (−ve) equation of state and it is not supported by observation.

Subsequently, Chakraborty and collaborators [35, 36, 37, 38] in a series of works have shown the stability criteria for different type of dark fluids and have presented the stability conditions (in tabular form) for different ranges of the equation of state parameter. The present work is an extension of the interacting dark fluid model. Here a particular realization of the non-conservation of the energy momentum tensor is used with a diffusion of dark matter in a fluid of dark energy. The change of the energy-density is proportional to the particle density. The resulting non-equilibrium thermodynamics is studied and stability conditions are explicitly determined. The paper is organized as follows: Sect. 2 deals with thermodynamical analysis of non-interacting cosmic fluids with constant equation of state . The stability criteria for non-interacting cosmic fluids with variable equation of state has been organized in Sect. 3 . Finally, the paper ends with a brief discussion and the conclusion from the result in dark energy range in Sect. 4.

## 2 Thermodynamic analysis of non-interacting diffusive cosmic fluids having constant equation of state parameter

*a*(

*t*) is called the scale factor of the universe as all physical distance is scaled with same factor “

*a*” due to homogeneity and isotropy of space-time. Now due to the diffusive nature of the fluids, they do not obey the matter conservation equation (\({T_i^{\mu \nu }}_{;\mu }= 0\)), rather they follow,

*k*is a constant. But for the whole universe,the total stress-energy tensor is conserved i.e. ( \(\sum _i{{T_i^{\mu \nu }}_{;\mu }= 0}\))

*t*and at present time \(t_0\),the physical volume is \(v_0\) with \(a(t_0)=1\). According to the 1st law of thermodynamics (which is an energy conservation equation),

*v*) and temperature (

*T*) and also considering \(d S_i\) to be an exact differential, one obtains from equation (8),

### 2.1 Determination of thermodynamic derivatives: \(C_p,C_v,K_S,K_T\)

Now, the motivation of the present work is to find the criteria of thermodynamic stability of the universe . So the conditions which may be satisfied by the parameters, indicate the thermodynamic features of the evolution of the expanding universe. For any system the thermodynamic derivatives i.e. the heat capacities,compressibility and isobaric expansibility determine whether the system is stable or not. So it will be very important to analyze these parameters for each fluid in the present context.

### 2.2 Stability conditions for cosmic fluids

Conditions for stability

Range of \(\omega _i \) | Stability condition |
---|---|

\(\omega _i \ge 0\) | \( \left[ \frac{\partial \left\{ \ln \left( 1+\frac{\gamma _i}{\rho _{i 0}}A _i \right) \right\} }{\partial \ln T} \right] _{p_i} \ge \frac{1+\omega _i}{\omega _i } \) |

\(-1\le \omega _i \le 0\) | \( -(1+\omega _i) \le \left[ \frac{\partial \left\{ \ln \left( 1+\frac{\gamma _i}{\rho _{i 0}}A _i \right) \right\} }{\partial \ln T} \right] _{p_i} \text{ and } ~~~~\left[ \frac{\partial \left\{ \ln \left( 1+\frac{\gamma _i}{\rho _{i 0}}A _i \right) \right\} }{\partial \ln T} \right] _{p_i} \le \frac{1+\omega _i}{\omega _i } ~~\text{ simultaneously, }~~~~ \text{ which } \text{ is } \text{ not } \text{ possible } \text{ hence } \text{ unstable } \) |

\(-2 \le \omega _i \le -1\) | Unstable |

\(\omega _i \le -2 \) | \(\left[ \frac{\partial \left\{ \ln \left( 1+\frac{\gamma _i}{\rho _{i 0}}A _i \right) \right\} }{\partial \ln T} \right] _{p_i} \ge -(1+\omega _i) \) |

Conditions for stability criteria

Range of \(\omega _i \) | Stability condition |
---|---|

\(\omega _i \ge 0\) | \( \frac{d\ln \omega _i}{d \ln T}\le \frac{2+\omega _i}{\omega _i}~~ \text{ and } \left[ \frac{\partial \left\{ \ln \left( 1+\frac{\gamma _i}{\rho _{i 0}}A _i \right) \right\} }{\partial \ln T} \right] _{p_i} \ge max \left[ -(1+\omega _i),\left( \frac{1+\omega _i }{\omega _i}-\frac{d \ln \omega _i}{d \ln T}\right) \right] \) |

\(-1\le \omega _i \le 0\) | \(\frac{d\ln \omega _i}{d \ln T}\ge \frac{2+\omega _i}{\omega _i} ~~\text{ and }~~\left[ \frac{\partial \left\{ \ln \left( 1+\frac{\gamma _i}{\rho _{i 0}}A _i \right) \right\} }{\partial \ln T} \right] _{p_i} \ge max \left[ -(1+\omega _i) ,\left( \frac{1+\omega _i }{\omega _i}-\frac{d \ln \omega _i}{d \ln T}\right) \right] ~ \text{ if }~\frac{d \ln \omega _i}{d \ln T} \le \frac{1+3\omega _i +2\omega _i^2}{\omega _i^2}. \text{ Again }-(1+\omega _i) \le \left[ \frac{\partial \left\{ \ln \left( 1+\frac{\gamma _i}{\rho _{i 0}}A _i \right) \right\} }{\partial \ln T} \right] _{p_i} \le \left( \frac{1+\omega _i }{\omega _i}-\frac{d \ln \omega _i}{d \ln T}\right) ~~ \text{ if }~~ \frac{d \ln \omega _i}{d \ln T} \ge \frac{1+3\omega _i +2\omega _i^2}{\omega _i^2}~~~~~~~~~ \) |

\(\omega _i \le -1 \) | \( \frac{d\ln \omega _i}{d \ln T}\le \frac{2+\omega _i}{\omega _i} ~~~~\text{ and } ~~~\left[ \frac{\partial \left\{ \ln \left( 1+\frac{\gamma _i}{\rho _{i 0}}A _i \right) \right\} }{\partial \ln T} \right] _{p_i} \ge max \left[ -(1+\omega _i) ,\left( \frac{1+\omega _i }{\omega _i}-\frac{d \ln \omega _i}{d \ln T}\right) \right] \) |

## 3 Stability criteria with variable equation of state parameter

### 3.1 Derivation of the relation between physical equilibrium temperature (T) and temperature at constant volume \((T_a)\)

*L*is a constant . Now combining (45) and (46) yields

### 3.2 Thermodynamic derivatives

Conclusion on the result in dark energy range (\(-0.33<\omega _i<0 \))

Nature of dark energy | Constraint on diffusion parameter \(\gamma _i\) | Restriction on \(\frac{d\ln \omega _i}{d \ln T} \) from graphs |
---|---|---|

Constant equation of state parameter | Always unstable | Always unstable |

Variable equation of state parameter | \(\left[ \frac{\partial \left\{ \ln \left( 1+\frac{\gamma _i}{\rho _{i 0}}A _i \right) \right\} }{\partial \ln T} \right] _{p_i} \ge max \left[ -(1+\omega _i) ,\left( \frac{1+\omega _i }{\omega _i}-\frac{d \ln \omega _i}{d \ln T}\right) \right] \) | \(\frac{2+\omega _i}{\omega _i} \le \frac{d \ln \omega _i}{d \ln T} \le \frac{1+3\omega _i+2\omega _i^2}{\omega _i^2}\) |

## 4 Brief discussions

A detailed thermodynamic study of non-interacting cosmic fluids having diffusive nature has been organized in the present work. As a result, individual fluid does not obey the energy conservation relation, rather the non-conservation is proportional to the corresponding diffusive current. Each component of the cosmic fluids is assumed to have barotropic equation of state with constant or variable equation of state parameter. The stability conditions are expressed in the form of inequalities for different ranges of equation of state parameter. Also, the stability conditions for both constant (Table 1) and variable (Table 2) equation of state parameter in the range \(-1<\omega _i <0\) (i.e. for dark energy region) have been presented graphically in Figs. 1 and 2 respectively. From Fig. 2, different restrictions on diffusion parameter \(\gamma _i\) are found depending on whether the values of \(\frac{d \ln \omega _i}{d \ln T}\) (i.e. relative variation of \(\omega _i\) with respect to the temperature T) belong to REGION-I or REGION-II . However due to expanding nature of the universe, the isobaric expansibility (\(\alpha _i\)) should always be positive. This condition restricts the function \(\frac{d \ln \omega _i}{d \ln T}\) only in REGION-II,which is acceptable for expanding universe (Table 3).

Further, it is interesting to note that, for constant equation of state parameter, (i.e. Table 1), the system cannot be thermodynamically stable for \(-1<\omega _i\le 0\). This result is very much similar to that of [34]. However, the present work is thermodynamically stable in phantom region in contrary with non-diffusive cosmic fluids [34]. Finally, it is found that diffusive fluid with variable equation of state parameter is thermodynamically stable for all possible values of \(\omega _i\) under some restrictions on the diffusion parameter \(\gamma _i\) and nature of variation of \(\omega _i\) with respect to temperature.

## Notes

### Acknowledgements

The authors are thankful to IUCAA, Pune, India for research facilities at Library. The author SM acknowledges UGC-JRF and PB acknowledges DST-INSPIRE (File no: IF160086) for awarding Research fellowship. Also SC acknowledges the UGC-DRS Program in the Department of Mathematics, Jadavpur University.

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