Kaluza-Klein bulk viscous fluid cosmological models and the validity of the second law of thermodynamics in $f(R, T)$ gravity

TL;DR

This paper explores Kaluza-Klein cosmological models with bulk viscous fluids in $f(R,T)$ gravity, analyzing thermodynamic laws, and finds conditions under which the second law of thermodynamics holds or is violated.

Contribution

It provides exact solutions for viscous fluid cosmological models in $f(R,T)$ gravity and examines the validity of thermodynamic laws in these models.

Findings

Second law of thermodynamics is violated when viscous coefficient is proportional to $rac{ ext{d}^2a/ ext{d}t^2}{ ext{d}a/ ext{d}t$.

Second law and generalized second law are satisfied when viscous coefficient is proportional to $rac{ ext{d}a}{a}$ or combined forms.

Statefinder parameters differ from $ ext{w}$CDM model, indicating distinct cosmological evolution.

Abstract

The authors considered the bulk viscous fluid in $f(R, T)$ gravity within the framework of Kaluza-Klein space time. The bulk viscous coefficient $(\xi)$ expressed as $\xi=\xi_0+\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$, where $\xi_0$, $\xi_1$ and $\xi_2$ are positive constants. We take $p=(\gamma-1)\rho$, where $0\le\gamma\le2$ as an equation of state for perfect fluid. The exact solutions to the corresponding field equations are given by assuming a particular model of the form of $f(R, T)=R+2f(T)$, where $f(T)=\lambda T$, $\lambda$ is constant. We studied the cosmological model in two stages, in first stage: we studied the model with no viscosity, and in second stage: we studied the model involve with viscosity. The cosmological model involve with viscosity is studied by five possible scenarios for bulk viscous fluid coefficient $(\xi)$. The total bulk viscous coefficient seems to be negative, when the bulk viscous coefficient is proportional to $\xi_2\frac{\ddot{a}}{\dot{a}}$, hence the second law of thermodynamics is not valid, however, it is valid with the generalized second law of thermodynamics. The total bulk viscous coefficient seems to be positive, when, the bulk viscous coefficient is proportional to $\xi=\xi_1\frac{\dot{a}}{a}$, $\xi=\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$ and $\xi=\xi_0+\xi_1\frac{\dot{a}}{a}+\xi_2\frac{\ddot{a}}{\dot{a}}$, so the second law of thermodynamics and the generalized second law of thermodynamics is satisfied throughout the evolution. We calculate statefinder parameters of the model and observed that, it is different from the $\wedge$CDM model. Finally, some physical and geometrical properties of the models are discussed.