Equilibrium thermodynamics in modified gravitational theories

TL;DR

This paper demonstrates that a consistent equilibrium thermodynamics framework can be established on the apparent horizon in a broad class of modified gravity theories, aligning horizon entropy with the area law and analyzing its evolution.

Contribution

It introduces a method to define an energy-momentum tensor that respects local conservation, enabling equilibrium thermodynamics in modified gravity theories with scalar fields.

Findings

Horizon entropy equals quarter of the area in these theories.

In flat backgrounds, entropy increases over time for viable models.

Equilibrium thermodynamics simplifies analysis by incorporating entropy production.

Abstract

We show that it is possible to obtain a picture of equilibrium thermodynamics on the apparent horizon in the expanding cosmological background for a wide class of modified gravity theories with the Lagrangian density $f(R, \phi, X)$, where $R$ is the Ricci scalar and $X$ is the kinetic energy of a scalar field $\phi$. This comes from a suitable definition of an energy momentum tensor of the "dark" component that respects to a local energy conservation in the Jordan frame. In this framework the horizon entropy $S$ corresponding to equilibrium thermodynamics is equal to a quarter of the horizon area $A$ in units of gravitational constant $G$, as in Einstein gravity. For a flat cosmological background with a decreasing Hubble parameter, $S$ globally increases with time, as it happens for viable $f(R)$ inflation and dark energy models. We also show that the equilibrium description in terms of the horizon entropy $S$ is convenient because it takes into account the contribution of both the horizon entropy $\hat{S}$ in non-equilibrium thermodynamics and an entropy production term.