Thermodynamics of the apparent horizon in massive cosmology

TL;DR

This paper explores the thermodynamics of the apparent horizon in a massive cosmological model, deriving a corrected entropy formula, verifying the first law, and analyzing the conditions under which the generalized second law holds.

Contribution

It introduces a modified entropy-area relation for the apparent horizon in massive cosmology and links it with the Friedmann equations and thermodynamic laws.

Findings

Corrected entropy formula derived for the apparent horizon.

First law of thermodynamics verified with volume change term.

Generalized second law holds under specific parameter conditions.

Abstract

Applying Clausius relation with energy-supply defined by the unified first law of thermodynamics formalism to the apparent horizon of a massive cosmological model proposed lately, the corrected entropic formula of the apparent horizon is obtained with the help of the modified Friedmann equations. This entropy-area relation, together with the identified internal energy, verifies the first law of thermodynamics for the apparent horizon with a volume change term for consistency. On the other hand, by means of the corrected entropy-area formula and the Clausius relation $\delta Q=T dS$, the modified Friedmann equations governing the dynamical evolution of the universe are reproduced with the known energy density and pressure of massive graviton. The integration constant is found to correspond to a cosmological term which could be absorbed into the energy density of matter. Having established the correspondence of massive cosmology with the unified first law of thermodynamics on the apparent horizon, the validity of the generalized second law of thermodynamics is also discussed by assuming the thermal equilibrium between the apparent horizon and the matter field bounded by the apparent horizon. It is found that, in the limit $H_c\rightarrow 0$ which recovers the Minkowski reference metric solution in the flat case, the generalized second law of thermodynamics holds if $\alpha_3+4\alpha_4<0$. Apart from that, even for the simplest model of dRGT massive cosmology with $\alpha_3=\alpha_4=0$, the generalized second law of thermodynamics could be violated.