Unified First Law and Thermodynamics of Apparent Horizon in FRW Universe

TL;DR

This paper explores the thermodynamic properties of the apparent horizon in FRW universe across different gravity theories, confirming the unified first law's applicability in Einstein and Lovelock theories but not in scalar-tensor gravity, indicating non-equilibrium thermodynamics.

Contribution

It demonstrates the applicability of the unified first law to the apparent horizon in various gravity theories, highlighting differences in thermodynamic behavior, especially in scalar-tensor gravity.

Findings

First law holds in Einstein gravity at the apparent horizon.

Lovelock theory yields the same entropy formula as black holes.

Scalar-tensor gravity shows non-equilibrium thermodynamics at the horizon.

Abstract

In this paper we revisit the relation between the Friedmann equations and the first law of thermodynamics. We find that the unified first law firstly proposed by Hayward to treat the "outer"trapping horizon of dynamical black hole can be used to the apparent horizon (a kind of "inner" trapping horizon in the context of the FRW cosmology) of the FRW universe. We discuss three kinds of gravity theorties: Einstein theory, Lovelock thoery and scalar-tensor theory. In Einstein theory, the first law of thermodynamics is always satisfied on the apparent horizon. In Lovelock theory, treating the higher derivative terms as an effective energy-momentum tensor, we find that this method can give the same entropy formula for the apparent horizon as that of black hole horizon. This implies that the Clausius relation holds for the Lovelock theory. In scalar-tensor gravity, we find, by using the same procedure, the Clausius relation no longer holds. This indicates that the apparent horizon of FRW universe in the scalar-tensor gravity corresponds to a system of non-equilibrium thermodynamics. We show this point by using the method developed recently by Eling {\it et al.} for dealing with the $f(R)$ gravity.