Thermodynamic Behavior of Friedmann Equation at Apparent Horizon of FRW Universe

TL;DR

This paper demonstrates that Friedmann equations at the apparent horizon of FRW universes can be expressed in a thermodynamic form, linking gravitational dynamics with horizon thermodynamics in Einstein, Gauss-Bonnet, and Lovelock gravities.

Contribution

It shows the universal thermodynamic form of Friedmann equations at the apparent horizon across multiple gravity theories, extending previous results in Einstein gravity.

Findings

Friedmann equations can be rewritten as $dE = TdS + WdV$ at the apparent horizon.

Entropy at the apparent horizon matches black hole entropy formulas.

The thermodynamic description extends to Gauss-Bonnet and Lovelock gravities.

Abstract

It is shown that the differential form of Friedmann equation of a FRW universe can be rewritten as a universal form $dE = TdS + WdV$ at apparent horizon, where $E$ and $V$ are the matter energy and volume inside the apparent horizon (the energy $E$ is the same as the Misner-Sharp energy in the case of Einstein general relativity), $W=(\rho-P)/2$ is the work density and $\rho$ and $P$ are energy density and pressure of the matter in the universe, respectively. From the thermodynamic identity one can derive that the apparent horizon has associated entropy $S= A/4G$ and temperature $T = \kappa / 2\pi$ in Einstein general relativity, where $A$ is the area of apparent horizon and $\kappa$ is the surface gravity at apparent horizon. We extend our procedure to the Gauss-Bonnet gravity and more general Lovelock gravity and show that the differential form of Friedmann equations in these gravities can also be rewritten to thee universal form $dE = TdS + WdV$ at the apparent horizon with entropy $S$ being given by expression previously known via black hole thermodynamics.