Thermodynamic Behavior of Friedmann Equation at Apparent Horizon of FRW Universe

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), $W=(\ddot{I}-P)/2$ is the work density and $\ddot{I}$ 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 $T=\hat{I}º/2\ddot{I}€$ in Einstein general relativity, where $A$ is the area of apparent horizon and $\hat{I}º$ 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.

v4: Revtex, 18 pages, to appear in PRD