Effective Pressure of the FRW Universe

Abstract

In this paper, we study the effective pressure of the $N$-dimensional FRW(Friedmann–Robertson– Walker) universe in Einstein gravity, Gauss–Bonnet gravity, and Lovelock gravity. The effective pressure is defined by $P_{eff}≔-dE/dV$, where $E=\rho V$ is the effective energy and $V$ is the volume of the FRW universe inside the apparent horizon. The effective pressure in Einstein gravity is always negative and its absolute value decreases with the horizon radius $R_A$. The effective pressure in Gauss–Bonnet gravity is different with the one in Einstein gravity only when $N\ge 6$, and in this case, the effective pressure is not always negative and has a minimum, which is very similar to the potential between molecules. The effective pressure in Lovelock gravity can have multiple zero-points and extreme points. The effective pressure in different dimensions has interesting relations. We also find that under certain condition, the effective pressure is equivalent with the ‘ordinary’ pressure $p$ of the perfect fluid, and this condition do not depend on the specific choice of gravitational theories.