First Law of Thermodynamics and Friedmann Equations of Friedmann-Robertson-Walker Universe

TL;DR

The paper addresses deriving the Friedmann equations for FRW cosmologies with arbitrary spatial curvature from the first law of thermodynamics applied to the apparent horizon. It first recovers the standard Einstein gravity relations using $S = \frac{A}{4G}$ and $T = \frac{1}{2\pi \tilde{r}_A}$, then extends the thermodynamic derivation to Gauss-Bonnet and Lovelock gravities by employing their horizon entropy expressions, yielding modified Friedmann relations. The resulting equations are $\dot H - \frac{k}{a^2} = -\frac{8\pi G}{n-1}(\rho+p)$ and $H^2 + \frac{k}{a^2} = \frac{16\pi G}{n(n-1)} \rho$ in Einstein gravity; $\left(1+2\tilde{\alpha}(H^2+ \frac{k}{a^2})\right)(\dot H- \frac{k}{a^2}) = -\frac{8\pi G}{n-1}(\rho+p)$ in Gauss-Bonnet gravity, together with $H^2+\frac{k}{a^2}+\tilde{\alpha}(H^2+\frac{k}{a^2})^2 = \frac{16\pi G}{n(n-1)} \rho$; and a Lovelock polynomial relation $\sum_{i=1}^m \hat{c}_i (H^2+\frac{k}{a^2})^i = \frac{16\pi G}{n(n-1)} \rho$ with a differential form $\sum_{i=1}^m i\hat{c}_i (H^2+\frac{k}{a^2})^{i-1}(\dot H-\frac{k}{a^2}) = -\frac{8\pi G}{n-1}(\rho+p)$. These results reinforce a thermodynamic/holographic perspective on cosmological dynamics and motivate further exploration of horizon thermodynamics in broader gravity theories.

Abstract

Applying the first law of thermodynamics to the apparent horizon of a Friedmann-Robertson-Walker universe and assuming the geometric entropy given by a quarter of the apparent horizon area, we derive the Friedmann equations describing the dynamics of the universe with any spatial curvature. Using entropy formulae for the static spherically symmetric black hole horizons in Gauss-Bonnet gravity and in more general Lovelock gravity, where the entropy is not proportional to the horizon area, we are also able to obtain the Friedmann equations in each gravity theory. We also discuss some physical implications of our results.