Corrected Entropy-Area Relation and Modified Friedmann Equations

TL;DR

This paper derives modified Friedmann equations from quantum-corrected entropy-area relations of the apparent horizon in FRW universes, revealing a positive logarithmic correction inconsistent with typical quantum geometry results.

Contribution

It introduces a new derivation of modified Friedmann equations incorporating quantum entropy corrections, highlighting a discrepancy in the sign of the logarithmic term.

Findings

Derived entropy expression matches corrected entropy-area relation at large horizon area.

The modified Friedmann equations do not predict a bounce solution.

The logarithmic correction coefficient is positive, conflicting with standard quantum gravity expectations.

Abstract

Applying Clausius relation, $\delta Q=TdS$, to apparent horizon of a FRW universe with any spatial curvature, and assuming that the apparent horizon has temperature $T=1/(2\pi \tilde {r}_A)$, and a quantum corrected entropy-area relation, $S=A/4G +\alpha \ln A/4G$, where $\tilde {r}_A$ and $A$ are the apparent horizon radius and area, respectively, and $\alpha$ is a dimensionless constant, we derive modified Friedmann equations, which does not contain a bounce solution. On the other hand, loop quantum cosmology leads to a modified Friedmann equation $H^2 =\frac{8\pi G}{3}\rho (1-\rho/\rho_{\rm crit})$. We obtain an entropy expression of apparent horizon of FRW universe described by the modified Friedmann equation. In the limit of large horizon area, resulting entropy expression gives the above corrected entropy-area relation, however, the prefactor $\alpha$ in the logarithmic term is positive, which seems not consistent with most of results in the literature that quantum geometry leads to a negative contribution to the area formula of black hole entropy.