Entropy and universality of Cardy-Verlinde formula in dark energy universe

TL;DR

The paper investigates entropy and Cardy-Verlinde (CV) universality in dark-energy FRW cosmologies, deriving general entropy expressions for a broad class of equations of state $p=w\rho$ and showing that a universal cosmological CV relation among holographic entropies holds for any matter content. It further extends these ideas to $f(R)$-modified gravity, demonstrating that the CV structure remains frame-independent between the Jordan and Einstein frames, and shows that black-hole entropy is enhanced in 1/$R$ gravity by a calculable factor, consistent with a modified CV description. Through hydrodynamical tests including a Kasner universe, the work probes the holographic bound on $\eta/s$ and finds potential violations in anisotropic settings, indicating the bound is not universal. Altogether, the results reinforce the holographic origin of cosmological entropy while highlighting the dependence of gravity-sector details on BH thermodynamics and the limits of universal viscosity bounds in early- universe contexts.

Abstract

We study the entropy of a FRW universe filled with dark energy (cosmological constant, quintessence or phantom). For general or time-dependent equation of state $p=wρ$ the entropy is expressed in terms of energy, Casimir energy, and $w$. The correspondent expression reminds one about 2d CFT entropy only for conformal matter. At the same time, the cosmological Cardy-Verlinde formula relating three typical FRW universe entropies remains to be universal for any type of matter. The same conclusions hold in modified gravity which represents gravitational alternative for dark energy and which contains terms growing at low curvature. It is interesting that BHs in modified gravity are more entropic than in Einstein gravity. Finally, some hydrodynamical examples testing new shear viscosity bound, which is expected to be the consequence of the holographic entropy bound, are presented for the early universe in the plasma era and for the Kasner metric. It seems that the Kasner metric provides a counterexample to the new shear viscosity bound.