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

Abstract

We study the entropy of a FRW universe filled with dark energy (cosmological constant, quintessence, or phantom). For the general or time-dependent equation of state $p=w\ensuremath{\rho}$ the entropy is expressed in terms of energy, Casimir energy, and w. The corresponding expression is reminiscent of the 2D conformal field theory (CFT) entropy only for conformal matter. At the same time, the cosmological Cardy-Verlinde formula relating three typical FRW universe entropies remains universal for any type of matter. The same conclusions hold in modified gravity, which represents the gravitational alternative for dark energy and which contains terms that increase at low curvature. It is interesting that black holes in modified gravity are more entropic than those in Einstein gravity. Finally, some hydrodynamical examples testing the 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.