Eur. Phys. J. C 69, 563-574 (2010)
https://doi.org/10.1140/epjc/s10052-010-1425-0

Cardy–Verlinde formula in FRW Universe with inhomogeneous generalized fluid and dynamical entropy bounds near the future singularity

¹ Department of Energy and Process Engineering, Norwegian University of Science and Technology, 7491, Trondheim, Norway
² Department of Physics, Nagoya University, Nagoya, 464-8602, Japan
³ Instituciò Catalana de Recerca i Estudis Avançats (ICREA), L. Companys 23, 08010, Barcelona, Spain
⁴ Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB, Facultat de Ciencies, Torre C5-Par-2a pl, 08193, Bellaterra (Barcelona), Spain

iver.h.brevik@ntnu.no

Received: 16 July 2010
Published online: 7 September 2010

Abstract

We derive a formula for the entropy for a multicomponent coupled fluid, which under special conditions reduces to the Cardy–Verlinde form relating the entropy of a closed FRW universe to its energy together with its Casimir energy. The generalized fluid obeys an inhomogeneous equation of state. A viscous dark fluid is included, and also modified gravity is included in terms of its fluid representation. It is demonstrated how such an expression reduces to the standard Cardy–Verlinde formula corresponding to the 2d CFT entropy in some special cases. The dynamical entropy bound for a closed FRW universe with dark components is obtained. The universality of the dynamical entropy bound near a future singularity (of all known four types), as well as near the Big Bang singularity, is investigated. It is demonstrated that, except from some special cases of Type II and Type IV singularities, the dynamical entropy bound is violated near the singularity even if quantum effects are taken into account. The dynamical entropy bound seems to be universal for the case of a regular universe, including the asymptotic de Sitter universe.