Transition between phantom and non-phantom phases with time dependent cosmological constant and Cardy-Verlinde formula

TL;DR

This paper explores the transition between phantom and non-phantom phases of the universe with a time-dependent cosmological constant, analyzing singularities and the applicability of the Cardy-Verlinde formula under inhomogeneous equations of state.

Contribution

It introduces a model with a variable cosmological constant proportional to the square of the Hubble parameter and examines the resulting phase transitions and entropy relations.

Findings

Universe can transition from non-phantom to phantom phase leading to singularities.

Phantom universe ends in a type III singularity, unlike the big rip in constant cosmological constant models.

Cardy-Verlinde formula matches the universe's entropy only for specific parameter choices.

Abstract

We investigate the transition phenomenon of the universe between a phantom and a non-phantom phases. Particular attention is devoted to the case in which the cosmological constant depends on time and is proportional to the square of the Hubble parameter. Inhomogeneous equations of state are used and the equation of motion is solved. We find that, depending on the choice of the input parameters, the universe can transit from the non-phantom to the phantom phase leading to the appearance of singularities. In particular, we find that the phantom universe ends in the singularity of type III, unlike the case without variable cosmological constant in which the phantom phase ends exclusively in the big rip (singularity of type I). The Cardy-Verlinde formula is also introduced for inhomogeneous equation of state and we find that its equivalence with the total entropy of the universe, coming from the Friedmann equations, occurs only for special choice of the input parameter $m$ at the present time.