Brane cosmological evolution in a bulk with cosmological constant

TL;DR

This paper derives a general cosmological evolution equation for a 3-brane universe in a five-dimensional bulk with a cosmological constant, allowing analysis of various matter equations of state without explicit dependence on the fifth dimension.

Contribution

It provides a first integral of Einstein's equations for brane cosmology, enabling the study of brane evolution under arbitrary matter content without metric dependence on the extra dimension.

Findings

Standard cosmological evolution can be recovered after an early phase.

A small deviation from perfect bulk-brane tension balance induces an effective late-time cosmological constant.

Explicit solutions are given for the case of constant equation of state w.

Abstract

We consider the cosmology of a ``3-brane universe'' in a five dimensional (bulk) space-time with a cosmological constant. We show that Einstein's equations admit a first integral, analogous to the first Friedmann equation, which governs the evolution of the metric in the brane, whatever the time evolution of the metric along the fifth dimension. We thus obtain the cosmological evolution in the brane for any equation of state describing the matter in the brane, without needing the dependence of the metric on the fifth dimension. In the particular case $p = w \rho$, $(w = constant)$, we give explicit expressions for the time evolution of the brane scale factor, which show that standard cosmological evolution can be obtained (after an early non conventional phase) in a scenario \`a la Randall and Sundrum, where a brane tension compensates the bulk cosmological constant. We also show that a tiny deviation from exact compensation leads to an effective cosmological constant at late time. Moreover, when the metric along the fifth dimension is static, we are able to extend the solution found on the brane to the whole spacetime.