Brane cosmological evolution in a bulk with cosmological constant
Pierre Binetruy, Cedric Deffayet, Ulrich Ellwanger, David Langlois
TL;DR
This work builds a bridge between 5D brane-world cosmology and standard 4D cosmology by deriving a first integral of the 5D Einstein equations that governs brane evolution independent of the fifth dimension. It shows how a Randall–Sundrum–type tuning between brane tension and bulk cosmological constant can reproduce standard cosmology at low energies, while a small mismatch yields an effective late-time cosmological constant. The authors also obtain explicit 5D bulk solutions for a stabilized extra dimension and provide analytic brane-scale factor evolution for common equations of state, highlighting transitions from non-conventional to conventional expansion and potential late-time acceleration. They further discuss constraints from nucleosynthesis and demonstrate how a C-term behaves like extra radiation, constraining the model's parameters.
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 $p = w ρ$, $(w = constant)$, 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 à 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.