Transition from big crunch to big bang in brane cosmology

Abstract

We consider branes $N=I \times \mathcal{S}_0$, where $\mathcal{S}_0$ is an $n$-dimensional space form, not necessarily compact, in a Schwarzschild-AdS$_{(n+2)}$ bulk $\mathcal{N}$. The branes have a big crunch singularity. If a brane is an ARW space, then, under certain conditions, there exists a smooth natural transition flow through the singularity to a reflected brane $\hat{N}$, which has a big bang singularity and which can be viewed as a brane in a reflected Schwarzschild-AdS$_{(n+2)}$ bulk $\hat{\mathcal{N}}$. The joint branes $N \cup \hat{N}$ can thus be naturally embedded in $\mathbb{R}^2 \times \mathcal{S}_0$, hence there exists a second possibility of defining a smooth transition from big crunch to big bang by requiring that $N \cup \hat{N}$ forms a $C^\infty$-hypersurface in $\mathbb{R}^2 \times \mathcal{S}_0$. This last notion of a smooth transition also applies to branes that are not ARW spaces, allowing a wide range of possible equations of state.