Transition from big crunch to big bang in brane cosmology

TL;DR

This paper explores a mathematical model in brane cosmology where a universe with a big crunch singularity can smoothly transition into a big bang, using specific geometric and topological conditions in higher-dimensional bulk spaces.

Contribution

It introduces a framework for a smooth transition from big crunch to big bang in brane cosmology, extending previous models to include non-ARW spaces and broader equations of state.

Findings

Existence of smooth transition flow through singularity under certain conditions.

Construction of reflected branes with big bang singularity.

Embedding of joint branes as smooth hypersurfaces in higher-dimensional space.

Abstract

We consider branes $N=I\times\so$, where $\so$ is an $n$\ndash dimensional space form, not necessarily compact, in a Schwarzschild-AdS_{(n+2)} bulk $\mc N$. 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 $\hat{\mc N}$. The joint branes $N\uu \hat N$ can thus be naturally embedded in $R^2\times \so$, hence there exists a second possibility of defining a smooth transition from big crunch to big bang by requiring that $N\uu\hat N$ forms a $C^\infty$-hypersurface in $R^2\times \so$. 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.