Branes, moduli spaces and smooth transition from big crunch to big bang

TL;DR

This paper demonstrates a mathematical framework where branes in a Schwarzschild-AdS bulk can smoothly transition from a big crunch to a big bang, using a transformed scalar field equation that remains regular across the singularity.

Contribution

It introduces a method to reformulate scalar field equations on branes, enabling smooth solutions across singularities and showing a possible geometric and physical transition from big crunch to big bang.

Findings

Smooth solutions across the singularity at r=0

Existence of a smooth hypersurface connecting big crunch and big bang

Mathematical model supporting a smooth cosmological transition

Abstract

We consider branes $N$ in a Schwarzschild-$\text{AdS}_{(n+2)}$ bulk, where the stress energy tensor is dominated by the energy density of a scalar fields map $\f:N\ra \mc S$ with potential $V$, where $\mc S$ is a semi-Riemannian moduli space. By transforming the field equation appropriately, we get an equivalent field equation that is smooth across the singularity $r=0$, and which has smooth and uniquely determined solutions which exist across the singularity in an interval $(-\e,\e)$. Restricting a solution to $(-\e,0)$ \resp $(0,\e)$, and assuming $n$ odd, we obtain branes $N$ \resp $\hat N$ which together form a smooth hypersurface. Thus a smooth transition from big crunch to big bang is possible both geometrically as well as physically.