Solutions of the wave equation bounded at the Big Bang

Abstract

By solving a singular initial value problem, we prove the existence of solutions of the wave equation ${\square }_g\varphi =0$ which are bounded at the Big Bang in the Friedmann-Lemaitre-Robertson-Walker cosmological models. More precisely, we show that given any function $A\in H^3(\Sigma )$ (where $\Sigma ={\mathbb{R}}^n,{\mathbb{S}}^n$ or ${\mathbb{H}}^n$ models the spatial hypersurfaces) there exists a unique solution $\varphi$ of the wave equation converging to $A$ in $H^1(\Sigma )$ at the Big Bang, and whose time derivative is suitably controlled in $L^2(\Sigma )$.

11 pages, 1 figure; v2: minor changes; v3: references updated, matches final published version