Solutions of the wave equation bounded at the Big Bang

TL;DR

This paper proves the existence of solutions to the wave equation in cosmological models that remain bounded at the Big Bang, using a singular initial value problem approach.

Contribution

It introduces a method to construct solutions of the wave equation that are bounded at the Big Bang in FLRW models, extending previous understanding of wave behavior near singularities.

Findings

Existence of solutions bounded at the Big Bang.

Unique solutions converging to prescribed initial data.

Control over the solutions' derivatives in L^2 norm.

Abstract

By solving a singular initial value problem, we prove the existence of solutions of the wave equation $\Box_g\phi=0$ 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 $\phi$ of the wave equation converging to $A$ in $H^1(\Sigma)$ at the Big Bang, and whose time derivative is suitably controlled in $L^2(\Sigma)$.