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 g φ = 0 which are bounded at the Big Bang in the Friedmann-Lemaître-Robertson-Walker cosmological models. More precisely, we show that given any function A ∈ H 3 (Σ) (where Σ = R n , S n or H n models the spatial hypersurfaces) there exists a unique solution φ of the wave equation converging to A in H 1 (Σ) at the Big Bang, and whose time derivative is suitably controlled in L 2 (Σ).

Wave equations on silent big bang backgrounds

“…The subject of these notes is linear systems of wave equations on cosmological backgrounds. There are several previous results on this topic; cf., e.g., [2,34,48,46,1,20,6,45] and references cited therein. As far as the study of the singularity is concerned, the assumptions made in these notes are less restrictive than the ones made in most of these references.…”

Blow-up of waves on singular spacetimes with generic spatial metrics

“…As shown in [10], it does not even hold in the FLRW setting that any wave exhibits blow-up towards the Big Bang hypersurface. To furthermore show that the blow-up of highest possible order is actually generic, we will establish an open condition on the initial data (ψ(t 0 , •), ∂ t ψ(t 0 , •)) on a hypersurface M t 0 = {t 0 } × M for small enough t 0 > 0 such that A does not become zero pointwise, see Theorem 14 and Corollary 15.…”

Decay of solutions of the wave equation in cosmological spacetimes -- a numerical analysis

“…It has been settled, in particular, that the decay of the energy associated to waves, caused by the redshift effect, is only one of the many factors that influence the propagation, together for instance with the non-compactness of the space sections and the dispersive properties of the background spacetime. While we are interested in the asymptotics of solutions for large times, several approaches to study their behaviour near the Big Bang singularity have also been developed in the last years [2,7,9,15]. We will often refer to the companion paper [13], where explicit expressions and decay rates for the solutions of (1) in a class of FLRW spacetimes are provided by use of analytical techniques.…”