Explicit formulas and decay rates for the solution of the wave equation in cosmological spacetimes

TL;DR

This paper derives explicit solutions and decay rate formulas for the wave equation in various cosmological spacetimes, extending previous results and analyzing their properties related to Huygens principle.

Contribution

It introduces a method to transform wave solutions in FLRW spacetimes to simpler models, providing new explicit formulas and decay rate insights for multiple cosmological geometries.

Findings

Explicit formulas for wave solutions in FLRW spacetimes

Decay rate analysis and conjectures for flat and hyperbolic cases

Extension of previous formulas to new cosmological models

Abstract

We obtain explicit formulas for the solution of the wave equation in certain Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes. Our method, pioneered by Klainerman and Sarnak, consists in finding differential operators that map solutions of the wave equation in these FLRW spacetimes to solutions of the conformally invariant wave equation in simpler, ultra-static spacetimes, for which spherical mean formulas are available. In addition to recovering the formulas for the dust-filled flat and hyperbolic FLRW spacetimes originally derived by Klainerman and Sarnak, and generalizing them to the spherical case, we obtain new formulas for the radiation-filled FLRW spacetimes and also for the de Sitter, anti-de Sitter and Milne universes. We use these formulas to study the solutions with respect to the Huygens principle and the decay rates, and to formulate conjectures about the general decay rates in flat and hyperbolic FLRW spacetimes. The positive resolution of the conjecture in the flat case is seen to follow from known results in the literature.