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

We obtain explicit formulas for the solution of the wave equation, as a function of the initial data, 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. Moreover, we use these formulas to obtain the decay rates for the solutions of the wave equation, formulate a conjecture for the decay rates in a certain family of flat FLRW spacetimes, and prove this conjecture for a large subfamily.