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The decay of solutions to the Klein-Gordon equation is studied in twoexpanding cosmological spacetimes, namely the de Sitter universe in flatFriedmann-Lema\^{i}tre-Robertson-Walker (FLRW) form, and the cosmologicalregion of the Reissner-Nordstr\"om-de Sitter (RNdS) model. Using energymethods, for initial data with finite higher order energies, decay rates forthe solution are obtained. Also, a previously established decay rate of thetime derivative of the solution to the wave equation, in an expanding de Sitteruniverse in flat FLRW form, is improved, proving Rendall's conjecture. Asimilar improvement is also given for the wave equation in the cosmologicalregion of the RNdS spacetime.

Decay of solutions to the Klein-Gordon equation on some expanding cosmological spacetimes