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

TL;DR

This paper investigates how solutions to the Klein-Gordon equation decay over time in specific expanding cosmological spacetimes, providing decay rates and improving previous results, including Rendall's conjecture.

Contribution

The paper derives decay rates for Klein-Gordon solutions in de Sitter and RNdS spacetimes and improves existing decay estimates for wave equations, confirming Rendall's conjecture.

Findings

Decay rates for Klein-Gordon solutions in de Sitter and RNdS spacetimes

Improved decay estimates for wave equations in these spacetimes

Confirmation of Rendall's conjecture for the wave equation in de Sitter universe

Abstract

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