Linear waves on the expanding region of Schwarzschild-de Sitter spacetimes: forward asymptotics and scattering from infinity

Abstract: We study solutions to the linear wave equation on the cosmological region of Schwarzschild-de Sitter spacetimes. We show that all sufficiently regular finite-energy solutions to the linear equation possess a particular finite-order asymptotic expansion near the future boundary. Specifically, we prove that several terms in this asymptotic expansion are identically zero. This is accomplished with new weighted higher-order energy estimates that capture the global expansion of the cosmological region. Furthermore we prove existence and uniqueness of scattering solutions to the linear wave equation on the expanding region. Given two pieces of scattering data at infinity, we construct solutions that have the same asymptotics as forward solutions. The proof involves constructing asymptotic solutions to the wave equation, as well as a new weighted energy estimate that is suitable for the backward problem. This scattering result extends to a large class of expanding spacetimes, including the Kerr de Sitter family.