Nonlinear Scattering Theory for Asymptotically de Sitter Vacuum Solutions in All Even Spatial Dimensions

Abstract

The purpose of this paper is to establish a definitive quantitative nonlinear scattering theory for asymptotically de Sitter solutions of the Einstein vacuum equations in $(n+1)$ dimensions with $n\geq4$ even, which are determined by small scattering data at the spacelike asymptotic boundaries $\mathscr{I}^-$ and $\mathscr{I}^+.$ The case of even spatial dimension $n$ poses significant challenges compared to its odd counterpart and was left open by the previous works in the literature. Here, scattering theory is understood to mean existence and uniqueness of scattering states, asymptotic completeness, and the existence of an invertible scattering map with quantitative control on its norm. The existence and uniqueness of scattering states imply that for any small asymptotic data there exists a unique global solution to the Einstein equations, which remains close to the de Sitter metric. Asymptotic completeness is the converse statement, showing that any such solution induces asymptotic data at $\mathscr{I}^-$ and at $\mathscr{I}^+.$ For sufficiently small asymptotic data, we construct the scattering map $\mathscr{S}$ taking data at $\mathscr{I}^-$ to data at $\mathscr{I}^+,$ and we show that the map $\mathscr{S}$ is locally invertible and locally Lipschitz at the de Sitter data, with respect to a Sobolev-type norm. The scattering map result is sharp and avoids any "derivative loss", in the sense that we measure the smallness of asymptotic data at $\mathscr{I}^-$ and $\mathscr{I}^+$ using the same Sobolev norm. The proof of the sharp result requires a detailed analysis of the Einstein equations involving a geometric Littlewood-Paley decomposition of the solution, carried out in our companion paper.