Systems of Wave Equations on Asymptotically de Sitter Vacuum Spacetimes in All Even Spatial Dimensions

TL;DR

This paper develops quantitative estimates for systems of wave equations, including Einstein vacuum equations, on asymptotically de Sitter spacetimes in even dimensions, crucial for understanding nonlinear scattering behavior.

Contribution

It provides the first detailed quantitative estimates for wave systems on these backgrounds, advancing the nonlinear scattering theory for Einstein vacuum solutions.

Findings

Established sharp top order estimates for the scattering map.

Proved quantitative bounds for wave systems on asymptotically de Sitter spacetimes.

Included Einstein vacuum equations with nonlinear terms as inhomogeneous factors.

Abstract

This is the second paper of a two part work that establishes a definitive quantitative nonlinear scattering theory for asymptotically de Sitter vacuum solutions $(M,g)$ in $(n+1)$ dimensions with $n\geq4$ even, which are determined by small scattering data at $\mathscr{I}^{\pm}.$ In this paper we prove quantitative estimates for systems of wave equations on the $(M,g)$ backgrounds. The systems considered include the Einstein vacuum equations commuted with suitable time-dependent vector fields, where we treat the nonlinear terms as general inhomogeneous factors. The estimates obtained are essential in establishing sharp top order estimates for the scattering map of the Einstein vacuum equations, taking asymptotic data at $\mathscr{I}^-$ to asymptotic data at $\mathscr{I}^+$.