Stability of de Sitter Space and Expansion at the Conformal Boundary

TL;DR

This paper provides a new proof of the nonlinear stability of de Sitter space in higher dimensions, detailing the precise asymptotic expansion at the conformal boundary and establishing a correspondence with scattering data.

Contribution

It introduces a novel approach to analyze the stability and boundary behavior of de Sitter space, including the smoothness conditions depending on spatial dimension parity.

Findings

Conformal boundary expansion is smooth in odd dimensions.

In even dimensions, smoothness depends on the vanishing of the obstruction tensor.

A one-to-one correspondence between solutions near de Sitter space and boundary scattering data is established.

Abstract

Using an approach similar to arXiv:2409.15460, we give a new proof of the nonlinear stability of de Sitter space as a solution to the Einstein vacuum equations with positive cosmological constant in $n+1$ dimensions, with $n\geq3$. Using the gauge freedom of the equations, we are able to prove a precise expansion of the perturbed spacetime at the conformal boundary. In $n=$ odd spatial dimensions, the conformally rescaled metric is smooth up to the future conformal boundary and in $n=$ even spatial dimensions it is smooth if and only if the obstruction tensor of the boundary metric vanishes; if not, then the conformally rescaled metric is log smooth at the boundary. These results also hold for asymptotically de Sitter spaces. Using the results of Fefferman and Graham (1985, Conformal invariants), arXiv:0710.0919, arXiv:1705.09674 and arXiv:2311.02739, the structure of our expansion allows us to establish a 1-1 correspondence between solutions to the Einstein vacuum equations close to de Sitter space and scattering data prescribed on the conformal boundary in general dimension.