Einstein spaces as attractors for the Einstein flow

TL;DR

This paper proves that small perturbations of certain Einstein spacetimes in higher dimensions evolve globally and asymptotically approach Einstein geometries, demonstrating stability and causal completeness in an expanding universe context.

Contribution

It establishes a global existence and stability result for perturbations of higher-dimensional Einstein spacetimes, extending previous cosmological models.

Findings

Perturbed spacetimes are causally geodesically complete in the expanding direction.

The geometry converges to an Einstein geometry in the moduli space.

Decay rates depend on the stability properties of the Einstein geometry.

Abstract

In this paper we prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of $n+1$-dimensional, $n \geq 3$, spatially compact spacetimes which generalizes the $k=-1$ Friedmann--Robertson--Walker vacuum spacetime. Our results demonstrate causal geodesic completeness of the perturbed spacetimes, in the expanding direction, and show that the scale-free geometry converges towards an element in the moduli space of Einstein geometries, with a rate of decay depending on the stability properties of the Einstein geometry.