Einstein spaces as attractors for the Einstein flow

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.