  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.
