  In this paper we prove a global existence theorem, in the direction of
cosmological expansion, for sufficiently small perturbations of a family of
spatially compact variants of the $k=-1$ Friedmann--Robertson--Walker vacuum
spacetime. We use a special gauge defined by constant mean curvature slicing
and a spatial harmonic coordinate condition, and develop energy estimates
through the use of the Bel-Robinson energy and its higher order
generalizations. In addition to the smallness condition on the data, we need a
topological constraint on the spatial manifold to exclude the possibility of a
non--trivial moduli space of flat spacetime perturbations, since the latter
could not be controlled by curvature--based energies such as those of
Bel--Robinson type. Our results also demonstrate causal geodesic completeness
of the perturbed spacetimes (in the expanding direction) and establish precise
rates of decay towards the background solution which serves as an attractor
asymptotically.
