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, spatially compact spacetimes, which generalizes the k = −1 Friedmann–Lemaˆıtre-Robertson–
Walker vacuum spacetime. This work extends the result from [3].The background spacetimes we consider are Lorentz cones over
negative Einstein spaces of dimension n 3.
We use a variant of the constant mean curvature, spatially harmonic (CMCSH) gauge introduced in [2]. An important difference
from the 3+1 dimensional case is that one may have a nontrivial moduli space of negative Einstein geometries. This makes it necessary
to introduce a time-dependent background metric, which is used to define the spatially harmonic coordinate system that goes into the gauge.
Instead of the Bel-Robinson energy used in [3], we here use an expression analogous to the wave equation type of energy introduced in [2] for the Einstein equations in CMCSH gauge. In order to prove energy estimates, it turns out to be necessary to assume
stability of the Einstein geometry. Further, for our analysis it is necessary to have a smooth moduli space. Fortunately, all known examples of negative Einstein geometries satisfy these conditions.
We give examples of families of Einstein geometries which have nontrivial moduli spaces. A product construction allows one to generate new families of examples.
Our results demonstrate causal geodesic completeness of the perturbed spacetimes, in the expanding direction, and show that
the scale-free geometry converges toward an element in the moduli space of Einstein geometries, with a rate of decay depending on
the stability properties of the Einstein geometry.