Einstein spaces as attractors for the Einstein flow

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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-Lemaître-Robertson-Walker vacuum spacetime. This work extends the result from Future complete vacuum spacetimes. The background spacetimes we consider are Lorentz cones over negative Einstein spaces of dimension $n \ge 3$. We use a variant of the constant mean curvature, spatially harmonic (CMCSH) gauge introduced in Elliptic-hyperbolic systems and the Einstein equations. 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 Future complete vacuum spacetimes, we here use an expression analogous to the wave equation type of energy introduced in Elliptic-hyperbolic systems and the Einstein equations 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.