Attractors of the `n+1' dimensional Einstein-$Λ$ flow

Abstract

Here we prove a global existence theorem for sufficiently small however fully nonlinear perturbations of a family of background solutions of the $`n+1$' vacuum Einstein equations in the presence of a positive cosmological constant $Λ$. With the advent of dark energy driven accelerated expansion of the universe, it is of fundamental importance in mathematical cosmology to include a positive cosmological constant, the simplest form of the dark energy in the vacuum Einstein equations. Such Einsteinian evolution is here designated as the `Einstein-$Λ$' flow. We study the background solutions of this `Einstein-$Λ$' flow in $`n+1$' dimensional spacetimes in constant mean curvature spatial harmonic gauge, $n\geq3$ and establish both linear and non-linear stability of such solutions. In the cases of number of spatial dimensions being strictly greater than $3$, the finite dimensional Einstein muduli spaces form the center manifold of the dynamics. A suitable shadow gauge condition \cite{andersson2011einstein} is implemented in order to treat these cases. In addition, the autonomous character of the suitably re-scaled Einstein flow breaks down as a consequence of including $Λ(>0)$. We construct a Lyapunov function (controlling a suitable norm of the small data) similar to a wave equation type energy for the non-linear non-autonomous evolution of the small data and prove its decay in the direction of cosmological expansion. Our results demonstrate the future stability and geodesic completeness of the perturbed spacetimes, and show that the scale-free geometry converges to an element of the Einstein moduli space (a point for $n=3$ and a finite dimensional space for $n>3$), which has significant consequence for the cosmic topology while restricting to the case of $n=3$.