Instability of Spatially Homogeneous Solutions in the Class of T2-Symmetric Solutions to Einstein’s Vacuum Equations

In the subject of cosmology, spatially homogeneous solutions are often used to model the universe. It is therefore of interest to ask what happens when perturbing into the spatially inhomogeneous regime. To this end, we, in the present paper, study the future asymptotics of solutions to Einstein’s vacuum equations in the case of T 2 -symmetry. It turns out that in this setting, whether the solution is spatially homogeneous or not can be characterized in terms of the asymptotics of one variable appearing in the equations; there is a monotonic function such that if its limit is finite, then the solution is spatially homogeneous and if the limit is infinite, then the solution is spatially inhomogeneous. In particular, regardless of how small the initial perturbation away from spatial homogeneity is, the resulting asymptotics are very different. Using spatially homogeneous solutions as models is therefore, in this class, hard to justify.