Formation of quiescent big bang singularities

Abstract

Hawking's singularity theorem says that cosmological solutions arising from initial data with positive mean curvature have a past singularity. However, the nature of the singularity remains unclear. We therefore ask: If the initial hypersurface has sufficiently large mean curvature, does the curvature necessarily blow up towards the singularity? In case the eigenvalues of the expansion-normalized Weingarten map are everywhere distinct and satisfy a certain algebraic condition (which in 3+1 dimensions is equivalent to them being positive), we prove that this is the case in the CMC Einstein-non-linear scalar field setting. More specifically, we associate a set of geometric expansion-normalized quantities to any initial data set with positive mean curvature. These quantities are expected to converge, in the quiescent setting, in the direction of crushing big bang singularities. Our main result says that if the mean curvature is large enough, relative to an appropriate Sobolev norm of these geometric quantities, and if the algebraic condition is satisfied, then a quiescent (as opposed to oscillatory) big bang singularity with curvature blow-up forms. This provides a stable regime of big bang formation without requiring proximity to any particular class of background solutions. An important recent result by Fournodavlos, Rodnianski and Speck demonstrates stable big bang formation for all the spatially flat and spatially homogeneous solutions to the Einstein-scalar field equations satisfying the algebraic condition. Here we obtain analogous stability results for any solution inducing data at the singularity, in the sense introduced by the third author, in particular generalizing the aforementioned result. Moreover, we are able to prove both future and past global non-linear stability of a large class of spatially locally homogeneous solutions.