Formation of quiescent big bang singularities
Hans Oude Groeniger, Oliver Petersen, Hans Ringström
TL;DR
This work establishes a robust, quiescent regime for big bang singularities in Einstein–non-linear scalar field cosmologies with admissible potentials by introducing expansion-normalized quantities and the Fournodavlos–Rodnianski–Speck (FRS) framework. The authors prove that, for large initial mean curvature and a non-degenerate spectrum of the expansion-normalized Weingarten map satisfying a Kasner-type algebraic condition, the past evolution converges to a quiescent singularity with curvature blow-up, characterized by Kasner-like exponents and well-defined asymptotics. The paper develops a scaffold-based bootstrap argument in a constant-mean-curvature, vanishing-shift gauge with a Fermi–Walker frame, showing both past global existence and precise asymptotics, and extends stability results to data induced on the singularity as well as broad classes of spatially locally homogeneous solutions. Consequently, the results yield stable big bang formation for large families of spacetimes, including past and future global non-linear stability for many Bianchi-type models with potentials, thereby generalizing and unifying several prior stability results. The approach provides a flexible, background-independent regime for understanding quiescent singularities and offers a rigorous bridge between initial data on the singularity and the corresponding spacetime developments.
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.