Hans Oude Groeniger: Formation of quiescent big bang singularities

Singularities are a natural feature of general relativity. In cosmology this was shown by Hawking in his famous singularity theorems, though the only conclusion obtained regarding the nature of the singularity is geodesic incompleteness. An important assumption in the theorem is a lower bound on the mean curvature. The question arises: how much mileage can we get out of assuming large mean curvature? Does the curvature necessarily blow up toward the initial singularity?

We show that, in the CMC Einstein-nonlinear scalar field setting, this is indeed the case, under two important assumptions. Firstly, we require an algebraic condition on the eigenvalues of expansion-normalized Weingarten map K, and we require the eigenvalues to be distinct. Secondly, we require bounds on a set of geometric expansion-normalized quantities constructed from the initial data. Notwithstanding technicalities, depending on these two assumptions there exists a threshold for the mean curvature, which, if surpassed, guarantuees a quiescent big bang singularity. By this is meant: the past global existence of the development until the blowup of the Kretschmann scalar, and convergence of the eigenvalues of K. We also obtain asymptotics for the eigenvalues of K as well as for expansion-normalized quantities relating to the scalar field. If time permits we discuss some classes of initial data to which the theorem may be applied. This is joint work with Oliver Petersen and Hans Ringström.