On the past maximal development of near-FLRW data for the Einstein scalar-field Vlasov system

TL;DR

This paper proves the stable formation of a Big Bang singularity in near-FLRW solutions of the Einstein scalar-field Vlasov system without symmetry assumptions, showing detailed asymptotic behavior and stability in the collapsing direction.

Contribution

It provides the first stability result for Einstein equations with Vlasov matter in the collapsing direction without symmetry assumptions, including detailed asymptotic analysis.

Findings

Stable Big Bang formation with Kretschmann scalar blow-up

Vlasov distribution remains close to FLRW solution asymptotically

Distribution concentrates in preferred velocity directions

Abstract

We show that the maximal globally hyperbolic development of near-FLRW initial data for the Einstein scalar-field Vlasov system exhibits stable Big Bang formation in the collapsing direction. The solutions exhibit stable Kretschmann scalar blow-up, causing the spacetime to become causally geodesically past incomplete, and are asymptotically velocity term dominated. This is the first stability result for the Einstein equations in the collapsing spacetime direction in presence of Vlasov matter that does not rely on any symmetry assumptions. Furthermore, the Vlasov distribution remains close to that of the FLRW solution as a function on the co-mass shell, and so does its momentum support if one assumes it to be close to that of the FLRW distribution initially. On the other hand, the leading order terms in components of the Vlasov energy-momentum tensor exbihit an offset in asymptotic order controlled by the perturbation size, and when viewed on the mass shell, the distribution asymptotically concentrates in certain preferred velocity directions. To ensure that this behaviour is sufficiently mitigated by the scalar field, we crucially exploit a scaling hierarchy between horizontal and vertical derivatives in the commuted Vlasov equation.