On the past maximal development of near-FLRW data for the Einstein scalar-field Vlasov system
David Fajman, Liam Urban
TL;DR
The paper analyzes the Einstein–scalar-field–Vlasov system near FLRW spacetimes and proves a robust stability result for the contracting (Big Bang) direction without symmetry. The authors develop a two-tier energy/bootstrap framework, combining spacetime–scalar-field energies with a hierarchical Vlasov energy scheme that distinguishes horizontal and vertical derivatives on the mass shell via Sasaki geometry. The main findings are a stable Big Bang with Kretschmann scalar blow-up of order $t^{-4}$, AVTD dynamics, and a Bang footprint for the Vlasov distribution that remains close to its FLRW counterpart on the co-mass shell, with velocity concentration along preferred directions on the mass shell. The work also shows that, for $ appa>0$, a stable Big Crunch is obtainable in the future, highlighting the generality of quiescent Big Bang formation in cosmological ESFV spacetimes and marking the first past-stability result for the Einstein equations with Vlasov matter in the contracting regime without symmetry assumptions. The results significantly advance the understanding of gravitational collapse with kinetic matter and provide a rigorous stability framework compatible with the expected AVTD/“quiescent” behaviour in near-FLRW cosmologies.
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.