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

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, i.e., stable Kretschmann scalar blow-up, causing the spacetime to become past incomplete. This is the first stability result towards the collapsing direction in the presence of Vlasov matter. Furthermore, while the asymptotic behaviour of Vlasov matter is close to that of the FLRW solution up to a small change in asymptotic order controlled by initial data, generically, the distribution function itself asymptotically concentrates in certain preferred directions that are teleologically determined. 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 Vlasov equation.