Abstract
We prove the nonlinear stability in the contracting direction of Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions to the Einstein-scalar field equations in n >= 3 spacetime dimensions that are defined on spacetime manifolds of the form (0, t(0)] x Tn-1, t(0) > 0. Stability is established under the assumption that the initial data is synchronized, which means that on the initial hypersurface Sigma = {t(0)} x Tn-1, the scalar field tau = exp(root 2(n-2)/n-1 phi) is constant, that is, Sigma = tau(-1)({t(0)}). As we show that all initial data sets that are sufficiently close to FRLW ones can be evolved via the Einstein-scalar field equation into new initial data sets that are synchronized, no generality is lost by this assumption. By using tau as a time coordinate, we establish that the perturbed FLRW spacetime manifolds are of the form M = boolean OR(t is an element of(0,t0])tau(-1)({t}) congruent to (0, t(0)] x T(n-)1, the perturbed FLRW solutions are asymptotically pointwise Kasner as t SE arrow 0, and a big bang singularity, characterised by the blow up of the scalar curvature, occurs at tau = 0. An important aspect of our past stability proof is that we use a hyperbolic gauge reduction of the Einstein-scalar field equations. As a consequence, all of the estimates used in the stability proof can be localized and we employ this property to establish a corresponding localized past stability result for the FLRW solutions.