Localized Big Bang Stability for the Einstein-Scalar Field Equations

Abstract

We prove the nonlinear stability in the contracting direction of Friedmann–Lemaître–Robertson–Walker (FLRW) solutions to the Einstein-scalar field equations in \(n\ge 3\) spacetime dimensions that are defined on spacetime manifolds of the form \((0,t_0]\times \mathbb {T}{}^{n-1}\), \(t_0>0\). Stability is established under the assumption that the initial data is synchronized, which means that on the initial hypersurface \(\Sigma = \{t_0\}\times \mathbb {T}{}^{n-1}\), the scalar field \(\tau = \exp \bigl (\sqrt{\frac{2(n-2)}{n-1}}\phi \bigr ) \) 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 \(M = \bigcup _{t\in (0,t_0]}\tau ^{-1}(\{t\})\cong (0,t_0]\times \mathbb {T}{}^{n-1}\), the perturbed FLRW solutions are asymptotically pointwise Kasner as \(\tau \searrow 0\), 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.