In \(n\)-spacetime dimensions, the Einstein–scalar field equations with an exponential potential are given by
where
with \(V_0 \in \{-1,0,1\}\) and \(s\in \mathbb {R}\). The Kasner–scalar field spacetimes are a distinguished family of solutions to the Einstein-scalar field equations that are past geodesically incomplete. In these solutions, past-directed timelike geodesics terminate at a spacelike big bang singularity characterised by curvature blow-up. Remarkable progress has been made recently on establishing the past stability of these solutions and their big bang singularities. The first major breakthrough was achieved by Fournodavlos-Rodnianski-Speck, who proved stability over the full sub-critical range of Kasner exponents in the case of a vanishing potential, i.e. \(V_0 = 0\). Subsequently, Oude Groeniger-Petersen-Ringström established past stability for the Kasner-scalar field spacetimes with non-vanishing potentials, \(|V_0| \neq 0\), under the condition \(s < \sqrt {\frac {8(n-1)}{n-2}}\). In both settings, perturbations of these spacetimes terminate in the past at quiescent, generically anisotropic big bang singularities that are characterised by curvature blow-up.
For the parameter choices \(V_0 = -1\) and \(s > \sqrt {\frac {8(n-1)}{n-2}}\), the Einstein-scalar field equations admit a distinct family of isotropic solutions with big bang singularities, known as ekpyrotic FLRW spacetimes. In this talk, I will present a new proof of big bang stability for this family. A remarkable feature of perturbations of these solutions is that, unlike perturbations of the Kasner-scalar field family, anisotropies are dynamically suppressed, and the spacetimes isotropise as they approach quiescent, spacelike big bang singularities characterised by curvature blow-up.