Big bang stability and isotropisation for the Einstein-scalar field equations in the ekpyrotic regime
Florian Beyer, David Garfinkle, James Isenberg, Todd A. Oliynyk
Abstract
It has been shown that, in spacetime dimensions $n\geq 3$, that the Kasner-scalar field solutions to the Einstein-scalar fields equations with potential $V_0 e^{-s φ}$, where $s<s_c=\sqrt{\frac{8(n-1)}{n-2}}$ and $V_0\in \mathbb{R}$, are nonlinearly stable to the past and terminate at a quiescent big bang singularity over the full range of sub-critical Kasner exponents. In particular, the spatially homogeneous and isotropic solutions, the Friedman-Lemaitre-Robertson-Walker (FLRW) spacetimes, to the Einstein-scalar field equations are stable in this sense for $s<s_c$ and $V_0\in\mathbb{R}$. While perturbations of the sub-critical Kasner-scalar field family of solutions, including the FLRW solutions, are asymptotically velocity term dominated (AVTD) near the big bang, they do not in general isotropise near the big bang singularity. Rather, they remain highly anisotropic, even for small perturbations of the isotropic FLRW solutions. In this article, we establish the stability of FLRW solutions to the Einstein-scalar field equations for the potential parameter values $s>s_c$ and $V_0<0$. Such scalar field potentials are known in the literature as \textit{ekpyrotic}. In particular, we prove that the FLRW solutions to the Einstein-scalar field equations are nonlinearly stable to the past and terminate at a quiescent, crushing AVTD big bang singularity. A distinguishing property of these perturbed spacetimes is that they isotropise towards the big bang.