Past stability of critical-FLRW Einstein–scalar field solutions

Determine the past nonlinear stability of the spatially flat critical Friedman–Lemaître–Robertson–Walker (FLRW) solutions to the n-dimensional Einstein–scalar field equations (ESF.1–ESF.2) with exponential potential V(φ)=V0 e^{-s φ} at the critical steepness s=±s_c, where s_c=√(8(n−1)/(n−2)), for the normalisations V0∈{−1,0}. Specifically, establish whether these critical-FLRW fixed points (x1=x2=1 when s=s_c; x1=x3=−1 when s=−s_c) are past attractors and nonlinearly stable against small perturbations in the contracting direction.

Background

The paper analyzes the one-dimensional dynamical system governing spatially flat FLRW solutions of the Einstein–scalar field equations with exponential potentials and identifies three fixed points corresponding to regimes: Kasner (s<s_c), critical (s=±s_c), and ekpyrotic (s>s_c). The authors prove nonlinear past stability for the ekpyrotic case and recall existing results for Kasner, but explicitly note that the critical case remains unresolved.

In the state-space analysis (Figure 1 and Table 1), the critical points occur when x1 equals one of the Kasner points, indicating degenerate behavior at s=±s_c. The question is whether these critical FLRW configurations are past attractors and nonlinearly stable under perturbations as t→0.