Conformal Weyl tensor regularity at the big bang for ekpyrotic perturbations

Ascertain whether the conformal Weyl tensor W_{ijkl} associated with the conformal metric g_{ij} extends continuously to the big bang hypersurface τ=0 for the perturbed positive ekpyrotic-FLRW solutions constructed in Theorem 1.1 to the Einstein–scalar field equations with potential V(φ)=−e^{-s φ} (s>s_c). In particular, determine the regularity of W_{ijkl} across τ=0 under the conformal rescaling \bar g_{ij}=e^{2Φ} g_{ij} with e^{2Φ}=e^{2Φ_0} τ^{2(1−\mathring{R})/(n−2)}, and clarify whether the resulting singularity is isotropic in the strong sense of Goode–Wainwright.

Background

The main theorem establishes past nonlinear stability for perturbations of the positive ekpyrotic-FLRW solution, proves isotropisation, and shows Ricci dominance. The authors employ a conformal representation \bar g=e^{2Φ} g and demonstrate that the conformal geometry extends at least continuously to τ=0.

However, the analysis does not settle the behavior of the conformal Weyl tensor at the singularity. The derived estimates are not strong enough to ensure that all components of W_{ijkl} compensate for the τ^{-2} factor in the relation between physical and conformal Weyl tensors, leaving the precise regularity of W_{ijkl} at τ=0 open.