The Maximal Development of Near-FLRW Data for the Einstein-Scalar Field System with Spatial Topology S³

Abstract

The Friedmann–Lemaître–Robertson–Walker (FLRW) solution to the Einstein-scalar field system with spatial topology S³ models a universe that emanates from a singular spacelike hypersurface (the Big Bang), along which various spacetime curvature invariants blow up, only to re-collapse in a symmetric fashion in the future (the Big Crunch). In this article, we give a complete description of the maximal developments of perturbations of the FLRW data at the chronological midpoint of the FLRW evolution. We show that the perturbed solutions also exhibit curvature blowup along a pair of spacelike hypersurfaces, signifying the dynamic stability of the Big Bang and the Big Crunch. Moreover, we provide a sharp description of the asymptotic behavior of the solution up to the singularities, showing in particular that various time-rescaled solution variables converge to regular tensorfields on the singular hypersurfaces that are close to the corresponding time-rescaled FLRW tensorfields. Our proof crucially relies on L²-type approximate monotonicity identities in the spirit of the ones we used in our joint works with I. Rodnianski, in which we proved similar results for nearly spatially flat solutions with spatial topology T³. In the present article, we rely on new ingredients to handle nearly round spatial metrics on S³, whose curvatures are order-unity near the initial data hypersurface. In particular, our proof relies on (i) the construction of a globally defined spatial vectorfield frame adapted to the symmetries of a round metric on S³; (ii) estimates for the Lie derivatives of various geometric quantities with respect to the elements of the frame; and (iii) sharp estimates for the asymptotic behavior of the FLRW solution’s scale factor near the singular hypersurfaces.