Stable Big Bang Formation in Near-FLRW Solutions to the Einstein-Scalar Field and Einstein-Stiff Fluid Systems

We prove a stable singularity formation result for solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Our results apply to small perturbations of the spatially flat FLRW solution with topology $(0,\infty) \times \mathbb{T}^3.$ The FLRW solution models a spatially uniform scalar-field/stiff fluid evolving in a spacetime that expands towards the future and that has a "Big Bang" singularity at $\lbrace 0 \rbrace \times \mathbb{T}^3,$ where its curvature blows up. We place data on a Cauchy hypersurface $\Sigma_1'$ that are close to the FLRW data induced on $\lbrace 1 \rbrace \times \mathbb{T}^3.$ We study the perturbed solution in the collapsing direction and prove that its basic features closely resemble those of the FLRW solution. In particular, we construct constant mean curvature-transported spatial coordinates for the perturbed solution covering $(t,x) \in (0,1] \times \mathbb{T}^3$ and show that it also has a Big Bang at $\lbrace 0 \rbrace \times \mathbb{T}^3,$ where its curvature blows up. The blow-up confirms Penrose's Strong Cosmic Censorship hypothesis for the "past-half" of near-FLRW solutions. The most difficult aspect of the proof is showing that the solution exists for $(t,x) \in (0,1] \times \mathbb{T}^3,$ and to this end, we derive energy estimates that are allowed to mildly blow-up as $t \downarrow 0.$ To close these estimates, we use the most important ingredient in our analysis: an $L^2-$type energy approximate monotonicity inequality that holds for near-FLRW solutions. In the companion article "A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation," we used the approximate monotonicity to prove a stability result for solutions to linearized versions of the equations. The present article shows that the linear stability result can be upgraded to control the nonlinear terms.