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

Understand

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.