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We prove a stable singularity formation result for solutions to theEinstein-scalar field and Einstein-stiff fluid systems. Our results apply tosmall perturbations of the spatially flat FLRW solution with topology$(0,\infty) \times \mathbb{T}^3.$ The FLRW solution models a spatially uniformscalar-field/stiff fluid evolving in a spacetime that expands towards thefuture 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 Cauchyhypersurface $\Sigma_1'$ that are close to the FLRW data induced on $\lbrace 1\rbrace \times \mathbb{T}^3.$ We study the perturbed solution in the collapsingdirection and prove that its basic features closely resemble those of the FLRWsolution. In particular, we construct constant mean curvature-transportedspatial 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'sStrong Cosmic Censorship hypothesis for the "past-half" of near-FLRW solutions.  The most difficult aspect of the proof is showing that the solution existsfor $(t,x) \in (0,1] \times \mathbb{T}^3,$ and to this end, we derive energyestimates that are allowed to mildly blow-up as $t \downarrow 0.$ To closethese estimates, we use the most important ingredient in our analysis: an$L^2-$type energy approximate monotonicity inequality that holds for near-FLRWsolutions. In the companion article "A regime of linear stability for theEinstein-scalar field system with applications to nonlinear Big Bangformation," we used the approximate monotonicity to prove a stability resultfor solutions to linearized versions of the equations. The present articleshows that the linear stability result can be upgraded to control the nonlinearterms.

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