|Location:||MSRI: Simons Auditorium|
I will discuss some results that I recently obtained in collaboration with Igor Rodnianski. Our main result is a proof of stable Big Bang formation in small perturbations of the well-known Friedmann-Lema^tre-Robertson-Walker (FLRW) solution to the Einstein-scalar eld system. To study near-FLRW solutions, we place data on a Cauchy hypersurface 1 that are close to the FLRW data (at time 1) as measured by a Sobolev norm. We then study the global behavior of the perturbed solution in the collapsing direction. We rst show that the spacetime region of interest can be foliated by a family of spacelike Cauchy hypersurfaces t; t 2 (0; 1]; of constant mean curva- ture 1 3 t 1: We then analyze the behavior of the solution as t # 0 and provide a detailed description of its asymptotics. Our main conclusion is that the perturbed solution remains globally close to the FLRW solution and has approximately monotonic behavior. In particular, the perturbed solution has a Big Bang singularity at 0: More precisely, as t # 0; various curvature invariants uniformly blow-up and the volume of t collapses to 0: These blow-up results demonstrate the validity of Penrose's Strong Cosmic Censorship conjecture for the past half of the perturbed spacetimes. We have also shown that the same results hold for the sti uid matter model.
From the point of view of analysis, our main results can be viewed as a proof of stable blow-up for an open set of solutions to a highly nonlinear elliptic-hyperbolic system. The most important aspect of our analysis is our identification of a new L2 type energy almost-monotonicity inequality that holds for the solutions under consideration. I will discuss, in particular, two aspects of our proof that are connected to the approximate monotonicity and that may be of general interest: i) I will exhibit two gauges in which the approximate monotonicity is visible: constant mean curvature-transported spatial coordinates gauge, and a closely related one-parameter family of gauges involving well-chosen parabolic PDEs for the lapse function; ii) I will show how to derive L2 type energy estimates for
solutions to the Einstein equations directly in transported spatial coordinates.