Stable Big Bang Formation in Near-Flrw Solutions to the Einstein-Stiff Fluid System

Thursday, March 21, 2013 -
3:00pm to 4:00pm
I will discuss some results that I recently obtained in collaboration with Igor Rodnianski.  The results concern small perturbations of the well-known Friedmann-Lema\^{\i}tre-Robertson-Walker solution to the Einstein-stiff fluid system (stiff FLRW). A stiff fluid is a perfect fluid verifying the equation of state $p = \rho.$ The stiff FLRW solution is a special case of a family of spatially homogeneous, isotropic solutions that arise in cosmology. It models a stiff fluid evolving in a spacetime that expands as $t \to \infty$ and that collapses as $t \downarrow 0.$ In particular, the stiff FLRW solution contains a ``Big Bang'' singularity at $\Sigma_0 := \lbrace t = 0 \rbrace.$ To study the perturbed solutions, we place data on a Cauchy hypersurface  $\Sigma_1$ that are close to the stiff FLRW data (at time $1$) as measured by a Sobolev norm. No symmetry assumptions are made on the data. We then study the global behavior of the perturbed solution in the \emph{collapsing} direction. We first show that the spacetime region of interest can be foliated by a family of spacelike Cauchy hypersurfaces $\Sigma_t,$ $t \in (0,1],$ of constant mean curvature $- \frac{1}{3} t^{-1}.$ We then analyze the behavior of the solution as $t \downarrow 0$ and provide a detailed description of its asymptotics. Our main conclusion is that the perturbed solution remains globally close to the stiff FLRW solution and has approximately monotonic behavior. In particular, the perturbed solution also has a Big Bang singularity at $\Sigma_0.$ More precisely, as $t \downarrow 0,$ various curvature invariants uniformly blow-up and the volume of $\Sigma_t$ 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. 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 $L^2-$type \emph{energy almost-monotonicity inequality} that holds for the solutions under consideration.
Jared Speck
Event Location: 
Fine Hall 314