A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation

Abstract

We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on (0, infinity) x T-3. The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially anisotropic spacetime that expands towards the future and that has a “Big Bang” singularity at t = 0. We place initial data for the linearized system along t = 1 similar or equal to T-3 and study the linear solution’s behavior in the collapsing direction t down arrow 0. Our first main result is the proof of an approximate L-2 monotonicity identity for the linear solutions. Using it, we prove a linear stability result that holds when the background Kasner solution is sufficiently close to the Friedmann-Lemaitre-Robertson-Walker (FLRW) solution. In particular, we show that as t down arrow 0, various timerescaled components of the linear solution converge to regular functions defined along t = 0. In addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMC-transported spatial coordinates gauge can be viewed as a limiting version of a family of parabolic gauges for the lapse variable; an approximate monotonicity identity and corresponding linear stability results also hold in the parabolic gauges, but the corresponding parabolic PDEs are locally well posed only in the direction t down arrow 0. Finally, based on the linear stability results, we outline a proof of the following result, whose complete proof will appear elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing direction t down arrow 0 under small perturbations of its data at t = 1.