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 $(0,\infty) \times \mathbb{T}^2$. 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 $\{t = 1\}\simeq \mathbb{T}^3$ and study the linear solution's behavior in the collapsing direction $t\downarrow 0$. 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-Lemaître-Robertson-Walker (FLRW) solution. In particular, we show that as $t\downarrow 0$, various time-rescaled 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 $t\downarrow 0$. 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 $t\downarrow 0$ under small perturbations of its data at $\{t = 1\}$.