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

TL;DR

This paper proves linear stability of Kasner solutions in Einstein-scalar field systems near the Big Bang singularity, showing convergence of solutions and motivating gauges for future nonlinear stability analysis.

Contribution

It establishes linear stability and approximate monotonicity identities for Einstein-scalar field equations near Kasner solutions, paving the way for nonlinear stability proofs.

Findings

Linear solutions exhibit approximate $L^2$ monotonicity.

Linear stability holds near FLRW solutions as $t o 0$.

Time-rescaled components converge to regular functions at the singularity.

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 $(0,\infty) \times \mathbb{T}^3$. 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 $\lbrace t = 0 \rbrace$. We place initial data for the linearized system along $\lbrace t = 1 \rbrace \simeq \mathbb{T}^3$ 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-Lema\^{\i}tre-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 $\lbrace t = 0 \rbrace$. 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 $\lbrace t = 1 \rbrace$.