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,∞)×T3. 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}≃T3 and study the linear solution's behavior in the collapsing
direction t↓0. Our first main result is the proof of an
approximate L2 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↓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↓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↓0 under small perturbations of its data at {t=1}.Comment: 73 page