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We linearize the Einstein-scalar field equations, expressed relative toconstant mean curvature (CMC)-transported spatial coordinates gauge, aroundmembers of the well-known family of Kasner solutions on $(0,\infty) \times\mathbb{T}^3$. The Kasner solutions model a spatially uniform scalar fieldevolving in a (typically) spatially anisotropic spacetime that expands towardsthe future and that has a "Big Bang" singularity at $\lbrace t = 0 \rbrace$. Weplace initial data for the linearized system along $\lbrace t = 1 \rbrace\simeq \mathbb{T}^3$ and study the linear solution's behavior in the collapsingdirection $t \downarrow 0$. Our first main result is the proof of anapproximate $L^2$ monotonicity identity for the linear solutions. Using it, weprove a linear stability result that holds when the background Kasner solutionis sufficiently close to the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW)solution. In particular, we show that as $t \downarrow 0$, varioustime-rescaled components of the linear solution converge to regular functionsdefined along $\lbrace t = 0 \rbrace$. In addition, we motivate the preferreddirection of the approximate monotonicity by showing that the CMC-transportedspatial coordinates gauge can be viewed as a limiting version of a family ofparabolic gauges for the lapse variable; an approximate monotonicity identityand corresponding linear stability results also hold in the parabolic gauges,but the corresponding parabolic PDEs are locally well-posed only in thedirection $t \downarrow 0$. Finally, based on the linear stability results, weoutline a proof of the following result, whose complete proof will appearelsewhere: the FLRW solution is globally nonlinearly stable in the collapsingdirection $t \downarrow 0$ under small perturbations of its data at $\lbrace t= 1 \rbrace$.

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