  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 $\lbrace t
= 1 \rbrace$.
