  We prove the nonlinear stability in the contracting direction of
Friedmann-Lema\^itre-Robertson-Walker (FLRW) solutions to the Einstein-scalar
field equations in $n\geq 3$ spacetime dimensions that are defined on spacetime
manifolds of the form $(0,t_0]\times \mathbb{T}^{n-1}$, $t_0>0$. Stability is
established under the assumption that the initial data is
\textit{synchronized}, which means that on the initial hypersurface $\Sigma=
\{t_0\}\times \mathbb{T}^{n-1}$ the scalar field $\tau=
\exp\bigl(\sqrt{\frac{2(n-2)}{n-1}}\phi\bigr) $ is constant, that is,
$\Sigma=\tau^{-1}(\{t_0\})$. As we show that all initial data sets that are
sufficiently close to FRLW ones can be evolved via the Einstein-scalar field
equation into new initial data sets that are \textit{synchronized}, no
generality is lost by this assumption. By using $\tau$ as a time coordinate, we
establish that the perturbed FLRW spacetime manifolds are of the form $M =
\bigcup_{t\in (0,t_0]}\tau^{-1}(\{t\})\cong (0,t_0]\times \mathbb{T}^{n-1}$,
the perturbed FLRW solutions are asymptotically pointwise Kasner as $\tau
\searrow 0$, and a big bang singularity, characterised by the blow up of the
scalar curvature, occurs at $\tau=0$. An important aspect of our past stability
proof is that we use a hyperbolic gauge reduction of the Einstein-scalar field
equations. As a consequence, all of the estimates used in the stability proof
can be localized and we employ this property to establish a corresponding
localized past stability result for the FLRW solutions.
