Future stability of solutions of the Einstein-nonlinear scalar field system with decelerated expansion

We study solutions to the Einstein equations coupled to a nonlinear scalar field with exponential potential. This system admits Friedmann-Lema\^itre-Robertson-Walker solutions undergoing decelerated expansion, with $\mathbb{T}^3$ spatial topology and scale factor $a(t) = t^p$ for $1/3 < p < 1$. For each $p \in (2/3,1)$, we prove that the corresponding FLRW spacetime is future-stable as a solution to the Einstein-nonlinear scalar field system. Given initial data on a spacelike hypersurface that is sufficiently close to the FLRW data, we show the resulting solution is future-causal geodesically complete, and remains close to the FLRW solution for all time. Moreover, we show the perturbed metric components and scalar field converge to spatially homogeneous functions as $t \rightarrow \infty$. A key feature of our analysis is the decomposition of the metric and scalar field perturbations into their spatial averages and oscillatory remainders with zero average.