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-Lemaître-Robertson-Walker solutions undergoing decelerated expansion, with ${\mathbb{T}}^3$ spatial topology and scale factor $a(t)=t^p$ for $1\mathrm{/}3<p<1$. For each $p\in (2\mathrm{/}3,1)$, 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 $t\to \mathrm{\infty }$. 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.