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

TL;DR

This paper proves the future stability of certain decelerating FLRW solutions to the Einstein-nonlinear scalar field system with exponential potential, showing that small perturbations remain close and converge to homogeneous states over time.

Contribution

It establishes the future stability and geodesic completeness of decelerated FLRW solutions with exponential scalar fields, a new result for this class of cosmological models.

Findings

Solutions are future-causally geodesically complete.

Perturbed solutions remain close to FLRW solutions over time.

Scalar field and metric perturbations converge to homogeneous functions.

Abstract

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.