Boundedness and decay of waves on spatially flat decelerated FLRW spacetimes

TL;DR

This paper analyzes the behavior of solutions to the linear wave equation on decelerated FLRW spacetimes, establishing energy bounds, decay estimates, and pointwise decay, which are crucial for understanding wave propagation in cosmological models.

Contribution

It introduces a novel application of twisted $t$-weighted multiplier vector fields and $r^p$-weighted energy estimates to prove decay and boundedness of waves on decelerated FLRW backgrounds.

Findings

Established uniform energy bounds for waves.

Derived integrated local energy decay estimates.

Proved pointwise decay estimates with optimality in certain regimes.

Abstract

We study the linear wave equation on a class of spatially homogeneous and isotropic Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetimes in the decelerated regime with spatial topology $\mathbb{R}^3$. Employing twisted $t$-weighted multiplier vector fields, we establish uniform energy bounds and derive integrated local energy decay estimates across the entire range of the decelerated expansion regime. Furthermore, we obtain a hierarchy of $r^p$-weighted energy estimates \`a la the Dafermos-Rodnianski $r^p$-method, which leads to energy decay estimates. As a consequence, we demonstrate pointwise decay estimates for solutions and their derivatives. In the wave zone, this pointwise decay is optimal in the "radiation" and "sub-radiation" cases, and almost optimal around the radiation case.