Boundedness and decay of waves on spatially flat decelerated FLRW spacetimes

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.