Boundedness and Decay of Waves on Spatially Flat Decelerated FLRW Spacetimes

Abstract We study the linear wave equation on a class of spatially homogeneous and isotropic Friedmann–Lemaǐtre–Robertson–Walker (FLRW) spacetimes in the decelerated regime with spatial topology $$\mathbb {R}^3$$ 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 $$r^p$$ r p -weighted energy estimates à la the Dafermos–Rodnianski $$r^p$$ 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.