Decay conjecture for the wave equation in flat FLRW universes

Let $\phi$ be a solution of the wave equation in a flat FLRW universe with scale factor $a(t)=t^p$, where $p\geq 0$. Assume that the initial data $\phi_0(x)\coloneqq\phi(t_0,x)$ and $\phi_1(x)\coloneqq\partial_t\phi(t_0,x)$ are sufficiently regular and belong to appropriate Sobolev spaces.

Decay conjecture in the flat case.

$$\|\phi(t,\cdot)\|_{L^{\infty}(\mathbb{R}^3)}\lesssim\begin{cases}(1+t)^{-1},&0\leq p\leq\frac23,\\(1+t)^{-3(1-p)},&\frac23\leq p<1,\end{cases}$$

or, equivalently,

$$\|\phi(t,\cdot)\|_{L^{\infty}(\mathbb{R}^3)}\lesssim\begin{cases}(1+t)^{-1},&w\geq 0,\\(1+t)^{-\frac{3w+1}{w+1}},&-\frac13<w\leq 0.\end{cases}$$

Moreover,

$$\|\phi(t,\cdot)\|_{L^{\infty}(\mathbb{R}^3)}\lesssim(\log t)^{-\frac32}$$

for $p=1$ (that is, $w=-\frac13$), and there is no decay for $p>1$ (that is, $-1<w<-\frac13$).

The conjecture gives the expected $L^{\infty}$ decay rates across flat FLRW cosmologies. It is stated as a conjectural extension of the rates known in particular dust- and radiation-filled cases; the supplied text later states that it holds for $0\leq p\leq1$, while the $p>1$ regime has no decay.