A note about late-time wave tails on a dynamical background

Consider a spherically symmetric spacetime generated by a self-gravitating massless scalar field $\phi$ and let $\psi$ be a test (nonspherical) massless scalar field propagating on this dynamical background. Gundlach, Price, and Pullin \cite{gpp2} computed numerically the late-time tails for different multipoles of the field $\psi$ and suggested that solutions with compactly supported initial data decay in accord with Price's law as $t^{-(2\ell+3)}$ at timelike infinity. We show that in the case of the time-dependent background Price's law holds only for $\ell=0$ while for each $\ell\geq 1$ the tail decays as $t^{-(2\ell+2)}$.