Late--time Kerr tails: ``up" and ``down" excitations

We revisit the question of Kerr spacetime late--time scalar--field tails numerically, specifically the excitation of ``up" and ``down" modes. Specifically, an ``up" mode is an excited $(\ell,m)$ mode because of an initial $(\ell',m)$ mode for $\ell>\ell'$. The definition of a ``down" mode is commensurate. We propose to generalize the Barack--Ori formula for the decay rate of any tail multipole given a generic initial data set, to the contribution of any initial multipole mode. Our proposal leads to a much simpler expression for the late--time power law index. Specifically, we propose that the late--time decay rate of a kinematically allowed $Y_{\ell m}$ spherical harmonic multipole moment because of an initial $Y_{\ell' m}$ multipole is independent of the azimuthal number $m$, and is given by $t^{-n}$, where $n=\ell'+\ell+1$ for $\ell<\ell'$ and $n=\ell'+\ell+3$ for $\ell\ge\ell'$. The independence of $m$ may be surprising because of the explicit dependence of the Green's function on $m$. The much greater complexity of the usual Hod formula is an artifact of the additional requirement of describing the slowest damped mode.