How pure is the tail of gravitational collapse ?

Waves propagating in a curved spacetime develop tails. In particular, it is well established that the late-time dynamics of gravitational collapse is dominated by a power-law decaying tail of the form $Mt^{-(2l+3)}$, where $M$ is the black-hole mass. It should be emphasized, however, that in a typical evolution scenario there is a considerable time window in which the signal is no longer dominated by the black-hole quasinormal modes, but the leading order power-law tail has not yet taken over. Higher-order terms may have a considerable contribution to the signal at these intermediate times. It is therefore of interest to analyze these higher-order corrections to the leading-order power-law behavior. We show that the higher-order contamination terms die off at late times as $M^2t^{-4}\ln(t/M)$ for spherical perturbations, and as $M^2t^{-(2l+4)}\ln^2(t/M)$ for non-spherical $(l\neq0)$ perturbations. These results imply that the leading-order power-law tail becomes "pure" (namely, with less than 1% contamination) only at extremely late times of the order of $10^4M$.