The numerical investigation of wave propagation in the asymptotic domain of Kerr spacetime has only recently been possible thanks to the construction of suitable hyperboloidal coordinates. The asymptotics revealed an apparent puzzle in the decay rate
s of scalar fields: the late-time rates seemed to depend on whether finite distance observers are in the strong field domain or far away from the rotating black hole, an apparent phenomenon dubbed splitting. We discuss far-field splitting in the full field and near-horizon splitting in certain projected modes using horizon-penetrating, hyperboloidal coordinates. For either case we propose an explanation to the cause of the splitting behavior, and we determine uniquely decay rates that previous studies found to be ambiguous or immeasurable. The far-field splitting is explained by competition between projected modes. The near-horizon splitting is due to excitation of lower multipole modes that back excite the multipole mode for which splitting is observed. In both cases splitting is an intermediate effect, such that asymptotically in time strong field rates are valid at all finite distances. At any finite time, however, there are three domains with different decay rates whose boundaries move outwards during evolution. We then propose a formula for the decay rate of tails that takes into account the inter--mode excitation effect that we study.
The signal-to-noise ratio (SNR) for black hole quasinormal mode sources of low-frequency gravitational waves is estimated using a Monte Carlo approach that replaces the all-sky average approximation. We consider an eleven dimensional parameter space
that includes both source and detector parameters. We find that in the black-hole mass range $Msim 4$-$7times 10^6M_{odot}$ the SNR is significantly higher than the SNR for the all-sky average case, as a result of the variation of the spin parameter of the sources. This increased SNR may translate to a higher event rate for the Laser Interferometer Space Antenna (LISA). We also study the directional dependence of the SNR, show at which directions in the sky LISA will have greater response, and identify the LISA blind spots.
