Gravitational-wave echoes from spinning exotic compact objects: Numerical waveforms from the Teukolsky equation

We present numerical waveforms of gravitational-wave echoes from spinning exotic compact objects (ECOs) that result from binary black hole coalescence. We obtain these echoes by solving the Teukolsky equation for the ${\ensuremath{\psi}}_{4}$ associated with gravitational waves that propagate toward the horizon of a Kerr spacetime, and process the subsequent reflections of the horizon-going wave by the surface of the ECO, which lies right above the Kerr horizon. The trajectories of the infalling objects are modified from Kerr geodesics, such that the gravitational waves propagating toward future null infinity match those from merging black holes with comparable masses. In this way, the corresponding echoes can be used to approximate to those from nonspinning comparable-mass mergers. For boundary conditions at the ECO surface, we adopt recent work using the membrane paradigm, which relates ${\ensuremath{\psi}}_{0}$ associated with the horizon-going wave and ${\ensuremath{\psi}}_{4}$ of the wave that leaves the ECO surface. We obtain ${\ensuremath{\psi}}_{0}$ of the horizon-going wave from ${\ensuremath{\psi}}_{4}$ using the Teukolsky-Starobinsky relation. The echoes we obtain turn out to be significantly weaker than those from previous studies that generate echo waveforms by modeling the ringdown part of binary black hole coalescence waveforms as originating from the past horizon.