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Gravitational-wave echoes from numerical-relativity waveforms via space-time construction near merging compact objects

We propose a new approach toward reconstructing the late-time near-horizon geometry of merging binary black holes, and toward computing gravitational-wave echoes from exotic compact objects. A binary black-hole merger spacetime can be divided by a time-like hypersurface into a Black-Hole Perturbation (BHP) region, in which the space-time geometry can be approximated by homogeneous linear perturbations of the final Kerr black hole, and a nonlinear region. At late times, the boundary between the two regions is an infalling shell. The BHP region contains late-time gravitational-waves emitted toward the future horizon, as well as those emitted toward future null infinity. In this region, by imposing no-ingoing wave conditions at past null infinity, and matching out-going waves at future null infinity with waveforms computed from numerical relativity, we can obtain waves that travel toward the future horizon. In particular, the Newman-Penrose $ψ_0$ associated with the in-going wave on the horizon is related to tidal deformations measured by fiducial observers floating above the horizon. We further determine the boundary of the BHP region on the future horizon by imposing that $ψ_0$ inside the BHP region can be faithfully represented by quasi-normal modes. Using a physically-motivated way to impose boundary conditions near the horizon, and applying the so-called Boltzmann reflectivity, we compute the quasi-normal modes of non-rotating ECOs, as well as gravitational-wave echoes. We also investigate the detectability of these echoes in current and future detectors, and prospects for parameter estimation.