Ringdown overtones, black hole spectroscopy, and no-hair theorem tests
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Abstract
Validating the black-hole no-hair theorem with gravitational-wave\nobservations of compact binary coalescences provides a compelling argument that\nthe remnant object is indeed a black hole as described by the general theory of\nrelativity. This requires performing a spectroscopic analysis of the\npost-merger signal and resolving the frequencies of either different angular\nmodes or overtones (of the same angular mode). For a nearly-equal mass binary\nblack-hole system, only the dominant angular mode ($l=m=2$) is sufficiently\nexcited and the overtones are instrumental to perform this test. Here we\ninvestigate the robustness of modelling the post-merger signal of a binary\nblack hole coalescence as a superposition of overtones. Further, we study the\nbias expected in the recovered frequencies as a function of the start time of a\nspectroscopic analysis and provide a computationally cheap procedure to choose\nit based on the interplay between the expected statistical error due to the\ndetector noise and the systematic errors due to waveform modelling. Moreover,\nsince the overtone frequencies are closely spaced, we find that resolving the\novertones is particularly challenging and requires a loud ringdown signal.\nRayleigh's resolvability criterion suggests that in an optimistic scenario a\nringdown signal-to-noise ratio larger than $\\sim 30$ (achievable possibly with\nLIGO at design sensitivity and routinely with future interferometers such as\nEinstein Telescope, Cosmic Explorer, and LISA) is necessary to resolve the\novertone frequencies. We then conclude by discussing some conceptual issues\nassociated with black-hole spectroscopy with overtones.\n