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Abstract

Validating the black-hole no-hair theorem with gravitational-wave
observations of compact binary coalescences provide a compelling argument that
the remnant object is indeed a black hole as described by the general theory of
relativity. This requires performing a spectroscopic analysis of the
post-merger signal and resolving the frequencies of either different angular
modes or overtones (of the same angular mode). For a nearly-equal mass binary
black-hole system, only the dominant angular mode ($l=m=2$) is sufficiently
excited and the overtones are instrumental to perform this test. Here we
investigate the robustness of modelling the post-merger signal of a binary
black hole coalescence as a superposition of overtones. Further, we study the
bias expected in the recovered frequencies as a function of the start time of a
spectroscopic analysis and provide a computationally cheap procedure to choose
it based on the interplay between the expected statistical error due to the
detector noise and the systematic errors due to waveform modelling. Moreover,
since the overtone frequencies are closely spaced, we find that resolving the
overtones is particularly challenging and requires a loud ringdown signal.
Rayleigh's resolvability criterion suggests that --~in an optimistic
scenario~-- a ringdown signal-to-noise ratio larger than $\sim 30$ (achievable
possibly with LIGO at design sensitivity and routinely with future
interferometers such as Einstein Telescope, Cosmic Explorer, and LISA) is
necessary to resolve the overtone frequencies. We then conclude with
discussions on some conceptual issues associated with black-hole spectroscopy
with overtones.