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Validating the black-hole no-hair theorem with gravitational-waveobservations of compact binary coalescences provides a compelling argument thatthe remnant object is indeed a black hole as described by the general theory ofrelativity. This requires performing a spectroscopic analysis of thepost-merger signal and resolving the frequencies of either different angularmodes or overtones (of the same angular mode). For a nearly-equal mass binaryblack-hole system, only the dominant angular mode ($l=m=2$) is sufficientlyexcited and the overtones are instrumental to perform this test. Here weinvestigate the robustness of modelling the post-merger signal of a binaryblack hole coalescence as a superposition of overtones. Further, we study thebias expected in the recovered frequencies as a function of the start time of aspectroscopic analysis and provide a computationally cheap procedure to chooseit based on the interplay between the expected statistical error due to thedetector noise and the systematic errors due to waveform modelling. Moreover,since the overtone frequencies are closely spaced, we find that resolving theovertones is particularly challenging and requires a loud ringdown signal.Rayleigh's resolvability criterion suggests that in an optimistic scenario aringdown signal-to-noise ratio larger than $\sim 30$ (achievable possibly withLIGO at design sensitivity and routinely with future interferometers such asEinstein Telescope, Cosmic Explorer, and LISA) is necessary to resolve theovertone frequencies. We then conclude by discussing some conceptual issuesassociated with black-hole spectroscopy with overtones.

Ringdown overtones, black hole spectroscopy, and no-hair theorem tests