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

Validating the black hole no-hair theorem with gravitational-wave observations of compact binary coalescences provides 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 postmerger 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 performing this test. Here we investigate the robustness of modeling the postmerger 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 modeling. 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 the Einstein Telescope, Cosmic Explorer, and LISA) is necessary to resolve the overtone frequencies. We then conclude by discussing some conceptual issues associated with black hole spectroscopy with overtones.