Striking the right tone: toward a self-consistent framework for measuring black hole ringdowns

The ringdown portion of a binary black hole merger consists of a sum of modes, each containing an infinite number of tones that are exponentially damped sinusoids. In principle, these can be measured as gravitational-waves with observatories like LIGO/Virgo/KAGRA, however in practice it is unclear how many tones can be meaningfully resolved. We investigate the consistency and resolvability of the overtones of the quadrupolar $\ell = m = 2$ mode by starting at late times when the gravitational waveform is expected to be well-approximated by the $\ell m n = 220$ tone alone. We present a Bayesian inference framework to measure the tones in numerical relativity data. We measure tones at different start times, checking for consistency: we classify a tone as stably recovered if and only if the 95\% credible intervals for amplitude and phase at time $t$ overlap with the credible intervals at all subsequent times. We test a set of tones including the first four overtones of the fundamental mode and the 320 tone and find that the 220 and 221 tones can be measured consistently with the inclusion of additional overtones. The 222 tone measurements can be stabilised when we include the 223 tone, but only in a narrow time window, after which it is too weak to measure. The 223 tone recovery appears to be unstable, and does not become stable with the introduction of the 224 tone. We find that $N=3$ tones can be stably recovered simultaneously. However, when analysing $N \geq 4$ tones, the amplitude of one tone is consistent with zero. Thus, within our framework, one can identify only $N=3$ tones with non-zero amplitude that are simultaneously stable.