High-overtone fits to numerical relativity ringdowns: beyond the dismissed n=8 special tone

In general relativity, the remnant object originating from an uncharged black hole merger is a Kerr black hole. The approach to this final state is reached through the emission of a late train of radiation known as the black hole ringdown. The ringdown morphology is described by a countably infinite set of damped sinusoids, whose complex frequencies are solely determined by the final black hole's mass and spin. Recent results advocate that ringdown waveforms from numerical relativity can be fully described from the peak of the strain onwards if quasi-normal mode models with $N_{max}=7$ overtones are used. In this work we extend this analysis to models with $N_{max}\geq 7$ up to $N_{max}=16$ overtones by exploring the parameter bias on the final mass and final spin obtained by fitting the nonprecessing binary black hole simulations from the SXS catalogue. To this aim, we have computed the spin weight $-2$ quasi-normal mode frequencies and angular separation constants for the special $(l=m=2, n=8,9)$ overtones for the Kerr spacetime. We find that a total of $N_{max}\sim 6$ overtones are on average sufficient to model the ringdown starting at the peak of the strain, although about $21\%$ of the cases studied require at least $N_{max}\sim 12$ overtones to reach a comparable accuracy on the final state parameters. Considering the waveforms from an earlier or later point in time, we find that a very similar maximum accuracy can be reached in each case, occurring at a different number of overtones $N_{max}$. We provide new error estimates for the SXS waveforms based on the extrapolation and the resolution uncertainties of the gravitational wave strain. Finally, we observe substantial instabilities on the values of the best-fit amplitudes of the tones beyond the fundamental mode and the first overtone, that, nevertheless, do not impact significantly the mass and spin estimates