Black Hole Spectroscopy: Systematic Errors and Ringdown Energy Estimates

The relaxation of a distorted black hole to its final state provides important tests of general relativity within the reach of current and upcoming gravitational wave facilities. In black hole perturbation theory, this phase consists of a simple linear superposition of exponentially damped sinusoids (the quasinormal modes) and of a power-law tail. How many quasinormal modes are necessary to describe waveforms with a prescribed precision? What error do we incur by only including quasinormal modes, and not tails? What other systematic effects are present in current state-of-the-art numerical waveforms? These issues, which are basic to testing fundamental physics with distorted black holes, have hardly been addressed in the literature. We use numerical relativity waveforms and accurate evolutions within black hole perturbation theory to provide some answers. We show that (i) a determination of the fundamental $l=m=2$ quasinormal mode to within $1\%$ or better requires the inclusion of at least the first overtone, and preferably of the first two or three overtones; (ii) a determination of the black hole mass and spin with precision better than $1\%$ requires the inclusion of at least two quasinormal modes for any given angular harmonic mode $(\ell,\,m)$. We also improve on previous estimates and fits for the ringdown energy radiated in the various multipoles. These results are important to quantify theoretical (as opposed to instrumental) limits in parameter estimation accuracy and tests of general relativity allowed by ringdown measurements with high signal-to-noise ratio gravitational wave detectors.