The landscape of stellar mass black hole spectroscopy with the Einstein Telescope and Cosmic Explorer

Black hole spectroscopy is the measurement of the quasi normal mode spectrum in the gravitational waves emitted by a remnant black hole from a binary merger. Detecting multiple excited modes provides an excellent opportunity to test the dynamics of general relativity in the strong field regime and the hypothesis that the final object relaxes to a Kerr black hole. Third generation detectors will be a quantum leap for black hole spectroscopy thanks to a large increase in the ringdown signal-to-noise ratio (SNR), which will allow to identify routinely subdominant excited modes besides the dominant one. We use a simulated catalog of merging stellar mass binary black holes to investigate the prospects of black hole spectroscopy with ground based third generation detectors, with particular emphasis on the Einstein Telescope working in synergy with a Cosmic Explorer detector. We study the cumulative distribution in the ringdown SNR and quantify the precision with which the quasi normal modes will be measured during the operational time of the detectors. Overall, we find that third generation detectors will allow to detect several thousands of ringdown events per year, with O(10) golden events per year with SNR larger than 100. Next, we find that about a thousand events per year will alow to constrain deviations from general relativity in the spectrum with better than 10 per cent precision in the individual events. Finally, we discuss the prospects for constraining deviations from general relativity when stacking together the population of the detected events.