Landscape of stellar-mass black-hole spectroscopy with third-generation gravitational-wave detectors

Gravitational-wave black-hole spectroscopy provides a unique opportunity to test the strong-field regime of gravity and the nature of the final object formed in the aftermath of a merger. Here we investigate the prospects for black-hole spectroscopy with third-generation gravitational-wave detectors, in particular the Einstein Telescope in different configurations, possibly in combination with Cosmic Explorer. Using a state- of-the-art population model for stellar-origin binary black holes informed by LIGO-Virgo-KAGRA data, we compute the average number of expected events for precision black-hole spectroscopy using a Fisher-matrix analysis. We perform our analysis on the dominant mode (2, 2, 0) and a set of subdominant modes [(3, 3, 0), (2, 1, 0), (4, 4, 0)] using amplitude and phase fits corresponding to the aligned spin configurations. We find that the Einstein Telescope will measure two independent quasinormal modes within O(1)% (resp O(10%) relative uncertainty for at least O(1) [resp O(500)] events per year, with similar performances in the case of a single triangular configuration or two L-shaped detectors with same arm length. A 15-km arm-length configuration would improve rates by roughly a factor of two relative to a 10-km arm-length configuration. When operating in synergy with Cosmic Explorer the rates will improve significantly, reaching few-percent accuracy for O(100) events per year.