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The Einstein Telescope (ET), a proposed third-generation gravitational waveobservatory, would enable tests of the no-hair theorem by looking at thecharacteristic frequencies and damping times of black hole ringdown signals. Inprevious work it was shown that with a single $500 - 1000\,M_\odot$ black holeat distance $\lesssim 6$ Gpc (or redshift $z \lesssim 1$), deviations of a fewpercent in the frequencies and damping times of dominant and sub-dominant modeswould be within the range of detectability. Given that such sources may berelatively rare, it is of interest to see how well the no-hair theorem can betested with events at much larger distances and with smaller signal-to-noiseratios, thus accessing a far bigger volume of space and a larger number ofsources. We employ a model selection scheme called TIGER (Test Infrastructurefor GEneral Relativity), which was originally developed to test generalrelativity with weak binary coalescence signals that will be seen insecond-generation detectors such as Advanced LIGO and Advanced Virgo. TIGER iswell-suited for the regime of low signal-to-noise ratio, and information from apopulation of sources can be combined so as to arrive at a stronger test. Byperforming a range of simulations using the expected noise power spectraldensity of Einstein Telescope, we show that with TIGER, similar deviations fromthe no-hair theorem as considered in previous work will be detectable withgreat confidence using $\mathcal{O}(10)$ sources distributed uniformly inco-moving volume out to 50 Gpc ($z \lesssim 5$).

Testing the no-hair theorem with black hole ringdowns using TIGER