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

The Einstein Telescope, a proposed third-generation gravitational-wave observatory, would enable tests of the no-hair theorem by looking at the characteristic frequencies and damping times of black hole ringdown signals. In previous work it was shown that with a single $500--1000\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ black hole at a distance $\ensuremath{\lesssim}6$ Gpc (or redshift $z\ensuremath{\lesssim}1$), deviations of a few percent in the frequencies and damping times of dominant and subdominant modes would be within the range of detectability. Given that such sources may be relatively rare, it is of interest to see how well the no-hair theorem can be tested with events at much larger distances and with smaller signal-to-noise ratios, thus accessing a far bigger volume of space and a larger number of sources. We employ a model-selection scheme called TIGER (Test Infrastructure for GEneral Relativity), which was originally developed to test general relativity with weak binary coalescence signals that will be seen in second-generation detectors, such as Advanced LIGO and Advanced Virgo. TIGER is well suited for the regime of low signal-to-noise ratios, and information from a population of sources can be combined so as to arrive at a stronger test. By performing a range of simulations using the expected noise power spectral density of the Einstein Telescope, we show that with TIGER, similar deviations from the no-hair theorem (such as those considered in previous works) will be detectable with great confidence using $\mathcal{O}(10)$ sources distributed uniformly in a comoving volume out to $50\text{ }\text{ }\mathrm{Gpc}\text{ }(z\ensuremath{\lesssim}5)$.