Bayesian model selection for testing the no-hair theorem with black hole ringdowns

In this paper we examine the extent to which black hole quasinormal modes (QNMs) could be used to test the no-hair theorem with future ground- and space-based gravitational-wave detectors. We model departures from general relativity (GR) by introducing extra parameters which change the mode frequencies or decay times from their values in GR. With the aid of Bayesian model selection, we assess the extent to which the presence of such a parameter could be inferred, and its value estimated. We find that it is harder to measure the departure of the mode decay times from their GR values than it is with the mode frequencies. The Einstein Telescope (ET, a third generation ground-based detector) could detect departures of as little as 8% in the frequency of the dominant QNM mode of a $500{M}_{\ensuremath{\bigodot}}$ black hole, out to a maximum range of $\ensuremath{\simeq}6\text{ }\text{ }\mathrm{Gpc}$ ($z\ensuremath{\simeq}0.91$). The New Gravitational Observatory (NGO, an ESA space mission to detect gravitational waves) can detect departures of $\ensuremath{\sim}0.6%$ in a ${10}^{8}{M}_{\ensuremath{\bigodot}}$ black hole to a luminosity distance of 50 Gpc ($z\ensuremath{\simeq}5.1$), and departures of $\ensuremath{\sim}10%$ in a ${10}^{6}{M}_{\ensuremath{\bigodot}}$ black hole to a luminosity distance of $\ensuremath{\simeq}6\text{ }\text{ }\mathrm{Gpc}$. In this exploratory work we have made a specific choice of source position (overhead), orientation (inclination angle of $\ensuremath{\pi}/3$) and mass ratio of progenitor binary (${m}_{1}/{m}_{2}=2$). A more exhaustive Monte Carlo simulation that incorporates progenitor black hole spins and a hierarchical model for the growth of massive black holes is needed to evaluate a more realistic picture of the possibility of ET and NGO to carry out such tests.