In this paper we examine the extent to which black hole quasi-normal 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 _ { \odot } black hole , out to a maximum range of \simeq 6 Gpc ( z \simeq 0.91 ) .
The New Gravitational Observatory ( NGO , an ESA space mission to detect gravitational waves ) can detect departures of \sim 0.6 \% in a 10 ^ { 8 } M _ { \odot } black hole to a luminosity distance of 50 Gpc ( z \simeq 5.1 ) , and departures of \sim 10 \% in a 10 ^ { 6 } M _ { \odot } black hole to a luminosity distance of \simeq 6 Gpc .
In this exploratory work we have made a specific choice of source position ( overhead ) , orientation ( inclination angle of \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 .