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Overtones or higher harmonics? Prospects for testing the no-hair theorem with gravitational wave detections

In light of the current (and future) gravitational wave detections, more sensitive tests of general relativity can be devised. Black hole spectroscopy has long been proposed as a way to test the no-hair theorem, that is, how closely an astrophysical black hole can be described by the Kerr geometry. We use simulations from the Simulating eXtreme Spacetimes project to assess the detectability of one extra quasinormal mode in the ringdown of a binary black hole coalescence, with numbers $(\ell,m,n)$ distinct from the fundamental quadrupolar mode (2,2,0). Our approach uses the complex waveform as well as the time derivative of the phase in two prescriptions that allow us to estimate the point at which the ringdown is best described by a single mode or by a sum of two modes. By scaling all amplitudes to $t_{\rm peak}+10M$, our results for non-spinning binaries indicate that for mass ratios of 1:1 to 5:1 the first overtone (2,2,1) will always have a larger excitation amplitude than the fundamental modes of other harmonics, making it a more promising candidate for detection. Even though the (2,2,1) mode damps about three times faster than the higher harmonics and its frequency is very close to that of the (2,2,0) mode, its larger excitation amplitude still guarantees a more favorable scenario for detection. In particular, for equal-mass binaries the ratio of the amplitude of the first overtone (2,2,1) to the fundamental mode (2,2,0) will be $> 0.65$, whereas the corresponding ratio for the higher harmonics will be $< 0.05$. For binaries with mass ratios larger than 5:1 we find that the modes (2,2,1), (2,1,0) and (3,3,0) should have comparable amplitude ratios in the range 0.3 - 0.4. The expectation that the (2,2,1) mode should be more easily detectable than the (3,3,0) mode is confirmed with an extension of the mode resolvability analysis for larger mass ratios.