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 numerical relativity simulations from the Simulating eXtreme Spacetimes project (SXS) to assess the detectability of one extra quasinormal mode in the ringdown of a binary black hole coalescence, with numbers $(\ensuremath{\ell},m,n)$ distinct from the fundamental quadrupolar mode (2,2,0). Our approach uses the information from the complex waveform as well as from the time derivative of the phase in two different 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 a fiducial time ${t}_{\mathrm{peak}}+10M$ (${t}_{\mathrm{peak}}$ is the time of maximum waveform amplitude), our results for nonspinning binaries indicate that, for mass ratios of $1\ensuremath{\mathbin:}1$ to approximately $5\ensuremath{\mathbin:}1$, the first overtone (2,2,1) will always have a larger excitation amplitude than the fundamental modes of the other harmonics (2,1,0), (3,3,0), and (4,4,0), making it a more promising candidate for detection. Even though the (2,2,1) mode damps about 3 times faster than the fundamental 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, as we show in a preliminary Rayleigh $\mathrm{criterion}+\mathrm{Fisher}$ matrix mode resolvability analysis of a simulation with nonzero spin consistent with GW150914. In particular, for nonspinning equal-mass binaries, the ratio of the amplitude of the first overtone (2,2,1) to the fundamental mode (2,2,0) will be $\ensuremath{\gtrsim}0.65$, whereas the corresponding ratio for the higher harmonics will be $\ensuremath{\lesssim}0.05$. For nonspinning binaries with mass ratios larger than $5\ensuremath{\mathbin:}1$, 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 nonspinning cases with larger mass ratios, keeping the mass of the final black hole compatible with GW150914.