Relevance of precession for tests of the black hole no hair theorems

The multipole moments of black holes in general relativity obey certain consistency relations known as the no hair theorems. The details of this multipolar structure are imprinted into the gravitational waves emitted by binary black holes, particularly if the binary is precessing. If black holes do not obey the vacuum field equations of general relativity, then the no hair theorems may be broken, and the observed gravitational waves will be modified, thus providing an important test of the no hair theorems. Recently, analytic solutions to the precession dynamics and inspiral waveforms were computed within the context of binaries possessing nonaxisymmetric mass quadrupole moments, which are parametrized by a modulus qₘ and phase aₘ with m=1, 2 the azimuthal spherical harmonic number. Here, we use a Fisher analysis to study plausible constraints one may obtain on generic, nonaxisymmetry quadrupole configurations using current and future ground-based detectors. For nonprecessing binaries, we generically find that no meaningful constraints can be placed with current detectors on the nonaxisymmetry parameters (qₘ,aₘ) due to the presence of strong degeneracies with other waveform parameters, while with next generation detectors, only weak constraints are possible. For precessing configurations, the exact value of the uncertainty is strongly dependent on the sky location, system orientation relative to the line of sight, and initial inclination angle of the orbital angular momentum. After averaging over these parameters, we find that with GWTC-3-like events, one should be able to plausibly constraint nonaxisymmetric mass quadrupole deviations to Δqₘ~10^-2 for LIGO at design sensitivity, and Δqₘ~10^-4 for the same sources with Einstein Telescope and Cosmic Explorer.