Spin it as you like: the (lack of a) measurement of the spin tilt distribution with LIGO-Virgo-KAGRA binary black holes

While much has been learned about black holes by analyzing the latest LVK catalog, GWTC-3, a measurement of the astrophysical distribution of the black hole spin orientations remains elusive. This is usually probed by measuring the cosine of the tilt angle ($\cos\tau$) between each black hole spin and the orbital angular momentum, $\cos\tau=+1$ being perfect alignment. Abbott et al. has modeled the $\cos\tau$ distribution as a mixture of an isotropic component and a Gaussian component with mean fixed at +1 and width measured from the data. We want to verify if the data require the existence of such a peak at $\cos\tau=+1$. We use various alternative models for the astrophysical tilt distribution and measure their parameters using the LVK GWTC-3 catalog. We find that a) Augmenting the LVK model such that the mean $\mu$ of the Gaussian is not fixed at +1 returns results that strongly depend on priors. If we allow $\mu>+1$ then the resulting astrophysical $\cos\tau$ distribution peaks at +1 and looks linear, rather than Gaussian. If we constrain $-1\leq \mu\leq+1$ the Gaussian component peaks at $\mu=0.48^{+0.46}_{-0.99}$ (median and 90% symmetric credible interval). Two other 2-component mixture models yield $\cos\tau$ distributions that either have a broad peak centered at $0.19^{+0.22}_{-0.18}$ or a plateau that spans the range [-0.5, +1], without a clear peak at +1. b) All of the models we considered agree on the fact that there is no excess of black hole tilts at around -1. c) While yielding quite different posteriors, the models considered in this work have Bayesian evidences that are the same within error bars. We conclude that the current dataset is not sufficiently informative to draw any model-independent conclusions on the astrophysical distribution of spin tilts, except that there is no excess of spins with negatively aligned tilts.