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

The growing set of gravitational-wave sources is being used to measure the properties of the underlying astrophysical populations of compact objects, black holes and neutron stars. Most of the detected systems are black hole binaries. While much has been learned about black holes by analyzing the latest LIGO-Virgo-KAGRA (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 ($cos{ au}$) between each black hole spin and the orbital angular momentum, $cos{ au}=+1$ being perfect alignment. The LVK has modeled the $cos{ au}$ distribution as a mixture of an isotropic component and a Gaussian component with mean fixed at $+1$ and width measured from the data. In this paper, we want to verify if the data require the existence of such a peak at $cos{ au}=+1$. We use various models for the astrophysical tilt distribution and 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 $cos{ au}$ distribution peaks at $+1$ and looks linear, rather than Gaussian. If we constrain $-1leq mu leq +1$ the Gaussian component peaks at $mu=0.47^{+0.47}_{-1.04}$ (median and 90\% symmetric credible interval). Two other 2-component mixture models yield $cos{ au}$ distributions that either have a broad peak centered at $0.20^{+0.21}_{-0.18}$ 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.