Not all spins of black-hole binaries formed in clusters are isotropic and not all spins of black-hole binaries formed in isolation are aligned

Spin directions are promising observables for distinguishing between the formation channels of merging stellar-mass binary black holes with gravitational-wave observations. In this work, we challenge the standard expectation that binaries formed via dynamical interactions in stellar clusters have isotropically distributed spins and that binaries formed in isolation have spins aligned with the orbital angular momentum. For dynamically formed binaries, we account for observational findings suggesting that the spins of stars in clusters may exhibit a preferred degree of alignment and construct a simple geometrical description of the resulting black-hole spin directions. This results in an analytical joint distribution for the polar angles of first-generation black-hole binaries, whose key feature is a correlation between the two spin-orbit tilts. We show that, under reasonable assumptions and independently of the initial spin configuration, spins are fully randomized after a single black-hole merger, implying that binaries formed through hierarchical mergers have isotropically distributed spin orientations. For isolated binaries, we introduce a simple model for spin evolution as a function of the natal kick imparted during the first supernova explosion, while accounting for tidal interactions and mass-ratio reversal. The resulting distribution is also fully analytic. We investigate the impact of relativistic spin precession, as well as that of the finite escape speed of clusters, on our geometrically derived distributions. Our results provide a simple analytical framework for modeling the spin directions of merging black holes in gravitational-wave population fits.