Spin Doctors: How to diagnose a hierarchical merger origin

Gravitational-wave observations provide the unique opportunity of studying black hole formation channels and histories -- but only if we can identify their origin. One such formation mechanism is the dynamical synthesis of black hole binaries in dense stellar systems. Given the expected isotropic distribution of component spins of binary black hole in gas-free dynamical environments, the presence of anti-aligned or in-plane spins with respect to the orbital angular momentum is considered a tell-tale sign of a merger's dynamical origin. Even in the scenario where birth spins of black holes are low, hierarchical mergers attain large component spins due to the orbital angular momentum of the prior merger. However, measuring such spin configurations is difficult. Here, we quantify the efficacy of the spin parameters encoding aligned-spin (₄₅₅) and in-plane spin (₏) at classifying such hierarchical systems. Using Monte Carlo cluster simulations to generate a realistic distribution of hierarchical merger parameters from globular clusters, we can infer mergers' ₄₅₅ and ₑ₌ ₏. The cluster populations are simulated using Advanced LIGO-Virgo sensitivity during the detector network's third observing period and projections for design sensitivity. Using a "likelihood-ratio"-based statistic, we find that 2\% of the recovered population by the current gravitational-wave detector network has a statistically significant ₏ measurement, whereas no ₄₅₅ measurement was capable of confidently determining a system to be anti-aligned with the orbital angular momentum at current detector sensitivities. These results indicate that measuring spin-precession through ₏ is a more detectable signature of a hierarchical mergers and dynamical formation than anti-aligned spins.