Kicking time back in black-hole mergers: Ancestral masses, spins, birth recoils and hierarchical-formation viability of GW190521

Pair-instability supernova (PISN) prevents black-hole formation from stellar collapse within the approximate mass range $M\in [65,130]M_{\odot }$. However, such black holes may form hierarchically through merging ancestral black holes, whose properties determine those of the child'' one: mass, spin, and recoil velocity. Crucially, the child will leave its host environment if its birth recoil’’ exceeds the corresponding escape velocity, preventing further mergers. We exploit relations between the final recoil and spin of quasi-circular black-hole mergers to obtain posterior probability distributions for the hypothetical ancestral masses, spins and birth recoils of the component black holes of GW190521. To this, we present a Bayesian framework applicable to existing estimates for the components of black-hole merger observations. We consider both the quasi-circular (generically spinning) analysis performed by the LIGO-Virgo-KAGRA collaboration and the eccentric (aligned-spin) one performed by Romero-Shaw et. al. We evaluate the probability $p_{2g}$ that the GW190521 components inferred by these analyses formed from the merger of stellar-origin black holes and were retained by their environment. For the primary component, which populates the PISN gap, such scenario is strongly suppressed if GW190521 happened in a Globular Cluster with $p_{2g}\sim {10}^{-3}$ unless it was quasi-circular and its ancestors had aligned-spins, uncharacteristic of hierarchical formation channels, or small spins, which yields $p_{2g}\simeq {10}^{-2}$. If GW190521 was eccentric, we obtain $p_{2g}\simeq 0.1$ for any host other than an AGN, and zero for a Globular Cluster. If GW190521 was quasi-circular, a Nuclear-Star Cluster origin is possible with $p_{2g}\in (\sim 0.4\sim ,0.8)$