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 $p_{2g} \in (\sim 0.4 \sim ,0.8)$