Coalescence Forensics: Weighing the Hosts of Hierarchical Binary Black Hole Mergers

We present a novel framework to infer the mass of clusters that host hierarchical binary black hole (BBH) mergers detected with gravitational-waves (GWs), on a single event basis. We show that the requirement that a second-generation (2G) remnant be retained, and subsequently undergo a dynamical encounter, places strong constraints on the mass of the cluster. Using a Plummer model as a readily interpretable baseline, we derive analytic scaling relations between the peak of the inferred host mass posterior, the GW-driven recoil velocity of the remnant, and the parameters that determine the structure of the host. We then perform exact numerical marginalization over thermal and recoil velocities, angles, and cluster structure parameters, to infer the host-mass posterior. We apply our framework to putative hierarchical mergers GW241011 and GW241110, and infer the masses of their hosts on a single-event basis. We find that these are consistent with either heavy globular clusters or nuclear star clusters, with inferred masses spanning $10^{5.7 - 7.7} M_{\odot}$ at $68%$ confidence depending on the 2G recoil velocity distribution used.

💡 Research Summary

The paper introduces a novel Bayesian framework for inferring the mass of the stellar system that hosts a hierarchical binary‑black‑hole (BBH) merger, using information from a single gravitational‑wave (GW) detection. The key insight is that a second‑generation (2G) black‑hole remnant must satisfy two necessary conditions to participate in a 1G+2G merger: (i) it must be retained by the host cluster after the recoil imparted by the first merger, and (ii) it must subsequently undergo a dynamical encounter with another object that contains at least one first‑generation (1G) black hole.

The authors adopt the Plummer sphere as a tractable analytic model for globular clusters (GCs) and non‑cusp nuclear star clusters (NSCs). The Plummer density profile $\rho(r)=\frac{3M}{4\pi b^{3}}(1+r^{2}/b^{2})^{-5/2}$ defines the total mass $M$ and scale radius $b$, while the escape speed $v_{\rm esc}(r)=\sqrt{-2\Phi(r)}$ follows from the potential $\Phi(r)=-GM/\sqrt{r^{2}+b^{2}}$. Assuming an isotropic Maxwell‑like velocity distribution truncated at $v_{\rm esc}$, the authors derive an exact expression for the retention probability $P_{\rm R}(M,v_{r})$, which is the fraction of recoil velocities $v_{r}$ that satisfy $|\mathbf{v}+\mathbf{v}{r}|<v{\rm esc}(r)$. By averaging over angles and the thermal velocity distribution, they obtain a simple scaling $v_{r}^{2}<\frac{3}{4}v_{\rm esc}^{2}$, leading to an analytic retention radius $r_{R}$ and a closed‑form approximation $P_{\rm R}\approx