Stellar mass black holes in star clusters: gravitational wave emission and detection rates

We investigate the dynamics of stellar-mass black holes (BH) in star clusters focusing on the dynamical formation of BH-BH binaries, which are very important sources of gravitational waves (GW). We examine the properties of these BH-BH binaries through direct N-body computations of Plummer clusters, having initially N(0) <= 10^5 low mass stars and a population of stellar mass BHs, using the state-of-the-art N-body integrator “NBODY6”. We find that the stellar mass BHs segregate rapidly into the cluster core to form a central dense sub-cluster of BHs in which BH-BH binaries form via 3-body encounters. While most of the BH binaries finally escape from the cluster by recoils due to super-elastic encounters with the single BHs, we find that for clusters with N(0) >= 5 X 10^4, typically a few of them dynamically harden to the extent that they can merge via GW emission within the cluster. Also, for each of such clusters, there are a few escaped BH binaries that merge within a Hubble time, most of the mergers happening within a few Gyr of cluster evolution. These results imply that the intermediate-aged massive clusters constitute the most important class of star clusters that can produce dynamical BH-BH mergers at the present epoch. The BH-BH merger rates obtained from our computations imply a significant detection rate (~ 30/yr) for the “Advanced LIGO” GW detector that will become operative in the near future. Finally, we briefly discuss our ongoing development on this work incorporating the formation of BHs in star clusters from stellar evolution. In particular, we highlight the effect of stellar metallicity on the BH sub-cluster driven expansion of a star cluster’s core.

💡 Research Summary

This paper presents a comprehensive study of the dynamical evolution of stellar‑mass black holes (BHs) in star clusters and the consequent production of gravitational‑wave (GW) sources. Using direct N‑body simulations with the state‑of‑the‑art integrator NBODY6, the author investigates how BHs segregate, form a dense BH sub‑cluster, generate BH‑BH binaries through three‑body encounters, and ultimately either merge inside the cluster or escape and merge later. The simulations adopt Plummer models with half‑mass radii ≤ 1 pc, containing up to 10⁵ low‑mass stars (0.5–1 M☉) and a population of equal‑mass (10 M☉) BHs. Two retention scenarios are explored: full retention and 50 % retention after supernova natal kicks.

The results show that mass‑segregation driven by the Spitzer instability causes the BHs to sink to the cluster core within ~50 Myr, forming a compact BH core with densities of order 10⁴ M☉ pc⁻³. In this environment, three‑body interactions efficiently produce BH‑BH binaries. These binaries harden via repeated super‑elastic encounters (Heggie’s law). As the binaries harden, recoil velocities increase; many single BHs and binaries are ejected from the core, while a few binaries become sufficiently tight (small semi‑major axis and/or high eccentricity) that their GW merger time, given by Peters’ formula, drops to ≤ 10 Myr. In the simulated clusters with initial particle numbers N₀ ≥ 5 × 10⁴, typically one or two such in‑cluster mergers occur over a Hubble time, while a comparable number of escaped binaries have merger times < 3 Gyr. The average merger rate per cluster is ≈ 0.4 Gyr⁻¹ for in‑cluster events and ≈ 0.9 Gyr⁻¹ for escaped binaries, yielding a total of ≈ 1.3 Gyr⁻¹ per cluster.

To translate these rates into observable detection rates, the author adopts a space density of massive, intermediate‑age clusters of ρ_cl ≈ 3.5 h³ Mpc⁻³ (h = 0.73) as derived from observations of young populous clusters. Multiplying by the per‑cluster merger rate gives an estimated Advanced LIGO detection rate of roughly 30 BH‑BH mergers per year. This dynamical channel therefore dominates over the contribution from isolated primordial binaries, which is predicted to be an order of magnitude lower.

The paper also explores the influence of stellar metallicity on the BH sub‑cluster dynamics. Two otherwise identical clusters are evolved with metallicities Z = Z☉ and Z = 0.01 Z☉, using a full Kroupa initial mass function and stellar evolution recipes. Lower metallicity leads to more massive BH remnants (average ≈ 14 M☉ versus ≈ 7 M☉ at solar metallicity). The heavier BHs produce stronger heating of the cluster core, causing a more pronounced expansion of the core radius and a higher density BH core. Consequently, the low‑metallicity cluster forms three escaping BH‑BH binaries that will merge within a Hubble time, compared to only one in the solar‑metallicity case. This demonstrates that metallicity is a key parameter controlling the efficiency of dynamical BH‑BH binary formation and the resulting GW merger rate.

In summary, the study establishes that (1) BHs rapidly form a dense central sub‑cluster in massive star clusters, (2) three‑body encounters efficiently generate BH‑BH binaries, (3) dynamical hardening and recoil lead to a mixture of in‑cluster mergers and escaped binaries that merge later, (4) intermediate‑age massive clusters (initial mass ≳ 3 × 10⁴ M☉, ages of a few Gyr) are the most prolific present‑day sources, and (5) metallicity strongly modulates the BH mass spectrum and the subsequent GW event rate. The author notes that future work will incorporate realistic BH mass spectra from stellar evolution, primordial binaries, and external tidal fields to refine the predictions further.