Stellar-mass black holes in star clusters: implications for gravitational wave radiation

We study the dynamics of stellar-mass black holes (BH) in star clusters with particular attention to the formation of BH-BH binaries, which are interesting as sources of gravitational waves (GW). We examine the properties of these BH-BH binaries through direct N-body simulations of star clusters using the GPU-enabled NBODY6 code. We perform simulations of N <= 10^5 Plummer clusters of low-mass stars with an initial population of BHs. Additionally, we do several calculations of star clusters confined within a reflective boundary mimicking only the core of a massive cluster. We find that stellar-mass BHs with masses ~ 10 solar mass segregate rapidly into the cluster core and form a sub-cluster of BHs within typically 0.2 - 0.5 pc radius, which is dense enough to form BH-BH binaries through 3-body encounters. While most BH binaries are ejected from the cluster by recoils received during super-elastic encounters with the single BHs, few of them harden sufficiently so that they can merge via GW emission within the cluster. We find that for clusters with $N \ga 5\times 10^4$, typically 1 - 2 BH-BH mergers occur within them during the first ~ 4 Gyr of evolution. Also for each of these clusters, there are a few escaping BH binaries that can merge within a Hubble time, most of the merger times being within a few Gyr. These results indicate that intermediate-age massive clusters constitute the most important class of candidates for producing dynamical BH-BH mergers. Old globular clusters cannot contribute significantly to the present-day BH-BH merger rate since most of the mergers from them would have occurred earlier. In contrast, young massive clusters are too young to produce significant number of BH-BH mergers. Our results imply significant BH-BH merger detection rates for the proposed “Advanced LIGO” GW detector. (Abridged)

💡 Research Summary

The paper investigates the dynamical evolution of stellar‑mass black holes (BHs) in star clusters with a focus on the formation and fate of BH–BH binaries, which are prime sources of gravitational‑wave (GW) radiation. Using the GPU‑accelerated direct N‑body code NBODY6, the authors performed a suite of simulations of Plummer‑model clusters containing up to 10⁵ low‑mass stars and an initial population of ≈10 M⊙ black holes. In addition to full‑cluster runs, they carried out “core‑only” simulations with reflective boundaries to achieve higher resolution in the dense central region of massive clusters.

The simulations show that massive BHs rapidly sink to the cluster centre via dynamical friction, forming a compact BH sub‑cluster of radius 0.2–0.5 pc within 10–30 Myr. Within this high‑density environment, three‑body encounters efficiently produce BH–BH binaries. These binaries initially have relatively wide separations but quickly harden through super‑elastic encounters with surrounding single BHs. The recoil imparted during each hardening encounter often exceeds the cluster escape speed, causing most binaries to be ejected. Nevertheless, a small fraction of binaries remain bound long enough to harden to the point where gravitational‑wave emission dominates their evolution, leading to mergers inside the cluster within the first ≈4 Gyr.

Quantitatively, clusters with N ≳ 5 × 10⁴ stars typically experience 1–2 in‑cluster BH–BH mergers over the first four gigayears. In addition, several binaries are ejected with orbital parameters that allow them to merge within a Hubble time; most of these have merger times of a few gigayears. The merger rate therefore scales strongly with cluster mass: low‑mass clusters produce few binaries, while intermediate‑mass, intermediate‑age clusters (∼1–4 Gyr old) are the most prolific producers of dynamical BH–BH mergers today. Old globular clusters (>10 Gyr) would have generated the bulk of their mergers early in their histories, making them negligible contributors to the present‑day GW detection rate. Conversely, very young massive clusters have not yet had time to form and harden sufficient numbers of binaries.

The authors discuss the implications for GW observatories. Their results suggest that Advanced LIGO (and similar detectors) could observe tens to hundreds of BH–BH merger events per year originating from dynamical channels in intermediate‑age massive clusters, a rate comparable to or exceeding that expected from isolated binary evolution. The study also highlights the importance of high‑performance GPU computing for resolving the small‑scale dynamics of BH sub‑clusters, which are essential for accurate predictions of merger rates.

In conclusion, the paper provides a comprehensive, first‑principles demonstration that dynamical interactions in dense star‑cluster cores can produce a substantial population of merging BH binaries. It identifies intermediate‑age massive clusters as the dominant contemporary source of such events, while clarifying why both ancient globular clusters and very young clusters contribute little to the current GW detection budget. Future work is suggested to explore the effects of metallicity, initial BH mass spectra, and external tidal fields on the merger statistics, thereby refining predictions for the next generation of GW observations.