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Formation and Growth of IMBHs in Dense Star Clusters: Lessons from N-body and MOCCA Monte Carlo Simulations

Published online by Cambridge University Press:  15 October 2026

Abbas Askar Open the ORCID record for Abbas Askar [Opens in a new window] ,
Marcelo C. Vergara  and
Sohaib Ali
Abbas Askar*
Affiliation:
Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Warsaw, Poland
Marcelo C. Vergara
Affiliation:
Astronomisches Rechen-Institut, Zentrum für Astronomie, University of Heidelberg, Mönchhofstrasse 12-14, 69120, Heidelberg, Germany
Sohaib Ali
Affiliation:
Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Warsaw, Poland Institute of Astronomy, Nicolaus Copernicus University, Jurija Gagarina 11, 87-100 Toruó, Poland
*
email: askar@camk.edu.pl
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Abstract

Dense star clusters are promising nurseries for the formation and growth of intermediate-mass black holes (IMBHs; $\rm\sim10^2$–$\rm 10^5\,\rm M_\odot$), with increasing observational evidence pointing to their presence in massive clusters and stripped dwarf-galaxy nuclei. During the early evolution of compact clusters, massive stars can rapidly segregate to the center, where frequent collisions may trigger the runaway growth of a very massive star (VMS). This object can subsequently collapse to form an IMBH or merge with a stellar-mass black hole. We use direct N-body and Monte Carlo simulations of clusters with initial core densities between $10^6$ and $4\times10^8\,\rm M_\odot\,\mathrm{pc}^{-3}$ and total masses of $\rm 5.9\times10^5$ and $\rm1.3\times10^6\,M_\odot$. These models show that IMBHs of $\rm10^3$–$\rm10^4\,M_\odot$ can form within $\lesssim5$ Myr through the runaway collision channel. At later times, the IMBHs continue to grow through mergers with black holes, stars, and compact remnants, providing predictions testable with future gravitational-wave and transient surveys.

Keywords

stars: black holesgalaxy: globular clusters: generalmethods: numericalstars: kinematics and dynamicsgravitational waves

Information

Type
Contributed Paper
Information
Proceedings of the International Astronomical Union , Volume 21 , Symposium S398: Compact Objects and Binaries in Dense Star Clusters , September 2025 , pp. 264 - 267
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Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of The International Astronomical Union

1. Introduction

Intermediate-mass black holes (IMBHs; tilde 10 squared $\sim10^2$ – 10 Superscript 5 Baseline normal upper M Subscript circled dot $\rm 10^5\,M_\odot$ ) occupy the gap between stellar-mass BHs (sBHs; less than or equivalent to 100 normal upper M Subscript circled dot $\rm \lesssim100\,M_\odot$ ) and supermassive BHs (SMBHs; greater than or equivalent to 10 Superscript 6 Baseline normal upper M Subscript circled dot $\rm \gtrsim10^6\,M_\odot$ ). Observational evidence for their existence is mounting. In particular, fast-moving stars in the core of Omega Centauri are consistent with the presence of an IMBH (Häberle et al. Reference Häberle2024), while additional candidates have been found in dwarf galaxies and extragalactic clusters (Lin et al. Reference Lin, Strader and Carrasco2018; Pechetti et al. Reference Pechetti2022). The first confirmed detections of low-mass IMBHs ( tilde 10 squared normal upper M Subscript circled dot $\rm \sim10^2\,M_\odot$ ) have come from binary BH mergers observed in gravitational waves (GWs) by the LIGO–Virgo-KAGRA (LVK) collaboration, including GW190521 and GW231123 (LVK Scientific Collaboration et al. 2025).

Dense clusters, with short relaxation times and frequent stellar encounters, are favorable sites for seeding and growing IMBHs. Several pathways have been proposed (see Askar, Baldassare & Mezcua Reference Askar, Baldassare and Mezcua2024, for a detailed review and references therein), including runaway stellar collisions that form a very massive star (VMS) which may later collapse or merge with a sBH (e.g., Portegies Zwart & McMillan Reference Portegies Zwart and McMillan2004; Freitag, Gürkan & Rasio Reference Freitag, Gürkan and Rasio2006; Giersz et al. Reference Giersz, Leigh, Hypki, Lützgendorf and Askar2015; Di Carlo et al. Reference Di Carlo2021; Fujii et al. Reference Fujii, Wang, Tanikawa, Hirai and Saitoh2024; Rantala, Naab & Lahén Reference Rantala, Naab and Lahén2024; Sharma & Rodriguez Reference Sharma and Rodriguez2025; Vergara et al. Reference Vergara, Askar and Kamlah2025a). Here we briefly summarize results from complementary N-body and MOCCA Monte Carlo simulations of compact, centrally concentrated star clusters. We examine the formation of a VMS through collisional runaway, the conditions for IMBH seeding, and its subsequent growth through stellar and compact-object mergers, while highlighting caveats and implications for gravitational-wave and high-energy observations.

2. First study: Million-body compact cluster model

We model the evolution of an extremely dense, isolated cluster with upper N equals 10 Superscript 6 $N=10^6$ single stars and no primordial binaries at metallicity upper Z equals 0.01 $Z=0.01$ , using Nbody6++GPU (Vergara et al. Reference Vergara, Askar and Kamlah2025a). The cluster has normal upper M equals 5.87 times 10 Superscript 5 Baseline normal upper M Subscript circled dot $\rm M = 5.87\times10^5\,M_\odot$ , a King (Reference King1966) profile with normal upper W 0 equals 6 $\rm W_0=6$ , and half-mass radius normal r Subscript normal h Baseline equals 0.1 $\rm r_h=0.1$ pc. Stars are initialized on the zero-age main sequence with masses between 0.08 $0.08$ and 150 normal upper M Subscript circled dot $\rm 150\,M_\odot$ , drawn from a two-component Kroupa (Reference Kroupa2001) initial mass function (IMF). For comparison, we also ran a MOCCA simulation with the same initial modelFootnote †. These conditions yield a core density rho Subscript normal c Baseline equals 4 times 10 Superscript 8 Baseline normal upper M Subscript circled dot Baseline normal p normal c Superscript negative 3 $\rm \rho_c=4\times10^8\,M_\odot\,\mathrm{pc}^{-3}$ and half-mass relaxation time of 8.6 Myr. Massive stars segregate on much shorter timescales, driving high encounter rates that lead to runaway collisions and formation of a VMS of greater than or equivalent to 5 times 10 Superscript 4 Baseline normal upper M Subscript circled dot $\rm \gtrsim5\times10^4\,M_\odot$ within less than or equivalent to 5 $\lesssim5$ Myr.

Both simulations were evolved to 5 Myr. By t asymptotically equals 4.5 $t\simeq4.5$ Myr, the runaway produced a VMS that collapsed into an IMBH at the cluster center in both runs. The VMS grew to greater than or equivalent to 5 times 10 Superscript 4 Baseline normal upper M Subscript circled dot $\rm \gtrsim5\times10^4\,M_\odot$ through successive collisions, most within the first 2–3 Myr (Fig. 1a). Stellar evolution followed updated SSE/BSE prescriptions (Hurley, Tout & Pols Reference Hurley, Tout and Pols2002; Kamlah et al. Reference Kamlah2022), with improved treatment of VMS properties, rejuvenation, and mass loss (Vergara et al. Reference Vergara, Askar and Kamlah2025a).

Figure 1.

(a) VMS/IMBH mass growth up to 5 Myr from NBODY6++GPU (solid) and MOCCA (dashed). The star marks IMBH formation. (b) Mass distribution of stars colliding with the VMS during runaway growth, from NBODY6++GPU (black) and MOCCA (gray).

Two graphs depict the growth and collision data of very massive stars and intermediate-mass black holes.

The two approaches agree on the overall growth pathway and timing of IMBH formation, but the N-body run produced nearly twice as many VMS collisions, mainly with low-mass stars. At higher masses, the distributions agree more closely, with both methods yielding on the order of 10 cubed $10^3$ collisions involving stars above 10 normal upper M Subscript circled dot $\rm 10\,M_\odot$ (Fig. 1b). These differences arise from MOCCA’s probabilistic treatment of collisions, which tends to underestimate the frequency of low-mass mergers in this type of cluster. As a result, IMBH formation occurs slightly earlier ( tilde 4.5 $\sim4.5$ Myr vs. tilde 4.7 $\sim4.7$ Myr in the N-body run), owing to weaker rejuvenation of the VMS. As a cautionary point, collisions in the rapid early growth phase (first 2 Myr) occur on 200–1000 yr timescales, shorter than the thermal timescale of the VMS ( tilde 10 Superscript 4 $\sim10^4$ yr). Consequently, the structure of the runaway product before collapse remains uncertain, and its inferred mass should be considered an upper limit.

3. Second study: MOCCA simulations of dense clusters

We also ran a suite of MOCCA simulations of dense clusters with upper N equals 2 times 10 Superscript 6 $N=2\times10^6$ objects (10% binaries), metallicity upper Z equals 0.0005 $Z=0.0005$ , total mass normal upper M equals 1.29 times 10 Superscript 6 Baseline normal upper M Subscript circled dot $\rm M=1.29\times10^6\,M_\odot$ , and circular orbits at 10 kpc. We adopted King (Reference King1966) models with normal upper W 0 element of StartSet 5 comma 8 EndSet $\rm W_0\in\{5,8\}$ and varied normal r Subscript normal h $\rm r_h$ to span rho Subscript normal c Baseline equals 8.7 times 10 Superscript 5 $\rm \rho_c=8.7\times10^5$ – 4.4 times 10 Superscript 8 Baseline normal upper M Subscript circled dot Baseline normal p normal c Superscript negative 3 $\rm 4.4\times10^8\,M_\odot\,\mathrm{pc}^{-3}$ (Table 1). Stellar and binary physics included rapid SN/fallback (Fryer et al. Reference Fryer2012), metallicity-dependent winds (Belczynski et al. Reference Belczynski2010), and updated low-Z massive-star evolution (Tanikawa et al. Reference Tanikawa, Yoshida, Kinugawa, Takahashi and Umeda2020). In BH–star collisions, 25% of stellar mass was accreted; MS–MS collisions followed BSE rejuvenation; and GW recoil kicks were applied with natal BH spins sampled uniformly in left bracket 0 comma 0.1 right bracket $[0,0.1]$ .

Table 1.

Key initial conditions and outcomes of the MOCCA models. All runs have upper N equals 2 times 10 Superscript 6 $N=2\times10^6$ objects (10% binaries), upper Z equals 0.0005 $Z=0.0005$ , normal upper M equals 1.29 times 10 Superscript 6 Baseline normal upper M Subscript circled dot $\rm M=1.29\times10^6\,M_\odot$ , and stars initialized on the ZAMS with masses 0.08 $0.08$ – 150 upper M Subscript circled dot $150\,M_\odot$ from a two-component Kroupa (Reference Kroupa2001) IMF. Here normal upper W 0 $\rm W_0$ is the King concentration parameter and normal upper T Subscript normal r normal h $\rm T_{rh}$ is the initial half-mass relaxation time.

A table comparing initial conditions and outcomes of MOCCA models.
Table 1 Long description

A table comparing initial conditions and outcomes of MOCCA models. The table has 5 rows and 5 columns. Column headers are Model, r_h (pc), W_0, rho_core (M_sun pc^-3), T_rh (Myr), and IMBH?. Row 1: Model, rh0.25-W08; r_h (pc), 0.25; W_0, 8; rho_core (M_sun pc^-3), 4.38 x 10^8; T_rh (Myr), 42.7; IMBH?, (2.762 x 10^4 M_sun). Row 2: Model, rh0.5-W08; r_h (pc), 0.50; W_0, 8; rho_core (M_sun pc^-3), 5.86 x 10^7; T_rh (Myr), 121.0; IMBH?, (1.537 x 10^4 M_sun). Row 3: Model, rh0.5-W05; r_h (pc), 0.50; W_0, 5; rho_core (M_sun pc^-3), 4.34 x 10^6; T_rh (Myr), 121.0; IMBH?, x. Row 4: Model, rh1.0-W08; r_h (pc), 1.00; W_0, 8; rho_core (M_sun pc^-3), 7.36 x 10^6; T_rh (Myr), 342.3; IMBH?, (6.244 x 10^3 M_sun). Row 5: Model, rh2.0-W08; r_h (pc), 2.00; W_0, 8; rho_core (M_sun pc^-3), 8.67 x 10^5; T_rh (Myr), 965.6; IMBH?, x.

IMBHs form only in the three densest models ( rho Subscript normal c Baseline greater than or equivalent to 10 Superscript 7 Baseline normal upper M Subscript circled dot Baseline normal p normal c Superscript negative 3 $\rm\rho_c\gtrsim10^7\,M_\odot \, \mathrm{pc}^{-3}$ ; Table 1), underscoring the strong dependence on central density (Sharma & Rodriguez Reference Sharma and Rodriguez2025) and half-mass relaxation time. As shown in Fig. 2, a VMS forms within the first few Myr and seeds an IMBH through a VMS-BH collision at tilde 3 $\sim3$ Myr. Over the next 0.5 Gyr, the IMBH grows by a factor of tilde 3 $\sim3$ –4, primarily through mergers with sBHs, supplemented by mergers with stars and other compact remnants. IMBH-sBH mergers both drive this growth and rapidly deplete the sBH population, with 524, 390, and 178 such events in models rh0.25–W08, rh0.5–W08, and rh1.0–W08, respectively. These mergers produce “light IMRIs” (LIMRIs), most of which have characteristic strains and frequencies within the sensitivity band of next-generation GW detectors such as the Einstein Telescope. At later times, tidal disruption events contribute an additional tilde 10 cubed $\sim10^3$ – 10 Superscript 4 Baseline normal upper M Subscript circled dot $\rm 10^4\,M_\odot$ to the IMBH, with massive MS stars dominating early and white dwarfs at later epochs (Fig. 2b).

Figure 2.

Formation and growth of IMBHs in MOCCA models. Left: rapid seeding via VMS–sBH collisions and early sBH mergers. Right: long-term growth from MS stars, WDs, and BHs.

Two line graphs depict the formation and growth of intermediate-mass black holes in star clusters.

4. Conclusions and Outlook

Our N-body and MOCCA studies show that collisional runaway in extremely dense clusters can produce IMBHs within a few Myr, though growth histories depend on collision prescriptions, rejuvenation, and the internal structure of the VMS. IMBHs form only in the most compact models ( rho Subscript normal c Baseline greater than or equivalent to 5 times 10 Superscript 6 Baseline normal upper M Subscript circled dot Baseline normal p normal c Superscript negative 3 $\rm\rho_c \gtrsim 5\times10^6\,M_\odot\,\mathrm{pc}^{-3}$ and upper T Subscript r h Baseline less than or equivalent to 500 normal upper M normal y normal r $T_{rh} \lesssim 500\,\mathrm{Myr}$ ), underscoring their sensitivity to cluster density and relaxation time. While both methods agree on the overall pathway, they differ in the number of collisions and the timing of collapse, reflecting their respective approximations. These results assume monolithic cluster formation, which may not be universal. Key uncertainties remain, including the structure and evolution of the VMS and the use of simplified stellar evolution prescriptions. Continued cross-validation of N-body and Monte Carlo approaches (Vergara et al. Reference Vergara, Askar and Flammini Dotti2025b), coupled with improved VMS physics, will refine predictions and guide searches for IMBH signatures in tidal disruptions and GW sources such as LIMRIs.

Acknowledgements

We thank M. Giersz, R. Spurzem, F. Flammini Dotti, A. Kamlah, D. Schleicher, M. Arca Sedda, A. Hypki, J. Hurley, P. Berczik, A. Escala, N. Hoyer, N. Neumayer, X. Pang, A. Tanikawa, R. Cen, T. Naab, G. Wiktorowicz, L. Hellström, and J. Szyndler. AA and SA acknowledge support from project No. 2021/43/P/ST9/03167 co-funded by the Polish National Science Center (NCN) and the EU Horizon 2020 Marie Skłodowska-Curie grant agreement No. 945339. For the purpose of Open Access, the authors have applied for a CC-BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission. MCV acknowledges funding through ANID (Doctorado acuerdo bilateral DAAD/62210038) and DAAD (funding program number 57600326). MCV acknowledges the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD).

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Figure 0

Figure 1. (a) VMS/IMBH mass growth up to 5 Myr from NBODY6++GPU (solid) and MOCCA (dashed). The star marks IMBH formation. (b) Mass distribution of stars colliding with the VMS during runaway growth, from NBODY6++GPU (black) and MOCCA (gray).

Figure 1

Table 1. Key initial conditions and outcomes of the MOCCA models. All runs have N=2×106$N=2\times10^6$ objects (10% binaries), Z=0.0005$Z=0.0005$, M=1.29×106M⊙$\rm M=1.29\times10^6\,M_\odot$, and stars initialized on the ZAMS with masses 0.08$0.08$–150M⊙$150\,M_\odot$ from a two-component Kroupa (2001) IMF. Here W0$\rm W_0$ is the King concentration parameter and Trh$\rm T_{rh}$ is the initial half-mass relaxation time.Table 1 long description.

Figure 2

Figure 2. Formation and growth of IMBHs in MOCCA models. Left: rapid seeding via VMS–sBH collisions and early sBH mergers. Right: long-term growth from MS stars, WDs, and BHs.