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Supermassive black hole growth in hierarchically merging nuclear star clusters

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that supermassive black hole seeds formed in compact nuclear star clusters at high redshift, and that growing them along galaxy merger histories reproduces observed black hole–galaxy scaling relations.

desk verdict A transparent, code-backed plausibility argument for NSC-born SMBH seeds, with the main caveat being an unverified compact-radius assumption and some tuned parameters. read the letter →

arxiv 2412.15334 v1 pith:BQPPJDEI submitted 2024-12-19 astro-ph.GA astro-ph.HEgr-qc

Konstantinos Kritos , Ricarda S. Beckmann , Joseph Silk , Emanuele Berti , Sophia Yi , Marta Volonteri , Yohan Dubois , Julien Devriendt This is my paper · ORCID
classification astro-ph.GAastro-ph.HEgr-qc
keywords supermassiveblackholeseedsnuclearstarclustersintermediate-massholesmergersgalaxymergertreesgravitationalwavebackgroundstidaldisruptioneventsdwarfgalaxies
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the seeds of supermassive black holes did not need exotic massive objects: they could have grown from ordinary stellar-mass black holes in nuclear star clusters, the dense stellar systems found at galaxy centers. The authors couple a semianalytic model of nuclear cluster evolution to realistic galaxy merger trees and show that runaway collisions, repeated black hole mergers, gas accretion, and star consumption can grow ten-solar-mass black holes into seeds of thousands to hundreds of thousands of solar masses within a gigayear. What makes the claim worth attention is that this ordinary-seed scenario reproduces observed black hole–galaxy scaling relations and predicts testable populations: intermediate-mass black holes in dwarf galaxies, ejected and wandering black holes, tidal disruption events, and a gravitational-wave merger rate accessible to space-based observatories.

What carries the argument

The carrying object is Nuce, a semianalytic code for nuclear cluster evolution that relies on Hénon's principle, the balanced evolution by which energy generation in the cluster core drives gradual expansion. Nuce evolves cluster structure and the central black hole through four growth channels: runaway stellar collisions, Bondi gas accretion capped at the Eddington limit, repeated mergers of stellar-mass black holes in a core-collapsed subsystem, and loss-cone consumption of stars. Coupled to galaxy merger trees from the NewHorizon simulation, the code decides which nuclear star clusters coalesce after major galaxy mergers and which black hole binaries merge, using published fitting formulae for remnant mass, spin, and gravitational-wave recoil; whether the merger remnant stays in the nucleus is set by comparing the kick velocity to the cluster escape velocity.

What would settle it

A high-redshift census of star-forming galaxies at redshifts 6 to 10 that finds nuclear star clusters are rarely as compact as about one parsec would undercut the model's seed channel; the compact clusters JWST has already found would need to be representative rather than rare exceptions.

Watch

Extended reading notes

Core claim

At the paper's center is the claim that galactic nuclei assemble their supermassive black holes hierarchically from light seeds. Each protogalaxy above a critical stellar mass forms a nuclear star cluster; inside compact clusters, stellar-mass black holes formed from massive stars undergo repeated mergers, with an escape-velocity threshold of roughly 300 kilometers per second, and are boosted by gas accretion to produce seeds of intermediate mass. When the seeding model is evolved along galaxy merger trees, the local population contains about 0.13 intermediate-mass black holes per cubic megaparsec, an occupation fraction near 80 percent in dwarf galaxies below one billion solar masses, and a black hole mass versus galaxy stellar mass relation with slope around 1.1. Supermassive black holes grow further during major galaxy mergers, those that grow by coherent gas accretion emerge with near-maximal spins, and gravitational-wave recoil occasionally ejects them into off-nuclear wandering or isolated states.

Load-bearing premise

The model assumes that almost every small protogalaxy above about ten million solar masses formed a dense nuclear star cluster at high redshift with sub-parsec scale sizes, and that such compact clusters are common enough; if they are not, the seed-assembly engine loses its fuel.

Editorial extensions

If this is right

  • If the scenario is right, supermassive black hole seeds can form from ordinary stellar-mass black holes, removing the need for massive Population III or primordial seeds.
  • The local universe should contain roughly 0.13 intermediate-mass black holes per cubic megaparsec, about half nuclear and half satellite, concentrated in dwarf galaxies.
  • Space-based gravitational-wave observatories should see a massive black hole merger rate of about 5.3 events per year out to redshift 5, dominated by intermediate-mass black hole binaries.
  • Nuclear star clusters would naturally produce a high tidal disruption event rate density of 10,000 to 100,000 per year per cubic gigaparsec at redshifts below 5.
  • Nuclear supermassive black holes grown by coherent gas accretion should appear highly spinning, while intermediate-mass black holes should show a bimodal spin distribution depending on formation channel.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A test the paper does not run is to compare its predicted near-maximal spins for supermassive black holes against X-ray reflection measurements with selection effects accounted for; if accretion is chaotic rather than coherent, spins would be lower and the growth channel would need revision.
  • Because the simulated volume is small and the merger trees stop at redshift 0.25, the model says little about the rare, highly luminous quasars seen at redshift above 6; applying the same seeding prescription to larger-volume trees would show whether an additional massive-seed channel is needed.
  • The predicted black hole occupation fraction of about 80 percent in dwarf galaxies is effectively an upper bound under the assumption that every galaxy above the mass threshold forms a nuclear cluster; patchy cluster formation would lower the implied number densities proportionally.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. This paper presents a semi-analytic model, NSC-tree, that couples the Nuce nuclear star cluster evolution code to galaxy merger trees extracted from the NewHorizon simulation. The model assumes that each protogalaxy above a critical stellar mass forms a nuclear star cluster with specified initial mass, gas fraction, and a log-uniform initial half-mass radius prior, and that stellar-mass black holes in these clusters grow through runaway collisions, repeated black-hole mergers, gas accretion, and stellar consumption to form massive black hole seeds. These seeds then grow and merge hierarchically during major galaxy mergers, producing predictions for the local MBH-Mstar relation, black hole spins, occupation fractions, number densities of IMBHs and SMBHs, NSC properties, and rates of massive black hole mergers, stellar-mass black hole captures, extreme-mass-ratio inspirals, and tidal disruption events. The headline claim is that the seeds of supermassive black holes formed in nuclear star clusters via stellar black hole mergers at early epochs, and that this scenario can reproduce local scaling relations and yields a LISA-detectable massive black hole merger rate.

Significance. If the central scenario is correct, the paper offers a concrete pathway from ordinary stellar-mass black holes to SMBH seeds without invoking exotic massive seed formation, and it produces a rich set of falsifiable predictions: local IMBH number densities, occupation fractions, TDE rates, EMRI rates, and LISA-detectable merger rates. The manuscript is transparent about its simplifications, provides a public code, and makes an explicit connection to a high-resolution cosmological simulation via merger trees. However, the model's headline outputs are not fully independent predictions: the essential initial conditions for high-redshift NSCs are unconstrained, and the authors state in the Conclusions that many free parameters were chosen to give a crude fit to local scaling laws. These issues must be addressed before the claim that the NSC paradigm explains SMBH seeding can be regarded as robust.

major comments (4)
  1. [§5 and §4.2] The log-uniform prior on the initial half-mass radius, rh,0 in [0.1, 10] pc, is load-bearing but observationally unconstrained at high redshift. The seed-formation thresholds in §2.3.1 (0.2 tau_rh,0 < 3 Myr for runaway collisions and v_esc > 300 km/s for repeated BH mergers) imply that only the compact tail, roughly rh,0 < 0.2-0.5 pc for N ~ 1e6-1e7, contributes to seed formation. Under the fiducial prior, this tail contains roughly 15-35% of the clusters, but if the true high-redshift NSC radius distribution is centered at a few pc with only a small compact tail, the seed-forming fraction drops by about an order of magnitude, and the predicted z=0.25 IMBH density of 0.13 Mpc^-3 falls below the 0.02 Mpc^-3 lower end of the observationally allowed range from Greene et al. (2020). The authors themselves flag this in §4.1: 'Compact NSCs on sub-pc scales could theoretically form... This assumption is essential for assembling SMBH seeds in the NSC paradigm.' I request a quantitative sensitivity test that reweights or replaces the radius prior with a distribution informed by local NSC radii and by the few available high-z cluster observations, and that reports how the IMBH density, occupation fraction, and merger rates change.
  2. [§2.2 and §3.1.3] The Conclusions state explicitly that 'Our code has many free parameters, that we have chosen to give a crude fit to local scaling laws involving NSCs.' Consequently, headline outputs such as the MBH-Mstar slope in Eq. (1), the occupation fractions in Fig. 6, and the IMBH number density in Table 2 are partly calibrated rather than independent predictions. The hyperparameter tests in §4.2 show that the slope of the MBH-Mstar relation varies from about 0.35 (fg,max = 0.1) to about 1.13 (fstar,max = 0.1), so the fiducial slope of 1.10 is not robust to plausible parameter choices. Please specify which hyperparameters were tuned, against which observables and with what tolerance, and then separate calibrated statements from genuinely predicted quantities. A small grid or emulator-based exploration of the parameter region that still matches the calibration targets would greatly strengthen the paper.
  3. [§2.4 and §3.3.1] The assumption that every protogalaxy forms one NSC in a single burst when its stellar mass reaches Mcr = 1e7 Msun directly produces the near-unity NSC and IMBH occupation fractions at stellar masses below about 1e8 Msun. The authors acknowledge in §3.1.3 that 'Our predictions for high occupation fractions are impacted by our choice to form an NSC in every galaxy with stellar mass larger than 1e7 Msun.' Because the occupation fraction is a key comparison with Greene et al. (2020) and Nguyen et al. (2019), this assumption needs to be relaxed or bracketed, for example by introducing an NSC formation efficiency with a physically motivated dependence on galaxy mass or environment, and by modeling the ex-situ contribution from inspiraling globular clusters, which is currently neglected.
  4. The model ignores time delays between galaxy mergers, NSC mergers, and massive BH binary mergers. The estimate in §4.1, tau_har + tau_gw ~ 44 Myr, addresses only the hardening and gravitational-wave phase and explicitly drops the dynamical-friction delay tau_df. Since the predicted LISA merger-rate redshift distribution in Fig. 10 and the detectability statement in §3.3.1 depend on when massive BH binaries actually coalesce, the instantaneous-merger approximation could bias the z-distribution of events. A simple log-normal delay prescription, or at least a comparison of the 44 Myr estimate with the median inter-merger time in the NewHorizon trees, would show whether this simplification is benign for the paper's LISA predictions.
minor comments (4)
  1. [§1] The phrase 'strong HHEII emission' contains a typo and should read 'strong He II emission.'
  2. [§3.2.1] The sentence 'Most of these NSCs have half-mass radii larger than 50M⊙' should read 'larger than 50 pc,' consistent with the detectability threshold defined in §3.2.
  3. [§4.1] The word 'super-Eddigton' should be 'super-Eddington.'
  4. [§2.4] The description of the major/minor merger classification is confusing: 'if the asymmetry in the galaxy masses is smaller' appears to describe a minor merger, but the sentence then states the NSC becomes an ultra-compact dwarf. Please rephrase to make the threshold Qth and the two outcomes unambiguous.

Circularity Check

2 steps flagged · score 6.0 of 10

Partially circular: several headline 'predictions' (MBH-Mstar relation, NSC/BH occupation fractions) are explicitly partly fitted to, or built into, the assumed NSC scaling laws and universal NSC formation, so the claimed consistency with observations is partly by construction.

  1. fitted input called prediction [Section 5 (Conclusions), paragraph beginning 'Our code has many free parameters']
    "Our code has many free parameters, that we have chosen to give a crude fit to local scaling laws involving NSCs. We caution the reader that this is in the same spirit as other discussions of semi-analytical models of galaxy formation, e.g., Nadler et al. (2023)."

    The paper presents the MBH-Mstar relation (Eq. 1), the MBH-MNSC correlation, the occupation fractions, and the local number densities as model predictions in Sec. 3, and then compares them with observed scaling relations. However, the model's hyperparameters were explicitly chosen to 'give a crude fit to local scaling laws involving NSCs.' Since those local NSC scaling laws overlap with the observational benchmarks used in Sec. 3.1-3.2, the agreement is partly the target of the fit rather than an independent confirmation. The paper also declines a comprehensive hyperparameter study in Sec. 4.2, so the robustness of the 'prediction' over the parameter space is not quantified.

  2. self definitional [Section 3.1.3 (Occupation fraction), final paragraph]
    "Our predictions for high occupation fractions are impacted by our choice to form an NSC in every galaxy with stellar mass larger than 10^7 M_sun."

    The NSC occupation fraction fNSC is presented as a model output and compared with Virgo observations, but by construction every galaxy with Mstar > Mcr = 10^7 M_sun hosts an NSC. Consequently, the high massive-BH occupation fractions in dwarfs (e.g., ~80% with MBH > 10^3 M_sun) largely follow from this input assumption, because galaxies without NSCs cannot host nuclear IMBHs in the model. The quoted sentence concedes that the high occupation fractions are 'impacted by our choice' to assume universal NSC formation; this makes the occupation-fraction 'prediction' partly self-definitional rather than an emergent test of the model.

full rationale

The core Nuce evolution plus NewHorizon merger-tree calculation is not itself circular: the seed-formation thresholds (runaway collisions requiring 0.2 tau_rh,0 < 3 Myr, repeated BH mergers requiring vesc > 300 km/s), the Bondi/Eddington accretion prescription, and the GW-recoil retention criterion are stated as explicit physical inputs, and the resulting growth curves in Fig. 2-3 are computed rather than assumed. The self-citations to Kritos et al. (2024a,b) for Nuce and spin evolution are not load-bearing in a circular sense here, because the present paper spells out the relevant equations and assumptions, and the code is public. The compact-NSC prior (log-uniform rh,0 over 0.1-10 pc) is admittedly essential for seed formation, but it is an openly flagged physical assumption rather than a derived prediction; the paper even labels it 'essential' in Sec. 4.1. The genuine circularity is narrower and located in the presentation of results: the paper states that its free parameters were chosen to give a crude fit to local NSC scaling laws, and that the high NSC occupation fraction follows from a deliberate assumption that every galaxy above 10^7 M_sun forms an NSC. Therefore, the agreement of the model's MBH-Mstar slope, occupation fractions, and NSC abundance with observed scaling relations is partly a consequence of those choices, not an independent confirmation of the seeding scenario. This is partial circularity in the headline 'prediction' claims, not a full reduction of the derivation to its inputs; the seed-formation mechanism itself retains independent dynamical content.

Assumptions & free parameters 8 free parameters · 8 assumptions · 0 invented entities

The model's central claim rests on a set of chosen hyperparameters (Table 1), a hand-picked initial radius distribution, and several physical prescriptions carried over from prior work. No new particles, forces, or exotic objects are introduced; wandering and ejected BHs are known astrophysical outcomes. The most sensitive choice is the compact NSC initial condition, which the authors themselves flag as essential but unconfirmed.

free parameters (8)
  • Mcr (critical stellar mass for NSC formation) = 1e7 solar masses
    Fiducial hyperparameter in Table 1; every galaxy above this mass forms an NSC, driving NSC and BH occupation fractions. Lowering to 1e6 solar masses reduces BH masses.
  • Qth (major merger mass ratio threshold) = 0.1
    Fiducial hyperparameter in Table 1; sets which galaxy mergers produce NSC-NSC and BH-BH mergers. Tested at 0.01 with modest effect.
  • tau_ge (gas expulsion timescale) = 100 Myr
    Fiducial hyperparameter in Table 1; controls how long Bondi gas accretion can grow the BH. Lowering to 10 Myr suppresses SMBH formation.
  • fg,max (maximum initial gas fraction in NSC) = 1.0
    Fiducial hyperparameter in Table 1; higher gas gives more accretion and higher BH spins. Lowering to 0.1 flattens the MBH-Mstar relation.
  • fstar,max (maximum initial NSC mass fraction of galaxy) = 1.0
    Fiducial hyperparameter in Table 1; sets seed masses. Lowering to 0.1 brings the MBH-Mstar slope closer to observations.
  • Initial half-mass radius prior = log-uniform 0.1 to 10 pc
    Chosen by hand in Sec. 2.2; sub-pc clusters are essential for efficient seed formation, as the authors note in Sec. 4.1.
  • Runaway collision seed mass fraction = 1e-3 of initial stellar mass
    Fixed prescription in Sec. 2.3.1 following Portegies Zwart and McMillan (2002); sets the initial seed mass in clusters meeting the runaway condition.
  • Direct collapse seed spin = 0
    Arbitrarily set to zero in Sec. 2.3.2; affects spin evolution but not masses.
assumptions (8)
  • domain assumption Henon's principle and balanced evolution govern NSC core evolution.
    Used throughout Nuce (Sec. 2.2, Kritos et al. 2024a) to compute core collapse and expansion; not re-derived here.
  • ad hoc to paper Each protogalaxy forms one NSC in a single burst when its stellar mass reaches Mcr.
    Sec. 2.2; drives NSC occupation and seed production. The authors admit monolithic formation is a simplification in Sec. 4.1.
  • domain assumption Initial NSC gas is ionized hydrogen with cs = 10 km/s and is removed exponentially with timescale tau_ge.
    Sec. 2.2; gas retention is the dominant growth control for massive BHs.
  • domain assumption Runaway stellar collisions form a seed BH of mass 1e-3 Mstar,0 if 0.2 tau_rh,0 < 3 Myr.
    Sec. 2.3.1; based on prior simulations but taken as an instantaneous prescription.
  • domain assumption Gas accretion is Bondi-like, capped at Eddington, and coherently aligned with BH spin.
    Sec. 2.3.1 and Sec. 3.1.2; coherent accretion forces high spins for SMBHs.
  • domain assumption The final parsec problem is assumed solved, so BH binaries merge promptly after galaxy mergers.
    Sec. 4.1, citing Khan et al. (2013); time delays are neglected except for an order-of-magnitude estimate.
  • ad hoc to paper Massive BHs and NSCs do not feed back onto the galaxy merger tree.
    Sec. 4.1; the model uses NewHorizon merger histories without BH or NSC feedback.
  • domain assumption Merger trees retain only galaxies above 1e7 solar masses and stop at z = 0.25.
    Sec. 2.1; affects number densities and the mass range of host galaxies.

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Cite this review

Pith. "Pith review of Supermassive black hole growth in hierarchically merging nuclear star clusters." pith.science (2026). https://pith.science/paper/BQPPJDEI

@misc{pith2026241215334,
  author       = {Pith},
  title        = {Pith review of: Supermassive black hole growth in hierarchically merging nuclear star clusters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BQPPJDEI}},
  note         = {Machine review of arXiv:2412.15334}
}
read the original abstract

Supermassive black holes are prevalent at the centers of massive galaxies, and their masses scale with galaxy properties, increasing evidence suggesting that these trends continue to low stellar masses. Seeds are needed for supermassive black holes, especially at the highest redshifts explored by the James Webb Space Telescope. We study the hierarchical merging of galaxies via cosmological merger trees and argue that the seeds of supermassive black holes formed in nuclear star clusters via stellar black hole mergers at early epochs. Observable tracers include intermediate-mass black holes, nuclear star clusters, and early gas accretion in host dwarf galaxies, along with a potentially detectable stochastic gravitational wave background, ejection of intermediate and supermassive black holes, and consequences of a significant population of tidal disruption events and extreme-mass ratio inspirals.

Figures

Figures reproduced from arXiv: 2412.15334 by the authors.

Figure 1
Figure 1. Example galaxy merger tree showing the galaxy stellar mass evolution of the main progenitor (black), as well as all other progenitors (colored lines) that reach a mass of M⋆ > 107 M⊙ before the galaxy merger. Colored markers mark galaxy mergers. included in this analysis, we omit discussing this further. Interested readers are referred to Beckmann et al. (2023). Halos and galaxies were identified in the simulation u… view at source ↗
Figure 2
Figure 2. The time evolution of BH mass (MBH, upper panel) and dimensionless spin (χBH, lower panel). Different colors correspond to various initial conditions for (N⋆,0, rh,0, η⋆), as shown in the legends. 2.3.2. Time evolution of MBH and χBH In this subsection we test the massive BH growth model in NSCs without using the merger trees described in Sec. 2.1. In [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Masses of the BH seeds MBH formed through the combined processes of successive mergers and gas accretion within 1 Gyr in NSCs as a function of the initial number of stars N⋆,0 and initial half-mass radius rh,0. Each point corresponds to an NSC simulation with Nuce. A few pairs of the star formation efficiency (η⋆) and gas expulsion time scale (τge) have been assumed and shown in the title of each panel. The colored … view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: The mass of nuclear massive BHs and the corre￾sponding stellar mass of their host galaxies at redshift z = 0.25 from the catalog of Greene et al. (2020). The red points with error bars show BH mass estimates in nearby galaxies, while the red triangles are upper limits …
Figure 6
Figure 6. Figure 6: Occupation fraction of massive BHs with mass > 103 M⊙ (gray dashed), > 106 M⊙ (gray dash-dotted), and > 108 M⊙ (gray dotted), as well as the NSC occupation fraction (fNSC, thick gray solid). The linear model of Greene et al. (2020) for the MBH > 105 M⊙ occupation fract…
Figure 7
Figure 7. Figure 7: Local massive BH mass function, decomposed into satellite (dash-dot), nuclear (dashed), and ejected (dotted) BH components. The thick solid line corresponds to the total number density. The lower limit estimates “Linear” and “NSC” from Greene et al. (2020) for the nucl…
Figure 8
Figure 8. Figure 8: The BH mass vs. the total mass of the host NSC for the population of massive BHs at redshift z = 0.25. The hollow circles correspond to NSCs whose half-mass radius has expanded past 50 pc. We also plot observational data from Greene et al. (2020), including claimed det…
Figure 9
Figure 9. Figure 9: Left: Present-day structural properties of NSCs (gray points) in the half-mass radius (rh) versus NSC mass (MNSC) plane. Also shown are Milky-Way globular clusters (blue crosses) and the NSC catalog of Georgiev et al. (2016) (red points with error bars). Right: same as…
Figure 10
Figure 10. Figure 10: Source-frame merger rate density of massive BHs as a function of redshift (gray). We decompose this rate into IMBH-IMBH (red-dashed), IMBH-SMBH (green-dash￾dotted), and SMBH-SMBH (blue-dotted) merger contribu￾tions. The inset shows the cumulative observed merger rate …
Figure 11
Figure 11. Figure 11: Massive BH binary mergers in our generated catalog in the plane of merger redshift z and total source￾frame binary mass MMBH,tot. The color shows the signal-to￾noise ratio (SNR) of each merger event in the LISA detector. The point size represents √q, where q = m1/m2 >…
Figure 12
Figure 12. Figure 12: Source-frame rate density of stellar-mass BH￾BH captures (solid blue), extreme-mass ratio inspirals (solid green), and TDEs (dashed red) in NSCs as a function of red￾shift. Their corresponding observer-frame cumulative num￾bers [Rc(< z)] and the number density of NSCs…

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Reference graph

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Reviewed August 11, 2026 · model on record in the stance chip above.