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REVIEW 4 major objections 7 minor 157 references

The supermassive black hole population from seeding via collisions in Nuclear Star Clusters

T0 review · 4 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Collisions between stars in compact nuclear star clusters can seed black holes that grow to ~10^9 solar masses, and the predicted local black-hole mass function matches observations above 10^8 solar masses.

desk verdict A transparent SAM exploration of NSC collision seeding that produces plausible high-mass BHMFs, but the entire seed population rests on an unvalidated radius rescaling that should be reframed as proof-of-concept. read the letter →

arxiv 2412.08280 v2 pith:WANR4IJM submitted 2024-12-11 astro-ph.GA

M. Liempi , D.R.G. Schleicher , A. Benson , A. Escala , M.C. Vergara This is my paper · ORCID
classification astro-ph.GA
keywords galaxies:evolutionformationnucleiGalaxy:centerquasars:supermassiveblackholesnuclearstarclustersholeseedingstellarcollisions
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 supermassive black holes can be seeded by runaway stellar collisions inside nuclear star clusters that form in situ from gas funneled to galaxy centers. Implementing this channel in the semi-analytic model Galacticus, the authors show that only initially compact clusters — those a factor 2 to 10 smaller than the present-day radius relation — pass the critical-mass threshold and produce seeds of roughly $5\times10^2$ to $1.7\times10^5\,M_\odot$. Those seeds then accrete and grow to black holes as massive as $\sim10^9\,M_\odot$ by $z=0$, producing a black-hole mass function that tracks the observed one above $10^8\,M_\odot$ and overpredicts it below. The nuclear star cluster population formed in the same runs has a mass-function shape comparable to observations and overlaps the observed scaling relations, though quantitative agreement is still limited. If this channel is real, it offers a single origin for the coexistence of nuclear star clusters and supermassive black holes in galaxies of roughly $10^{10}\,M_\odot$.

What carries the argument

The load-bearing object is the critical mass $M_{\rm crit}(r_{\rm NSC})$ from Eq. (17), defined by setting the stellar collision timescale $t_{\rm coll}$ equal to the cluster's mass-weighted age $t_H$; clusters above this mass are expected, via runaway collisions, to convert 10–50% of their mass into a central massive object. The model gathers NSC mass from the in-situ star formation reservoir and imposes the seed condition $M_{\rm NSC}\ge M_{\rm crit}$ and $M_{\rm NSC}\ge M_{\rm threshold}$, with seed mass $M_\bullet = 0.5\,M_{\rm NSC}$; the radius rescaling $\epsilon_r$ is what moves otherwise stable clusters across the threshold.

What would settle it

Measure the radii of young nuclear star clusters at high redshift (or infer their initial radii from the present-day expansion of young massive clusters) and check whether compact clusters with $r_{\rm NSC}\lesssim0.1$ pc exist; if no NSC population is 2–10 times more compact than the local $r_{\rm NSC}$–$M_{\rm dyn}$ relation at the epochs when seeds must form, the critical-mass condition $M_{\rm NSC}\ge M_{\rm crit}$ is essentially never met and the predicted black-hole population collapses.

Watch

Extended reading notes

Core claim

The paper's central claim is that a collision-based seeding channel operating in nuclear star clusters can account for the observed population of supermassive black holes above $10^8\,M_\odot$. The formation criterion is the critical mass $M_{\rm crit}(r_{\rm NSC})$ of Eq. (17), the cluster mass at which the stellar collision timescale equals the mass-weighted age of the cluster; when $M_{\rm NSC}^{\rm stellar}\geq M_{\rm crit}$ and the cluster is well-sampled ($M_{\rm NSC}^{\rm stellar}\geq M_{\rm threshold}$), a seed of mass $M_\bullet = \epsilon_\bullet M_{\rm NSC}^{\rm stellar}$ with $\epsilon_\bullet=0.5$ replaces the initial $10\,M_\odot$ black hole and begins accreting. Because the adopted present-day radius relation $r_{\rm NSC}=r_0\sqrt{M_{\rm dyn}^{\rm NSC}/10^6\,M_\odot}$ gives radii too large for collisions to matter, the model rescales radii by $\epsilon_r$, with $\epsilon_r=0.1$–$0.5$; these compact clusters form seeds from $5\times10^2$ to $1.7\times10^5\,M_\odot$, with the heaviest seeds appearing in the most compact systems and growing to a few times $10^9\,M_\odot$ by $z=0$. The resulting mass function agrees with the local SMBH mass function of Vika et al. (2009) above $10^8\,M_\odot$.

Load-bearing premise

Every predicted black hole assumes that nuclear star clusters were initially a factor 2 to 10 more compact than the present-day radius relation implies, because with no radius rescaling ($\epsilon_r=1$) no model forms a single seed.

Editorial extensions

If this is right

  • If the channel is correct, the low-mass end of the SMBH population is overproduced by the model: below $10^8\,M_\odot$ the predicted mass function exceeds the Vika et al. (2009) data by factors of 10–100, so other physics (feedback, accretion suppression, or additional seeds) must trim that population.
  • The heaviest seeds ($\sim10^5\,M_\odot$) require the most compact clusters ($\epsilon_r=0.1$) and form late in these halos, at $z\sim0.35$; accordingly the channel contributes mainly to the local SMBH population rather than to the $z\sim6$ quasar population.
  • The fraction of galaxies hosting both an NSC and an SMBH is set by $\epsilon_r$: models with $\epsilon_r=0.1$ give occupation fractions of 54–90%, while $\epsilon_r=0.5$ gives a few percent, making NSC compactness a predictor of SMBH occupation.
  • The predicted scaling relations $M_{\rm NSC}$–$M_{\rm galaxy}$ and $M_{\rm NSC}$–$\sigma$ overlap the observed ones regardless of $A_{\rm res}$, implying the in-situ channel alone can reproduce the shape of these relations even though the normalization depends on gas transfer efficiency.

Reading between the lines

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

  • The paper does not follow this up, but its parameter grid implies a sharp observable dichotomy: if collision seeding is the main channel, SMBH occupation should be nearly complete in the most compact NSCs and nearly absent in the least compact ones, at fixed host stellar mass — a prediction that can be tested by resolved kinematical surveys of nearby galactic nuclei.
  • A testable extension is to compare the predicted seed-mass distribution with gravitational-wave merger rates: heavy seeds of $\sim10^5\,M_\odot$ formed late would leave a distinctive imprint on the mass spectrum of binary black hole mergers detectable by future gravitational-wave observatories.
  • The model's reliance on $\epsilon_r$ could be turned into a measurement: fitting the observed local black-hole mass function with $\epsilon_r$ as a free parameter would constrain how much early expansion typical NSC progenitors undergo, connecting the seeding channel to cluster-formation theory.
  • If the overprediction below $10^8\,M_\odot$ is real rather than an artifact of the comparison sample, it suggests that many low-mass galaxies should contain black holes of $10^5$–$10^7\,M_\odot$ that current surveys might have missed; targeted X-ray or variability searches in dwarf galaxies would test this.
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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 / 7 minor

Summary. The paper implements an in-situ Nuclear Star Cluster (NSC) formation model in the semi-analytic galaxy formation code Galacticus, and couples it to a collision-based supermassive black hole (SMBH) seeding recipe. NSC gas is fed from the spheroid at a rate A_res times the spheroid star formation rate, NSC sizes follow the observed r_NSC-M_dyn relation with an ad hoc rescaling factor epsilon_r, and a BH seed forms when the NSC stellar mass exceeds a critical collision mass Mcrit; the seed mass is set to 0.5 times the NSC stellar mass and subsequently grows by Bondi/disk accretion. The authors run a grid over A_res, epsilon_r, and a minimum NSC mass threshold, and compare the predicted NSC scaling relations, NSC mass function, BHMF, and NSC-SMBH coexistence fractions with local observations. The headline claim is that compact NSCs (epsilon_r = 0.1-0.5) form BH seeds of roughly 5x10^2 to 1.7x10^5 solar masses that grow to SMBHs up to about 10^9 solar masses, and that the predicted z=0 BHMF is comparable to the Vika et al. (2009) observational BHMF above 10^8 solar masses.

Significance. If the central claim holds, the paper provides a useful proof-of-concept that collision-based seeding in dense NSCs can produce a cosmologically relevant SMBH population within a semi-analytic framework. The work is transparent: the parameter grid is clearly specified, the implementation is modular, and Appendix A presents convergence tests for both the NSC and BH mass functions. The significance is limited, however, by three structural features: every SMBH-forming model requires an ad hoc uniform compression of NSC radii, the 'observed' NSC mass function used for validation is itself derived by assuming a constant M_NSC/M_gal scaling rather than being a direct census, and the heaviest seeds in this implementation form only at low redshift (z ~ 0.2-0.35), so the channel as modeled does not directly address high-redshift SMBH formation. These caveats are partially acknowledged in the manuscript, but they materially weaken the strength of the headline BHMF comparison.

major comments (4)
  1. [Sec. 2.2.1, Eq. (2), Table 2] The entire seed-forming parameter space depends on the rescaling r_NSC -> epsilon_r r_NSC. With epsilon_r = 1, none of the models forms any BH seed (Table 2 lists no seed for A1, B1, C1, D1, and their epsilon_r = 1 counterparts), and the heaviest seeds, up to ~1.7x10^5 M_sun, occur only for epsilon_r = 0.1. The headline BHMF comparison is therefore conditional on the assumption that a substantial part of the NSC population has radii 2-10 times smaller than the mean relation of Eq. (2) at all masses and all times. The cited motivation, Banerjee and Kroupa (2017), concerns the early expansion of young massive clusters, not a permanent uniform compression of the whole NSC population. The authors should validate this assumption, for example by comparing the model's predicted z=0 NSC radius distribution with the observed scatter in the r_NSC-M_dyn relation, or by implementing a physically motivated, time-dependent NSC radius evolution and showing how the BHMF responds.
  2. [Sec. 3.4, Sec. 4, Table B.1] The central claim of a 'comparable' BHMF above 10^8 M_sun is not stated consistently. Section 3.4 says the BHMF from Vika et al. (2009) is comparable at masses above 10^8 M_sun, while Section 4 states that at higher masses the model tends to underpredict the observed population. The quantitative summary in Table B.1 reports average deviations with ±1 dex scatter over the full mass range, but no goodness-of-fit statistic restricted to the claimed 10^8-10^9 M_sun range is provided. Given that the low-mass BHMF is overpredicted by roughly two orders of magnitude at 10^6-10^7 M_sun, the paper should either report a quantitative agreement measure for the high-mass range (for example, chi-square or a Kolmogorov-Smirnov test over 10^8-10^9 M_sun) or consistently describe the high-mass behavior as an underprediction.
  3. [Sec. 3.2] The 'observed' NSC mass function used for validation is not a direct NSC census; it is derived from the GAMA galaxy stellar mass function by assuming a constant scaling M_NSC = 10^-3 M_galaxy (Georgiev et al. 2016). This proxy presupposes a linear scaling between NSC mass and galaxy mass, which is precisely the kind of relation the model is being tested against. The agreement in shape therefore cannot validate the absolute normalization or the low-mass behavior of the model. The manuscript already acknowledges the lack of a direct NSC sample; I recommend supplementing the comparison with direct NSC samples, for example the volume-limited samples of Sanchez-Janssen et al. (2019) or the sample of Pechetti et al. (2020), with selection effects accounted for, or at minimum clearly labeling the derived mass function as a conditional proxy rather than an observational benchmark.
  4. [Sec. 3.4, Appendix B] The paper attributes the low-mass BHMF overprediction to an excess of galaxies predicted by Galacticus relative to the Baldry et al. (2012) sample, but Appendix B reports a galaxy stellar mass function overprediction of only 0.5-0.6 dex, i.e., a factor of 3-4, while the BHMF is overpredicted by about 10^2 at 10^6-10^7 M_sun. A factor-of-100 discrepancy cannot be explained by the stated galaxy normalization offset alone. The authors should quantify how much of the BHMF excess follows from the galaxy mass function offset, from the epsilon_r-driven seeding recipe, and from the accretion model, for example by rerunning model A4 with weights that match the observed galaxy mass function or by decomposing the BHMF excess into occupation fraction and seed mass contributions.
minor comments (7)
  1. [Sec. 3.4] In the text 'Phi_bullet ~ 5 [Mpc^-1 dex^-1]' the unit should read Mpc^-3 dex^-1.
  2. [Sec. 3.2] The sentence 'the most massive NSC is is NGC 4461' contains a duplicated verb, and 'There is a clearly deviation' should be 'There is a clear deviation'.
  3. [Fig. 6 caption] The non-seeding condition is written as 'Mthreshold >= M_NSC'; since the seeding condition is M_NSC >= Mthreshold, the non-seeding condition should be M_NSC < Mthreshold, i.e., 'Mthreshold > M_NSC'.
  4. [Fig. 5 caption] The caption lists three values of epsilon_r (0.5, 0.2, 0.1) for the two models D3 and D4 shown, and the phrase 'Mthreshold = 10^3 M_sun 10^3 M_sun' repeats the mass; the caption should be cleaned up.
  5. [Table 2] The G4 row is missing its seed-mass range; Section 3.4 states that G4 forms SMBHs up to ~10^9 M_sun, so the blank entry appears to be an omission.
  6. [Sec. 3.1] The chi-square values mentioned in the text and in Figure 1 are not quoted in the text, so the reader cannot assess the fit quality or the degrees of freedom used.
  7. [Sec. 2.2.2] The fate of the initial 10 M_sun BH seed is not described once the new collision-formed seed replaces it; the text should clarify whether it merges with the new seed or remains as a separate object.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the BHMF comparison is an external benchmark, and the seed recipe is transferred from independent prior simulations rather than fitted to the target observable.

full rationale

The central prediction is the z=0 black hole mass function compared with Vika et al. (2009) in Section 3.4 and Fig. 7. Vika et al. (2009) is not among the calibration datasets listed in Section 2 (Leauthaud et al. 2012; Baldry et al. 2012; Muzzin et al. 2013; Kormendy & Ho 2013), so the comparison is not a fitted variable dressed as a prediction. Seed formation is set by the critical-mass condition (Eq. 17) and the seed mass relation M_dot = epsilon_dot M_NSC (Eq. 18), with epsilon_dot = 0.5 taken from the published numerical simulations of Vergara et al. (2023); that efficiency is not re-fit against the Vika et al. mass function. Final SMBH masses up to ~1e9 Msun are produced by later Bondi/Eddington-limited accretion (Eqs. 19-21) rather than being a direct rescaling of the seed masses, so the high-mass end of the BHMF is not a restatement of Eq. 18. The epsilon_r radius rescaling in Section 2.2.1 is presented as a free-parameter exploration, and the paper explicitly acknowledges in Section 4 that it does not explore the NSC mass-radius correlation from first principles but adopts the observed correlation from Neumayer et al. (2020) and varies it through epsilon_r. That is an unvalidated modeling assumption and a legitimate correctness risk, particularly because no seeds form for epsilon_r = 1 and heavy seeds form only for epsilon_r = 0.1, but it is not circular: the model is not validated against the same input that sets epsilon_r. Self-citations to Escala (2021) and Vergara et al. (2023) are load-bearing for the physical framework, but they are independent prior computational/observational results and are not used to define the target observable or to forbid alternative seeding channels. No derivation step in the paper reduces by construction to its own input.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim depends on several adopted scaling laws and efficiency parameters. The most fragile are the radius rescaling, the seeding criterion, and the fixed seed mass fraction, none of which are independently constrained within this paper.

free parameters (5)
  • A_res = varied 1e-4 to 1e-1
    Efficiency of gas transfer from spheroid star formation to the NSC reservoir (Eq. 1). It sets the NSC mass scale and therefore the BH seed population.
  • epsilon_r = grid: 1, 0.5, 0.2, 0.1
    Rescaling factor for the NSC radius (Section 2.2.1). Seed formation occurs only for values below 1, so this parameter is load-bearing.
  • epsilon_dot = 0.5
    Fraction of NSC stellar mass converted into the BH seed at seeding (Eq. 18). This value is assumed, not derived.
  • M_threshold = 1e3 or 1e4 solar masses
    Minimum NSC stellar mass required to form a seed, introduced to avoid nonphysical undersampled clusters. Its value delays seed formation in the higher-threshold models.
  • Accretion enhancement factors alpha = alpha_spheroid = 5, alpha_NSC = 5, alpha_CGM = 6
    Fixed multiplicative boosts on Bondi-Hoyle-Lyttleton accretion rates (Eq. 21). They help seeds grow to 10^9 solar masses and are chosen by hand from the Galacticus setup.
assumptions (6)
  • domain assumption The in-situ NSC gas supply rate is proportional to the spheroid star formation rate with constant efficiency A_res (Eq. 1).
    Borrowed from Antonini et al. (2015). It controls NSC growth and hence when and whether a seed can form.
  • ad hoc to paper The NSC radius follows the observed size-mass relation r_NSC = r0 sqrt(M_dyn / 10^6 M_sun) (Eq. 2), with an additional ad hoc rescaling by epsilon_r.
    Seed formation depends critically on compactness; without the epsilon_r rescaling no model forms any BH seed.
  • domain assumption NSCs are composed of equal-mass solar-type stars with M_star = 1 M_sun and R_star = 1 R_sun (Section 2.2.1).
    This enters the collision cross section and Safronov number. A realistic IMF would alter the collision timescale.
  • domain assumption A BH seed forms when the NSC stellar mass exceeds Mcrit from the condition t_coll = t_H, and the seed mass is epsilon_dot M_NSC (Eqs. 17 and 18).
    This is the core seeding criterion from Escala (2021) and Vergara et al. (2023). The 50 percent conversion efficiency is an adopted value.
  • domain assumption BH growth follows radiatively efficient thin-disk accretion with Bondi-Hoyle-Lyttleton rates and fixed alpha factors (Eqs. 19 to 21).
    The growth from light seeds to 10^9 solar masses depends on this standard but simplified accretion prescription.
  • domain assumption Galacticus best-match parameters calibrated to galaxy stellar mass functions and the BH-bulge relation are adopted from Knebe et al. (2018).
    The population weights and underlying galaxy formation physics are inherited from a previous calibration rather than re-fit in this paper.

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Pith. "Pith review of The supermassive black hole population from seeding via collisions in Nuclear Star Clusters." pith.science (2026). https://pith.science/paper/WANR4IJM

@misc{pith2026241208280,
  author       = {Pith},
  title        = {Pith review of: The supermassive black hole population from seeding via collisions in Nuclear Star Clusters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WANR4IJM}},
  note         = {Machine review of arXiv:2412.08280}
}
abstract

The coexistence of nuclear star clusters (NSCs) and supermassive black holes (SMBHs) in galaxies with stellar masses $\sim 10^{10}~$M$_\odot$, the scaling relations between their properties and properties of the host galaxy (e.g., $M_{NSC}^{stellar}-M_{galaxy}^{stellar}$, $M_{BH}-M_{galaxy}^{stellar}$), and the fact that NSCs seem to take on the role of SMBHs in less massive galaxies and vice versa in the more massive ones, suggest that the origin of NSCs and SMBHs is related. In this study, we implement an 'in-situ' NSC formation scenario, where NSCs are formed in the center of galaxies due to star formation in the accumulated gas. We explore the impact of the free parameter $A_{res}$ which regulates the amount of gas transferred to the NSC reservoir, playing a crucial role in shaping the cluster's growth. Simultaneously, we include a BH seed formation recipe based on stellar collisions within NSCs in the Semi-Analytical Model (SAM) Galacticus to explore the resulting population of SMBHs. We determine the parameter space of the NSCs that form a BH seed and find that in initially more compact NSCs the formation of these BH seeds is more favorable, leading to the formation of light, medium and heavy BH seeds which finally reach masses up to $\sim 10^9$~M$_\odot$ and is comparable with the observed SMBH mass function at masses above $10^8$~M$_\odot$. Additionally, we compare the resulting population of NSCs with a derived NSC mass function from the stellar mass function of galaxies from the GAMA survey at $z<0.06$ finding a well agreement in shape terms. We also find a considerable overlap in the observed scaling relations between the NSC mass, and the host galaxy stellar mass and velocity dispersion which is independent of the value of $A_{res}$. However, the chi-square analysis suggests that the model requires further refinement to achieve better quantitative agreement.

Figures

Figures reproduced from arXiv: 2412.08280 by the authors.

Figure 1
Figure 1. Left panel: Stellar mass of the NSC as a function of the stellar mass of the galaxy, for models A1, D1 and G1. Right panel: Stellar mass of the NSCs as a function of the stellar velocity dispersion of the galaxy for the same models. Magenta dots correspond to observed NSCs fol￾lowing [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. NSC stellar mass function for models listed in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Radius of the NSC (rNSC) multiplied by the respective efficiency ϵr , velocity dispersion (σ) at rNSC, age of the system (tH), and the red￾shift in function of the stellar mass of the NSC for models A2 (rose dots), A3 (blue dots), and A4 (red dots), where Mthreshold = 103 M⊙ and Ares = 10−1 for all the models. The values of ϵr are 0.5, 0.2, and 0.1 for models A2, A3 and A4 respectively, as listed in [PITH_FULL_IMAG… view at source ↗
Figures from the paper (4 more)
Figure 6
Figure 6. Figure 6: Same as [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Comparison between the BHMF determined by Vika et al. (2009) in gray dots with our semi analytic models at z = 0 for galaxies with stellar masses larger than 106 M⊙ and hosting NSCs with stellar masses above 103 M⊙. The gray dots include the ±1σ error bars, while the t…
Figure 8
Figure 8. Figure 8: Mstellar galaxy versus MBH/Mstellar NSC . The 2D PDF is shown as contour lines where the thick solid line mark the 1σ of the 2D PDFs. The best fit of the data is represented by dashed lines. From left to right, the contours indicate model A2 (light rose), A3 (blue), A4…
Figure 9
Figure 9. Figure 9: Same as [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

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