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Formation of supermassive stars and dense star clusters in metal-poor clouds exposed to strong FUV radiation

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper argues that supermassive stars can form below a metallicity threshold of about $10^{-3}\,Z_\odot$, with the most massive reaching $3$–$8\times10^4\,M_\odot$, and that above this threshold dense star clusters form instead.

desk verdict A plausible but underdetermined upward revision of the SMS metallicity threshold, built on one cloud realization and a two-point bracket. read the letter →

arxiv 2412.14900 v1 pith:LIGBJ5DF submitted 2024-12-19 astro-ph.GA astro-ph.SR

Sunmyon Chon , Kazuyuki Omukai This is my paper · ORCID
classification astro-ph.GAastro-ph.SR
keywords supermassivestarsdirectcollapseblackholesstarclusterformationmetallicitythresholdFUVradiationhydrodynamicssuper-competitiveaccretionPopulationIII
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 tries to show that the direct-collapse pathway to supermassive black holes does not require chemically pristine gas. Using radiation hydrodynamic simulations run for about two million years across metallicities $Z=10^{-6}$ to $10^{-2}\,Z_\odot$, it finds supermassive stars above $10^4\,M_\odot$ forming whenever $Z$ is at or below roughly $10^{-3}\,Z_\odot$, with final masses of $3$–$8\times10^4\,M_\odot$ for $[Z/H]\lesssim -3$. The key process is 'super-competitive accretion': despite sub-parsec fragmentation, the central massive stars capture most of the inflowing gas, by direct accretion or mergers. At $[Z/H] = -2$ this fails: fragmentation on 1–10 pc scales, aided by ionizing feedback, limits the most massive star to about $2000\,M_\odot$ and produces a bound dense cluster instead. A sympathetic reader should care because this raises the expected density of massive seed black holes toward the observed local number of supermassive black holes and offers a unified origin for supermassive stars and dense star clusters.

What carries the argument

The central mechanism is 'super-competitive accretion': after fragmentation creates many protostars and low-mass stars, the gravitational potential of the central massive stars steers the remaining gas into them, both through direct inflow and through mergers of inward-migrating fragments, so the gas supply is not split evenly among competitors. The surrounding physics that makes this work is the FUV radiation field, which dissociates H$_2$ and keeps the low-density gas warm near $10^4$ K, sustaining high infall rates of order $1\,M_\odot\,\mathrm{yr}^{-1}$; at higher densities, metal-line and dust cooling cause fragmentation, but on scales small enough that the fragments still fall into the center. The paper also models the stellar radiative feedback (ionizing radiation, H$_2$ dissociation, H$^-$ photodetachment, and dust heating) and shows that its main effect at low metallicity is to suppress low-mass star formation rather than to halt supermassive star growth.

What would settle it

A simulation of a different FUV-irradiated cloud at $[Z/H] = -3$ that fragments into cores separated by more than a few parsecs and fails to feed a central star past $10^4\,M_\odot$, or a high-redshift cluster with $[Z/H] > -3$ that demonstrably formed without a very massive star, would contradict the claim that the threshold is general.

Watch

Extended reading notes

Core claim

The central discovery is that supermassive stars with masses exceeding $10^4\,M_\odot$ can form in strongly FUV-irradiated clouds even when the gas is metal-enriched, as long as the metallicity stays below roughly $10^{-3}\,Z_\odot$. In the simulations, the most massive stars reach $6$–$8\times10^4\,M_\odot$ at $[Z/H]\lesssim -4$ and about $3\times10^4\,M_\odot$ at $[Z/H] = -3$, whereas at $[Z/H] = -2$ the most massive star barely exceeds $2000\,M_\odot$ and the system becomes a dense star cluster. The growth of the central stars is sustained by 'super-competitive accretion', in which the gravitational dominance of the central massive stars channels most of the gas reservoir toward them even while sub-parsec fragmentation produces many low-mass companions. Radiative feedback from the forming stars is ineffective at stopping this growth at low metallicity, because high accretion rates keep the surroundings dense enough to absorb ionizing photons, although dust heating does suppress low-mass star formation. These results revise the earlier expectation that metal enrichment at levels above about $10^{-5}\,Z_\odot$ would prevent direct-collapse seed formation.

Load-bearing premise

The load-bearing premise is that one simulated cloud, with its particular density, spin, turbulence, and tidal environment, stands in for all FUV-irradiated metal-poor clouds; if other real clouds differ enough, the $10^{-3}\,Z_\odot$ threshold and the super-competitive accretion mode could shift or disappear.

Editorial extensions

If this is right

  • If the threshold holds, supermassive star formation becomes viable in clouds with a prior episode of star formation, not only in pristine gas, increasing the expected number density of massive seed black holes.
  • Using the semi-analytic halo counts and a 5% collapse-success correction, the paper estimates a seed black hole density of $0.1$–$1\,\mathrm{Mpc}^{-3}$ for $[Z/H]\gtrsim -3$, comparable to the local supermassive black hole density.
  • At $[Z/H] = -2$, the outcome is a bound, compact cluster with a central very massive star of about $1000$–$2000\,M_\odot$ and a stellar surface density near $10^3\,M_\odot\,\mathrm{pc}^{-2}$, resembling young massive or globular clusters and matching the observed metallicity floor of Milky Way globular clusters around $[Z/H] \sim -2.5$.
  • The simulated remnant population contains four to six massive seed black holes per system plus a population of intermediate-mass black holes, and the massive black holes often form binaries whose gravitational-wave mergers could be detectable at high redshift.
  • The low-metallicity mass spectra are composite: a Chabrier-like low-mass population peaking near $1\,M_\odot$, produced and shaped by stellar irradiation, plus a supermassive component of $10^4$–$10^5\,M_\odot$ that dominates the total stellar mass.

Reading between the lines

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

  • Inference: because the simulations start from a single cloud realization, the $10^{-3}\,Z_\odot$ threshold may not be universal; clouds with different spin, turbulence, or tidal fields could shift the boundary by a factor of several.
  • Inference: if the seed black hole density really reaches $0.1$–$1\,\mathrm{Mpc}^{-3}$, the scenario may remove the need for super-Eddington accretion to explain the local supermassive black hole population.
  • Inference: nitrogen-rich compact clusters seen in the early universe could serve as observational tests; clusters with metallicities above the threshold that still show evidence of very massive stars would weaken the claim, while clusters below the threshold would support the supermassive-star nucleosynthesis route.
  • Inference: magnetic fields, which are not included here, tend to suppress disk fragmentation and could push the supermassive star formation channel to higher metallicities or to even larger final masses.
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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

3 major / 4 minor

Summary. The paper presents three-dimensional radiation hydrodynamics simulations of star formation in strongly FUV-irradiated, metal-poor clouds, with metallicities from [Z/H] = -6 to -2. The authors follow collapse from a single cosmological initial condition (the Spherical Cloud of Chon et al. 2016), including non-equilibrium primordial chemistry, simplified C+ and O line cooling, dust cooling and heating, and radiative feedback from accreting protostars. Using paired high-resolution/short-term and low-resolution/long-term runs, they find that SMSs of 3-8e4 Msun form for [Z/H] <= -3, that the [Z/H] = -3 case catches up to the SMS regime only after about 1e5 yr via accretion from a pc-scale dense filament, and that at [Z/H] = -2 the central star reaches only about 2e3 Msun while a compact star cluster forms. The paper interprets this as a metallicity threshold Z_th ~ 1e-3 Zsun for SMS formation, proposes super-competitive accretion as the responsible mechanism, and uses this threshold to argue that massive seed black holes may be abundant enough to match local SMBH number densities.

Significance. If the result holds, it is an important step beyond Paper I: it relaxes the direct-collapse requirement from essentially pristine gas to Z up to roughly 1e-3 Zsun, roughly an order of magnitude above the earlier threshold, and it gives a concrete mechanism (super-competitive accretion) through which fragmentation does not prevent SMS growth. The simulations are substantially improved over Paper I by including self-consistent thermal/chemical evolution, stellar radiative feedback, and two-million-year runs, and the resolution study in Appendix A directly checks the main mass-budget claim. The comparison with one-zone thermal tracks in Figure 4 is a useful sanity check. The claimed transition from SMS formation to dense cluster formation at [Z/H] = -2 is interesting and potentially relevant to globular cluster and JWST compact-cluster observations. However, the central threshold claim is currently supported by only one cloud realization and by a two-point metallicity bracket, and the cosmological abundance estimate relies on an extrapolated collapse fraction; these issues make the quantitative threshold and the SMBH abundance conclusion less secure than the abstract suggests.

major comments (3)
  1. [Section 2.1 and Section 3.1] All five metallicity runs start from the same Spherical Cloud initial condition, with the same density field, velocity field, angular momentum, turbulence, and tidal environment. The decisive difference between [Z/H] = -3 and [Z/H] = -2 is attributed to the scale at which metal-line cooling triggers fragmentation (pc-scale filament versus 1-10 pc cores), but this outcome is sensitive to the specific structure of this single cloud. In particular, the late-time SMS growth at [Z/H] = -3 requires a warm outer reservoir delivered by a pc-scale filament; whether such a configuration is generic for FUV-irradiated, mildly enriched halos is not demonstrated. The paper should either add variations of the initial turbulent field, rotation, or tidal truncation, or explicitly restrict the claim to this cloud class and state that the universal threshold is not yet established.
  2. [Section 3.1 and Figure 4] The claimed threshold near [Z/H] = -3 is bracketed only by the two metallicities [Z/H] = -3 and -2, a factor of ten in Z, with no run at an intermediate value such as [Z/H] = -2.5. This matters because the [Z/H] = -3 case is also the one where the simulation deviates most strongly from the one-zone model (the text attributes this to collapse faster than free-fall), so the success at [Z/H] = -3 is not backed by the same analytic expectation as the lower-metallicity cases. A run at [Z/H] = -2.5, or a quantitative analytic estimate of when 10-pc-scale fragmentation starts to starve the center, is needed before the abstract's 'approximately 1e-3 Zsun' threshold can be considered robust.
  3. [Section 4 and Figure 15] The cosmological conclusion that massive seed BHs reach number densities of 0.1-1 Mpc^-3 and can account for local SMBHs uses a fixed 5% success rate from Chon et al. (2016), applied to the candidate-halo sample of Chiaki et al. (2023). That 5% fraction was derived for the same cloud population from which the present initial condition is drawn, and it is not shown to apply to finite-metallicity, FUV-irradiated halos. Since the ordinate of Figure 15 is directly proportional to this correction factor, a metallicity-dependent or halo-dependent collapse fraction could shift the inferred abundance by orders of magnitude. The authors should either compute or bound f_coll for the finite-metallicity case, or present the abundance prediction as a conditional estimate with explicit sensitivity to this parameter.
minor comments (4)
  1. [Section 3.3.2 and Table 1] The text states that for [Z/H] = -2 the authors 'adopt a slope alpha of 1.5 instead of the value of 1 obtained from the fitting,' but Table 1 reports alpha = 1.09 for [Z/H] = -2; the discrepancy between 1 and 1.09 should be reconciled.
  2. [Appendix A, Figure A1 caption] The caption contains the typo 'reolution' for 'resolution' in the labels, which should be corrected.
  3. [Section 2.2] Metal-line cooling is limited to C+ and O, with C and O assumed fully ionized/neutral respectively; since the [Z/H] = -2 boundary is where the transition occurs, the absence of other coolants (e.g., Si+, Fe+) is a simplification that could shift the threshold. The text acknowledges the simplification, but a brief discussion of its expected direction and magnitude would help.
  4. [Section 2.5] The adiabatic cutoff at n_adia = 1e16 cm^-3 and sink formation at twice that density mean that the final SMS masses are determined partly by unresolved gas above this density. This is a standard opacity-limit treatment, and Appendix A shows total mass convergence between resolutions, but the paper should state more explicitly that the quoted final masses are the masses of the resolved sinks and could be affected by sub-grid accretion or disk physics.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity: the Z≈10^-3 Zsun threshold emerges from the hydrodynamics, and the self-citations enter only as independent prior simulation inputs in the cosmological extrapolation.

full rationale

The paper's central claim, that SMSs of ≳10^4 Msun form even at Z≲10^-3 Zsun, is not fitted or defined into existence. It is a computed outcome of radiation-hydrodynamic simulations with non-equilibrium chemistry, metal and dust cooling, and radiative feedback: the [Z/H]=-3 run grows a 3×10^4 Msun star through late accretion from a pc-scale dense filament, while the [Z/H]=-2 run fragments on 1-10 pc scales and yields only ~2000 Msun. This contrast is an emergent result, not an input. The thermal evolution is checked against external one-zone models (Fig. 4), the massive-star growth is shown to be resolution-insensitive (Appendix A), and the feedback on/off comparison brackets the outcome. The initial cloud is taken from the authors' own Chon et al. (2016) cosmological simulation, and the Section 4 number-density estimate uses Chiaki et al. (2023) with a 5% collapse fraction from Chon et al. (2016); these are prior simulation products used as inputs, not parameters tuned to force the present conclusion, so the self-citations are not circular in the load-bearing sense. The paper itself acknowledges unmodeled magnetic fields and mass loss as uncertainties. The main caveats—only two metallicity bracket points and a single initial condition—are robustness or underdetermination concerns, not reductions of the derivation to its own inputs. No circular step with a quotable equation-to-equation equivalence was found.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central threshold emerges from simulations rather than being fitted, so the circularity burden is low. The explicitly fitted IMF parameters are used only to reconstruct low-mass populations and do not set SMS masses. The collision fraction of 0.05 is an adopted external success fraction from the authors' own earlier simulation, which adds mild self-citation to the cosmological implication but not to the central claim. No new physical entities are introduced; super-competitive accretion is a descriptive label for an emergent accretion mode.

free parameters (3)
  • IMF high-mass slope alpha = [Z/H]=-6:1.46, -5:1.51, -4:1.57, -3:1.57, -2:1.09
    Fitted to the high-resolution mass spectra via Eq. 11 and Table 1; used to reconstruct the unresolved low-mass population in long-term runs. Does not set the SMS masses.
  • IMF peak mass m0 = [Z/H]=-6:0.98, -5:0.93, -4:0.53, -3:0.53, -2:0.05 Msun
    Best-fit peak masses from the same tapered power-law fit; used only for low-mass reconstruction, not for the massive-star growth that determines SMS formation.
  • Seed BH collapse fraction f_coll = 0.05
    Adopted from Chon et al. (2016) and applied in Section 4 and Fig. 15 to convert candidate halo densities from Chiaki et al. (2023) into a seed BH number density. It is a prior simulation-based success fraction and is a mild self-citation in the cosmological extrapolation.
assumptions (5)
  • domain assumption Carbon and oxygen are entirely in the forms of C+ and O, and metal-line cooling is limited to [CII] and [OI].
    Invoked in Section 2.2. This simplification affects the thermal evolution and fragmentation scale, which directly sets the metallicity threshold between SMSs and clusters.
  • domain assumption Dust abundance, composition, and size distribution follow the Milky Way and scale linearly with metallicity.
    Section 2.2. Dust cooling drives sub-pc fragmentation and the behavior near [Z/H] = -4 and -3, so this scaling is load-bearing for the threshold.
  • domain assumption The cloud is exposed to an FUV radiation field of J21 = 1000 with a 10^4 K blackbody spectrum.
    Section 2.2. This is the standard direct-collapse strength, but the metallicity threshold and feedback behavior could depend on the assumed intensity.
  • ad hoc to paper Above n_adia = 10^16 cm^-3, gas evolves adiabatically, and sink particles form at 2 n_adia; no cooling is modeled past this density.
    Section 2.5. This numerical choice is standard in this group but is not independently calibrated here and could affect the final stellar masses.
  • domain assumption One-dimensional stellar evolution tables with f_acc = 0.75 and Mdot_crit = 0.04 Msun/yr describe protostar radii, luminosities, and feedback.
    Section 2.3. Radiative feedback strength depends on these prescriptions, especially the supergiant protostar phase, which is important at [Z/H] = -2.

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Pith. "Pith review of Formation of supermassive stars and dense star clusters in metal-poor clouds exposed to strong FUV radiation." pith.science (2026). https://pith.science/paper/LIGBJ5DF

@misc{pith2026241214900,
  author       = {Pith},
  title        = {Pith review of: Formation of supermassive stars and dense star clusters in metal-poor clouds exposed to strong FUV radiation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LIGBJ5DF}},
  note         = {Machine review of arXiv:2412.14900}
}
abstract

The direct collapse scenario, which predicts the formation of supermassive stars (SMSs) as precursors to supermassive black holes (SMBHs), has been explored primarily under the assumption of metal-free conditions. However, environments exposed to strong far-ultraviolet (FUV) radiation, which is another requirement for the direct collapse, are often chemically enriched to varying degrees. In this study, we perform radiation hydrodynamic simulations of star-cluster formation in clouds with finite metallicities, $Z=10^{-6}$ to $10^{-2} Z_{\odot}$, incorporating detailed thermal and chemical processes and radiative feedback from forming stars. Extending the simulations to approximately two million years, we demonstrate that SMSs with masses exceeding $10^4~M_\odot$ can form even in metal-enriched clouds with $Z \lesssim 10^{-3} Z_{\odot}$. The accretion process in these cases, driven by "super-competitive accretion," preferentially channels gas into central massive stars in spite of small (sub-pc) scale fragmentation. At $Z \simeq 10^{-2} Z_{\odot}$, however, enhanced cooling leads to intense fragmentation on larger scales, resulting in the formation of dense star clusters dominated by very massive stars with $10^3 M_{\odot}$ rather than SMSs. These clusters resemble young massive or globular clusters observed in the distant and local universe, exhibiting compact morphologies and high stellar surface densities. Our findings suggest that SMS formation is viable below a metallicity threshold of approximately $10^{-3} Z_{\odot}$, significantly increasing the number density of massive seed black holes to levels sufficient to account for the ubiquitous SMBHs observed in the local universe. Moreover, above this metallicity, this scenario naturally explains the transition from SMS formation to dense stellar cluster formation.

Figures

Figures reproduced from arXiv: 2412.14900 by the authors.

Figure 1
Figure 1. Growth histories of the most massive stars at the end of the simulations for different metallicities: [Z/H]= −2 (purple), [Z/H]= −3 (blue), [Z/H]= −4 (green), [Z/H]= −5 (red), and [Z/H]= −6 (yellow). The data are derived from the long-term, low-resolution runs. For [Z/H]= −2, the dashed line represents the case without stellar feedback, while for the other metallicities, the results with and without feedback are sim… view at source ↗
Figure 2
Figure 2. Projected density distributions at three different spatial scales for various metallicities, [Z/H]= −6 to −2 (from left to right), taken at the epoch when the first protostar forms. The snapshots are from the high-resolution runs, which resolve gas densities up to ∼ 1016 cm−3 . Each column corresponds to a different metallicity and shows the distribution at progressively smaller scales. Note that each panel uses a d… view at source ↗
Figure 3
Figure 3. Temperature distributions corresponding to the density distributions shown in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: The gas distributions on density-temperature phase plots at the epoch of the first protostar formation for different metallicities, [Z/H]= −6, −5, −4, −3, and −2, from left to right panels. The plots are derived from high-resolution runs. The color coding represents th…
Figure 5
Figure 5. Figure 5: The gas infall rate, estimated from the snapshot at the time of the first protostar formation, as a function of the enclosed mass within a given radius (see text) for different metallicities: [Z/H]= −2 (purple), [Z/H]= −3 (blue), [Z/H]= −4 (green), [Z/H]= −5 (red), and…
Figure 6
Figure 6. Figure 6: Projected density distributions at three different epochs following the formation of the first protostar for various metallicities: [Z/H]= −6, −5, −4, −3, and −2, from left to right. The top and middle rows show the distributions at 100 years and 1,000 years, respectiv…
Figure 7
Figure 7. Figure 7: Fraction of the mass of the most massive star in each simulation that was acquired through stellar mergers, as opposed to gas accretion, at the end of the respective runs. The top panel shows results from the short￾term, high-resolution runs at 104 years for all metall…
Figure 8
Figure 8. Figure 8: Cloud structures for different metallicity cases ([Z/H]=-6,..., -2 from left to right columns) at the onset of cloud evaporation due to stellar radiative feedback. The top panels display the projected density distributions, while the bottom panels show the correspondin…
Figure 9
Figure 9. Figure 9: Top panel: Cumulative mass distribution of stars with masses 𝑀∗ < 100 𝑀⊙ for different metallicities: [Z/H]= −2 (purple), [Z/H]= −3 (blue), [Z/H]= −4 (green), [Z/H]= −5 (red), and [Z/H]= −6 (yellow) at 𝑡 = 2 Myr taken from the long-term, low-resolution runs. Bottom pan…
Figure 10
Figure 10. Figure 10: The mass distributions at the end of the high-resolution simulation for [Z/H]= −6, −5, −4, −3, and −2 from left to right columns. The top and bottom panels show the distribution when we do not include and include stellar feedback. We also attach the time at the end of…
Figure 11
Figure 11. Figure 11: The same as [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: Distributions of gas density and temperature at 104 yr after the formation of the first protostar, same as in [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 14
Figure 14. Figure 14: Projected distributions of density (left), gas temperature (middle), and dust temperature (right) for the case of [Z/H]= −2, similar to [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]
Figure 15
Figure 15. Figure 15: The number density of massive seed BHs as a function of the threshold metallicity below which SMSs can form. The BH densities are derived from the dark matter-only semi-analytic calculations by Chiaki et al. (2023), corrected by assuming that only 5% of the candidate …
Figure 16
Figure 16. Figure 16: The mass function of remnant BHs in different metallicity environments. Stars with masses in the range 30𝑀⊙ < 𝑀∗ < 80𝑀⊙ or 𝑀∗ > 260 𝑀⊙ are assumed to directly collapse into BHs, following Heger & Woosley (2002). No mass loss processes are considered in this calculatio…

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    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

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