Pregalactic globular cluster formation

Jarrod Hurley; Jeremy Mould

REVIEW 4 major objections 5 minor 2 references

Pregalactic globular cluster formation

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read ~1000-solar-mass black holes from the QCD phase transition can act as pregalactic nuclei, accreting smaller black holes into a dark cluster that draws in gas and triggers globular cluster formation by z~100, with the nucleus persisting.

desk verdict A speculative but honest scenario paper that links QCD-transition PBHs to globular cluster formation; the N-body growth runs are real and the counter-evidence section is candid, but the gas-to-star transition is explicitly not simulated, which is exactly the link that needs to hold. read the letter →

arxiv 2509.02165 v1 pith:JWT7Y3Q6 submitted 2025-09-02 astro-ph.GA

Jeremy Mould , Jarrod Hurley This is my paper · ORCID

classification astro-ph.GA PACS 97.60.Lf98.20.Gm

keywords globularclustersprimordialblackholesQCDphasetransitionpregalacticstarformationBondi-Hoyleaccretionintermediate-massdark

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 proposes a new answer to an old puzzle: how globular clusters — the oldest star clusters in the Universe — formed before galaxies did. The authors argue that primordial black holes created at the QCD phase transition, with masses spread up to about 1000 solar masses, would act as pregalactic nuclei: the dominant black hole accretes its smaller companions into a dense 'dark cluster' that emits no light but creates a deep potential well. Gas falling into this well would reach densities around 10^-6 grams per cubic centimeter and become Jeans-unstable, cooling and fragmenting into a luminous globular cluster by redshift ~100. Once formed, the nuclear black hole stays, continuing to feed on a fraction of the mass lost by evolved stars. The paper stops short of simulating the gas: the link between the simulated dark cluster and a real star cluster is asserted from density and Jeans-length estimates, and the authors specify kinematic and X-ray observations that could detect — or rule out — such nuclei.

What carries the argument

The load-bearing object is the QCD-transition primordial black hole: a black hole of order 1000 solar masses formed when the Universe cooled through the QCD phase transition at T~220 MeV, the top of a mass distribution that spreads equal mass per decade. A dominant-mass nucleus accretes smaller PBHs at the Bondi-Hoyle rate dm/dt = πσρv, with the sound speed taken from prior work, in N-body simulations of 10^5 particles started near z=1300. What carries the argument toward a globular cluster is the 'dark cluster' — the tightly concentrated potential well (several thousand solar masses within ~10 AU) formed by the accreted PBHs. Gas drawn into this well reaches densities ~10^-6 g/cm^3, where t

What would settle it

Measure the stellar velocity dispersion in the cores of a large sample of the oldest metal-poor globular clusters: a 1000-solar-mass nucleus at 7.6 kpc would raise the dispersion to ~5.4 km/s within 2 arcseconds of the centre, and the paper notes M92's current 1σ upper limit of 980 solar masses sits right at that edge. If a survey of blue clusters at that sensitivity finds no central dark mass in the 10^3–10^4 solar-mass range, the mechanism is ruled out for those clusters; a clean complement is a deep stacked X-ray search showing no central sources where Bondi accretion predicts detectable lu

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Extended reading notes

Core claim

The paper's claim is that QCD-transition primordial black holes up to ~1000 solar masses can seed globular clusters. In gravity-only N-body runs of 100,000 particles, a dominant-mass nucleus grows by accreting smaller PBHs through Bondi-Hoyle capture — e.g., from 5000 to 25,000 solar masses by late times — producing a strongly concentrated dark cluster whose potential well deepens by an order of magnitude. The authors infer that baryons falling into this well reach densities ~10^-6 g/cm^3 with Jeans lengths of order 0.1 AU below 116 K, so star formation ignites around z~100 and a globular cluster forms around the persistent intermediate-mass black hole. Old metal-poor globular clusters are t

Load-bearing premise

Everything rests on what happens to the gas: the paper assumes baryons falling into the simulated dark cluster cool and fragment into a luminous globular cluster by redshift 100, but it does not simulate that step — if the gas fails to collapse and make stars, the model fails even if the black-hole accretion runs are correct.

Editorial extensions

If this is right

Metal-poor (blue) globular clusters, the oldest stellar systems, should harbour nuclear black holes of order 10^3–10^4 solar masses that persist today and keep accreting a small fraction of the mass shed by evolved stars.
High-resolution kinematics of cluster cores can now test this: a 1000-solar-mass nucleus would raise the velocity dispersion to ~5.4 km/s within 2 arcseconds of the centre of a cluster at 7.6 kpc, a signal within reach of current instruments on clusters like M92.
If accretion disks form around the nuclei, the clusters should emit weakly in X-rays or the far ultraviolet — or be hidden by dust and gas; present Chandra data for 75 of 81 Milky Way clusters already constrain such emission.
Supernovae in the first stellar generation would enrich pristine gas, and gas loss from the shallow potential would yield the low metal abundances and bimodal colour distribution observed in globular cluster systems.
Stellar-mass black holes captured and merged around these nuclei during and after cluster formation could contribute to the cosmic gravitational-wave background.

Reading between the lines

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

If the mechanism is right, the mass of the nuclear black hole should correlate with cluster age and metallicity — the oldest, most metal-poor clusters formed earliest when PBH densities were highest, so they should show the most massive nuclei; this correlation is not derived in the paper but is directly testable.
The model could turn globular clusters into indirect probes of the primordial black hole abundance: the observed incidence of nuclear dark masses in metal-poor clusters would calibrate the QCD-transition PBH fraction far below the ~1% dark-matter limit quoted in the paper, and even null results would set a competitive upper bound.
The paper's required endpoint — gas cooling to ~100 K and fragmenting by z~100 inside a dark-matter-dominated well — is precisely the regime of first-star formation; a natural extension would be a cosmological hydrodynamical simulation seeded with one of these dark clusters, turning the asserted step into a computed one.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know this paper is a scenario sketch, not a completed model. The new piece is the specific chain: a ~1000 M_sun PBH formed at the QCD transition acts as a nucleus, gravitationally accretes smaller PBHs into a dark cluster at z~1300, and then gas supposedly falls into that potential, cools, and fragments into a globular cluster by z~100. That specific proposal, with new N-body runs and an explicit prediction of nuclear PBHs in old GCs, is genuinely not in the cited literature. The authors deserve credit for running the simulations, for being explicit that the runs are dark-matter-only, and for writing a counter-evidence section that takes the strongest observational constraints (X-ray upper limits, kinematic limits like M92 and NGC 6362) seriously. They don't oversell: the abstract says “may,” and the conclusions call the model “currently viable” with a decisive test within reach.

The soft spot is the one the stress-test note names, and it's real. Section 3 says, in effect, we don't simulate the gas, we just outline what we expect. That is the load-bearing step. If the baryons don't actually cool and collapse into stars, you don't get a luminous GC, no matter how well the PBH accretion runs work. The paper's own Section 2 notes that the gas is hotter than the dark matter and has a significant relative velocity on small scales, which makes quiet infall and cooling less automatic than the outline implies. No cooling mechanism (H2 or otherwise) is shown to bring the gas from those conditions down to the T<116 K and densities used for the Jeans argument. On top of that, the baryonic mass estimate uses Run 82, one of the most favorable runs (5000 -> 25000 M_sun), while several other runs show almost no growth (e.g., Runs 75, 78, 79). There are no error bars, no control runs, and the code is available but unversioned. These are not fatal for a scenario paper, but they mean the quantitative part is shakier than the narrative suggests.

That said, the authors are transparent about the limits. They aren't hiding the unmodeled transition; they flag it explicitly. The work is clear thinking on its own terms, the citations are appropriate, and the dependence on their earlier results (Mould 2025, Mould & Batten 2025) is a dependency, not a circularity. The paper is for people working on GC formation and PBH phenomenology who want a testable alternative to the usual protogalactic formation channels. It deserves a serious referee, with the main request being a real baryonic simulation or at least a quantified cooling and fragmentation argument before the central claim is taken as supported. I'd accept it for review and let the referee push on that step.

Referee Report

4 major / 5 minor

Summary. The paper proposes that primordial black holes formed at the QCD phase transition, with a log-flat IMF up to ~1000 Msun, serve as gravitational seeds for globular clusters. N-body simulations of PBH accretion in an expanding universe show that a dominant 'nucleus' can grow in some configurations (e.g., Run 82 from 5000 to 25000 Msun). The authors assume the accompanying baryons (15% of the final PBH mass) fall into the dark matter potential well, cool, and fragment into a luminous globular cluster by z~100, and they argue that old GCs would then harbor nuclear PBHs. The paper surveys X-ray, kinematic, and CMB constraints and concludes that current observations do not rule out the scenario.

Significance. If the gas-to-star step could be established, this would be an important and falsifiable formation channel for globular clusters and would connect PBHs to an observable population. The paper's strengths are its relatively simple, reproducible N-body code (available at github/jrmould/darkmatter), its explicit engagement with kinematic and X-ray counter-evidence, and its specific predictions about nuclear PBHs and a possible gravitational wave background. The main weakness is that the decisive baryonic collapse and fragmentation step is not modeled; the paper itself says it 'outline[s] what we expect to happen.' The quantitative baryon budget from the fiducial run is also far below a typical GC mass. The scenario is conditional, not established.

major comments (4)

[Section 3] The central claim that PBH dark clusters become luminous GCs rests on an unmodeled baryonic transition. Section 3 states 'We do not follow this with further simulation, but outline what we expect to happen,' and Section 2 notes that gas is 'significantly hotter than the dark matter' and has 'significant net relative velocity on small scales.' No cooling mechanism (e.g., H2) is demonstrated to bring pristine gas to T<116 K, and the Jeans-length estimate is not tied to a fragmentation criterion. The simulations by themselves establish only dark clusters; the GC claim requires a hydrodynamical treatment or at least a quantitative cooling/condensation argument.
[Section 3 / Run 82] The baryon mass budget is orders of magnitude too small. The paper takes Run 82's final PBH mass of 25000 Msun and assigns 'accompanying baryons Omega_b/Omega_m = 15% of that mass,' i.e. ~3750 Msun. A globular cluster contains ~1e5-1e6 Msun in stars. The claim that 'GC sized gas clouds need little encouragement to collapse' does not follow unless the dark cluster subsequently accretes ~1e5 Msun of baryons from the surrounding medium; that accretion is not included in the runs or quantified.
[Table 1 / Section 2.1] The fiducial growth run is not representative. Table 1 shows substantial growth only in Runs 80, 81, 82, and 84; Runs 75, 76, 78, 79, 80a, 83, and 85 grow by factors below ~1.5 or not at all. The text adopts Run 82 for the baryon scaling and figures emphasize its growth. There are no error bars, no multiple realizations of the same parameters, and no convergence or resolution tests. The reader cannot assess whether the 'nucleus remains' conclusion is robust or specific to initial conditions.
[Section 4.2] The kinematic test in Section 4.2 is treated too optimistically. For the archetypal metal-poor cluster M92, Kamann et al. (2014) find a 1-sigma upper limit of 980 Msun and a 3-sigma limit of 2700 Msun on a central IMBH. The model's predicted final masses range from ~10^3 to 2.5e4 Msun; thus the 3-sigma M92 limit rules out the high-growth runs, and the 1-sigma limit is at the lower end of the predicted range. The paper's response (NGC 6362 is not metal-poor) does not address M92, which is directly relevant to 'oldest globular clusters.' A quantitative likelihood statement is needed.

minor comments (5)

[References] Reference list formatting: 'V olonteri', 'Senchnya', and 'Carr & Kuhnel 2021 arxiv 21100282' contain typos or incomplete fields; the text cites Harris (1986) while the reference entry is listed as Harris 1996.
[Figure 2] The caption says 'The scale is in pixels, which are effectively AU' but the pixel-to-AU conversion is not defined; please add units or a scale bar.
[Section 2.1] The text says '100000 particle n-body runs' but Table 1 lists Runs 80 and 82 as having 150000 particles; clarify the default particle number.
[Abstract/Conclusions] The abstract and conclusions emphasize 1000 Msun nuclei, but several simulations end at 25000 Msun. Specify whether the initial or final mass is the prediction for GC nuclei.
[Section 2.1] The accretion formula dm/dt = pi sigma rho v is not numbered; number the equations for easier reference in the text.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: PBH mass and IMF are external inputs; the N-body runs and falsifiable predictions are independent content.

full rationale

The paper's central claim is conditional on the prior existence of QCD-transition PBHs and a log-flat IMF, both taken from earlier work by the authors (Mould 2025; Mould & Batten 2025). These are external premises, not results derived in this paper. The N-body simulations are not calibrated to globular cluster observations; outputs such as final PBH masses are not used to adjust the initial nucleus mass or IMF. The gas-to-star transition is explicitly not simulated ('We do not follow this with further simulation, but outline what we expect to happen'), which is an acknowledged assumption/limitation, not a circular step. The counter-evidence section uses independent observational constraints (X-ray limits, kinematics) that could falsify the model. No equation in the paper reduces to its inputs by construction, and no fitted parameter is relabeled as a prediction. Self-citations appear, but they supply external inputs or peripheral remarks; they do not form an unverified justificatory chain for the central claim. Score 2 reflects a minor non-load-bearing self-citation, not circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the assumed existence and mass function of QCD-era PBHs (external), the simulation choice that only the nucleus accretes, and the unmodelled assumption that baryons captured by the dark cluster cool and form a globular cluster. The paper's own counter-evidence section shows that current observations neither confirm nor refute the nuclear PBH prediction.

free parameters (5)

Initial PBH nucleus mass m = 1000 M_sun fiducial; runs 2000, 3000, 5000, 10000 M_sun in Table 1
The nucleus mass is chosen to be the maximum of the assumed QCD mass function, not the typical mass; runs vary it widely, so the accretion outcome is sensitive to this choice.
PBH mass function endpoints m1, m2 and shape = Top-hat in log m, e.g. m1=1, m2=1000 (Run 75); alternate blue-tilted IMF in Run 80a
The IMF determines how much mass is available for accretion; it is assumed and only one alternate shape is tested. Mould and Batten (2025) is cited as external justification.
Dark matter fraction in PBHs (f') = Not fixed; listed as a parameter to explore in Section 2.2
The normalization of the PBH population is not constrained in this paper; Chen and Hall (2024) limits are discussed but not adopted in the simulations.
Simulation start redshift = z=1300; z=1100 for Run 75
Accretion is concentrated at early times in Figure 4, so the initial epoch sets the density and the final mass; no systematic scan is presented.
Gas collapse and star-formation threshold = z~100; Jeans criterion with T<116 K; baryon fraction 15% of run 82 nucleus
The baryonic part is not simulated; the conditions under which gas fragments into a GC are estimated by hand in Section 3.

assumptions (5)

domain assumption PBHs exist with masses reaching ~1000 M_sun from the QCD phase transition, with a log-flat mass function.
The whole model starts from this prior-literature assumption; the paper says 'Whether PBH exist or not remains an open question' (Section 1).
ad hoc to paper The dominant PBH in an overdensity accretes smaller PBHs via Bondi-Hoyle accretion, and only the nucleus accretes.
Imposed in the simulation setup; Table 1 caption states 'Only the nucleus accretes', which suppresses competing growth.
ad hoc to paper The dark-matter-only potential captures baryons that cool and form a globular cluster by z~100.
This is the unmodelled leap in Section 3: 'We do not follow this with further simulation, but outline what we expect to happen'.
standard math The standard n-body integration scheme (delta v = a dt, delta r = v dt) with adaptive timesteps is accurate enough at the reported resolution.
The integration is a simple Euler scheme with a softening at 10^-20; no convergence tests are shown (Section 2.1).
domain assumption The oldest, metal-poor (blue) globular clusters are the objects formed by this pregalactic mechanism.
Section 3.1 assigns blue clusters to the pregalactic channel and red clusters to a post-galactic channel without a quantitative comparison to the bimodal color distribution.

invented entities (1)

Pregalactic 'dark clusters' of PBHs (GC precursors)
purpose: Intermediate stage between a single 1000 M_sun nucleus and a luminous globular cluster; provides the potential well that later collects baryons for star formation.
The paper says it has 'simulated the formation of dark clusters of PBH' (Section 3.1), but these clusters have no specified direct observational signature; their existence is inferred only through the final GCs, which are the same objects the model aims to explain.

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Cite this review

Pith. "Pith review of Pregalactic globular cluster formation." pith.science (2026). https://pith.science/paper/JWT7Y3Q6

@misc{pith2026250902165,
  author       = {Pith},
  title        = {Pith review of: Pregalactic globular cluster formation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWT7Y3Q6}},
  note         = {Machine review of arXiv:2509.02165}
}

read the original abstract

The QCD phase transition in the early universe may provide primordial black hole nuclei for globular clusters. We consider the accretion and star formation that follow, once 1000 solar mass nuclei have formed. When such a nucleus has formed, it remains. Whether these are common in the oldest globular clusters is one decidedly challenging question for the model, which is, as yet, unanswered; another is a possible contribution to the cosmic gravitational radiation background.

Figures

Figures reproduced from arXiv: 2509.02165 by the authors.

**Figure 1.** Evolutionary tracks of PBHs. The horizontal axis is the temperature of the Universe and the vertical axis the Hawking temperature. The tracks become dashed lines as they pass into the matter dominated era and dotted in the dark energy dominated time. The vertical dotted lines are the present day (2.7K) and the QCD phase transition at 220 MeV, which intersects the green diagonal PBH birth line at M3. Galaxy formation… view at source ↗

**Figure 2.** Velocities of PBH particles towards the nucleus in blue, away in red. The scale is in pixels, which are effectively AU. the largest in the distribution, and zero velocity. Other PBH particles range between m1 and m2. The simulations are scale free in radius, but accretion is density dependent (dm/dt = πσρv), where σ is the cross section, the square of the Bondi radius, ρ the density and v the median velocity disper… view at source ↗

**Figure 3.** The distribution of PBH masses in run 77 (black) and run 80a (green). with the fate of the gas that falls into the potential ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Growth of nuclei with redshift by accretion. Colour coding is given in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: Radial profile of mass as a function of redshift in run 84. Redshifts are numbered. The y-axis is the number of particles per unit projected area normalized to 1; the x-axis is radius divided by effective radius. The effective radius contains half the particles. The st…

**Figure 6.** Figure 6: Potential well for run 84 at redshift 817 (solid curve). This has deepened markedly from the original potential of the nucleus alone (dashed curve). The gas and other particles are drawn into this potential well and, once the gas has cooled, e.g. by redshift 100, and …

**Figure 1.** Figure 1: 5 https://www.intel.com/content/www/us/en/developer/tools/oneapi/basetoolkit-download.html [PITH_FULL_IMAGE:figures/full_fig_p007_1.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

[1]

& Kaiser, D

Alonso-Monsalve, E. & Kaiser, D. 2023, PRL, 132.231402 Baade, W. 1944, ApJ, 100, 137 Batten, A. & Mould, J. 2025, submitted to MNRAS Beasley, M. et al. 2003, ApJ, 596, L187 Bicknell, G. & Henriksen, R. 1979, ApJ, 232, 670 Bird, S., Flynn, C., Harris, W. & Valtonen, M. 2013, AAS, 221, 30301 Bondi, H. & Hoyle, F. 1944, MNRAS, 104, 273 Brodie, J. & Strader, ...

arXiv 2023
[5]

See Mould (2025) for the code for Figure

It is available at github/jrmould/darkmatter and may be useful as a paral- lelization demonstration. See Mould (2025) for the code for Figure

work page 2025

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

& Kaiser, D

Alonso-Monsalve, E. & Kaiser, D. 2023, PRL, 132.231402 Baade, W. 1944, ApJ, 100, 137 Batten, A. & Mould, J. 2025, submitted to MNRAS Beasley, M. et al. 2003, ApJ, 596, L187 Bicknell, G. & Henriksen, R. 1979, ApJ, 232, 670 Bird, S., Flynn, C., Harris, W. & Valtonen, M. 2013, AAS, 221, 30301 Bondi, H. & Hoyle, F. 1944, MNRAS, 104, 273 Brodie, J. & Strader, ...

arXiv 2023

[2] [5]

See Mould (2025) for the code for Figure

It is available at github/jrmould/darkmatter and may be useful as a paral- lelization demonstration. See Mould (2025) for the code for Figure

work page 2025