The Formation of Globular Clusters

J. M. Diederik Kruijssen

REVIEW 1 major objections 6 minor 2 cited by

The Formation of Globular Clusters

T0 review · 1 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper argues that globular clusters are the natural outcome of high-pressure star formation in high-redshift galaxies, surviving today only after a phase of tidal-shock destruction and migration out of the galactic disk.

desk verdict A clear, well-illustrated review of the high-pressure GC formation framework, honest about its soft spots but too confident in calling it the 'standard paradigm'. read the letter →

arxiv 2501.16438 v1 pith:PVRGT2L6 submitted 2025-01-27 astro-ph.GA

J. M. Diederik Kruijssen This is my paper · ORCID

classification astro-ph.GA

keywords globularclustersyoungmassivetidalshocksclusterformationefficiencyinitialmassfunctiongalaxyassemblyhigh-redshiftstarJWST

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 review argues that globular clusters are not exotic survivors from special early-Universe conditions but the natural product of ordinary high-pressure star formation in gas-rich galaxies at redshifts z≈2–3. It assembles the case that young massive clusters form in these disks, undergo a phase of rapid tidal-shock destruction that removes most low-mass clusters, and become long-lived globular clusters only after migrating out of the disk. On this picture, the observed peaked mass function, metallicity bimodality, specific frequency, and kinematics of globular cluster systems all emerge from a single galaxy-formation framework. If true, globular clusters become quantitative fossils for reconstructing how galaxies assembled, and the next generation of telescopes can test the model from formation to the present day.

What carries the argument

The load-bearing object is the tidal-shock destruction timescale of Equation (3), $t_{\rm sh}=176\,{\rm Myr}\,(f_\Sigma/4)^{-1}(\rho_{\rm ISM}/({\rm M}_\odot\,{\rm pc}^{-3}))^{-3/2}(M/10^5\,{\rm M}_\odot)\,\phi_{\rm ad}^{-1}$, where $f_\Sigma$ is the GMC-to-ISM surface density ratio, $\rho_{\rm ISM}$ is the mean mid-plane ISM density, $M$ is the cluster mass, and $\phi_{\rm ad}$ is an adiabatic correction. This timescale scales roughly linearly with cluster mass, converting a power-law initial cluster mass function into a peaked present-day mass function with characteristic mass near $10^5\,{\rm M}_\odot$. The complementary ingredients are the pressure-dependent cluster formation efficiency, which sets how many bound clusters form, and the migration of surviving clusters out of the gas-rich disk, which marks their transition into long-lived globular clusters.

What would settle it

Measure the mass function of globular cluster populations at z≈0.5–1: the framework predicts the characteristic peak near $10^{5}$ solar masses should already be in place by z≈1. Finding a power-law mass function at those redshifts, or a proto-globular cluster mass function at high redshift that is already peaked at birth, would contradict the claim.

Watch

Extended reading notes

Core claim

The central claim is that globular clusters are the relics of regular star formation in high-redshift galaxies. Clusters form with a power-law-like double-Schechter initial mass function; the fraction of star formation in bound clusters rises with gas pressure, and the upper mass limit of the initial cluster mass function also rises with pressure, so high-pressure z≈2–3 disks naturally make young massive clusters up to $10^{6}$–$10^{8}$ solar masses. In the natal disk, tidal shocks from giant molecular clouds destroy clusters on a timescale that grows with cluster mass, so low-mass clusters are removed while the more massive ones survive. The present-day properties of globular cluster systems are then explained as the result of formation, destruction, and galaxy assembly, not as a record of a special formation event.

Load-bearing premise

The whole quantitative argument depends on how quickly tidal shocks from dense gas destroy clusters in the disk; that rate is calibrated on local cluster populations and depends on cluster densities and initial radii that cannot be reconstructed from today's globular clusters.

Editorial extensions

If this is right

If globular clusters are remnants of ordinary high-pressure star formation, the peaked globular cluster mass function is a destruction artifact, so its near-universal characteristic mass does not require a universal formation mechanism.
The metal-rich and metal-poor globular cluster populations emerge from galaxy assembly histories and destruction, rather than from two distinct formation channels.
Globular cluster specific frequency, spatial distribution, and kinematics become diagnostic of a galaxy's merger tree, allowing reconstruction of assembly histories for external galaxies.
The framework predicts that the characteristic mass of the globular cluster mass function should already be in place by z≈1, set by early tidal-shock destruction rather than by slow evaporation.
Models that include a cold interstellar medium and tidal-shock destruction should simultaneously reproduce local young massive cluster populations and the old globular cluster demographics of present-day galaxies.

Reading between the lines

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

Extension: the same destruction law implies a substantial population of dissolved cluster debris in galaxy halos, so measuring the abundance patterns of field halo stars in the Milky Way would test whether the number of destroyed clusters matches the model.
Extension: the framework could be calibrated directly by measuring the age distribution of young massive clusters in local high-pressure environments such as galaxy mergers and galactic centers, checking that the disruption timescale scales with ISM density as in Equation (3).
Extension: if migration out of the disk is required for long-term survival, galaxies with smooth, quiescent assembly histories should have relatively fewer globular clusters per unit stellar mass, a testable prediction for low-mass and ultra-diffuse galaxies.
Extension: the model predicts a near-universal globular cluster mass function peak established early; future intermediate-redshift surveys with 30m-class telescopes can verify whether the peak is already present by z≈1.

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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 read this as a review chapter, not a research paper. The author is the architect of the framework being reviewed, and the review essentially argues that globular clusters are the natural outcome of high-pressure star formation in z=2-3 disks, with tidal-shock destruction and later migration shaping the present-day population. If you want one up-to-date synthesis of that framework, including the recent JWST proto-GC results, this is a good one.

What is genuinely useful: the organizational framing in Section 6.1 (the 'trinity' of diagnostics: local YMC physics, high-redshift formation conditions, and pan-redshift demographics) is a sensible way to structure the field. The review also integrates the 2023-2024 JWST proto-GC observations and is transparent about the major open questions, including multiple populations and the model failures in low-redshift galaxies. The literature coverage is broad, and the description of competing theories is fair enough.

The main soft spot is exactly the one flagged by the stress-test note: the quantitative backbone of the whole picture, the tidal-shock destruction timescale of Eq. (3), depends on cluster densities that cannot be reconstructed from present-day properties. A factor-two uncertainty in initial half-mass radius changes the destruction timescale by about a factor of eight, which would either erase or preserve the low-mass end of the GCMF. The author does concede this in Section 5.2, so the review is honest about it, but the central claim that the peaked GCMF is a robust consequence of ordinary star formation is presented more strongly than the underlying parameters justify. This is a weakness of the framework, not a flaw in the review's execution.

I also agree with the reader that the 'standard paradigm' language overstates the degree of consensus, and the self-citation concentration is real, but it is the natural consequence of one person reviewing their own framework. That is not a reason to reject the review, only to read it as a partisan, well-argued synthesis rather than a neutral survey.

Who benefits: graduate students entering the field, and researchers in galaxy evolution who need a compact, current overview of the GC-formation picture with pointers to the key papers. It deserves peer review in the sense that a serious referee should check the physical claims and the balance of citations, and I would accept it with the request that the author soften the 'standard paradigm' phrasing and more explicitly separate the framework's successes from its parameter sensitivities. Worth a cite.

Recommendation: engage with it, but keep Eq. (3) in mind as a known, consequential uncertainty.

Referee Report

1 major / 6 minor

Summary. This review chapter argues that globular clusters (GCs) are the natural outcome of ordinary, high-pressure star formation in high-redshift galaxies. The proposed sequence is that GCs form as the massive tail of the young massive cluster population in gas-rich disks at z=2-3, are rapidly destroyed by tidal shocks from GMCs, and survive to the present only after migrating out of their natal disk (Figure 2). The paper reviews the physics of cluster formation (CFE, ICMF), cluster destruction (tidal shocks, evaporation), and the host-galaxy context, and surveys current observations of proto-GCs and future tests with JWST, ELT, and Euclid. It concludes with a list of open questions and argues that GCs can be used as quantitative tracers of galaxy assembly, including the identification of the Milky Way progenitor 'Kraken'.

Significance. If the framework holds, it would unify a wide range of GC demographics - mass function, metallicity bimodality, specific frequency, kinematics - within a single galaxy-formation picture, making GCs quantitative fossils of galaxy assembly. The review is comprehensive and provides useful, explicit caveats, including the acknowledged free-parameter dependence of the models (Section 2.3), the unresolved multiple-populations question (Section 6.4, question 1d), and the poorly-constrained cluster densities that enter tidal-shock destruction (Section 5.2). It also offers concrete falsifiable predictions, such as the timing of the proto-GC formation peak relative to the cosmic star formation peak and the early assembly of the GCMF peak at z~1. The main weakness is that the quantitative backbone of the GCMF prediction rests on a destruction timescale normalization that is calibrated locally and is sensitive to unconstrained initial cluster radii; the review states this caveat, but the abstract and early sections present the paradigm with more certainty than that caveat warrants.

major comments (1)

[Section 2.3, Table 1] The review's lists of successes and favored explanations in Section 2.3 and Table 1 are drawn almost exclusively from the author's own framework and close collaborations (Kruijssen 2015; E-MOSAICS series; Pfeffer et al.; Reina-Campos et al.; De Lucia et al.). For a review that aims to be a comprehensive point of reference, the absence of a critical intercomparison of independent model implementations is a significant imbalance. Including a discussion of where independent codes (e.g., FIRE, GRIFFIN, SMUGGLE, or other semi-analytic models) agree and disagree with the E-MOSAICS-based predictions would substantially strengthen the review's credibility, especially for the GCMF and metallicity-distribution claims.

minor comments (6)

[Section 4.1] Typo: 'gaseous environnment' should be 'gaseous environment'.
[Section 2.1] Typo: 'gravitional field' should be 'gravitational field'.
[Section 3.2] Notation conflict: 'M' is used for the Mach number in the pressure expression (P∝ρM^2) and for cluster mass in nearby equations; please introduce a different symbol or clarify.
[Section 5.2] The important caveat about uncertain cluster densities and initial radii should be introduced when Eq. (3) is first presented in Section 4, not only in Section 5.2, so that the reader is flagged to this sensitivity from the outset.
[Section 6.3] The quotation from Searle and Zinn (1978) contains 'contined'; if reproducing the original text, mark it as '[sic]' or correct it.
[Abstract] The abstract says 'James Webb Telescope'; the correct name is 'James Webb Space Telescope' (the full text uses JWST).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the review's central framework is anchored in external observations and independent simulations; heavy self-citation is not load-bearing.

full rationale

I walked the paper's derivation chain and found no step in which a prediction reduces by construction to its own input. The central claim that GCs are the natural outcome of high-pressure star formation is supported by external, non-circular anchors: direct ISM pressure measurements at z=2-3 (Section 2.2), observed YMC demographics in local mergers and starbursts (Section 3.2), high-redshift proto-GC observations with JWST (Section 6.2), and the E-MOSAICS simulations, which are a published, code-based suite comparing against external data (Section 2.3). Equation (3), the tidal-shock destruction timescale, is calibrated to reproduce age distributions of young cluster populations in the local Universe (Section 4.1), and is then used to derive the GCMF peak; this is an extrapolation over a Hubble time, not a tautology. The paper's Section 6.2 explicitly notes that no empirical input from proto-GC demographics was used when predicting their luminosity function, which is a genuine out-of-sample test. The main self-citation concentration is real but not load-bearing: the definition of a GC from Kruijssen (2015) in Section 1.2 is an explicit convention, not a derived result, and the Figure 2 architecture attributed to Kruijssen (2015) is independently instantiated by other groups' models cited throughout. The limitation conceded in Section 5.2, that tidal-shock effectiveness depends on poorly constrained initial cluster radii, weakens the robustness of the quantitative prediction but does not constitute circular reasoning. Given the external empirical anchors and out-of-sample predictions, the appropriate circularity score is low, with only minor deductions for the concentrated self-citation framing.

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

The framework summarized here rests on a set of modeling choices inherited from the author's prior papers: the survival-based definition of GCs, the log-normal ISM density PDF as the driver of the CFE, tidal-shock destruction by cold ISM substructure as the dominant low-mass cluster killer, and the assumption that z=2-3 ISM pressure measurements characterize GC birth environments. None of these is proven in this review; each is a load-bearing premise with external anchors that are partly observational and partly model-internal.

free parameters (5)

GMC-to-ISM surface density ratio fSigma = 4 (fiducial)
Sets the tidal-shock destruction timescale in Equation (3); the underlying Kruijssen (2015) calibration adopts fSigma = 4, and the result is absorbed into the 176 Myr normalization.
Adiabatic correction phi_ad = [0, 1]
Equation (3) interpolation factor between impulsive and adiabatic shock regimes; calibrated to N-body simulations in the cited literature and unobservable for individual GCs.
Normalization of the tidal-shock destruction timescale = 176 Myr
The amplitude in Equation (3); chosen to match observed age distributions of local young cluster populations (Section 4.1), and it directly sets how long GC progenitors can remain in the natal disk.
SFE per free-fall time = a few percent
Input to the CFE integral over the density PDF (Section 3.2); adopted from the star formation literature (Krumholz and Tan 2007; Leroy et al. 2017), it controls the mass budget of bound cluster formation.
ICMF truncation masses Mmin and Mmax = Mmin = 50-100 Msun in the solar neighborhood to 1e3-1e4 Msun in GC-forming dwarfs; Mmax = 1e4-1e8 Msun by environment
The double-Schechter ICMF (Equation 1) needs both truncations; the environmental scaling of Mmax carries the argument that high-pressure galaxies produce GC-mass clusters.

assumptions (5)

ad hoc to paper GCs are defined as gravitationally bound clusters that are not durably associated with the present star-forming disk of their host galaxy (Kruijssen 2015, Section 1.2).
This survival-based definition is the author's own and frames which objects count as GCs throughout the review; it is a definitional commitment, not an observed fact, and it shapes the framework's selection effects.
domain assumption The fraction of star formation in bound clusters (CFE) is set by the ISM density PDF, which is log-normal with width set by the Mach number, combined with a constant SFE per free-fall time (Section 3.2, Figure 4).
The Kruijssen (2012) theory that the review presents as established; the log-normal PDF and constant per-free-fall SFE are idealized descriptions of turbulent ISM star formation.
domain assumption Tidal shocks from GMCs and other ISM substructure dominate destruction of low-mass clusters in the natal disk, with a timescale given by Equation (3).
The central destruction mechanism of the framework; the review notes that simulations without cold ISM substructure cannot reproduce the observed GCMF (Section 5.2), indicating the mechanism is essential.
domain assumption Measured ISM pressures in z=2-3 galaxies (P/k of order 1e6-1e8 K cm^-3) represent the conditions under which GCs formed.
Section 2.2 makes this connection to anchor the framework in observations; the measurements characterize galaxy-averaged ISM properties, not individual GC birth sites.
domain assumption YMCs that survive the natal disk phase become long-lived GCs once they migrate out of the gas-rich disk, plausibly during galaxy mergers or tidal stripping (Figure 2b).
The migration-saves-the-cluster step is required to connect high-redshift formation to present-day halo GC populations; the review presents it as plausible but not directly observed.

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Cite this review

Pith. "Pith review of The Formation of Globular Clusters." pith.science (2026). https://pith.science/paper/PVRGT2L6

@misc{pith2026250116438,
  author       = {Pith},
  title        = {Pith review of: The Formation of Globular Clusters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVRGT2L6}},
  note         = {Machine review of arXiv:2501.16438}
}

read the original abstract

Globular clusters (GCs) are among the oldest and most luminous stellar systems in the Universe, offering unique insights into galaxy formation and evolution. While the physical processes behind their origin have long remained elusive, major theoretical and observational developments in the past decade have led to a new understanding of GCs as the natural outcome of high-pressure star formation in high-redshift galaxies. This review synthesizes recent advancements in our understanding of GC formation and aims to provide a comprehensive point of reference for leveraging the revolutionary capabilities of the current and upcoming generation of telescopes. The latest generation of GC models combine our understanding of their formation and destruction with advanced galaxy formation simulations. The next decade will provide the first-ever opportunity to test such models across their full evolutionary history, from GC formation at high redshift as seen with the James Webb Telescope, to snapshots of GC demographics at intermediate redshifts obtained with 30m-class telescopes, and eventually to the well-characterized GC populations observed at the present day. We identify the major questions that we should expect to address this way.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize ":" * " " *...

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o nig S , author Larsen SS , author Melinder J , author Sabbi E , author Smith LJ and author \

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