Direct N-body simulations of rotating and extremely massive Population III star clusters

Albrecht W. H. Kamlah; Ataru Tanikawa; Boyuan Liu; Francesco Flammini Dotti; Kai Wu; Manuel Arca Sedda; Marcelo C. Vergara; Nadine Neumayer; Rainer Spurzem

arxiv: 2603.29657 · v2 · submitted 2026-03-31 · 🌌 astro-ph.GA · astro-ph.CO· astro-ph.SR

Direct N-body simulations of rotating and extremely massive Population III star clusters

Kai Wu , Ataru Tanikawa , Francesco Flammini Dotti , Marcelo C. Vergara , Boyuan Liu , Albrecht W. H. Kamlah , Manuel Arca Sedda , Nadine Neumayer

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Rainer Spurzem

This is my paper

Pith reviewed 2026-05-13 23:28 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.COastro-ph.SR

keywords Population III starsstar clustersN-body simulationsblack hole mergersintermediate-mass black holescore collapsecluster rotationgravothermal catastrophe

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The pith

Direct N-body simulations show that initial bulk rotation in massive Population III star clusters drives earlier core collapse and more runaway black hole mergers that form intermediate-mass black holes.

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

The paper runs eight direct N-body simulations of extremely massive Population III star clusters containing 1.01 times 10 to the fifth stars each, including primordial binaries, differential rotation, tidal mass loss, updated stellar evolution formulae, and general-relativistic recoil kicks. It finds that higher initial rotation causes earlier core collapse, faster angular-momentum transport, more stellar collisions and black hole coalescences, and the growth of a central rotating subsystem of intermediate-mass black holes after pair-instability supernovae remove mass and angular momentum at roughly two to three million years. All clusters undergo the gravothermal-gravogyro catastrophe phase, and the results confirm two channels for galactic nucleus seed black holes: runaway mergers among black holes and among stars that later collapse into intermediate-mass black holes. Faster rotation also reduces the abundance of retained compact binaries while amplifying post-collapse expansion.

Core claim

Simulations of rotating Population III star clusters demonstrate that initial bulk rotation functions as a primary control parameter: faster rotation accelerates core collapse, mass segregation, and the gravothermal-gravogyro catastrophe, producing more gravitational runaway mergers and collisions that build intermediate-mass black holes both within and above the pair-instability gap. Post-collapse, each cluster hosts a rotating axisymmetric intermediate-mass black hole subsystem at its center surrounded by an expanding halo of lower-mass objects, after pair-instability events sharply reduce total mass and angular momentum.

What carries the argument

Initial bulk rotation as the primary control parameter that accelerates angular-momentum transport, gravothermal collapse, and post-collapse expansion in the gravothermal-gravogyro catastrophe, tracked through direct N-body integration.

If this is right

Faster initial rotation produces earlier core collapse and higher counts of coalescences and collisions.
Post-collapse clusters develop a rotating axisymmetric intermediate-mass black hole subsystem at the center and an expanding halo of lower-mass objects.
Pair-instability supernovae and compact-object formation around two to three million years sharply reduce total mass and angular momentum.
All Population III clusters experience the gravothermal-gravogyro catastrophe phase.
Initial bulk rotation lowers the retained abundance of compact binaries while favoring compact central intermediate-mass black hole subsystems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The rotation dependence implies that seed black hole formation efficiency in the early universe could vary strongly with the spin distribution of the first star clusters.
Models of supermassive black hole assembly that omit rotation may systematically underestimate merger-driven growth rates at high redshift.
Kinematic measurements of rotation in distant star clusters could serve as an indirect probe for the presence of central intermediate-mass black holes.
Varying the initial mass function or adding magnetic braking in follow-up runs would test how sensitive the reported merger counts remain to the stellar input physics.

Load-bearing premise

The updated fitting formulae for extremely massive metal-poor Population III stars together with the assumed initial mass function and primordial binary fraction accurately represent stellar evolution, mass loss, and binary properties at zero metallicity.

What would settle it

High-redshift observations that find no correlation between cluster rotation speed and either the rate of black hole mergers or the abundance of intermediate-mass black holes would contradict the reported dependence on initial bulk rotation.

read the original abstract

Aims. We present eight direct N-body simulations with NBODY6++GPU of extremely massive, initially rotating Population III star clusters with 1.01 x 10^5 stars. Methods. Our models include primordial binaries, a continuous initial mass function, differential rotation, tidal mass loss, updated fitting formulae for extremely massive metal-poor Population III stars, and general-relativistic merger recoil kicks. We assess their impact on cluster dynamics. Results. All runs form black holes below, within, and above the pair-instability gap, with multi-generation growth. Faster-rotating clusters core-collapse earlier; post-collapse clusters host a rotating, axisymmetric subsystem of intermediate-mass black holes (IMBHs) at the centre and an expanding halo of lower-mass objects. Pair-instability supernovae and compact-object formation at ~2-3 Myr sharply reduce total mass and a large fraction of the cluster's angular momentum. All Population III clusters in our simulations have the gravothermal-gravogyro catastrophe phase. Conclusions. We confirm two of the hypothesized formation channels of galactic nucleus seed black holes: gravitational runaway mergers of black holes and of Population III stars, which core-collapse into IMBHs thereafter. A higher initial star cluster bulk rotation correlates with earlier core collapse and, in the event counts reported here, with more coalescences and collisions, as well as lower retained (compact) binary abundances. Initial bulk rotation is a primary control parameter of cluster evolution: faster rotation accelerates early angular-momentum transport, gravothermal collapse, mass segregation, and amplifies post-collapse expansion, which also favours the formation of a compact central IMBH subsystem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Rotation speeds up core collapse and merger counts in these Pop III clusters, but the numbers after 2-3 Myr rest on the updated stellar fitting formulae.

read the letter

The main takeaway is that higher initial bulk rotation leads to earlier core collapse, higher merger and collision counts, and lower retained binary fractions in these extremely massive Population III clusters. It also shows the formation of a rotating central IMBH subsystem after the pair-instability phase. What is new is the inclusion of differential rotation and the updated fitting formulae for extremely massive metal-poor stars in direct N-body simulations of 10^5-star clusters. Prior work had looked at non-rotating cases, so this provides the first scan of rotation as a control parameter. The code handles binaries, tides, and GR kicks in a standard way, and all eight runs exhibit the expected gravothermal-gravogyro catastrophe with multi-generation black hole growth across the pair-instability gap. The work is well-executed in its direct integration approach. The central findings come straight from the time evolution of the stated initial conditions, without any self-referential fitting. The main soft spot is the reliance on the specific updated stellar evolution formulae. The headline correlations appear after the ~2-3 Myr phase where pair-instability supernovae remove mass and angular momentum. Any change in those inputs would affect the remnant masses and the subsequent dynamics. No error bars on the event counts or convergence tests with particle number are mentioned, and the data are not released publicly. That makes the quantitative claims harder to assess fully. This is aimed at researchers studying seed black hole formation channels for high-redshift quasars and gravitational-wave events. Readers interested in cluster dynamics at zero metallicity will find the rotation trends useful. I recommend sending it to peer review. The simulations add a clear new dimension even if the stellar inputs warrant close checking.

Referee Report

3 major / 2 minor

Summary. The paper reports eight direct N-body simulations with NBODY6++GPU of extremely massive (1.01×10^5 stars) Population III clusters that include primordial binaries, a continuous IMF, differential rotation, tidal mass loss, updated fitting formulae for metal-poor stellar evolution, and GR recoil kicks. All models undergo core collapse, form black holes across the pair-instability gap via multi-generation mergers, and develop a central rotating IMBH subsystem; faster initial bulk rotation is found to accelerate core collapse, increase coalescence/collision counts, and reduce retained compact binaries. The work claims to confirm two seed-BH formation channels and identifies initial rotation as a primary control parameter.

Significance. If robust, the results supply direct numerical support for rotation-driven acceleration of gravothermal collapse and runaway mergers in primordial clusters, thereby strengthening theoretical pathways to IMBHs and galactic-nucleus seeds. The use of a well-tested GPU-accelerated code together with updated Pop III fitting formulae constitutes a concrete advance over earlier analytic or lower-resolution studies.

major comments (3)

[Methods] Methods: no particle-number convergence tests are presented for N=1.01×10^5. Merger rates, core-collapse times, and post-PISN IMBH subsystem properties are known to be sensitive to resolution; without runs at N/2 and 2N the quantitative rotation–merger correlation cannot be shown to be converged.
[Results] Results (around the ~2–3 Myr PISN epoch): the reported dependence of collapse timing and event counts on initial rotation inherits the precise mass-loss rates and PISN thresholds from the adopted updated fitting formulae for extremely massive metal-poor stars. No sensitivity runs with alternative formulae (or ±10–20 % variations in wind/PISN parameters) are shown, leaving the headline correlation vulnerable to changes in those inputs.
[Results] Results: the event statistics (coalescences, collisions, retained binaries) are given as raw counts without error bars, Poisson uncertainties, or bootstrap estimates. With only eight simulations spanning a limited set of initial rotation rates, the strength and statistical significance of the claimed rotation dependence remain unquantified.

minor comments (2)

[Abstract] Abstract and §1: the range of initial bulk rotation rates (e.g., Ω/Ω_crit values) should be stated explicitly rather than described only qualitatively as “faster-rotating.”
[Figures] Figure captions: add the precise initial mass, half-mass radius, and binary fraction used for each run so that the models can be reproduced from the text alone.

Simulated Author's Rebuttal

3 responses · 2 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and indicate where revisions will be made to the manuscript.

read point-by-point responses

Referee: [Methods] Methods: no particle-number convergence tests are presented for N=1.01×10^5. Merger rates, core-collapse times, and post-PISN IMBH subsystem properties are known to be sensitive to resolution; without runs at N/2 and 2N the quantitative rotation–merger correlation cannot be shown to be converged.

Authors: We agree that convergence tests would strengthen the results. However, performing additional direct N-body simulations at N=5×10^4 and 2×10^5 is computationally prohibitive given the resources required for our current set of eight models. We have verified in lower-N test runs (not shown) that the qualitative trends with rotation persist, and will add a dedicated paragraph in the Methods section discussing the resolution limitations and referencing our prior convergence studies at smaller N. revision: partial
Referee: [Results] Results (around the ~2–3 Myr PISN epoch): the reported dependence of collapse timing and event counts on initial rotation inherits the precise mass-loss rates and PISN thresholds from the adopted updated fitting formulae for extremely massive metal-poor stars. No sensitivity runs with alternative formulae (or ±10–20 % variations in wind/PISN parameters) are shown, leaving the headline correlation vulnerable to changes in those inputs.

Authors: The updated fitting formulae are taken from the most recent literature on metal-poor stellar evolution. We acknowledge the sensitivity and will include a new paragraph in the Results section discussing the dependence on these inputs and citing the range of possible variations from alternative models. No new simulations will be added due to computational cost, but we will qualify the conclusions accordingly. revision: partial
Referee: [Results] Results: the event statistics (coalescences, collisions, retained binaries) are given as raw counts without error bars, Poisson uncertainties, or bootstrap estimates. With only eight simulations spanning a limited set of initial rotation rates, the strength and statistical significance of the claimed rotation dependence remain unquantified.

Authors: We will revise the Results section to include Poisson error bars on the event counts and add a discussion of the statistical limitations given the small number of realizations. We will also note that while the trends are clear across the rotation sequence, a larger ensemble would be needed for formal significance testing. revision: yes

standing simulated objections not resolved

Full particle-number convergence tests at N/2 and 2N
Sensitivity runs with alternative stellar evolution fitting formulae or parameter variations

Circularity Check

0 steps flagged

No circularity; results are forward N-body integrations from explicit initial conditions

full rationale

The paper performs direct N-body simulations of star clusters using NBODY6++GPU with fully specified initial conditions (1.01e5 stars, primordial binaries, continuous IMF, differential rotation, tidal mass loss, updated Pop III fitting formulae, GR recoil kicks). All reported outcomes—core-collapse timing, IMBH subsystem formation, merger/collision counts, angular-momentum loss at ~2-3 Myr, and the correlation between initial bulk rotation and earlier collapse—are direct numerical results of the time evolution. No parameters are fitted to the final statistics, no self-definitional relations are introduced, and no load-bearing step reduces to a self-citation or ansatz that is itself unverified within the paper. The derivation chain is therefore self-contained forward simulation rather than circular.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the accuracy of the stellar-evolution fitting formulae and the initial mass function at zero metallicity; these are taken from external literature rather than derived inside the paper.

free parameters (2)

initial bulk rotation rate
Different values are chosen for the eight runs to scan the effect; the exact angular-velocity profiles are not derived from first principles.
primordial binary fraction and period distribution
Assumed values for the initial binary population that affect merger rates.

axioms (2)

domain assumption Updated fitting formulae for extremely massive metal-poor Population III stars accurately capture mass loss and remnant masses
Invoked to determine when pair-instability supernovae occur and how much mass and angular momentum are removed at 2-3 Myr.
domain assumption General-relativistic merger recoil kicks are correctly modeled by the adopted prescription
Used to decide whether merger products remain bound to the cluster.

pith-pipeline@v0.9.0 · 5647 in / 1619 out tokens · 31313 ms · 2026-05-13T23:28:13.006075+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We present eight direct N-body simulations with NBODY6++GPU of extremely massive, initially rotating Population III star clusters... updated fitting formulae for extremely massive metal-poor Population III stars, and general-relativistic merger recoil kicks.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A higher initial star cluster bulk rotation correlates with earlier core collapse and... more coalescences and collisions.

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