Gravitational collapse in the expanding Universe

Nikodem Pop{\l}awski

REVIEW 2 major objections 5 minor 11 references

Gravitational collapse in the expanding Universe

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A spherical dust cloud can collapse to a galaxy only if its initial radius is below $(GM/H^2)^{1/3}$, the limit set by cosmic expansion.

desk verdict A correct but unoriginal LTB derivation of the known SdS static-radius bound; the galactic extrapolation is an overreach. read the letter →

arxiv 2501.06631 v1 pith:BIHZX7VO submitted 2025-01-11 gr-qc astro-ph.GA

Nikodem Pop{l}awski This is my paper · ORCID

classification gr-qcastro-ph.GA MSC 83C1583C7583F05 PACS 04.20.-q98.80.-k

keywords TolmanmetricgravitationalcollapsecosmologicalconstantHubbleparameterdustspheregalaxysizedarkenergy

topics Dark Energy

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 shows that in a universe with a positive cosmological constant, a spherical cloud of pressureless matter that starts from rest can collapse to form a galaxy or black hole only if its initial radius is smaller than a mass-dependent maximum. Using the Tolman metric, the author derives the inequality $b_0 \le (GM/H^2)^{1/3}$, where $M$ is the cloud's mass and $H$ is the Hubble parameter. If the initial radius exceeds this bound, the cosmic expansion overwhelms self-gravity and the cloud expands instead of collapsing. The relation gives a concrete, testable upper limit on galaxy size in a dark-energy-dominated universe.

What carries the argument

The central object is the Tolman metric, the general spherically symmetric dust solution in general relativity, and the derived boundary equation for the sphere's radius $b(\tau)$: $\dot b^2 = f(R_0) + r_g/b + \Lambda b^2/3$, where $r_g = 2GM$. The key step is a first-order perturbation around the initial radius $b_0$: substituting $b = b_0 + \delta b$ makes the right-hand side linear in $\delta b$, so the condition for $\delta b < 0$ reduces to a simple inequality that immediately yields the size bound. The cosmological constant is then linked to the observed Hubble parameter through $\Lambda = 3H^2$, converting the inequality into the mass-dependent size bound $b_0 \le (GM/H^2)^{1/3}$.

What would settle it

Numerically evolve Eq. (16) for a dust sphere with $b_0 = 2\,(GM/H^2)^{1/3}$ and $\dot b_0 = 0$: the derivation predicts the radius expands from the start. If such an evolution instead collapses, the sign in the perturbation argument is wrong. Likewise, finding an observed, relaxed galaxy with mass $M$ and radius larger than $(GM/H^2)^{1/3}$ would show the bound, as stated, is not a universal limit.

Watch

Extended reading notes

Core claim

The central claim is that a dust sphere of mass $M$ initially at rest in a spacetime with cosmological constant $\Lambda$ can collapse only if its radius satisfies $b_0 \le (3 r_g/(2\Lambda))^{1/3}$, which becomes $b_0 \le (GM/H^2)^{1/3}$ after substituting $\Lambda = 3H^2$. The derivation works from the Tolman metric, which yields an evolution equation for the boundary radius, $\dot b^2 = f(R_0) + r_g/b + \Lambda b^2/3$. Setting the cloud initially at rest fixes $f(R_0) = -r_g/b_0 - \Lambda b_0^2/3$, and expanding the equation to first order in a small displacement $\delta b$ shows that the radius decreases initially exactly when $\Lambda b_0 < 3 r_g/(2 b_0)$. The paper interprets this inequality as setting the largest radius of a spherical galaxy that can form from such dust-like matter in the expanding Universe.

Load-bearing premise

The load-bearing assumption is that a galaxy can be represented as a spherically symmetric cloud of pressureless dust that is initially at rest; if the cloud has angular momentum, internal pressure, or an initial radial velocity, the derived maximum radius need not apply.

Editorial extensions

If this is right

A spherical, pressureless cloud of mass $M$ that is initially at rest will not collapse if its radius starts above $(GM/H^2)^{1/3}$; expansion wins instead.
The bound acts as a cosmological limit on galaxy size: in a dark-energy-dominated universe, no such collapsing dust sphere can form a galaxy larger than this scale.
For the Milky Way, with $M \sim 10^{12}\,M_\odot$ and $H \sim 10^{-18}\,\mathrm{s}^{-1}$, the limit is roughly $10^{22}\,\mathrm{m}$, about an order of magnitude larger than the observed radius.
Because torsion vanishes in vacuum, the same bound holds in Einstein–Cartan theory, where the collapse equations reduce to those of general relativity.
Since the present Universe has not yet reached pure exponential expansion, the bound with the current $H$ is a conservative upper limit on sizes at formation.

Reading between the lines

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

A direct observational test would be to search for a relation between galaxy mass and maximum radius: if the bound is physically robust, no relaxed, isolated galaxy should exceed $R_{\max} \approx (GM/H^2)^{1/3}$, with $M$ including dark matter.
The inequality resembles a Jeans-type criterion set by dark energy and could be compared with structure-formation simulations to see whether the most massive halos respect it.
Because the derivation uses only the boundary equation, it may apply to inhomogeneous density profiles and to clouds with total mass $M$ regardless of internal distribution, as long as the boundary evolves as a Tolman dust sphere.
Real galaxies have angular momentum, pressure, and non-rest initial conditions, so the bound should be read as a strong idealization; the true maximum for realistic collapse could differ and might be probed by numerical relativity.

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Formalized claims in Lean

Claim #1: The central claim is that a dust sphere of mass $M$ initially at rest in a spacetime with cosmological constant $\Lambda$ can collapse only if its radius satisfies $b_0 \le (3 r_g/(2\Lambda))^{1/3}$, which becomes $b_0 \le (GM/H^2)^{1/3}$ after substituting $\Lambda = 3H^2$. The derivation works from the Tolman metric, which yields an evolution equation for the boundary radius, $\dot b^2 = f(R_0)

/-- @claim 1 The central claim is that a dust sphere of mass $M$ initially at rest in a spacetime with cosmological constant $\Lambda$ can collapse only if its radius satisfies $b_0 \le (3 r_g/(2\Lambda))^{1/3}$, which becomes $b_0 \le (GM/H^2)^{1/3}$ after substituting $\Lambda = 3H^2$. The derivation works from the Tolman metric, which yields an evolution equation for the boundary radius, $\dot b^2 = f(R_0)  -/
def central_claim : Prop :=

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

The bottom line: this is a correct derivation of a standard result, dressed up as a claim about galaxies. The algebra from (14) to (20) is sound, and the paper is self-contained, but the advertised bound on galaxy radius is not supported by the model.

What it does well: the Tolman–Bondi setup is clean, the small-δb expansion in Eq. (18) is handled correctly, and the collapse condition b0 ≤ (3rg/2Λ)^{1/3} follows without hidden assumptions. With Λ = 3H^2, that becomes b0 ≤ (GM/H^2)^{1/3}, which is exactly the static radius of Schwarzschild–de Sitter spacetime. That result is standard in the GR literature, and the paper does not cite it, so the novelty is low. But as a self-contained derivation of a known bound, it is fine.

The soft spot is the jump from a pressureless dust sphere initially at rest to a galaxy. The model assumes a comoving boundary, no pressure, no rotation, and initial rest. Real galaxies are virialized systems supported by orbital motion and velocity dispersion; nothing in the paper connects the dust-shell turn-around radius b0 to the outer radius of such a system. The sentence 'b0 is related to rg corresponding to the entire mass' is an assertion, not a derivation. So the abstract overstates the result: (20) bounds a specially prepared dust cloud, not galaxies in general. The Milky Way estimate and the unrelated Einstein–Cartan paragraph do not repair that gap.

For whom: a reader interested in LTB collapse formulas might find this a useful pedagogical note, but they should not take the galaxy claim at face value. The paper deserves a serious referee—it is not wrong in its core derivation—but it needs revision to restrict the interpretation to dust spheres and to cite the existing static-radius results.

Recommendation: send to peer review, with the expectation that the author will reframe the claim and remove the extraneous Einstein–Cartan material.

Referee Report

2 major / 5 minor

Summary. The paper studies the gravitational collapse of a spherically symmetric, pressureless dust sphere in the presence of a cosmological constant, using the Tolman metric. It derives an equation for the boundary radius b(τ) and, under the assumption that the sphere is initially at rest, obtains the inequality b0 ≤ (3rg/2Λ)^{1/3}. Using the relation Λ=3H^2, this is rewritten as b0 ≤ (GM/H^2)^{1/3}, which the paper interprets as the largest radius of a galaxy. A numerical estimate for the Milky Way is given, followed by an unrelated paragraph on Einstein-Cartan theory.

Significance. The derivation from Eq. (14) to Eq. (19) is algebraically correct, self-contained, and free of fitted parameters; it reproduces the known static radius of a point mass in de Sitter spacetime. The paper's novelty and importance, however, rest on the leap from an initially static dust sphere to a real galaxy. That step is not derived, and the central claim in the title and abstract therefore overreaches. If a rigorous connection between the dust-sphere threshold and the equilibrium radius of a stellar system could be supplied, the paper would be a useful contribution; in its present form, the significance is limited to a dust-collapse result.

major comments (2)

[Largest size of a galaxy, Eq. (20)] Equation (20) is presented as an upper limit on the radius of a spherical galaxy, with the justification that 'b0 is related to rg corresponding to the entire mass of the galaxy.' The preceding calculation, however, describes a pressureless dust sphere in comoving coordinates, where every fluid element is at rest at τ=0. A real galaxy is not such a system: its stars are supported by orbital motion and velocity dispersion, which are absent from the model. The quoted sentence asserts the connection between the initial dust-cloud radius b0 and the equilibrium radius of a stellar system; it does not derive it. No equation links the dust-shell radius to the radius of a virialized galaxy, so the abstract's claim that the largest radius of a galaxy is (GM/H^2)^{1/3} is not a consequence of the Tolman-Bondi analysis. Please either weaken the claim to 'the maximum initial radius of an initially static pressureless dust sphere that can begin to collapse' or provide a separate argument, for example using the effective potential for test particles in de Sitter spacetime, that establishes the bound for realistic galaxies.
[Gravitational collapse, Eq. (15)] The derivation of (19) relies crucially on the initial condition ˙b(0)=0, used to fix f(R0) via Eq. (15). This assumption is not physically justified for galaxy formation. If the initial radial velocity is inward, a cloud with b0 larger than the right-hand side of (19) can still collapse; if the initial velocity is outward, a cloud below the bound may expand. Thus the inequality (19) is not a universal bound on the initial radius of a collapsing dust sphere, but only on an initially static one. The paper does not state this limitation clearly; the abstract and the section 'Largest size of a galaxy' should make explicit that the bound applies only under the zero-initial-velocity condition.

minor comments (5)

[Final paragraph before Acknowledgments] The paragraph on Einstein-Cartan theory, torsion, and quantum electrodynamics is unrelated to the main result and reads as an assertion of unrelated research topics; it should be removed or moved to a separate paper.
[General] The paper would benefit from a reference to the standard literature on the maximum size of bound structures in the presence of a cosmological constant, such as the static radius in Schwarzschild-de Sitter spacetime; this would place Eq. (20) in proper context.
[Largest size of a collapsing sphere, Eq. (18)] The text says 'For small τ', but the expansion is in δb, not τ; please clarify that the linearization is valid for small deviations from b0, i.e., near the initial time.
[Largest size of a collapsing sphere, Eq. (16)] The statement 'The entire sphere will collapse, δb < 0' is based on the leading-order term; a more rigorous presentation would analyze the effective potential in Eq. (16) to show that the sign of the initial acceleration determines the subsequent fate globally.
[Abstract and title] There are minor typographical issues: 'collapseof' in the abstract is missing a space, and the author's name appears with a '(suppress)' artifact in the header, which is a LaTeX conversion issue that should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dust-sphere bound follows algebraically from the Tolman dust equations with no fitted parameters or load-bearing self-citations.

full rationale

The derivation is self-contained. The paper starts from the Tolman metric and the Einstein equations, specializes to a pressureless dust fluid (p=0, synchronous comoving coordinates), and obtains the dynamical equation r˙^2 = f(R) + F(R)/r + (1/3)Λr^2. The surface radius b(τ) satisfies Eq. (14), and the initial-rest condition fixes f(R0) = -rg/b0 - (1/3)Λb0^2. Substituting this into (14) and linearizing about the initial radius gives (18), whose sign yields the threshold b0 ≤ (3rg/(2Λ))^{1/3}. With rg = 2GM and Λ = 3H^2, this becomes Eq. (20), b0 ≤ (GM/H^2)^{1/3}. No parameter is fitted to the target; H is an external observational input, and the condition f(R0) is fixed by the initial rest assumption, not by the desired bound. The self-citation [3] is used only for intermediate equations that are in fact derived in the text from (8) and (9), so it is not load-bearing. The main interpretive step—identifying b0 with the largest radius of a galaxy—is an unproven physical assumption (a correctness concern), but it is not a circular reduction: Eq. (20) is not assumed as input and does not reduce by construction to the galaxy claim. The Einstein–Cartan paragraph is unrelated to the bound and does not affect the derivation chain.

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

The paper introduces no free parameters and no new entities. It relies on standard LTB equations and several physical modeling assumptions, the most fragile being the initial rest condition and the dust-sphere model of a galaxy.

assumptions (6)

standard math The Tolman metric is the most general spherically symmetric metric in comoving coordinates.
Eq (1), standard result in general relativity.
domain assumption The matter is a pressureless ideal fluid (dust) with p=0.
Assumed in the section 'Dustlike sphere with a cosmological constant'.
ad hoc to paper The sphere is initially at rest, ṫ(0)=0.
Used to derive Eq (15) and determine f(R0); not physically justified for galaxies.
domain assumption The cosmological constant is related to the Hubble parameter by Λ=3H^2.
Section 'Largest size of a galaxy'; standard for a de Sitter universe, not exact in the current universe.
standard math The mass of the sphere determines the Schwarzschild radius via rg = κ∫ρr^2r'dR.
Eq (11), standard result in LTB collapse.
ad hoc to paper Real galaxies can be approximated by a collapsing dust sphere.
Stated in the final section without justification; real galaxies have angular momentum, dark matter, and complex formation.

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Cite this review

Pith. "Pith review of Gravitational collapse in the expanding Universe." pith.science (2026). https://pith.science/paper/BIHZX7VO

@misc{pith2026250106631,
  author       = {Pith},
  title        = {Pith review of: Gravitational collapse in the expanding Universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BIHZX7VO}},
  note         = {Machine review of arXiv:2501.06631}
}

abstract

We use the Tolman metric to describe gravitational collapse of a sphere of a fluid without pressure in spacetime with the Hubble parameter $H$ related to the cosmological constant. We show that the largest radius of a galaxy formed from such a fluid with mass $M$ is given by $(GM/H^2)^{1/3}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

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Pith tools

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

[1] [1]

The dynamics of gravitational collapse of a dustlike sphere in the pre sence of a cosmological constant is determined by equation (8)

(15) 3 Largest size of a collapsing sphere . The dynamics of gravitational collapse of a dustlike sphere in the pre sence of a cosmological constant is determined by equation (8). That equation can be solved analytically for Λ = 0 [1–3 ], giving also an approximated solution for Λ > 0 if b0 ≪ (rg/Λ)1/ 3. If b0 is suﬃciently large, on the order of magnitud...

work page

[2] [2]

(18) The entire sphere will collapse, δb < 0, if 2Λ b0/3 < r g/b2 0, which gives b0 ≤ ( 3rg 2Λ ) 1/ 3

(16) For small τ (near the initial time τ = 0), putting b(τ) = b0 + δb(τ), δb ≪ b0 (17) into (16) and omitting terms of higher order in a small quantity δb gives ˙b2 = ( 2 3Λb0 − rg b2 0 ) δb. (18) The entire sphere will collapse, δb < 0, if 2Λ b0/3 < r g/b2 0, which gives b0 ≤ ( 3rg 2Λ ) 1/ 3 . (19) Largest size of a galaxy . The observed Universe is dom...

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Lemaˆ ıtre, Ann

G. Lemaˆ ıtre, Ann. Soc. Sci. Brux. A 53, 51 (1933); R. C. Tolman, Proc. Natl. Acad. Sci. USA 20, 169 (1934); J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939); H. Bondi, Mon. Not. R. Astron. Soc. 107, 410 (1947); G. C. Omer, Proc. Natl. Acad. Sci. USA 53, 1 (1965)

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L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, 1975). 4

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N. Pop/suppress lawski, Zh. Eksp. Teor. Fiz.159, 448 (2021); J. Exp. Theor. Phys. 132, 374 (2021); in: Proceedings of the Sixteenth Marcel Grossmann Meeting on General Relativity , ed. R. Ruﬃni and G. Vereshchagin, part B, p. 1327 (World Scie ntiﬁc, 2023); in: Regular Black Holes. Towards a New Paradigm of Gravitationa l Collapse , ed. C. Bambi, p. 485 (S...

work page 2021

[6] [6]

G. C. McVittie, Mon. Not. R. Astron. Soc. 92, 500 (1932); A. K. Raychaudhuri, Proc. Phys. Soc. 88, 545 (1966); M. Demia´ nski and J. P. Lasota, Nat. Phys. Sci. 241, 53 (1973); R. A. Sussman, Gen. Relativ. Gravit. 17, 251 (1985); J. Sultana and C. C. Dyer, Gen. Relativ. Gravit. 37, 1347 (2005)

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A. G. Riess et al ., Astrophys. J. 977, 120 (2024)

work page 2024

[8] [8]

Cartan, C

´E. Cartan, C. R. Acad. Sci. 174, 593 (1922); E. Schr¨ odinger,Space-time Structure (Cambridge University Press, 1954); T. W. B. Kibble, J. Math. Phys. 2, 212 (1961); D. W. Sciama, Rev. Mod. Phys. 36, 463 (1964); Rev. Mod. Phys. 36, 1103 (1964); F. W. Hehl and B. K. Datta, J. Math. Phys. 12, 1334 (1971); F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J ...

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