Observation of critical phenomena and self-similarity in the gravitational collapse of radiation fluid

Charles R. Evans; Jason S. Coleman

REVIEW

Observation of critical phenomena and self-similarity in the gravitational collapse of radiation fluid

Charles R. Evans, Jason S. Coleman

General Relativity and Quantum Cosmology · Submitted 23 Feb 1994 · 13 references

Pith's verdict: A founding critical-collapse paper whose β = 0.36 has held up, but whose headline claim about approaching the CSS solution rests on a single visual overlay.

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CitationCharles R. Evans, Jason S. Coleman. "Observation of critical phenomena and self-similarity in the gravitational collapse of radiation fluid." arXiv:gr-qc/9402041 (1994). https://pith.science/paper/KQCEQN5Z

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Editor's note

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The thing worth knowing: this is the paper that made critical collapse analytically tractable. Choptuik's scalar field and the axisymmetric wave case both gave discrete self-similarity, which is hard to work with. Radiation fluid gives continuous self-similarity, so the critical solution reduces to ODEs with a similarity exponent n fixed as an eigenvalue by regularity at ξ = 0, ξ = ∞ and the sonic point — not fitted to the PDE run. That is the real contribution and it is clean. The setup is fully specified: gauge, equations (2)–(5), the initial data family in (6). Anyone can rerun it. β ≈ 0.36 has since been reproduced independently, and the intermediate-asymptotics framing borrowed from the modified blast-wave problem is the right way to think about the scaling window r₀ ≪ r ≪ r₁ ≈ K|η−η_c|^β. That paragraph is the most useful one in the paper. The β measurement itself is the solid part: Fig. 3 is a power-law fit over two orders of magnitude in mass, with two resolutions shown. Where it is soft: the abstract's distinctive claim [abstract:s3] — that near-critical evolutions approach that solution — is supported by one metric function, at one time, overlaid visually (Fig. 4 caption). The coordinate ξ = r/C(−t)^n carries a free constant C and a free time origin, so this is effectively a two-parameter fit to a single curve. V(ξ), Ω(ξ) and N(ξ) are never compared, and no residual is reported as a function of (−t) or of η−η_c. The stress-test note is right about this. The authors themselves write that linear stability analysis “should reveal” the attractor property — asserted, not shown. That does not sink β, but it is exactly the step separating this paper from prior work, so it deserves more than a caption. Two smaller items. Ref. [4], the ODE solution, is unpublished and by one of the present authors, so independence is not checkable. And Ref. [9] concedes the solution “may be equivalent” to Ori–Piran exploding models, which would trim the novelty of the solution itself, though not the collapse result. Also, η_c appears as 1.0188 in text and 1.018828234 in the Fig. 3 caption, with no uncertainty on β or η_c, and “doubtlessly” universality from 0.36 versus 0.37 with no error bars on either side. Minor, and easy to fix. One correction to the reader's report: the classifier says “no limitations found.” That is wrong — the manuscript does flag non-asymptotic flatness (a → 1.07, m/r → 0.0596) and leaves stability to future work. Credit for that. Who should read it: anyone in critical collapse or numerical relativity, and anyone wanting a clean type-2 self-similarity problem to teach from. Send it out.

Referee report

Significance

If the results hold, this is the first fluid system shown to exhibit critical collapse, and the first in which the critical solution is continuously rather than discretely self-similar. That matters for two reasons the manuscript states well. First, a CSS critical solution is analytically far more tractable than the echoing solutions of scalar and gravitational-wave collapse, so it opens a route to the linear-stability calculation that would actually predict β rather than measure it. Second, the value β ≈ 0.36 agrees with the 0.37 reported for scalar and vacuum collapse, which is a genuinely non-trivial universality hint across three matter models. The identification of the phenomenon with intermediate asymptotics in the Barenblatt–Zel'dovich sense is a real conceptual contribution, and it yields a falsifiable statement: self-similarity should be broken on the scale r1 ≈ K|η − ηc|^β, so it is visible only when r1 ≪ r0. Credit is also due for the two discretization scales (3.3×10⁻³ and 1.0×10⁻³) shown in Fig. 3, and for the fact that n is an eigenvalue fixed by regularity conditions rather than fitted to the PDE evolution — that is what makes the comparison with the ODE solution a预测

Major objections (4)

[Fig. 4 and abstract, sentence 3] The abstract's distinctive claim — that near-critical evolutions asymptotically approach the CSS solution — rests on Fig. 4, which overlays a single metric function a(ξ) on a(r) at a single time (τ = −0.5). The similarity coordinate ξ = r/C(−t)^n carries a free dimensional constant C and a free time origin, so this is effectively a two-parameter fit of one curve over a limited radial range; a family of profiles can be brought into coincidence this way without the evolution actually sitting on the attractor. The remaining self-similar variables V(ξ), Ω(ξ) and N(ξ) are never compared. Please report a quantitative residual: for each of a, V, Ω and N, the deviation from the CSS profile as a function of central proper time τ, for at least one near-critical model, and state explicitly how C and the time origin are fixed (e.g. by matching at the sonic point ξ = 1, or by the asymptotic value m/r → 0.0596). The manuscript itself concedes the point when it writes that linear stability analysis 'should reveal' the perturbative response; as written the attractor property is asserted rather than demonstrated.
[Ref. [4] and the paragraph beginning 'A successful attempt was made...'] Every quantitative feature of the comparison target — n ≈ 1.1485, a → 1.07 as ξ → ∞, m(r)/r → 0.0596, Ω → 9.56×10⁻³ — is quoted from Ref. [4], which is listed as 'to be submitted to Phys. Rev. D (1993)' and is by one of the present authors. The reader of this Letter therefore cannot check the object against which the PDE evolution is being compared. I am not asking for a published proof to be reproduced; I am asking that a Letter whose central claim is an asymptotic approach to a specific solution be self-contained enough to identify that solution. A compact statement would suffice: the reduced ODE system, the three regularity conditions (ξ = 0, ξ = ∞, and the ingoing acoustic characteristic at ξ = 1), and how the over-determination fixes n as an eigenvalue. Please also confirm the status of Ref. [4] at proof.
[Eq. (7) and Fig. 3] The value β ≈ 0.36 is quoted without an uncertainty, without fit residuals, and without a statement of the number of models or the range in η − ηc actually fitted. The Fig. 3 caption gives ηc = 1.018828234, K = 3.27 and β = 0.36, while the text gives ηc ≃ 1.0188; nine significant figures is not credible without a stated convergence criterion for the bisection. This matters because the paper's universality claim — that β is 'numerically indistinguishable' from the 0.37 of Refs. [1] and [2] — is a physics claim about agreement between matter models, and at two significant figures it is only as strong as the error bar attached to it. Please give the fit residuals, an estimate of the uncertainty in β (from the fit and from the discretization), and the range of η − ηc over which the power law was verified.
[Paragraph beginning 'The fluid is initially everywhere out of equilibrium'] The manuscript asserts that 'Only a precisely-critical model would approach the self-similar solution and retain this dependence as r → 0, t → 0', but no evolution at η = ηc is presented; all statements about the precisely-critical spacetime are extrapolations from near-critical runs. Either show the closest-to-critical evolution obtained and how far in τ it retains the self-similar form, or soften the sentence to a conjecture. The related claim that the critical solution contains a naked singularity is supported only by the asymptotic behaviour α ~ r^{0.129}; it would help to say whether this redshift divergence holds along all outgoing null rays from the central singularity or only along the central observer's worldline, since that distinction is what a cosmic-censorship argument would turn on.

Minor objections (6)

[Eq. (6)] As printed, the coefficient 12π^{−3/2} in Eq. (6) is inconsistent with the stated total mass M = ηr0/2 by a factor of 24; the normalization works out only if the coefficient is (1/2)π^{−3/2}, so the fraction appears to have been lost in typesetting. Please confirm the intended coefficient.
[Fig. 3 caption] The caption refers to 'a power law of the form (8)', but the power law is Eq. (7) in the text; the equation numbering should be made consistent.
[Fig. 2 and the mass-extraction procedure] The black hole mass is read off from the kink in m(r) where r ≃ 2m(r). A sentence on the algorithm used to locate the kink and on the resulting uncertainty in MBH would strengthen the power-law fit, particularly at the low-mass end where MBH ≃ 0.001 M_ADM.
[Fig. 1] The similarity coordinate ξ = r/C(−t)^n contains a dimensional constant C that is never given a value or a normalization convention. Please state the convention used for the plotted ξ, or note that the figure is drawn in units with C = 1.
[Paragraph beginning 'Black-hole mass MBH is found...'] The sentence that the agreement of β with Refs. [1] and [2] 'doubtlessly' reflects a deep property of the gravitational field equations is stronger than two-significant-figure agreement can support; 'plausibly' or 'suggesting' would be proportionate.
[Refs. [3] and [4]] Both are cited as submitted rather than published; the reference list should be updated to reflect final publication status if this is possible before the Letter appears.

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The paper

The paper claims

A collapsing sphere of radiation fluid passes through a critical point, and the black holes that form just above it obey a power law whose exponent matches the one seen in scalar-field and gravitational-wave collapse.

The reading

This paper asks whether the threshold of black hole formation in a collapsing sphere of radiation fluid behaves like a genuine critical point, and whether the evolutions just at that threshold settle into a self-similar spacetime. Evans and Coleman evolve a one-parameter family of initial data — a Gaussian energy-density profile initially at rest, labelled by the dimensionless strength $M/r_0$ — and bisect for the value $M/r_0$ that separates complete dispersal from black hole formation. Just above threshold the hole's mass follows $M_{BH} = K( M/r_0 - (M/r_0)_c)^{0.36}$ over two orders of magnitude in mass, an exponent numerically indistinguishable from those reported for scalar-field and gravitational-wave collapse. Near threshold the inner region collapses while a strong inward-moving rarefaction wave eats it away from outside, holding $m(r)/r$ nearly constant, and the metric and fluid profiles are shown approaching those of a separately computed self-similar solution with similarity exponent $n = 1.1485$. If this holds, the onset of black hole formation is a critical phenomenon in a model simple enough that the critical solution itself can be written down and perturbed around.

What carries the argument

The load-bearing object is the continuously self-similar ansatz $ξ = r/C(-t)^n$ with $a(ξ)$, $U^r(ξ)$, $Ω(ξ)$, $N(ξ)$, whose generator $w = (1/n)t∂/∂t + r∂/∂r$ satisfies $£_w g = 2g$ and $£_w T = 0$. Because the similarity exponent $n$ is not fixed by dimensional analysis, this is a type-2 self-similarity: $n$ must be found as the eigenvalue that lets the ODE system pass regularly through the origin, spatial infinity, and the sonic point at $ξ = 1$. That solution supplies the target against which the PDE evolutions are compared, and it supplies the scaling $r_1 = K|η-η_c|^β$ that sets how large the self-similar window can be.

Extended reading notes

Core claim

The paper reports that spherically symmetric collapse of a radiation fluid ($p = (4/3-1)$ times the energy density) exhibits critical phenomena at the threshold of black hole formation, and that near-critical evolutions approach a continuously self-similar solution. The self-similar solution is a type-2 similarity ansatz: with $a(ξ)$, $U^r(ξ)$, $Ω(ξ)$ and $N(ξ)$ functions of $ξ = r/C(-t)^n$, the Einstein and fluid equations reduce to ordinary differential equations that are regular at $ξ = 0$, at $ξ = ∞$, and across the ingoing acoustic characteristic at $ξ = 1$, which over-determines the system unless $n$ is an eigenvalue; that eigenvalue is $n = 1.1485$. The solution is not asymptotically flat ($a → 1.07$, $m(r)/r → 0.0596$, $Ω → 9.56×10^{-3}$), so it can only represent the behaviour of the central region $r ≪ r_0$, $|t| ≪ |t_0|$. The numerical evolutions confirm this: supercritical models give black hole masses fitting a power law with exponent $β = 0.36$, and a near-critical model's metric function is shown at a series of times, equally spaced in $τ = log(-T)$, shifting to progressively smaller scales and matching the overlaid self-similar $a(ξ)$ curve. Self-similarity is broken at a scale $r_1 = K|η-η_c|^β$, so it is visible only in the window $r_1 ≪ r ≪ r_0$ — the structure hydrodynamics calls intermediate asymptotics, with the modified Sedov-Taylor blast wave as the model case.

Load-bearing premise

The claim that near-critical evolutions approach the self-similar solution rests on a companion calculation listed as unpublished and by one of the present authors, so the comparison target is not independently available to the reader and the conclusion depends on that solution being both correct and the actual attractor of the evolution.

Editorial extensions

If this is right

The exponent $β = 0.36$ agrees with the values found in scalar-field collapse and in axisymmetric gravitational-wave collapse, which supports the claim that the exponent is a property of the gravitational field equations rather than of the matter model.
A precisely critical evolution ends in a curvature singularity at $r = 0$, $t = 0$, with $R_{μν}R^{μν} ∼ T^{-4}$, but the lapse behaves as $α ∼ r^{1-1/n} ∼ r^{0.129}$ there, so the singularity sits behind an infinite redshift rather than being directly visible.
Self-similarity is only apparent between the initial-data scale $r_0$ and the breaking scale $r_1 = K|η-η_c|^β$, so any observation of the critical solution requires tuning the parameter closely enough that $r_1 ≪ r_0$.
Because the critical solution is a known set of ODEs, linear stability analysis around it becomes tractable and may yield an independent estimate of $β$.
The measured black hole masses extend down to about $0.001$ of the initial total mass, and the power law holds across that range at two different numerical resolutions.

Reading between the lines

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

If $β$ is truly universal across matter models, then varying the equation of state away from radiation ($γ$ near $4/3$) should shift the similarity exponent $n$ while leaving $β$ nearly fixed; that is a sharper test than another fluid run at $γ = 4/3$.
The infinite-redshift shroud at the critical point suggests a checkable relationship: the exponent $1 - 1/n$ in the lapse should track $n$ as the equation of state is varied, which would tie the observability of the near-critical region directly to the similarity eigenvalue.
Since the self-similar solution is not asymptotically flat, the physical critical spacetime must be truncated at some large radius; whether that truncation leaves any imprint on the fitted $β$ or only on the outer expansion is left open by the paper and seems worth checking numerically.
The intermediate-asymptotics framing suggests the same window structure should appear in any collapse model with a one-parameter family and a scale-free inner solution, so systems with a sonic point and a single conserved scale are the natural next place to look for this behaviour.

Original abstract

We observe critical phenomena in spherical collapse of radiation fluid. A sequence of spacetimes $\cal{S}[\eta]$ is numerically computed, containing models ($\eta\ll 1$) that adiabatically disperse and models ($\eta\gg 1$) that form a black hole. Near the critical point ($\eta_c$), evolutions develop a self-similar region within which collapse is balanced by a strong, inward-moving rarefaction wave that holds $m(r)/r$ constant as a function of a self-similar coordinate $\xi$. The self-similar solution is known and we show near-critical evolutions asymptotically approaching it. A critical exponent $\beta \simeq 0.36$ is found for supercritical ($\eta>\eta_c$) models.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,363 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We observe critical phenomena in spherical collapse of radiation fluid. A sequence of spacetimes $\cal{S}[\eta]$ is numerically computed, containing models ($\eta\ll 1$) that adiabatically disperse and models ($\eta\gg 1$) that form a black hole. Near the critical point ($\eta_c$), evolutions develop a self-similar region within which collapse is balanced by a strong, inward-moving rarefaction wave that holds $m(r)/r$ constant as a function of a self-similar coordinate $\xi$. The self-similar solution is known and we show near-critical evolutions asymptotically approaching it. A critical exponent $\beta \simeq 0.36$ is found for supercritical ($\eta>\eta_c$) models.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We observe critical phenomena in spherical collapse of radiation fluid.	result 0.86	0.93	0.59	asserted 1.00
s1	A sequence of spacetimes $\cal{S}[\eta]$ is numerically computed, containing models ($\eta\ll 1$) that adiabatically disperse and models ($\eta\gg 1$) that form a black hole.	method 0.80	0.92	0.08	asserted 1.00
s2	Near the critical point ($\eta_c$), evolutions develop a self-similar region within which collapse is balanced by a strong, inward-moving rarefaction wave that holds $m(r)/r$ constant as a function of a self-similar coordinate $\xi$.	result 0.98	0.86	0.33	asserted 1.00
s3	The self-similar solution is known and we show near-critical evolutions asymptotically approaching it.	result 0.65	0.85	0.22	asserted 1.00
s4	A critical exponent $\beta \simeq 0.36$ is found for supercritical ($\eta>\eta_c$) models.	result 1.00	0.95	0.30	asserted 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

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sources read

arXiv listing abstract v1

not examined

full text
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relations to other papers

levels run

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S sections
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R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Self-similar radiation fluid collapse. Fluid and met

**Figure 3.** FIG. 3. Critical behavior of black hole mass. For

**Figure 4.** FIG. 4. Intermediate asymptotic approach of a near-critica

Figures reproduced from arXiv:gr-qc/9402041 by the authors.

References

Works cited (13)

[4]

C. R. Evans, to be submitted to Phys. Rev. D (1993)

View on Pith 1993
[9]

L. I. Sedov, Similarity and Dimensional Methods in Mechanics, 10th Edit ion (CRC Press: Boca Raton), 1993. The self-similar solution described here may be equivalent to the ‘exploding’ models of Ref. [11]

View on Pith 1993
[1]

M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993)

View on Pith 1993
[2]

A. M. Abrahams and C. R. Evans, Phys. Rev. Lett. 70, 2980 (1993)

View on Pith 1993
[3]

C. Gundlach, R. H. Price and J. Pullin, submitted to Phys. Rev. D (1 993)

View on Pith
[5]

A. M. Abrahams and C. R. Evans, submitted to Phys. Rev. D (199 3)

View on Pith
[6]

D. M. Eardley, in Texas/PASCOS 92: Relativistic Astrophysics and Particle C osmology, eds. C. W. Akerlof and M. A. Srednicki (N. Y. Acad. Sci., New York), 1993

View on Pith 1993
[7]

D. M. Eardley, Commun. Math. Phys. 37, 287 (1974)

View on Pith 1974
[8]

We use natural units in which G = c = 1

View on Pith
[10]

G. Guderley, Luftfahrtforschung 19, 302 (1942). See also the discussion of this prob- lem in G. B. Whitham, Linear and Nonlinear Waves, (Wiley: New York), 1974. A spherically-symmetric imploding shock wave asymptotically approach es a self-similar form and “remembers” about the initial conditions only a single numbe r related to the strength of the initia...

View on Pith 1942
[11]

A. Ori and T. Piran, Phys. Rev. D 42, 1068 (1990)

View on Pith 1990
[12]

D. M. Eardley, talk at Numerical Astrophysics: A Symposium in Ho nor of James R. Wil- son, at the University of Illinois, 1982 (unpublished); See e.g., J. M. B ardeen and T. Piran, Phys. Rep. 96, 205 (1983); L. I. Petrich, S. L. Shapiro and S. A. Teukolsky, Phys . Rev. D 33, 2100 (1986); R. F. Stark and T. Piran, Phys. Rev. Lett. 55, 891 (1985). 8

View on Pith 1983
[13]

G. I. Barenblatt and Ya. B. Zel’dovich, in Ann. Rev. Fluid Mech., 4, 285 (1972). See their discussion of the modiﬁed blast wave problem on p. 287. 9 FIGURES FIG. 1. Self-similar radiation ﬂuid collapse. Fluid and met ric variables are plotted versus self-similarity coordinate ξ = r/C (−t)n (with n ≃ 1.1485): (1) radial metric function a(ξ) (solid), (2) ve...

View on Pith Pith review arXiv 1972

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Purely quadratic non-Gaussianity from tachyonic instability allows narrow curvature spectra to exponentially suppress primordial black hole overproduction via correlation coefficient ρ approaching -1 while retaining s...
Primordial Black Hole Formation from the Upward Step Model: Avoiding Overproduction
astro-ph.CO 2024-12 conditional novelty 5.0 of 10

In upward-step inflation, a non-Gaussian cutoff in curvature perturbations sharply suppresses type-I primordial black hole abundance for h above about 5.9, easing PTA overproduction constraints.
On the inflationary interpretation of the nHz gravitational-wave background
astro-ph.CO 2023-08 conditional novelty 5.0 of 10

A CMB-consistent single-field inflation model with a tuned ultra-slow-roll phase produces a scalar-induced gravitational-wave signal at nHz frequencies that is marginally compatible with, though slightly below, the NA...
Primordial Black Holes from Multifield Inflation with Nonminimal Couplings
hep-th 2022-05 conditional novelty 5.0 of 10

A supersymmetric two-field inflation model with nonminimal couplings can generate a spike in the curvature power spectrum large enough to seed primordial black holes, but only after tuning one coupling to about seven digits.
Hunt for Light Primordial Black Hole Dark Matter with Ultra-High-Frequency Gravitational Waves
astro-ph.CO 2022-05 conditional novelty 5.0 of 10

Resonant LC circuit detectors come within roughly one order of magnitude of the loudest single primordial black hole merger signal near 30 MHz, and a Gaussian beam detector could probe the stochastic background from a...
Primordial black holes in peak theory with a non-Gaussian tail
astro-ph.CO 2021-09 conditional novelty 5.0 of 10

Applying peak theory to an exponential-tail non-Gaussian curvature distribution yields a strongly enhanced primordial black hole abundance and a mass spectrum with a characteristic hard cutoff at a maximal mass.
Confronting the primordial black hole scenario with the gravitational-wave events detected by LIGO-Virgo
astro-ph.CO 2021-08 conditional novelty 5.0 of 10

A binned reconstruction of the primordial black hole mass function from GWTC-3 is well fit by a broad red-tilted curvature power spectrum, and the resulting stochastic gravitational-wave background should be detectabl...
Two populations of LIGO-Virgo black holes
astro-ph.CO 2020-12 conditional novelty 5.0 of 10

Fitting LIGO-Virgo GWTC-2 mergers with a combined astrophysical plus primordial black hole model disfavours a primordial-only population and tightens the primordial black hole dark matter fraction to below 0.2 percent...
Testing Stochastic Gravitational Wave Signals from Primordial Black Holes with Optical Telescopes
astro-ph.CO 2020-10 conditional novelty 5.0 of 10

Existing Subaru HSC microlensing data are consistent with, and four additional hours of observation could exclude at 95% CL, the broad primordial-black-hole mass function proposed to explain the NANOGrav gravitational...
NANOGrav Hints to Primordial Black Holes as Dark Matter
astro-ph.CO 2020-09 conditional novelty 5.0 of 10

A flat spectrum of curvature perturbations normalized so that primordial black holes make up all dark matter also yields second-order gravitational waves whose amplitude and flat slope match the NANOGrav 12.5-year signal.
Did NANOGrav see a signal from primordial black hole formation?
astro-ph.CO 2020-09 conditional novelty 5.0 of 10

The NANOGrav common-spectrum signal is consistent with gravitational waves produced when large curvature perturbations collapse into primordial black holes, at an amplitude that also matches supermassive black hole se...
Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation
astro-ph.CO 2019-04 conditional novelty 5.0 of 10

Accounting for the non-linear relation between density and curvature perturbation suppresses primordial black hole formation, requiring roughly twice the power spectrum amplitude for the same abundance.
Spin Distribution of Primordial Black Holes
gr-qc 2017-04 conditional novelty 5.0 of 10

Applying critical-collapse scaling laws for a rotating radiation fluid, the authors derive a spin distribution for primordial black holes and find that nearly all form with dimensionless spin below about 0.4.
Constraints on primordial black holes from Galactic gamma-ray background
astro-ph.CO 2016-04 conditional novelty 5.0 of 10

The Galactic gamma-ray background constrains primordial black holes with initial mass just above the evaporating mass, and a critical-collapse mass function allows the largest black holes to be dark matter without vio...
Primordial black hole formation in bulk-viscous cosmology
gr-qc 2026-06 unverdicted novelty 4.0 of 10

Bulk viscosity raises the critical collapse threshold for primordial black holes by an amount comparable to the viscosity strength and increases the resulting black hole masses.
CMB Bounds on Primordial Black Holes via Radiation Capture
astro-ph.CO 2026-06 unverdicted novelty 4.0 of 10

Radiation capture by primordial black holes heavier than 10^15 solar masses modifies cosmic expansion enough for Planck CMB data to bound their abundance below 10 percent.
Primordial non-Gaussianity as a saviour for PBH overproduction in SIGWs generated by Pulsar Timing Arrays for Galileon inflation
astro-ph.CO 2023-10 conditional novelty 4.0 of 10

Galileon inflation with local non-Gaussianity f_NL around minus six yields primordial black holes at the right abundance and scalar-induced gravitational waves consistent with NANOGrav 15 and EPTA, without overproduci...
Critical Phenomena in Gravitational Collapse
gr-qc 2025-07 unverdicted novelty 3.0 of 10

An authoritative review of critical collapse, updated with results on nonspherical vacuum collapse and rigorous PDE blowup.
Inflation and Primordial Black Holes
astro-ph.CO 2023-01 accept novelty 3.0 of 10

The paper is a pedagogical review that classifies inflationary mechanisms for producing primordial black holes into single-field non-slow-roll enhancement and multi-field tachyonic or gauge-field amplification.
PBH formation from spherically symmetric hydrodynamical perturbations: a review
gr-qc 2021-11 conditional novelty 3.0 of 10

The review consolidates the spherical collapse formalism, numerical implementation, and shape-dependent thresholds and masses for primordial black hole formation.
Primordial Black Holes as Dark Matter: Recent Developments
astro-ph.CO 2020-06 unverdicted novelty 3.0 of 10

Primordial black holes in specific mass ranges could account for some or all dark matter while resolving structure-formation and seed problems in standard cosmology.
Primordial Black Holes - Perspectives in Gravitational Wave Astronomy -
astro-ph.CO 2018-01 accept novelty 3.0 of 10

Gravitational-wave observations of binary black hole mergers provide a new, independent probe of primordial black holes, complementing electromagnetic searches.
Primordial Black Holes as Dark Matter Candidates
astro-ph.CO 2021-10 accept novelty 2.0 of 10

Compiling all published constraints leaves only two mass ranges, roughly 10^17 to 10^23 grams and 10 to 100 solar masses, in which primordial black holes could supply most of the dark matter.

How later papers used it

98 later papers cite it

Evidence: some. 98 of the citing papers share no author with it, and 25 of those carry a confident label from Pith's citation classifier.

Pith read 5 of the 127 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its data or values.

Uses its data or values 1 of 21 papers with this label

Used this paper's critical scaling law formula to give the masses of primordial black holes.

Primordial black holes and scalar-induced gravitational waves from the generalized Brans-Dicke theory (2022) · 0.88 · paraphrased: the licence does not allow quoting

How the literature uses it

84 arXiv papers cite this in 104 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 59 sentences without a confident label

Builds on it 1 paper

Niemeyer & Jedamzik [ref] demonstrated that near-critical behaviour applies to PBH formation, extending the results of Evans & Coleman [ref] to an asymptotic FLRW background.

Cosmology using numerical relativity · Primordial black holes · 0.92

Uses its method or tool 2 papers

By substituting [ref] into [ref] and integrating out $X$, $ℙ(𝒞_ℓ)$ can be rewritten as a single-variable integral over $Y$ [ref] , [equation] Using $𝒞_ℓ$, we can express the PBH mass spectrum as follows [ref] : [equation] which accounts for critical collapse effects [ref] : [equation] The relation between the current PBH abundance $f(M)$ as a fraction of dark matter and their initial abundance $β(M)$ is given by [ref] : [equation]

Primordial Black Hole Formation from the Upward Step Model: Avoiding Overproduction · Basic setup for extended Press-Shecheter formalism · 0.96
In [ref] an alternative procedure to compute $γ$ was proposed based on a renormalization group analysis of critical gravitational collapse.

Critical gravitational collapse: towards a holographic understanding of the Regge region · Introduction · 0.83

Uses its data or values 17 papers

The masses of primordial black holes in equation [ref] obey the critical scaling law with the formula [ref] [equation] where $κ=3.3$ for the real space top-hat window function and $γ=0.36$ in the radiation domination [ref] .

Primordial black holes and scalar-induced gravitational waves from the generalized Brans-Dicke theory · primordial black holes · 0.88
We use $δ_c ≃0.42$ [ref] (see also Refs. [ref] ), $k=3.3$ [ref] and $γ=0.36$ [ref] .

Primordial Black Holes from Thermal Inflation · Primordial black holes · 1.00
The PBH mass $m$ is related to the density contrast by the critical collapse, namely $m=M_H κ(δ_m-δ_c)^γ$ [ref] with $κ=3.3$ and $γ=0.36$ [ref] .

Confronting the primordial black hole scenario with the gravitational-wave events detected by LIGO-Virgo · Introduction · 0.97

Compares with it 2 papers

It is worth noting that the approximation ([ref]) bears resemblance to the one found in type I critical gravitational collapse [ref] .

Type I critical dynamical scalarization and descalarization in Einstein-Maxwell-scalar theory · Dynamical critical behaviors of scalarization · 0.98
The PBH mass in Eq. [ref] should be compared with the one obtained from the collapse of horizon-size overdensities [ref] , in which case we have [equation] with $γ∼( 1 )$.[Here, for simplicity, we neglect the effect of critical collapse ( Refs. [ref] ), which leads to a PBH mass spectrum.] Hence, $M_PBH$ is much larger than that of our proposed mechanism; their ratio being [equation]

Primordial Black Holes from Confinement · Mechanism · 1.00

Cites it as background 23 papers

It is well known that black-hole formation is associated with critical phenomena [ref] and the application of this to PBH formation has been studied by various authors [ref] .

Primordial Black Holes With Multi-Modal Mass Spectra · Critical Collapse · 0.98
Depending on the matter model, the critical solution turns out to be either discretely self-similar (DSS, for example for the scalar field considered by Choptuik) or continuously self-similar (CSS, for example for radiation fluids [ref] ).

Critical phenomena in the collapse of gravitational waves · Introduction · 0.95
Therefore, we remind ourselves that black-hole formation is associated with critical phenomena [ref] — the application of which to PBH formation has been studied by various authors ( Refs. [ref] ).

Extreme-Value Distributions and Primordial Black-Hole Formation · Introduction · 0.92

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [4]

C. R. Evans, to be submitted to Phys. Rev. D (1993)

View on Pith 1993

[2] [9]

L. I. Sedov, Similarity and Dimensional Methods in Mechanics, 10th Edit ion (CRC Press: Boca Raton), 1993. The self-similar solution described here may be equivalent to the ‘exploding’ models of Ref. [11]

View on Pith 1993

[3] [1]

M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993)

View on Pith 1993

[4] [2]

A. M. Abrahams and C. R. Evans, Phys. Rev. Lett. 70, 2980 (1993)

View on Pith 1993

[5] [3]

C. Gundlach, R. H. Price and J. Pullin, submitted to Phys. Rev. D (1 993)

View on Pith

[6] [5]

A. M. Abrahams and C. R. Evans, submitted to Phys. Rev. D (199 3)

View on Pith

[7] [6]

D. M. Eardley, in Texas/PASCOS 92: Relativistic Astrophysics and Particle C osmology, eds. C. W. Akerlof and M. A. Srednicki (N. Y. Acad. Sci., New York), 1993

View on Pith 1993

[8] [7]

D. M. Eardley, Commun. Math. Phys. 37, 287 (1974)

View on Pith 1974

[9] [8]

We use natural units in which G = c = 1

View on Pith

[10] [10]

G. Guderley, Luftfahrtforschung 19, 302 (1942). See also the discussion of this prob- lem in G. B. Whitham, Linear and Nonlinear Waves, (Wiley: New York), 1974. A spherically-symmetric imploding shock wave asymptotically approach es a self-similar form and “remembers” about the initial conditions only a single numbe r related to the strength of the initia...

View on Pith 1942

[11] [11]

A. Ori and T. Piran, Phys. Rev. D 42, 1068 (1990)

View on Pith 1990

[12] [12]

D. M. Eardley, talk at Numerical Astrophysics: A Symposium in Ho nor of James R. Wil- son, at the University of Illinois, 1982 (unpublished); See e.g., J. M. B ardeen and T. Piran, Phys. Rep. 96, 205 (1983); L. I. Petrich, S. L. Shapiro and S. A. Teukolsky, Phys . Rev. D 33, 2100 (1986); R. F. Stark and T. Piran, Phys. Rev. Lett. 55, 891 (1985). 8

View on Pith 1983

[13] [13]

G. I. Barenblatt and Ya. B. Zel’dovich, in Ann. Rev. Fluid Mech., 4, 285 (1972). See their discussion of the modiﬁed blast wave problem on p. 287. 9 FIGURES FIG. 1. Self-similar radiation ﬂuid collapse. Fluid and met ric variables are plotted versus self-similarity coordinate ξ = r/C (−t)n (with n ≃ 1.1485): (1) radial metric function a(ξ) (solid), (2) ve...

View on Pith Pith review arXiv 1972