Dynamics of Primordial Black Hole Formation

J.C. Niemeyer (UC Chicago); K. Jedamzik (MPA Garching)

REVIEW

Dynamics of Primordial Black Hole Formation

J.C. Niemeyer (UC Chicago), K. Jedamzik (MPA Garching)

Astrophysics · Submitted 21 Jan 1999 · 23 references

Pith's verdict: A solid numerical study that puts the PBH formation threshold near 0.7 and shows critical-collapse scaling survives in an expanding FRW background; the headline comparison to 1/3 is the shakiest part of the framing.

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CitationJ.C. Niemeyer (UC Chicago), K. Jedamzik (MPA Garching). "Dynamics of Primordial Black Hole Formation." arXiv:astro-ph/9901292 (1999). https://pith.science/paper/NWISGXYG

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Editor's note

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Two things here are worth your time. First, a numerically computed PBH formation threshold, δ_c ≈ 0.7 in the ΔM/M_h sense, bracketed across three perturbation shapes (0.67 Mexican hat, 0.70 Gaussian, 0.71 polynomial). Second, a clean demonstration that the Choptuik/Evans–Coleman mass scaling carries over to an expanding FRW background, with γ ≈ 0.36 recovered without being fitted for. That γ independently matches Evans and Coleman's 0.36 and Koike et al.'s 0.3558, which is the kind of check that makes me trust the rest of the numbers more than I otherwise would. The Hernandez–Misner null slicing is the right tool for following the evolution long after horizon formation, and the code is validated against the exact flat FRW solution to better than 1e-3. The fluid bounce in near-critical collapse, and the argument that earlier work missed it because its initial data sat too far from threshold, is specific and plausible. Credit earned there. On the soft spots, in order of how much they bother me. The comparison to Carr's 1/3 is the weakest link: the paper asserts that value is 'evaluated at the time of horizon crossing' but never shows it is the same uniform-Hubble ΔM/M_h quantity they compute. Section V's remark that only the scaling relation is reparameterization-invariant, not δ_c itself, makes that gap more pointed rather than less. I read this as an overstated headline rather than a wrong number, since the abstract's own 'if δ is defined as ΔM/M_h' qualifier shows they know the threshold is definition-dependent. Second, K is time-step dependent by their own admission, and the resolution floor at M_bh ≈ 0.1 means the near-critical regime carrying both the bounce and the scaling fit is the least resolved part of the calculation. They say so and call for an adaptive mesh; fair, but it caps how hard you can lean on K. Third, the mass-function payoff leans on reference [9], their own then-unpublished companion submission, which is a real dependency rather than a flaw. Fourth, three hand-picked shapes with a 0.67–0.71 spread is evidence of robustness within that set, not proof of universality across what inflation actually produces. This is for anyone doing PBH abundance or small-scale power spectrum constraints, and for numerical relativists interested in critical collapse in cosmological backgrounds. Send it to referees; the central result holds up and the technical work is careful.

Referee report

Significance

The methodological contribution is real: the null-slicing scheme lets the authors follow the evolution for long times after event-horizon formation without encountering the coordinate singularities that limit the Misner–Sharp evolution, which is what makes the accretion analysis in §IV possible at all and distinguishes this work from the earlier calculations of [10]. The recovery of γ ≈ 0.36 in an asymptotically expanding FRW background is a nontrivial independent check — it agrees with the Evans–Coleman numerical value and the Koike–Hara–Adachi perturbative value of 0.3558 — and it is not guaranteed a priori that critical scaling survives the cosmological boundary conditions. The bounce mechanism is a physically motivated and falsifiable explanation for suppressed accretion, and the quoted K and γ are directly usable in the mass function of [9]. If the threshold holds in the stated gauge, every cosmological PBH abundance estimate and the associated small-scale spectral-index constraints shift substantially, so the result is worth publishing in a form that makes the gauge and the numerical uncertainties unambiguous.

Major objections (3)

[Abstract, sentence s2; §I and §III] The headline contrast between δ_c ≈ 0.7 and 'the generally employed δ_c ≈ 1/3' is not established as a like-for-like comparison. §III defines δ as the energy overdensity in uniform Hubble constant gauge averaged over a horizon volume, 'equivalent to the additional mass-energy inside R_h in units of M_h', whereas §I attributes the 1/3 to the closed-FRW ansatz of Carr [8] 'evaluated at the time of horizon crossing' without stating the gauge or normalization in which that number is defined. §V then shows explicitly that only the scaling relation is invariant under redefinitions of the control parameter, up to the coefficient K — that is, δ_c itself is definition-dependent. As written, a reader who inserts 0.7 into an abundance formalism calibrated to a comoving-gauge density contrast will obtain a wrong answer, and part of the claimed factor-of-two may be definitional rather than physical. Please state the gauge and normalization of Carr's 1/3, give the conversion factor if it differs from the definition used here, or restate the abstract's comparison so that the two numbers are explicitly commensurate. The numerical threshold obtained in the paper's own gauge is not in question.
[§II (time-step discussion) versus §V (Fig. 8) and §VI] §II states that 'Decreasing the time step generally causes a decrease of resulting black hole mass' and that 'only the coefficient K in Eq. (1) is affected by the time step variation, while the scaling exponent γ appears very robust.' §V and the Conclusions then quote K ≈ 2.4–12 as a physical result and note that K is 'needed to evaluate the PBH initial mass function derived in [9].' The authors have thus identified K as the one quantity in Eq. (1) that is not numerically converged, and the fit uses 'the smallest six black holes of each family' — precisely the runs closest to threshold and most sensitive to resolution and time step. The quoted K values therefore need a quantified uncertainty: please report K from at least two time steps, and ideally two grid resolutions, for one perturbation family, and propagate the resulting error bar into Fig. 8 and the Conclusions. Without this, the factor-of-five spread in K attributed to perturbation shape cannot be separated from numerical systematics, and the mass function of [9] inherits an unquantified systematic.
[§III (threshold values) and §V (Fig. 8)] The three thresholds are quoted to three digits (0.67, 0.70, 0.71) with no numerical uncertainty attached. The bracketing precision is set by the runs actually shown — for the Gaussian, 0.7006 (no black hole, Fig. 5) versus 0.7015 (M_bh = 0.37, Fig. 4) — i.e., roughly 1e-3 in δ, but the systematic from grid resolution, the time-step restriction, and the outer boundary imposed at R_max ≳ 9 R_h is not estimated anywhere. Because δ_c enters the PBH abundance exponentially, a systematic at even the few-percent level is consequential for the paper's stated purpose. Please quote δ_c at two resolutions for at least one family. Relatedly, the universality claim rests on three smooth, spherically symmetric profiles, two of which are mass-compensated by construction; the manuscript should state explicitly that sharper profiles, non-compensated large-scale perturbations, and non-spherical collapse are untested, and should say whether the 0.67–0.71 spread is intended as a measured systematic or as a null result consistent with a single universal threshold.

Minor objections (8)

[Fig. 4 caption and Fig. 8 caption] The Gaussian value δ = 0.7015 appears both as the illustrative near-critical run producing M_bh = 0.37 (Fig. 4) and as the fitted threshold δ_c = 0.7015 (Fig. 8). Eq. (1) requires M_bh → 0 as δ → δ_c, so these cannot both be exact; presumably the fitted δ_c lies slightly below 0.7015 and the coincidence is rounding. Please quote the fitted δ_c to enough digits to make the two statements consistent, or state whether the fit was constrained to pass through the bracketing threshold.
[§III] The sentence 'It is likely that previous studies [10] failed to observe this phenomenon because their initial conditions were insufficiently close to the black hole formation threshold' is a claim about other authors' calculations. Either support it, for example by showing that the bounce is absent at the amplitudes those studies employed, or soften it to an explicit possibility.
[§III and Fig. 6] The disappearance of the bounce at large amplitude is documented by a single overcritical run (M_bh = 2.75). Please state the value of δ at which the bounce vanishes, or say explicitly that the transition is bracketed only coarsely.
[Fig. 1] State whether the three profiles are plotted at the same δ or at the same amplitude A. Since A is not a common control parameter across Eqs. (6)–(8), the visual comparison of the 'critical perturbations' requires the normalization to be specified.
[§II] The values N = 500 and C = 12 in Eq. (3) are given without the convergence test that fixed them; one sentence on how these were chosen, and on how the results change at N = 250 or N = 1000, would let the reader judge the resolution claim.
[§IV and Fig. 7] The attribution of the Gaussian curve's secondary growth phase to 'the rise of the second wave crest of the strongly damped density oscillation at the black hole event horizon' is an interpretation of the plotted growth rate rather than a directly measured quantity; label it as such.
[§II] The redefinition of ρ_0 as a conserved tracer number density and of e as the energy per tracer is clear in motivation but deserves one explicit sentence on how e is initialized from ǫ_0 and how the '1 + e' → 'e' replacement is justified quantitatively at the densities used.
[§III] Typographical: 'cirumferential' should read 'circumferential' in the paragraph following Eq. (6).

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The paper

The paper claims

The density contrast needed to collapse a horizon-size fluctuation into a primordial black hole is about 0.7, not the 1/3 long assumed, and the same value holds for three different perturbation shapes.

The reading

This paper asks a bookkeeping question with large consequences: how big must a density fluctuation be, at the moment it enters the horizon in the radiation-dominated early universe, for gravity to beat pressure and crush it into a primordial black hole? The standard answer, inherited from a toy model that pastes a closed Friedmann universe onto a flat one, is roughly one third of the horizon mass in excess. The authors instead evolve the full general relativistic fluid equations in spherical symmetry, for three different perturbation shapes, and find the threshold nearer $0.7$. That number is the hinge on which estimates of how many primordial black holes exist, and therefore constraints on small-scale primordial power and on evaporating black holes, turn.

What carries the argument

The load-bearing technical choice is the Hernandez–Misner formulation of the spherically symmetric hydrodynamic equations, in which the time coordinate is constant along outgoing null rays. Because the grid points sit on null surfaces rather than on slices of constant time, the computation never has to cross the event horizon, so the authors can follow a newly formed black hole for arbitrarily long times without hitting a coordinate singularity — the thing earlier Misner–Sharp-based simulations could not do, and the reason they could not see the post-formation bounce or measure late accretion. Surrounding this sits the definition of the control parameter: $��$ is the excess gravitational mass inside the initial horizon volume relative to the unperturbed Friedmann solution, and the black hole mass is read off as the gravitational mass enclosed by the innermost shell whose lapse has fallen below $10^{-10}$.

Extended reading notes

Core claim

The paper's central claim is that the threshold for primordial black hole formation, defined as the excess mass within the initial horizon volume in uniform Hubble gauge, is $��_{c} = 0.67$ for a Mexican-hat profile, $0.70$ for a Gaussian, and $0.71$ for a sixth-order polynomial, i.e. $��_{c} �� 0.7$ rather than the generally employed $��_{c} �� 1/3$. Two further results ride along. First, black holes that form close to threshold are surrounded by a fluid bounce: material just outside the event horizon is pushed outward in a compression wave, followed by a rarefaction that shuts off accretion almost completely, whereas black holes far above threshold keep accreting at a slowly declining rate; in neither case is the accreted mass significant. Second, the near-critical scaling $M_{bh} = K M_h (�� - ��_{c})^{��}$ known from asymptotically flat collapse carries over to an expanding Friedmann background, with $�� = 0.34$ to $0.37$ and $K$ between about $2.4$ and $12$ depending on shape.

Load-bearing premise

The claim that $��_{c} �� 0.7$ is a number cosmology can use rests on three hand-picked, spherically symmetric density profiles bracketing the space of perturbations inflation actually produces at horizon crossing; the paper treats the narrow spread it finds across them as evidence of universality rather than as a measured systematic.

Editorial extensions

If this is right

Cosmological PBH abundance estimates built on $��_{c} �� 1/3$ are wrong by a large factor for any steeply falling fluctuation spectrum, since the number of fluctuations above threshold is exponentially sensitive to where the threshold sits.
Upper bounds on the spectral index of the primordial power spectrum on small scales, derived from PBH overproduction and from evaporating black holes, loosen correspondingly.
Accretion can be dropped from the calculation of final PBH masses for the perturbation types studied, so $M_{bh}$ is effectively fixed at formation.
The initial PBH mass function acquires a sub-horizon tail from critical scaling, with $�� 0.36$, changing which masses dominate the evaporation-era phenomenology.
The same numerical machinery can be pointed at the cosmological QCD transition to test whether a softened equation of state really lowers the formation threshold.

Reading between the lines

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

Because $��$ here is a mass excess inside the horizon volume in uniform Hubble gauge, thresholds quoted elsewhere under a different gauge or a linear extrapolation of the density contrast are not directly comparable; a reader comparing numbers across the literature should check the definition before concluding there is a disagreement.
The near-agreement of the threshold across one uncompensated and two mass-compensated profiles hints that $��_{c}$ is set mainly by the equation of state and the background expansion rather than by the perturbation's shape, which is what makes a QCD-era drop in $��_{c}$ a clean prediction rather than a model-dependent hope.
That the fitted exponent $��$ matches the asymptotically flat radiation-fluid value suggests the critical solution barely notices the expanding background, which would make the resulting mass function largely insensitive to the cosmology of the formation epoch.
All three families are spherically symmetric and non-rotating; if shape-robustness holds only within that class, realistic three-dimensional perturbations could move the threshold, and that is the obvious next computation.

Original abstract

We present a numerical investigation of the gravitational collapse of horizon-size density fluctuations to primordial black holes (PBHs) during the radiation-dominated phase of the Early Universe. The collapse dynamics of three different families of initial perturbation shapes, imposed at the time of horizon crossing, is computed. The perturbation threshold for black hole formation, needed for estimations of the cosmological PBH mass function, is found to be $\delta_{\rm c} \approx 0.7$ rather than the generally employed $\delta_{\rm c} \approx 1/3$, if $\delta$ is defined as $\Delta M/\mh$, the relative excess mass within the initial horizon volume. In order to study the accretion onto the newly formed black holes, we use a numerical scheme that allows us to follow the evolution for long times after formation of the event horizon. In general, small black holes (compared to the horizon mass at the onset of the collapse) give rise to a fluid bounce that effectively shuts off accretion onto the black hole, while large ones do not. In both cases, the growth of the black hole mass owing to accretion is insignificant. Furthermore, the scaling of black hole mass with distance from the formation threshold, known to occur in near-critical gravitational collapse, is demonstrated to apply to primordial black hole formation.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,768 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.31 · human participants 0.01 · controlled comparison 0.13 · randomized assignment 0.02 · states quantitative result 0.99 · compares with prior results 0.91 · 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 present a numerical investigation of the gravitational collapse of horizon-size density fluctuations to primordial black holes (PBHs) during the radiation-dominated phase of the Early Universe. The collapse dynamics of three different families of initial perturbation shapes, imposed at the time of horizon crossing, is computed. The perturbation threshold for black hole formation, needed for estimations of the cosmological PBH mass function, is found to be $\delta_{\rm c} \approx 0.7$ rather than the generally employed $\delta_{\rm c} \approx 1/3$, if $\delta$ is defined as $\Delta M/\mh$, the relative excess mass within the initial horizon volume. In order to study the accretion onto the newly formed black holes, we use a numerical scheme that allows us to follow the evolution for long times after formation of the event horizon. In general, small black holes (compared to the horizon mass at the onset of the collapse) give rise to a fluid bounce that effectively shuts off accretion onto the black hole, while large ones do not. In both cases, the growth of the black hole mass owing to accretion is insignificant. Furthermore, the scaling of black hole mass with distance from the formation threshold, known to occur in near-critical gravitational collapse, is demonstrated to apply to primordial black hole formation.

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 present a numerical investigation of the gravitational collapse of horizon-size density fluctuations to primordial black holes (PBHs) during the radiation-dominated phase of the Early Universe.	contribution 1.00	0.94	0.05	asserted 0.93
s1	The collapse dynamics of three different families of initial perturbation shapes, imposed at the time of horizon crossing, is computed.	contribution 0.65	0.95	0.03	asserted 0.98
s2	The perturbation threshold for black hole formation, needed for estimations of the cosmological PBH mass function, is found to be $\delta_{\rm c} \approx 0.7$ rather than the generally employed $\delta_{\rm c} \approx 1/3$, if $\delta$ is defined as $\Delta M/\mh$, the relative excess mass within the initial horizon volume.	result 1.00	0.95	0.58	asserted 1.00
s3	In order to study the accretion onto the newly formed black holes, we use a numerical scheme that allows us to follow the evolution for long times after formation of the event horizon.	method 0.98	0.94	0.03	asserted 0.99
s4	In general, small black holes (compared to the horizon mass at the onset of the collapse) give rise to a fluid bounce that effectively shuts off accretion onto the black hole, while large ones do not.	result 1.00	0.90	0.09	asserted 0.98
s5	In both cases, the growth of the black hole mass owing to accretion is insignificant.	result 1.00	0.92	0.15	asserted 1.00
s6	Furthermore, the scaling of black hole mass with distance from the formation threshold, known to occur in near-critical gravitational collapse, is demonstrated to apply to primordial black hole formation.	result 0.99	0.88	0.15	asserted 1.00

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

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
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. Shapes of the critical perturbations as imposed at th

**Figure 2.** FIG. 2. Time evolution of a near-critical Mexi

**Figure 4.** FIG. 4. Time evolution of a near-critical Gaus

**Figure 8.** FIG. 8. Black hole masses as a function of the distance to the fo

Showing 5 of the paper's 8 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/9901292 by the authors.

References

Works cited (23)

[9]

J. C. Niemeyer and K. Jedamzik, astro-ph/9709072, subm. to Phys. Rev. Lett. (1997)

View on Pith Pith review arXiv 1997
[10]

D. K. Nadezhin, I. D. Novikov, and A. G. Polnarev, Sov. As tron. 22, 129 (1978); G. V. Bicknell and R. N. Henriksen, Astrophys. J., 232, 670 (1979)

View on Pith 1978
[8]

B. J. Carr, Astrophys. J. 201, 1 (1975)

View on Pith 1975
[1]

Ya. B. Zeldovich, and I. D. Novikov, Sov. Astron. A. J. 10, 602 (1967)

View on Pith 1967
[2]

S. W. Hawking, M.N.R.A.S. 152, 75 (1971)

View on Pith 1971
[3]

D. Cline, D. A. Sanders, and W. Hong, Astrophys. J. 486, 169 (1997)

View on Pith 1997
[4]

T. Nakamura, M. Sasaki, T. Tanaka, and K. S. Thorne, Astro phys. J. Lett. 487, L139 (1997)

View on Pith 1997
[5]

B. J. Carr, J. H. Gilbert, and J. E. Lidsey, Phys. Rev. D. 50, 4853 (1994); A. M. Green and A. R. Liddle, Phys. Rev. D56, 6166-6174 (1997); A. R. Liddle and A. M. Green, gr-qc/980403 4 (1998)

View on Pith 1994
[6]

C. Alcock et al. (MACHO Collaboration), Astrophys. J. 471, 771 (1996); C. Renault et al. (EROS Collaboration), Astron . Astrophys. 324, L69 (1997)

View on Pith 1996
[7]

K. Jedamzik, Phys. Rev. D, 55, R5871 (1997)

View on Pith 1997
[11]

J. M. Bardeen, J. R. Bond, N. Kaiser, and A. S. Szalay, Ast rophys. J., 304, 15 (1986)

View on Pith 1986
[12]

as implemented by Baumgarte et al. [13] (we omit restating the full system of equations but inst ead refer to the equations published by Baumgarte et al. by a capital “B” followed by the respective equation number [ 14]). Based on the original equations by Misner and Sharp [15], Hernande z and Misner proposed to exchange the Misner–Sharp time variable, t,...

View on Pith
[13]

W. C. Hernandez and C. W. Misner, Astrophys. J., 143, 452 (1966)

View on Pith 1966
[14]

T. W. Baumgarte, S. L. Shapiro, and S. A. Teukolsky, Astr ophys. J. 443, 717 (1995)

View on Pith 1995
[15]

Furthermore, we use the same notation and geometrized u nits as [13]

View on Pith
[16]

C. W. Misner and D. H. Sharp, Phys. Rev. 136, B571 (1964)

View on Pith 1964
[17]

C. R. Evans and J. S. Coleman, Phys. Rev. Lett 72, 1782 (1994)

View on Pith 1994
[18]

T. Koike, T. Hara, and S. Adachi, Phys. Rev. Lett. 74, 5170 (1995)

View on Pith 1995
[19]

B. J. Carr and S. W. Hawking, ibid. 168, 399 (1974); G. V. Bicknell and R. N. Henriksen, Astrophys. J ., 225, 237 (1978)

View on Pith 1974
[20]

We deﬁne PBH mass as a function of time as the rest mass-en ergy interior to the event horizon at the proper time of an infalling ﬂuid element which just passes over the event hori zon. Since the lapse of proper time between the ﬂuid element ﬁrst entering the strong gravitational ﬁeld of the PBH and th e ﬂuid element crossing over the event horizon is gen...

View on Pith
[21]

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

View on Pith 1993
[22]

C. Gundlach, gr-qc/9712084 (1997); M. W. Choptuik, gr- qc/9803075 (1998)

View on Pith Pith review arXiv 1997
[23]

J. Yokoyama, gr-qc/9804041 (1998). 10

View on Pith Pith review arXiv 1998

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gr-qc 2025-09 unverdicted novelty 6.0 of 10

One-loop quantum vacuum polarization in Einstein-scalar critical collapse generates a horizon and finite mass gap, enforcing black hole formation even under arbitrary fine-tuning.
Isotropy, anisotropies and non-Gaussianity in the scalar-induced gravitational-wave background: diagrammatic approach for primordial non-Gaussianity up to arbitrary order
astro-ph.CO 2025-05 unverdicted novelty 6.0 of 10

Extends diagrammatic approach for scalar-induced gravitational waves to arbitrary-order local PNG, deriving semi-analytic spectra for energy density, anisotropies, bispectrum and trispectrum up to quartic terms.
Resilience of Primordial Black Hole Formation from Inflation with a Light Spectator Field
astro-ph.CO 2025-04 conditional novelty 6.0 of 10

A spectator scalar field with minimal coupling can replace ultra-slow-roll with a turn and tachyonic growth mechanism, producing primordial black holes while reducing fine-tuning.
Probing Primordial Black Hole Mergers in Clusters with Pulsar Timing Data
astro-ph.CO 2024-12 conditional novelty 6.0 of 10

Bayesian analysis of IPTA DR2 shows scalar-induced gravitational waves dominate and PBH merger backgrounds are strongly disfavored relative to an astrophysical SMBHB explanation.
Primordial black hole formation during the QCD phase transition: threshold, mass distribution and abundance
astro-ph.CO 2023-03 accept novelty 6.0 of 10

Full numerical simulations of PBH collapse during the QCD epoch yield a threshold, a critical scaling law, and a mass function peaking at 1.9 solar masses with a ~10^3 abundance enhancement over pure radiation.
Search for scalar induced gravitational waves in the International Pulsar Timing Array Data Release 2 and NANOgrav 12.5 years datasets
astro-ph.CO 2023-02 accept novelty 6.0 of 10

The IPTA DR2 common-spectrum signal is decisively better fit by supermassive black hole binaries than by scalar-induced gravitational waves once primordial black hole overproduction caps the curvature power spectrum; ...
Anatomy of single-field inflationary models for primordial black holes
astro-ph.CO 2022-05 conditional novelty 6.0 of 10

A two-parabola inflaton potential yields a closed-form curvature power spectrum whose peak position and height map onto PBH mass and abundance, and the resulting spectral-distortion bound excludes single-field PBHs he...
Statistics of coarse-grained cosmological fields in stochastic inflation
astro-ph.CO 2021-11 conditional novelty 6.0 of 10

A stochastic-inflation framework gives the one-point probability distribution of the curvature perturbation, density contrast and compaction function at any coarse-graining scale, and yields a primordial black hole ma...
Analytical thresholds for black hole formation in general cosmological backgrounds
gr-qc 2020-07 accept novelty 6.0 of 10

Numerical collapse simulations show that the threshold density for primordial black hole formation depends mainly on the fluid equation of state and one shape parameter, and an eight-parameter fit reproduces the thres...
Evidence for primordial black hole dark matter from LIGO/Virgo merger rates
astro-ph.CO 2020-07 conditional novelty 6.0 of 10

Primordial black holes formed during the QCD epoch, if they constitute all dark matter, would merge in clusters at rates comparable to those LIGO/Virgo has observed.
Turning in the landscape: a new mechanism for generating Primordial Black Holes
hep-th 2020-04 conditional novelty 6.0 of 10

A limited period of strong geodesic deviation in multi-field inflation amplifies the curvature power spectrum by the factor of roughly ten million needed to produce primordial black holes as dark matter.
Gravitational Wave Production right after a Primordial Black Hole Evaporation
astro-ph.CO 2020-03 conditional novelty 6.0 of 10

Sudden reheating from evaporating primordial black holes amplifies induced gravitational waves enough that future detectors could constrain the abundance of 10^3 to 10^5 gram black holes.
Primordial black holes and gravitational waves from parametric amplification of curvature perturbations
astro-ph.CO 2019-12 conditional novelty 6.0 of 10

A periodic ripple on the inflaton potential parametrically amplifies small-scale curvature perturbations, and for suitable parameters the resulting primordial black holes can account for all dark matter while sourcing...
Primordial Black Holes from Cosmic Domain Walls
astro-ph.CO 2019-08 conditional novelty 6.0 of 10

A two-field inflationary model with time-dependent domain wall tension produces primordial black holes with a narrow, spike-like mass function, potentially explaining all dark matter or LIGO merger events.
The Ineludible non-Gaussianity of the Primordial Black Hole Abundance
astro-ph.CO 2019-04 conditional novelty 6.0 of 10

The nonlinear density-curvature relation makes the PBH abundance unavoidably non-Gaussian, suppressing it by an amount equivalent to a factor 2 to 3 in the power spectrum amplitude.
Spin of Primordial Black Holes
astro-ph.CO 2019-01 conditional novelty 6.0 of 10

Primordial black hole spin vanishes at first order in primordial fluctuations and is estimated at second order to be about one percent of the maximum for fluctuation amplitudes near 0.1.
Planck scale black hole dark matter from Higgs inflation
astro-ph.CO 2018-10 conditional novelty 6.0 of 10

Primordial black holes from Higgs inflation can account for all dark matter only if they leave behind Planck-mass relics, which requires a shallow local minimum in the Higgs potential rather than a bare critical point.
PBH abundance from random Gaussian curvature perturbations and a local density threshold
astro-ph.CO 2018-05 conditional novelty 6.0 of 10

Combining peak theory for a Gaussian curvature perturbation with a nonlinear curvature-to-density map and a density threshold raises the predicted primordial black hole abundance by many orders of magnitude over Press...
Stochastic gravitational waves associated with the formation of primordial black holes
astro-ph.CO 2016-12 conditional novelty 6.0 of 10

At fixed primordial black hole abundance, non-gaussian curvature perturbations lower the RMS fluctuation and hence the second-order induced gravitational-wave background, allowing the PBH scenario to evade current pul...
Inflationary primordial black holes for the LIGO gravitational wave events and pulsar timing array experiments
astro-ph.CO 2016-11 accept novelty 6.0 of 10

Interpreting LIGO's black hole mergers as inflationary primordial black holes requires a very sharp peak in the scalar power spectrum at k ~ 10^6 Mpc^-1, steep enough to evade pulsar timing array and CMB mu-distortion limits.
Cosmological long-wavelength solutions and primordial black hole formation
gr-qc 2015-03 conditional novelty 6.0 of 10

The two standard numerical approaches to primordial black hole formation are shown to be equivalent long-wavelength solutions in different gauges, and the collapse threshold is bracketed between two analytic values se...
Massive Primordial Black Holes from Hybrid Inflation as Dark Matter and the seeds of Galaxies
astro-ph.CO 2015-01 conditional novelty 6.0 of 10

A mild waterfall phase of hybrid inflation produces a broad primordial black hole mass spectrum that can match the dark matter abundance and seed supermassive black holes, for a narrow window of potential parameters.
Separate Universes Do Not Constrain Primordial Black Hole Formation
astro-ph.CO 2010-12 conditional novelty 6.0 of 10

The Carr-Hawking no-separate-universe condition does not constrain density fluctuations; the maximum density amplitude is a geometric artifact, and larger fluctuations also form primordial black holes in dust.
Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity
gr-qc 1999-05 conditional novelty 6.0 of 10

Primordial black hole formation requires a non-linear initial metric perturbation (conformal factor above about 1.4), and both the threshold and the final black hole mass depend on whether the density peak sits in an ...
Universal Suppression of Gravitational Waves from Black Hole Evaporation Dynamics
astro-ph.CO 2026-06 unverdicted novelty 5.0 of 10

Black hole populations with finite-width mass distributions exhibit universal late-time evaporation-driven evolution that produces characteristic power-law suppression of induced gravitational waves, directly linking ...
Primordial black holes in excursion set theory: Formation probabilities, mass functions, and window functions
astro-ph.CO 2026-05 unverdicted novelty 5.0 of 10

In excursion set theory with colored noises, the low-mass tail of the PBH mass function differs from Carr's formula because correlated noises end the degeneracy of formation probabilities, though Carr's formula remain...
Implications for Pulsar Timing Arrays of Sub-solar Black Hole Detections: From LVK to Einstein Telescope and Cosmic Explorer
astro-ph.CO 2025-08 conditional novelty 5.0 of 10

A Bayesian analysis shows that a future sub-solar PBH detection would make the primordial SIGW interpretation of PTA data favored over the SMBH interpretation, but this preference is driven by the detection prior.
Primordial black holes through preheating instabilities in $\alpha$-attractor models
astro-ph.CO 2025-05 conditional novelty 5.0 of 10

In alpha-attractor models, preheating self-resonance can form primordial black holes of 10^2 to 10^7 g, with the Khlopov-Polnarev collapse formalism staying viable and the Press-Schechter formalism overproducing.
Gravitational waves induced from primordial black hole fluctuations: The effect of an extended mass function
astro-ph.CO 2022-07 conditional novelty 5.0 of 10

Extending the primordial-black-hole Poisson gravitational-wave calculation to an extended mass function yields a narrow allowed window for the running spectral index, alpha_s between about 3.316 and 3.355 times 10^-3,...
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.
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.
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...
NANOGrav results and LIGO-Virgo primordial black holes in axion-like curvaton model
astro-ph.CO 2020-11 conditional novelty 5.0 of 10

Primordial non-Gaussianity in the axion-like curvaton model suppresses induced gravitational waves enough that stellar-mass primordial black holes and the NANOGrav nanohertz signal can be explained by the same early-u...
Self-interacting Dark Matter from Primordial Black Holes
hep-ph 2020-10 conditional novelty 5.0 of 10

Thermalization and number-changing self-interactions in a dark sector fed by primordial black hole evaporation can raise the dark matter abundance by orders of magnitude and relax the free-streaming mass bound to roug...
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...
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...
Abundance of Primordial Black Holes in Peak Theory for an Arbitrary Power Spectrum
astro-ph.CO 2020-08 conditional novelty 5.0 of 10

A revised peak-theory procedure estimates primordial black hole mass spectra for broad power spectra by focusing on peaks of the Laplacian of the curvature perturbation and removing the environmental constant offset.
EFT Compatible PBHs: Effective Spawning of the Seeds for Primordial Black Holes during Inflation
astro-ph.CO 2019-12 conditional novelty 5.0 of 10

A time-dependent negative quartic term in the extended EFT-of-inflation dispersion relation amplifies the scalar power spectrum by up to 10^9 at selected scales, potentially seeding primordial black holes of 30 solar ...
Pulsar Timing Array Constraints on the Induced Gravitational Waves
astro-ph.CO 2019-07 conditional novelty 5.0 of 10

A LIGO-fitted primordial black hole population induces nanohertz gravitational waves that exceed current pulsar timing bounds unless the primordial curvature perturbation carries local non-Gaussianity with f_NL of order ten.
Abundance of primordial black holes with local non-Gaussianity in peak theory
astro-ph.CO 2019-06 conditional novelty 5.0 of 10

Adding local-type non-Gaussianity with |f_NL| of order one to a peak-theory PBH abundance calculation shifts the predicted abundance by several orders of magnitude, comparable to a few factors of change in the power-s...
Primordial black hole tower: Dark matter, earth-mass, and LIGO black holes
astro-ph.CO 2019-04 conditional novelty 5.0 of 10

A hand-tuned four-phase hilltop inflation model produces three peaks in the curvature power spectrum, yielding primordial black holes at stellar, asteroid, and earth masses in a single construction.

See all 93 Pith citations

How the literature uses it

144 arXiv papers cite this in 192 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 whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

The threshold for black hole formation, \delta_{c} \approx 0.7, is found to be nearly identical for all perturbation families if the control parameter, \delta, is chosen as the total excess mass within the initial horizon volume.

Numerical investigation of the threshold for primordial black hole formation · 0.87

Papers citing it · 127 sentences without a confident label

Contests it 1 paper

Disregard for this issue affected early work on this subject [ref] , as pointed out in Ref. [ref] .

Formalism for Primordial Black Hole Formation in Spherical Symmetry · Initial Conditions · contest 0.77

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 4 papers

We employ the Misner-Sharp [ref] and Hernandez-Misner [ref] formalisms, building upon the ideas of Niemayer and Jedamzik [ref] and Polnarev et al. [ref] .

Formalism for Primordial Black Hole Formation in Spherical Symmetry · Introduction · 0.87
By using the Press-Schechter theory and smoothing on the scale $M$ with a Gaussian, the fractional energy density of PBH in the Universe is [equation] where the critical density perturbation is assumed to be in the range $δ_c=0.07-0.7$ [ref] , the density contrast $δ$ is related with the primordial curvature perturbation $ζ$ as $δ=4ζ/9$ during radiation domination, and the power spectrum for the primordial scalar perturbation is [equation] where we choose $M_Pl^-2=8πG=1$, the slow-roll parameter $ϵ=1/2(V_ϕ/V)^2$, $V_ϕ=dV/dϕ$, and the $ϕ_*$ is the value of $ϕ$ at the time when a given scale $k_*$ exits the horizon.

Primordial black holes and second order gravitational waves from ultra-slow-roll inflation · Primordial black holes from inflation · 0.86
The fraction of horizon patches that collapse is [equation] and the present abundance, normalized to the dark matter, follows by the standard redshift factor [ref] , [equation] We take a collapse threshold $δ_c=0.414$, representative of the radiation-era value [ref] , and quantify the sensitivity to this choice below.

One Feature, Three Clocks: Phase-Locked Gravitational Waves, Primordial Black Holes, and Non-Gaussianity from Periodic Warm Inflation · Primordial black holes · 0.86

Uses its data or values 16 papers

In GR, the mass of a primordial black hole near the threshold obeys the critical scaling relation $M_PBH∝(δ-δ_c)^γ$, with $γ_GR≃0.36$ for a radiation fluid [ref] .

Primordial Black Hole Formation in $f(R)=R+\alpha R^2$ Gravity: Perturbative and Non-Perturbative Analysis · Einstein–Frame Formulation for the Non–Perturbative Regime · 0.84
Horizon patches with $δ>δ_c$ collapse to PBHs with mass [ref] M=M_HK(δ-δ_c)^γ_r, where $M_H$ is the horizon mass as a function of $R$, and we use $δ_c=0.55$, $K=10$ and $γ_r=0.36$ [ref] .

Pinning down the primordial black hole formation mechanism with gamma-rays and gravitational waves · PBHs from curvature perturbations · 0.92
Here $$ gives the minimum density contrast required for a perturbation to later collapse into an object of type $X∈{PBHs$, UCMHs$}$; $PBH∼1/3$ [ref] , while $UCMH$ is a scale-dependent quantity [ref] , although typically $𝒪(10^-2-10^-3)$ [ref] .

Number Counts and Non-Gaussianity · Non-Gaussian Probability Distributions · 0.87

Compares with it 1 paper

The horizon mass can be easily estimate as follows [equation] This estimation for $M_PBH$ also differs from the one in [ref] , where the critical scaling model is used instead [ref] .

Primordial black holes through preheating instabilities in $\alpha$-attractor models · PBH mass · 0.95

Cites it as support 2 papers

Numerical general relativistic simulations [ref] have demonstrated that the scaling of black hole mass with the proximity to the formation threshold, known to occur in near-critical gravitational collapse, also applies to PBH formation.

Primordial black holes through preheating instabilities in $\alpha$-attractor models · PBH mass · 0.96
We do not include the effect of the critical collapse [ref] for simplicity because Ref. [ref] implies that it does not modify the PBH abundance so much.

NANOGrav results and LIGO-Virgo primordial black holes in axion-like curvaton model · Introduction · 0.97

Cites it as background 40 papers

The population of primordial black holes that formed in the early Universe depends on the Hubble rate and the total mass within the Hubble horizon [ref] .

Pion condensation in the early Universe at nonvanishing lepton flavor asymmetry and its gravitational wave signatures · Introduction · 0.80
After these pioneering works, the collapse of initial perturbations in the form of primordial curvature fluctuations, was numerically studied in Refs. [ref] , followed some years afterwards by an extensive numerical analysis by one of us using an explicit Lagrangian hydrodynamics code developed and used in Refs. [ref] and, recently, by other authors in Refs. [ref] .

From Primordial Black Holes Abundance to Primordial Curvature Power Spectrum (and back) · Primordial black holes formation · 0.91
Our present considerations do not apply to such situation.] A rough criterion for the formation of PBHs was first proposed by Carr [ref] , and much numerical work has been done in search of a more accurate criterion [ref] .

PBH abundance from random Gaussian curvature perturbations and a local density threshold · Introduction · 0.96

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] [9]

J. C. Niemeyer and K. Jedamzik, astro-ph/9709072, subm. to Phys. Rev. Lett. (1997)

View on Pith Pith review arXiv 1997

[2] [10]

D. K. Nadezhin, I. D. Novikov, and A. G. Polnarev, Sov. As tron. 22, 129 (1978); G. V. Bicknell and R. N. Henriksen, Astrophys. J., 232, 670 (1979)

View on Pith 1978

[3] [8]

B. J. Carr, Astrophys. J. 201, 1 (1975)

View on Pith 1975

[4] [1]

Ya. B. Zeldovich, and I. D. Novikov, Sov. Astron. A. J. 10, 602 (1967)

View on Pith 1967

[5] [2]

S. W. Hawking, M.N.R.A.S. 152, 75 (1971)

View on Pith 1971

[6] [3]

D. Cline, D. A. Sanders, and W. Hong, Astrophys. J. 486, 169 (1997)

View on Pith 1997

[7] [4]

T. Nakamura, M. Sasaki, T. Tanaka, and K. S. Thorne, Astro phys. J. Lett. 487, L139 (1997)

View on Pith 1997

[8] [5]

B. J. Carr, J. H. Gilbert, and J. E. Lidsey, Phys. Rev. D. 50, 4853 (1994); A. M. Green and A. R. Liddle, Phys. Rev. D56, 6166-6174 (1997); A. R. Liddle and A. M. Green, gr-qc/980403 4 (1998)

View on Pith 1994

[9] [6]

C. Alcock et al. (MACHO Collaboration), Astrophys. J. 471, 771 (1996); C. Renault et al. (EROS Collaboration), Astron . Astrophys. 324, L69 (1997)

View on Pith 1996

[10] [7]

K. Jedamzik, Phys. Rev. D, 55, R5871 (1997)

View on Pith 1997

[11] [11]

J. M. Bardeen, J. R. Bond, N. Kaiser, and A. S. Szalay, Ast rophys. J., 304, 15 (1986)

View on Pith 1986

[12] [12]

as implemented by Baumgarte et al. [13] (we omit restating the full system of equations but inst ead refer to the equations published by Baumgarte et al. by a capital “B” followed by the respective equation number [ 14]). Based on the original equations by Misner and Sharp [15], Hernande z and Misner proposed to exchange the Misner–Sharp time variable, t,...

View on Pith

[13] [13]

W. C. Hernandez and C. W. Misner, Astrophys. J., 143, 452 (1966)

View on Pith 1966

[14] [14]

T. W. Baumgarte, S. L. Shapiro, and S. A. Teukolsky, Astr ophys. J. 443, 717 (1995)

View on Pith 1995

[15] [15]

Furthermore, we use the same notation and geometrized u nits as [13]

View on Pith

[16] [16]

C. W. Misner and D. H. Sharp, Phys. Rev. 136, B571 (1964)

View on Pith 1964

[17] [17]

C. R. Evans and J. S. Coleman, Phys. Rev. Lett 72, 1782 (1994)

View on Pith 1994

[18] [18]

T. Koike, T. Hara, and S. Adachi, Phys. Rev. Lett. 74, 5170 (1995)

View on Pith 1995

[19] [19]

B. J. Carr and S. W. Hawking, ibid. 168, 399 (1974); G. V. Bicknell and R. N. Henriksen, Astrophys. J ., 225, 237 (1978)

View on Pith 1974

[20] [20]

We deﬁne PBH mass as a function of time as the rest mass-en ergy interior to the event horizon at the proper time of an infalling ﬂuid element which just passes over the event hori zon. Since the lapse of proper time between the ﬂuid element ﬁrst entering the strong gravitational ﬁeld of the PBH and th e ﬂuid element crossing over the event horizon is gen...

View on Pith

[21] [21]

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

View on Pith 1993

[22] [22]

C. Gundlach, gr-qc/9712084 (1997); M. W. Choptuik, gr- qc/9803075 (1998)

View on Pith Pith review arXiv 1997

[23] [23]

J. Yokoyama, gr-qc/9804041 (1998). 10

View on Pith Pith review arXiv 1998