Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity

Masaru Shibata; Misao Sasaki

REVIEW

Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity

Masaru Shibata, Misao Sasaki

General Relativity and Quantum Cosmology · Submitted 18 May 1999 · 18 references

Pith's verdict: The constant-mean-curvature formulation is the real contribution and it holds up; the profile-dependence physics is right, but the abstract leans on the variable that maximizes the spread and on a mass asymptote the authors themselves say they didn't reach.

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CitationMasaru Shibata, Misao Sasaki. "Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity." arXiv:gr-qc/9905064 (1999). https://pith.science/paper/RVESMHCO

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Bottom line: the formulation is the contribution, and it's a real one. Shibata and Sasaki write the Einstein and hydro equations in constant mean curvature slicing so the same code runs from the superhorizon regime, where the perturbation equations have a clean analytic solution, straight through horizon entry to collapse. No gauge switch, no numerical matching. That's the piece later work built on, and the long-wavelength derivation in Sec. III is careful: the order counting in Eq. (3.1) is explicit and psi is left non-linear throughout, which is the whole point. The physics result, that the collapse threshold depends on the surroundings and not just the local peak, has held up. A peak inside a low-density region collapses more easily than one inside average density. That is why PBH abundance can't be computed from a purely local criterion, and it's the sentence worth citing from this paper [abstract:s5]. Now the soft spots, and I'd rank them differently than the stress-test does. The claim that profile dependence largely vanishes under the compaction criterion is too strong. C(r) is itself an integral over the profile, so spatial correlation still enters; it just gets repackaged into a better variable. What is fair is that the abstract leads with psi0 [abstract:s4], the variable that maximizes the apparent spread, and buries the compaction function in Sec. IV.C. That's a framing choice worth flagging. The 'final black hole mass' in the abstract is the weaker item. The authors say plainly in Sec. IV.C that the runs ended before a definite asymptote could be established and only 'strongly suggests' one. The abstract doesn't carry that hedge. Same for the 1.428 lower bound, which comes from a separate flat-spacetime time-symmetric setup and is called approximate. Minor: thresholds in Figs. 6 and 7 have no error bars, though the authors report only few-percent mass accuracy, and the whole spread rests on the two-parameter ansatz in Eq. (4.21). They acknowledge the restricted family but don't argue it spans realistic inflationary profiles. None of this breaks the qualitative conclusion, which is appropriately hedged in the body. Send it out; the abstract should be tightened, not the physics.

Referee report

Significance

The formulation is a genuine and durable contribution: the CMC slicing lets one carry a single set of variables from the superhorizon regime, where the growing adiabatic mode is known analytically, all the way through horizon entry into a fully non-linear collapse, with no gauge switching and no numerical matching. That removes a real technical obstacle that had made earlier threshold estimates crude, and the explicit order-counting in Eq. (3.1) makes the analytic side checkable. The qualitative physical finding, that a peak embedded in a low-density region collapses more easily than one embedded in an average-density region, is well motivated and is the kind of environmental dependence that a local, peak-height criterion for primordial black hole formation cannot capture. The paper is also candid about its own limitations in Sec. IV.C and Sec. V, and it offers a concrete, if approximate, alternative criterion (C(r)max ≈ 0.4 in Fig. 8) that can be tested against other codes. The Kodama-mass cross-check of the apparent-horizon mass is a good practice that many numerical relativity papers of the period omit.

Major objections (4)

[Abstract vs. Sec. IV.C and Fig. 5] The abstract states that 'the final black hole mass considerably depend[s] on the initial density (or metric) profile', but Sec. IV.C says of the same runs that 'the simulations had to be ended before we could draw a definite conclusion' and that the results only 'strongly suggest' that MAH approaches an asymptotic value. Fig. 5 shows MAH still rising at the termination of each run, and the quoted spread (from >8r0 for σ=∞ down to ~1.1r0 for σ=2) is a spread in the mass at whatever time each simulation stopped, not in a converged final mass. The abstract should either be rewritten to describe the mass at a fixed late time or at horizon formation, or the authors should extend the runs until MAH is demonstrably stationary. As written, one of the abstract's two quantitative supports is not established by the body of the paper.
[Sec. IV.C, Fig. 8, and the abstract's ψ0 framing] The abstract leads with the ψ0(t=0) threshold spread (1.79 down to ~1.4) as the evidence for profile dependence, but the paper's own Fig. 8 shows that when the compaction function C(r)max of Eq. (4.29) is used instead, the threshold is 'about 0.4 irrespective of σ', and the authors state that using C(r)max 'we can approximately neglect the dependence on σ'. That is a substantial internal qualification: much of the ψ0 spread reflects the choice of a purely local variable rather than the collapse physics, since ψ0 at the origin does not by itself know about the surrounding underdensity. The abstract should say this explicitly, and the discussion in Sec. IV.C should state which of the two criteria the authors regard as the physically meaningful one for cosmological applications. The qualitative conclusion survives, because C(r) is itself an integral over the profile and so still carries spatial-correlation information, but the abstract's framing overstates the robustness of the ψ0-based spread.
[Sec. IV.A and Sec. IV.C (numerical methodology)] The threshold values in Figs. 6 and 7 are the paper's central quantitative output, yet no convergence or resolution study is reported. The grid is specified (3000 points, geometric spacing ri = Δr(f^i-1)/(f-1)) but neither f nor Δr is given numerically, and the only accuracy statement is that MAH agrees with the Kodama mass 'to within a few percent' except near the apparent horizon at late times. Since the threshold is located by bracketing between forming and non-forming cases (e.g. Cδ=15 versus Cδ=13 in Fig. 3), the quoted thresholds and the shape of the critical line in Fig. 7 need an explicit statement of the resolution at the origin, a check that the threshold is stable under a change of resolution, and an estimate of the uncertainty in ψ0(t=0) at the threshold. Without that, the claimed variation across σ cannot be distinguished from numerical error in the weakest cases.
[Sec. IV.B, Eq. (4.21), and Sec. V] The conclusion that spatial correlation must be included rests entirely on the two-parameter family of Eq. (4.21), a Gaussian peak minus a broader negative Gaussian whose width is σ, with r0 fixed to unity and no derivation from an inflationary spectrum. The authors acknowledge in Sec. V that they 'restricted our attention to a model which contains only two parameters', but the abstract and the closing sentence of Sec. V state the spatial-correlation conclusion in general terms. The paper should either soften that statement to a statement about this family, or add at least one profile from a different functional family (for example a top-hat or a profile with a different compensation fraction at fixed σ) to show that the ordering of thresholds is not an artifact of the Gaussian-minus-Gaussian shape. The physically relevant question, whether the threshold depends on the compensated fraction of the profile rather than on σ per se, is left untested.

Minor objections (7)

[Abstract] The abstract says the maximum value of ψ0 at t=0 'should be larger than ~1.4', but ψ0 is defined in the text as the central value ψ(r=0), not a maximum, and the flat-universe case requires ~1.79. The abstract should state the range and identify which profile gives which value, so that the ~1.4 figure is not read as a universal threshold.
[Sec. II, Eq. (2.7)] As printed, ρ0 = 16πΓ²t² is dimensionally inconsistent with a density and with the stated a = af t^{2/3Γ}; the correct radiation-era relation should be checked and the equation corrected. Similarly, the coefficient '2116A2' in Eq. (4.3) appears garbled and should be verified against the derivation.
[Sec. IV.B] The iterative scheme for constructing the initial data says only that it is repeated 'until sufficient convergence is achieved'. Please state the convergence criterion and the typical number of iterations, since the initial data determine ψ0(t=0) and hence the threshold.
[Sec. IV.C] The statement that 'it seems possible to have a black hole whose mass is much smaller than r0 near the threshold' is not supported by any run shown; either present such a run or mark the statement clearly as expectation rather than result.
[Sec. I] The introduction criticizes the initial conditions of Refs. [11] for being contaminated by decaying modes, but no quantitative comparison of thresholds or masses with those works is given anywhere. A short comparison in Sec. IV.C, even at the level of order of magnitude, would make the claimed advantage concrete.
[Sec. I] Typographical: 'it is impossible to control the initial date' should read 'initial data'; the Fig. 8 caption reads 'if the maximum value is large than ~0.4'; and Ref. [1] gives 'D. K. Nadezin', presumably Nadezhin.
[Sec. IV.C, Fig. 5] The figure caption should state the value of Cδ used for each of the four panels, since the mass histories are otherwise not reproducible from the text.

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The paper

The paper claims

Whether a dense peak becomes a black hole depends on what surrounds it, not just how tall it is.

The reading

The paper asks how large a primordial density fluctuation must be, in the very early universe, before it collapses into a black hole. Its answer is that there is no single number: the threshold depends on the shape of the fluctuation's neighbourhood. A peak sitting in an average-density flat universe needs a metric distortion of about $1.79$ at the initial time, while a peak surrounded by a low-density region can collapse from as little as about $1.43$. To reach that conclusion the authors build a single computational scheme that covers both ends of the problem, an analytic treatment while the fluctuation is larger than the Hubble horizon and a full numerical relativity simulation once it is well inside. The practical stake is that primordial black hole abundance is inferred from the high-amplitude tail of a Gaussian distribution, so a shift in the threshold of a few tenths changes the predicted number of black holes by orders of magnitude.

What carries the argument

The load-bearing device is the constant mean curvature slicing condition $K=-3ȧ/a$, which makes the equations for the lapse $α$ and the conformal factor $ψ$ resemble their maximal-slicing counterparts in asymptotically flat space and lets the authors factor the cosmological expansion out of the dynamical variables. This is what allows one set of variables, one gauge, and one code to run continuously from the epoch when a fluctuation is far outside the horizon, where the long-wavelength expansion gives an exact analytic answer, to the epoch when it is far inside and a black hole forms. The analytic piece supplies the initial data; the numerical piece supplies the threshold and the final mass, with the Hamiltonian constraint and the Kodama mass used as accuracy checks on the apparent-horizon mass.

Extended reading notes

Core claim

The authors set up the Einstein and hydrodynamic equations in a Friedmann background using constant mean curvature time-slicing, chosen so that the geometric equations take a form close to maximal slicing and so that the same variables remain valid from the superhorizon epoch through collapse. In the long-wavelength limit they obtain an analytic solution in which the density contrast and the lapse perturbation are proportional to $t^{2-4/3Γ}$ while the conformal factor $ψ$ is frozen to leading order, and they use that solution to build initial data for a spherically symmetric, radiation-dominated ($Γ=4/3$) simulation. Scanning a two-parameter family of profiles, a Gaussian peak of amplitude $C_δ$ minus a broader negative Gaussian of width $σ$, they find that a black hole forms only when the initial metric perturbation is already non-linear, with $ψ_0$ at $t=0$ above roughly $1.4$ in the most favourable case. The threshold is not universal: it is about $1.79$ when the peak is surrounded by a flat, average-density universe ($σ=∞$), falls for smaller $σ$ where the peak sits inside a low-density region, and reaches about $1.43$ in the flat-spacetime time-symmetric limit, which they treat as approximately the smallest possible threshold. The final apparent-horizon mass varies just as strongly, exceeding $8r_0$ for $σ=∞$ but reaching only about $1.1r_0$ for $σ=2$. As a profile-independent alternative, they define a compaction function $C(r)=δM_K/[rψ(r)^2a]$ from the Kodama mass and report that its maximum is about $0.4$ at threshold for every $σ$ they tried.

Load-bearing premise

The claim that the threshold depends on the surrounding density rests on one two-parameter family of profiles, a Gaussian peak minus a broader negative Gaussian of width $σ$ with the scale $r_0$ fixed to unity, and the paper does not derive that family from an inflationary spectrum or show that other compensated shapes give the same spread in thresholds.

Editorial extensions

If this is right

Threshold estimates for primordial black hole formation that use a single amplitude criterion, independent of the surrounding density field, will misstate the abundance by large factors.
The number of primordial black holes predicted from an inflationary power spectrum must be computed from a joint distribution of amplitude and local environment, not from the one-point amplitude distribution alone.
Final black hole masses span a wide range for the same peak amplitude, so a claimed mass function cannot be inferred from the threshold alone.
The compaction function maximum, about $0.4$ at threshold across the profiles tested, offers a profile-independent collapse criterion in spherical symmetry.
The same slicing and variable choice can be carried over to non-spherical simulations, which the authors name as the next step.

Reading between the lines

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

If the environment matters this much in spherical symmetry, then in three dimensions the tidal field of neighbouring peaks is a second environmental variable of comparable weight, and the threshold should be treated as a functional of the surrounding density field rather than a number.
The reported near-constancy of the compaction maximum suggests the right invariant is a mass-excess-to-radius ratio at horizon crossing; a general-relativistic version of a Jeans-type criterion written in that variable would unify the $σ$ dependence the authors see.
Because the threshold shift is modest while the abundance is exponentially sensitive to it, the dominant uncertainty in primordial black hole dark matter bounds may be the shape of the fluctuation profile rather than the amplitude of the power spectrum.
The authors' remark that a peak surrounded by a void collapses most easily implies that primordial black hole formation should correlate spatially with underdense regions, which is in principle a signature distinguishable from a purely local threshold model.

Original abstract

We study formation of black holes in the Friedmann universe. We present a formulation of the Einstein equations under the constant mean curvature time-slicing condition. Our formalism not only gives us the analytic solution of the perturbation equations for non-linear density and metric fluctuations on superhorizon scales, but also allows us to carry out a numerical relativity simulation for black hole formation after the scale of the density fluctuations is well within the Hubble horizon scale. We perform a numerical simulation of spherically symmetric black hole formation in the radiation-dominated, spatially flat background universe for a realistic initial condition supplied from the analytic solution. It is found that the initial metric perturbation has to be non-linear (the maximum value of 3D conformal factor $\psi_0$ at $t=0$ should be larger than $\sim 1.4$) for a black hole to be formed, but the threshold amplitude for black hole formation and the final black hole mass considerably depend on the initial density (or metric) profile of the perturbation: The threshold value of $\psi_0$ at $t=0$ for formation of a black hole is smaller for a high density peak surrounded by a low density region than for that surrounded by the average density region of the flat universe. This suggests that it is necessary to take into account the spatial correlation of density fluctuations in the study of 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.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.60 · human participants 0.01 · controlled comparison 0.20 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.10 · 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 study formation of black holes in the Friedmann universe. We present a formulation of the Einstein equations under the constant mean curvature time-slicing condition. Our formalism not only gives us the analytic solution of the perturbation equations for non-linear density and metric fluctuations on superhorizon scales, but also allows us to carry out a numerical relativity simulation for black hole formation after the scale of the density fluctuations is well within the Hubble horizon scale. We perform a numerical simulation of spherically symmetric black hole formation in the radiation-dominated, spatially flat background universe for a realistic initial condition supplied from the analytic solution. It is found that the initial metric perturbation has to be non-linear (the maximum value of 3D conformal factor $\psi_0$ at $t=0$ should be larger than $\sim 1.4$) for a black hole to be formed, but the threshold amplitude for black hole formation and the final black hole mass considerably depend on the initial density (or metric) profile of the perturbation: The threshold value of $\psi_0$ at $t=0$ for formation of a black hole is smaller for a high density peak surrounded by a low density region than for that surrounded by the average density region of the flat universe. This suggests that it is necessary to take into account the spatial correlation of density fluctuations in the study of 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 study formation of black holes in the Friedmann universe.	contribution 0.69	0.90	0.03	asserted 0.70
s1	We present a formulation of the Einstein equations under the constant mean curvature time-slicing condition.	method 0.71	0.93	0.03	asserted 1.00
s2	Our formalism not only gives us the analytic solution of the perturbation equations for non-linear density and metric fluctuations on superhorizon scales, but also allows us to carry out a numerical relativity simulation for black hole formation after the scale of the density fluctuations is well within the Hubble horizon scale.	contribution 0.80	0.93	0.10	asserted 1.00
s3	We perform a numerical simulation of spherically symmetric black hole formation in the radiation-dominated, spatially flat background universe for a realistic initial condition supplied from the analytic solution.	method 0.79	0.96	0.06	asserted 0.99
s4	It is found that the initial metric perturbation has to be non-linear (the maximum value of 3D conformal factor $\psi_0$ at $t=0$ should be larger than $\sim 1.4$) for a black hole to be formed, but the threshold amplitude for black hole formation and the final black hole mass considerably depend on the initial density (or metric) profile of the perturbation: The threshold value of $\psi_0$ at $t=0$ for formation of a black hole is smaller for a high density peak surrounded by a low density region than for that surrounded by the average density region of the flat universe.	result 1.00	0.96	0.63	asserted 1.00
s5	This suggests that it is necessary to take into account the spatial correlation of density fluctuations in the study of primordial black hole formation.	interpretation 1.00	0.66	0.63	hedged possibility 0.99

What kind of paper the classifier reports this is: simulation study 0.98 · 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

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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. The spectrum shapes of the density fluctuation

**Figure 4.** FIG. 4. The same as Fig. 3, but for

**Figure 7.** It is found that the maximum value C(r)max is about 0.4 irrespective of σ. We note that for each σ, if the maximum value is large than the value shown in

Showing 3 of the paper's 7 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:gr-qc/9905064 by the authors.

References

Works cited (18)

[1]

For example, Ya. B. Zel’dovich and I. D. Novikov, Astron. Zh. 43, 758 (1966): S. W. Hawking, Mon. Not. R. Astron. Soc. 152, 75 (1971): B. J. Carr and S. W. Hawking, Mon. Not. R. Astron. Soc. 168, 399 (1974): S. W. Hawking, Nature, 248, 30 (1974): D. K. Nadezin, I. D. Novikov, and A. G. Polnarev, Sov. Astron. 22, 129 (1978)

View on Pith 1966
[2]

C. Alcock et al. (MACHO collaboration), Astrophys. J. 471, 771 (1996)

View on Pith 1996
[3]

For example, R. Kippenhahn and A. Weigert, Stellar Structure and Evolution (Springer, 1990)

View on Pith 1990
[4]

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

View on Pith 1997
[5]

J. Garcia-Bellido, A. Linde and D. Wands, Phys. Rev. D54, 6040 (1996)

View on Pith 1996
[6]

J. Yokoyama, Phys. Rev. D58, 083510 (1998)

View on Pith 1998
[7]

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

View on Pith 1975
[8]

P. Ivanov, Phys. Rev. D57, 7145 (1998)

View on Pith 1998
[9]

H. Kodama and M. Sasaki, Prog. Theor. Phys. supple- ment 78, 1 (1984)

View on Pith 1984
[10]

M. Sasaki and T. Tanaka, Prog. Theor. Phys. 99, 763 (1998)

View on Pith 1998
[11]

J.C. Niemeyer and K. Jedamzik, astro-ph/9901292; K. Jedamzik and J.C. Niemeyer, astro-ph/9901293

View on Pith Pith review arXiv
[12]

M. Shibata and H. Asada, Prog. Theor. Phys. 94, 11 (1995)

View on Pith 1995
[13]

M. Shibata, K. Nakao, T. Nakamura and K. Maeda, Phys. Rev. D 50, 708 (1994)

View on Pith 1994
[14]

L. Smarr and J. W. York, Phys. Rev. D 17, 1945 (1978)

View on Pith 1978
[15]

S. W. Hawking and G. F. R. Ellis, The large scale struc- ture of space-time (Cambridge University Press, 1973)

View on Pith 1973
[16]

H. Kodama, Prog. Theor. Phys. 63, 1217 (1980): We thank K. Nakao for giving us a note in which Kodama’s result is concisely described

View on Pith 1980
[17]

In particular, for 3D simulation in which the grid num- ber we can take is restricted, the minimum distortion gauge may not be appropriate for a black hole formation problem, because the coordinate spreads outward (i.e., βi > 0) when a matter is collapsing (i.e., vi < 0) and hence it is soon diﬃcult to keep the numerical accuracy around the collapsing mat...

View on Pith
[18]

G. V. Bicknell and R. N. Henriksen, Astrophys. J. 232, 670 (1979). 9 FIG. 1. The spectrum shapes of the density ﬂuctuation (exp(−k2/4) − exp(−k2σ2/4)) are shown for several σ. The solid line denotes the case for σ = ∞ in which the density peak is surrounded by a ﬂat universe, and the dotted lines denote the cases for 1 .5 ≤σ ≤ 8. FIG. 2. ψ0 att = 0 as a f...

View on Pith 1979

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Primordial Asymmetries, Primordial Equation of State & Primordial Black Holes
astro-ph.CO 2026-08 conditional novelty 6.0 of 10

Lepton flavour asymmetries alter the early-universe equation of state, reshaping the predicted primordial black hole mass spectrum and favouring a spectral index n_s <= 0.99 in current gravitational-wave merger data.
One Feature, Three Clocks: Phase-Locked Gravitational Waves, Primordial Black Holes, and Non-Gaussianity from Periodic Warm Inflation
astro-ph.CO 2026-06 unverdicted novelty 6.0 of 10

Periodic warm inflation imprints one log-periodic feature on the curvature spectrum that saturates asteroid-mass PBHs, generates dual-band GW backgrounds, and offsets the bispectrum phase by a quarter cycle fixed by s...
Memoirs of the curvaton: non-perturbative non-Gaussianity and supermassive primordial black holes
astro-ph.CO 2026-06 unverdicted novelty 6.0 of 10

Curvaton self-interactions in non-quadratic potentials produce a local non-Gaussian map that enables supermassive primordial black hole formation at peak amplitudes of order 10^{-5} while remaining consistent with μ-d...
Primordial Black Hole from Tensor-induced Density Fluctuation: First-order Phase Transitions and Domain Walls
astro-ph.CO 2026-05 unverdicted novelty 6.0 of 10

Tensor perturbations from FOPT and domain-wall sources are claimed to induce second-order scalar perturbations large enough to form primordial black holes, potentially all of the dark matter.
Vacuum bubbles from cosmic ripples
hep-ph 2026-04 conditional novelty 6.0 of 10

Over-densities in the early universe reduce the Euclidean action for vacuum decay, making false-vacuum bubbles nucleate earlier; under-densities do the opposite.
Numerical simulations of primordial black hole formation via delayed first-order phase transitions
gr-qc 2026-01 conditional novelty 6.0 of 10

Spherically symmetric numerical relativity shows false-vacuum domains from delayed first-order phase transitions form type B (baby-universe) or type A (direct-collapse) primordial black holes, separated by a robust t_...
Friction in Stochastic Inflation
astro-ph.CO 2025-11 accept novelty 6.0 of 10

Adding a constant drift to time-reversed stochastic inflation in a flat potential yields an exact curvature distribution with exponential tails; forward and reverse delta-N formalisms differ by a factor of two in the ...
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.
Curvature Perturbations from First-Order Phase Transitions: Implications to Black Holes and Gravitational Waves
hep-ph 2025-03 unverdicted novelty 6.0 of 10

Covariant analysis of curvature perturbations from first-order phase transitions reveals gauge-dependent overestimation of primordial black holes and gravitational waves in prior non-covariant calculations, leading to...
Complementary Probes of Warped Extra Dimension: Colliders, Gravitational Waves and Primordial Black Holes from Phase Transitions
hep-ph 2025-02 conditional novelty 6.0 of 10

In Randall-Sundrum warped extra dimension models, the supercooled radion phase transition can form primordial black holes that account for all of dark matter for IR scales 10 TeV to 10^4 TeV, with correlated gravitati...
The Irrelevance of Primordial Black Hole Clustering in the LVK mass range
astro-ph.CO 2025-02 accept novelty 6.0 of 10

Initial spatial clustering of primordial black holes is irrelevant for binary mergers in the LVK mass range because FIRAS spectral-distortion constraints limit the clustering scale to below the merger-relevant separat...
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.
Non-perturbative non-Gaussianity and primordial black holes
astro-ph.CO 2022-11 conditional novelty 6.0 of 10

For a broad class of non-Gaussian primordial curvature distributions, the excess primordial black hole abundance is generated by the intermediate part of the distribution rather than its exponential far tail.
Primordial non-gaussianity up to all orders: theoretical aspects and implications for primordial black hole models
astro-ph.CO 2022-11 conditional novelty 6.0 of 10

Non-Gaussianity in the curvature perturbation must be treated exactly rather than order by order when the power spectrum is broad; for narrow spectra the series converges only beyond quadratic order.
Primordial black holes from stochastic tunnelling
astro-ph.CO 2022-10 accept novelty 6.0 of 10

For a false-vacuum dip during inflation, the primordial black hole abundance scales as a flat well, an exponential, or a doubly exponential in the barrier height depending on the ratio of barrier height to width and t...
Simulations of PBH formation at the QCD epoch and comparison with the GWTC-3 catalog
astro-ph.CO 2022-09 conditional novelty 6.0 of 10

Simulating primordial black hole formation with the time-varying QCD equation of state lowers and broadens the predicted peak near one to three solar masses, and the resulting merger rates are broadly compatible with ...
From inflation to black hole mergers and back again: Gravitational-wave data-driven constraints on inflationary scenarios with a first-principle model of primordial black holes across the QCD epoch
astro-ph.CO 2022-09 conditional novelty 6.0 of 10

Joint inference on GWTC-3 with a QCD-shaped, power-spectrum-derived primordial black hole mass function limits stellar-mass PBHs to about 10^-3 of dark matter while permitting a small primordial fraction of the detect...
Revisiting small-scale fluctuations in $\alpha$-attractor models of inflation
astro-ph.CO 2021-12 conditional novelty 6.0 of 10

For alpha-attractor inflation with an inflection point, the Planck spectral-tilt bound forces any primordial black holes below 10^8 g and pushes the induced gravitational-wave peak to roughly 10 kHz or above.
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...
The Threshold for Primordial Black Hole Formation: a Simple Analytic Prescription
astro-ph.CO 2020-11 conditional novelty 6.0 of 10

Once nonlinear growth at horizon crossing is included, the threshold density contrast for primordial black hole formation is about twice the standard superhorizon value, and can be computed from the power spectrum alone.
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...
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.
Non-linear statistics of primordial black holes from gaussian curvature perturbations
astro-ph.CO 2019-12 conditional novelty 6.0 of 10

A full non-linear statistics of primordial black hole formation, using the compaction function with a shape-dependent threshold, finds that linear-overdensity estimates over-predict and curvature-based estimates under...
Gravitational Wave Anisotropies from Primordial Black Holes
astro-ph.CO 2019-09 conditional novelty 6.0 of 10

Local, scale-invariant primordial non-Gaussianity imprints roughly 10^-4 anisotropies and a 10^-8 reduced bispectrum on the gravitational wave background from primordial black holes, and isocurvature bounds then imply...
PBH in single field inflation: the effect of shape dispersion and non-Gaussianities
astro-ph.CO 2019-08 conditional novelty 6.0 of 10

In a barrier single-field inflation model, primordial black holes form preferentially from false-vacuum bubbles once f_NL exceeds about 3.5, and shape dispersion has little effect on the compaction threshold.
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.
The Initial Spin Probability Distribution of Primordial Black Holes
astro-ph.CO 2019-03 conditional novelty 6.0 of 10

PBHs born from collapsing density peaks acquire a typical dimensionless spin of about 10^-2 from first-order tidal torques, with the value suppressed for narrow power spectra.
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...
A Cosmological Signature of the Standard Model Higgs Vacuum Instability: Primordial Black Holes as Dark Matter
hep-ph 2017-10 conditional novelty 6.0 of 10

Primordial black holes seeded by Standard Model Higgs fluctuations during inflation could account for the observed dark matter, requiring no physics beyond the Standard Model.
Gravitational Wave signatures of inflationary models from Primordial Black Hole Dark Matter
astro-ph.CO 2017-07 conditional novelty 6.0 of 10

The stochastic gravitational wave background from PBH formation is computed for Gaussian and chi-squared curvature statistics, and shown to be detectable at PTA/SKA and LISA scales, with peak frequency tracking the PB...
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.
Constant roll and primordial black holes
gr-qc 2019-10 conditional novelty 5.5 of 10

A constant-roll inflationary stage with -3/2 < beta < 0 generates a blue-tilted curvature spectrum whose peak, with tuned parameters, can produce primordial black holes at dark-matter, LIGO-Virgo, or OGLE abundances.
Limits on Primordial Black Hole Evaporation from LUX-ZEPLIN
astro-ph.CO 2026-09 conditional novelty 5.0 of 10

Hawking-boosted dark matter from evaporating primordial black holes cannot explain the LZ 248 keV event, and the LZ low-energy null excludes f_PBH above about 10^-6 at M ~ 2x10^11 g.
Purely quadratic non-Gaussianity from tachyonic instability: Primordial black holes and scalar-induced gravitational waves
astro-ph.CO 2026-04 unverdicted novelty 5.0 of 10

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...
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.

See all 109 Pith citations

How later papers used it

192 later papers cite it

Evidence: substantial. 184 of the citing papers share no author with it, and 36 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 374 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

3 later papers by other authors are listed below: 3 used its method or tool.

Uses its method or tool 3 of 23 papers with this label

More precisely, we consider an optimized criterion for PBH formation which is given by a threshold for the compaction function proposed in Ref. [ref] .

PBH abundance from random Gaussian curvature perturbations and a local density threshold (2018) · Summary and Discussion · 0.89
Used the compaction function introduced in this paper as the criterion for primordial black hole formation.

Effect of nonlinearity between density and curvature perturbations on the primordial black hole formation (2019) · 0.80 · paraphrased: the licence does not allow quoting
Used the criterion proposed in this paper, based on the compaction function.

From Primordial Black Holes Abundance to Primordial Curvature Power Spectrum (and back) (2019) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

164 arXiv papers cite this in 324 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 · 238 sentences without a confident label

Contests it 3 papers

In this paper, we show that the Shibata-Sasaki compaction function introduced in Ref. [ref] is not what they intended.

Revisiting compaction functions for primordial black hole formation · Introduction · contest 0.87
However, it should be noted that, for the CMC slice, the contribution from the second term in the integrand of Eq. [ref] cannot be neglected[In Refs. [ref] , the second term in the integrand of Eq. [ref] has been erroneously dropped, and the authors got the wrong relation $𝒞_CMC=𝒞_SS$.

The basics of primordial black hole formation and abundance estimation · PBH formation criterion · contest 0.93
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

Uses its method or tool 18 papers

Therefore, in this section, we introduce the criterion for PBH formation assuming spherical symmetry originally proposed in Ref. [ref] .

PBH abundance from random Gaussian curvature perturbations and a local density threshold · perturbation variables and threshold for PBH formation · 0.84
This construction combines the long-wavelength treatment of nonlinear cosmological perturbations with Gaussian peak statistics and subsequent refinements accounting for the nonlinear curvature–density relation, the profile dependence of the collapse threshold, and critical scaling near the threshold [ref] .

Projected Constraints on Primordial Black Holes from Scalar-Induced Gravitational Waves with Taiji · PBH formation from the curvature spectrum · 0.94
Here as the criterion for the PBH formation, we use the Compaction function $𝒞(r)$ introduced in Ref. [ref] .

Effect of nonlinearity between density and curvature perturbations on the primordial black hole formation · Introduction · 0.80

Uses its data or values 13 papers

In the constant mean curvature slicing, the numerical simulation gives the threshold value of the PBH formation as $𝒞_m;CMC^th≃0.4$ during the radiation dominated eta [ref] .

Effect of nonlinearity between density and curvature perturbations on the primordial black hole formation · PBH formation criterion and Press-Schechter formalism · 1.00
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
Shibata and Sasaki [ref] obtained the threshold value $≃0.4$ for radiation $Γ=4/3$.

Revisiting compaction functions for primordial black hole formation · Shibata-Sasaki compaction function and its significance · 0.92

Compares with it 1 paper

Further, this analytical estimate of $δ_c$ is in rough agreement with the range specified from the numerical analysis in [ref] .

Primordial black holes dark matter from inflection point models of inflation and the effects of reheating · Critical value of the density contrast · 0.81

Cites it as support 1 paper

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 42 papers

Another quantity, the maximum value of the so-called compaction function has been proposed as the one giving a robust threshold of primordial black hole formation by Shibata and Sasaki [ref] through general relativistic numerical simulations.

Revisiting compaction functions for primordial black hole formation · Introduction · 0.84
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] [1]

For example, Ya. B. Zel’dovich and I. D. Novikov, Astron. Zh. 43, 758 (1966): S. W. Hawking, Mon. Not. R. Astron. Soc. 152, 75 (1971): B. J. Carr and S. W. Hawking, Mon. Not. R. Astron. Soc. 168, 399 (1974): S. W. Hawking, Nature, 248, 30 (1974): D. K. Nadezin, I. D. Novikov, and A. G. Polnarev, Sov. Astron. 22, 129 (1978)

View on Pith 1966

[2] [2]

C. Alcock et al. (MACHO collaboration), Astrophys. J. 471, 771 (1996)

View on Pith 1996

[3] [3]

For example, R. Kippenhahn and A. Weigert, Stellar Structure and Evolution (Springer, 1990)

View on Pith 1990

[4] [4]

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

View on Pith 1997

[5] [5]

J. Garcia-Bellido, A. Linde and D. Wands, Phys. Rev. D54, 6040 (1996)

View on Pith 1996

[6] [6]

J. Yokoyama, Phys. Rev. D58, 083510 (1998)

View on Pith 1998

[7] [7]

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

View on Pith 1975

[8] [8]

P. Ivanov, Phys. Rev. D57, 7145 (1998)

View on Pith 1998

[9] [9]

H. Kodama and M. Sasaki, Prog. Theor. Phys. supple- ment 78, 1 (1984)

View on Pith 1984

[10] [10]

M. Sasaki and T. Tanaka, Prog. Theor. Phys. 99, 763 (1998)

View on Pith 1998

[11] [11]

J.C. Niemeyer and K. Jedamzik, astro-ph/9901292; K. Jedamzik and J.C. Niemeyer, astro-ph/9901293

View on Pith Pith review arXiv

[12] [12]

M. Shibata and H. Asada, Prog. Theor. Phys. 94, 11 (1995)

View on Pith 1995

[13] [13]

M. Shibata, K. Nakao, T. Nakamura and K. Maeda, Phys. Rev. D 50, 708 (1994)

View on Pith 1994

[14] [14]

L. Smarr and J. W. York, Phys. Rev. D 17, 1945 (1978)

View on Pith 1978

[15] [15]

S. W. Hawking and G. F. R. Ellis, The large scale struc- ture of space-time (Cambridge University Press, 1973)

View on Pith 1973

[16] [16]

H. Kodama, Prog. Theor. Phys. 63, 1217 (1980): We thank K. Nakao for giving us a note in which Kodama’s result is concisely described

View on Pith 1980

[17] [17]

In particular, for 3D simulation in which the grid num- ber we can take is restricted, the minimum distortion gauge may not be appropriate for a black hole formation problem, because the coordinate spreads outward (i.e., βi > 0) when a matter is collapsing (i.e., vi < 0) and hence it is soon diﬃcult to keep the numerical accuracy around the collapsing mat...

View on Pith

[18] [18]

G. V. Bicknell and R. N. Henriksen, Astrophys. J. 232, 670 (1979). 9 FIG. 1. The spectrum shapes of the density ﬂuctuation (exp(−k2/4) − exp(−k2σ2/4)) are shown for several σ. The solid line denotes the case for σ = ∞ in which the density peak is surrounded by a ﬂat universe, and the dotted lines denote the cases for 1 .5 ≤σ ≤ 8. FIG. 2. ψ0 att = 0 as a f...

View on Pith 1979