Computations of primordial black hole formation

2); 3); ((1) SISSA; England; (3) Department of Physics; Ilia Musco (1); INFN; International School for Advanced Studies; Italy; (2) Department of Physics (Astrophysics); John C. Miller (1; Louisiana State University; Luciano Rezzolla (1

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Computations of primordial black hole formation

Ilia Musco (1), John C. Miller (1, 2), Luciano Rezzolla (1, 3); ((1) SISSA, International School for Advanced Studies, INFN, Italy; (2) Department of Physics (Astrophysics)

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University of Oxford, England; (3) Department of Physics, Louisiana State University, USA)

General Relativity and Quantum Cosmology · Submitted 13 Dec 2004 · 47 references

Pith's verdict: A careful, well-designed numerical study that lowers the PBH threshold to δc ≈ 0.43–0.47 for growing-mode initial data; the new number is plausible and independently consistent, but the initial-data prescription behind it is asserted rather than checked.

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arXiv IDgr-qc/0412063 (version 2)

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Citation2), 3); ((1) SISSA, England; (3) Department of Physics, Ilia Musco (1), INFN, International School for Advanced Studies, Italy; (2) Department of Physics (Astrophysics), John C. Miller (1, Louisiana State University, Luciano Rezzolla (1, University of Oxford, USA). "Computations of primordial black hole formation." arXiv:gr-qc/0412063 (2004). https://pith.science/paper/EQ5ZO2PG

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Worth your time if you work on PBH abundances or on small-scale power-spectrum constraints. The headline is real: with growing-mode initial data specified super-horizon, δc drops to about 0.43 (Mexican hat) and 0.47 (polynomial), against 0.67 and 0.71 with horizon-crossing data, while γ stays at 0.36–0.37. Since abundance goes as exp(-δc²/2σ²), that shift moves predicted PBH fractions by orders of magnitude, so it matters for the constraints people actually quote. The result also lands inside the 0.3–0.5 range Green et al. inferred from Shibata–Sasaki, which is genuine external support from a different method — that's the strongest thing in the paper's favour, and the authors are right to say it resolves the noted discrepancy with Niemeyer–Jedamzik [30]. The design is also correct: they first reproduce [30]'s numbers with [30]'s data, then change one thing. That is the right experiment, and they report the replication honestly rather than burying it. The sub-critical section is nice too — the second bounce being more violent than the first because it is a shell falling into near-vacuum is a clean physical explanation, not just a plot. And the technical remark in §2.2 about using Eq. 27 rather than Eq. 24 to keep DkR→0 synchronised with f→0 in non-vacuum surroundings is a real methodological contribution for anyone doing null-slicing collapse; that alone is worth citing. Soft spots, in order of weight. Eq. 34, δu = -δe/4, is described as 'roughly' from Padmanabhan and is never checked against an exact linear growing mode at the epoch used. In the super-horizon regime the peculiar velocity is typically suppressed relative to the density contrast, so an unsuppressed relation could over-specify the inflow and ease collapse, biasing δc low. Second, the decaying-mode explanation for [30]'s higher δc is asserted ('The reason for the changed values of δc is clear') rather than shown by decomposing [30]'s critical data. I'd push back on one part of the stress-test note: δ is defined the same way in both runs (Eq. 33, at horizon crossing), so this is not two definitions of δ — it is an unquantified difference in the velocity content of the initial data. Third, no error bars on δc or γ, no resolution study, and the Λ slopes (Eqs. 39–40) are fits with no residuals reported. Those are minor-to-moderate; the central comparison holds up. Send it out. A referee should ask for the growing-mode check and a resolution study, not for a different result.

Referee report

Significance

If the result holds, it matters: δc enters directly into PBH abundance estimates and into constraints on the small-scale spectral index, and a factor-1.5 reduction in δc translates into a large change in predicted abundances. The paper's design is the right one and deserves explicit credit: the authors first reproduce the earlier numbers with the earlier initial data using their own code (δc ≈ 0.67/0.71, γ ≈ 0.36–0.37), and only then change a single ingredient, the initial data prescription. That is a controlled numerical experiment rather than an assertion, and the resulting δc ≈ 0.43–0.47 is consistent with the independent Shibata-Sasaki-based range quoted by Green et al., which the authors correctly note resolves a previously flagged discrepancy. The sub-critical phenomenology (two bounces, geometric amplification of the inward wave, steep-fronted second compression) is a genuine and checkable finding. The methodological point in §2.2 — that Eq. (27), not Eq. (24), is required to synchronise DkR → 0 with f → 0 in non-vacuum surroundings — is a useful technical contribution that generalises beyond cosmology, e.g. to core-collapse black hole formation, and is worth stating more p

Major objections (4)

[§3.1, Eq. (34)] The headline shift in δc rests entirely on the initial-data prescription, and the velocity perturbation is the weakest link in it. Eq. (34), δu = −δe/4, is introduced as holding 'roughly' (with a citation to Padmanabhan) and is then applied without further comment at Ro/RH ≈ 5 with δ ~ 10^-2, i.e. deep in the super-horizon regime where the peculiar velocity relative to the Hubble flow is suppressed relative to the density contrast. No check is presented that Eq. (34) reproduces the exact general-relativistic growing mode at the epoch actually used, and no sensitivity study is reported. This is load-bearing: an over-specified inflow would artificially ease collapse and bias δc low, which is precisely the direction of the reported result. A cheap and decisive test is available: repeat one profile (Mexican hat is sufficient) with δu = 0 and with δu = −δe/2, and report δc for each. If δc moves by more than a few per cent across that range, the quoted δc ≈ 0.43 needs an explicit systematic uncertainty; if it does not move, the result is robust and the point is settled. Either outcome should be reported.
[§3.2] The causal explanation for the changed threshold is asserted rather than demonstrated. The text states 'The reason for the changed values of δc is clear': the earlier horizon-crossing initial data contained a decaying-mode component that rapidly disappeared, so the effective growing-mode amplitude was smaller than the nominal δ. That is a plausible and probably correct story, but the manuscript never decomposes the Niemeyer-Jedamzik critical data into growing and decaying parts, and never shows the quantitative mapping between the two δ measures. Without that, the comparison of 0.43 against 0.67 could in part reflect two different definitions or evaluation epochs of δ rather than a physical correction. The decisive test is straightforward with the machinery already in place: take the new growing-mode initial data and add a decaying-mode component of the amplitude that the earlier prescription would have contained, then show that δc rises back to ≈ 0.67. The manuscript should also state explicitly that δ in Eq. (33) is evaluated at horizon-crossing in both sets of runs and with the same compensation convention for the underdense region, so that the two numbers are directly comparable.
[§2.3 and §3.2] No numerical convergence or resolution study is reported anywhere in the paper, yet two of the headline numbers are resolution-sensitive by construction. The threshold δc is obtained by bracketing between super- and sub-critical runs, and the black-hole mass is extracted from an operational criterion (the maximum of 2M/R where 1 − 2M/R first falls below 10^-4) applied on a grid whose spacing is non-uniform and prescribed by a composite recipe. Neither the bracketing tolerance in δ nor the sensitivity of δc and MBH to grid resolution is stated. Given that the central claim is a shift in δc of order 0.2, a demonstration that δc for one profile is stable at the few-per-cent level across two or three resolutions, together with an explicit statement of the bisection tolerance, is needed before the revised threshold can be quoted as a physical number rather than a property of this grid.
[§3.3, Eqs. (39) and (40)] The vacuum-energy results are presented as linear laws, γ(Λ) ≈ γ(0) − 8.3y and δc(Λ) ≈ δc(0) + 0.98y, but the manuscript does not state how many values of y were computed, over what range, or with what residuals. The stated range 10^-3 to 10^-2 spans only a decade, which is barely enough to distinguish a line from a mild curve, and the coefficients are quoted to two significant figures with no uncertainty. Please report the individual (y, δc) and (y, γ) points, the number of runs behind each, and the fit residuals; if only two or three values of y were used, the linear form should be presented as an interpolation ansatz rather than as an established scaling. It would also help to state which of the two horizons defined in Appendix A is used to normalise Ro/RH = 5 when Λ ≠ 0, since the Hubble and particle horizons differ in that case.

Minor objections (7)

[§2.3] The citation to Miller & Rezzolla (1995) in the description of the grid organisation is given as [41], but [41] is Miller & Sciama (1980); Miller & Rezzolla (1995) is [44]. The same numbering slip should be checked wherever the code lineage is described.
[§3.1 and §3.2] Section 3.1 states that the Gaussian profile is 'not very satisfactory for cosmological perturbations' and that the authors 'concentrate here on the first two types', yet §3.2 quotes δc ≈ 0.71 for the Gaussian alongside the polynomial. Either include the Gaussian in the presentation (a scaling curve, or at least a statement of how many runs) or drop the number.
[Appendix A, Eq. (A.1)] The left-hand side of the Friedmann equation is printed as (ȧ/a) and should be (ȧ/a)^2. Eq. (A.8) also carries a stray period before the comma at the end of the displayed expression.
[References] Two author names are misspelled: [27] 'Pondurets' should read 'Podurets', and [13] 'Anshordi' should read 'Afshordi'.
[Figure 1] Figure 1 shows only the Mexican-hat scaling curve. Since the polynomial case gives a distinctly different δc ≈ 0.47 and is used in the conclusions, adding the polynomial curve (or stating explicitly that it is omitted for legibility) would let the reader see that the exponent is common to both profiles while the threshold is not.
[Figure 3 caption] The parenthesis in the caption is unbalanced: '(δ − δc) = −3.0 × 10^-3)'. The same expression appears in §3.4 and should be made consistent between text and caption.
[§3.4, item (iv)] The statement that the second compression wave 'is not quite a shock' is made without a quantitative criterion. Since the presence or absence of a shock is one of the physically interesting outcomes of the near-critical runs, please state what measure (e.g. the steepness of the density jump relative to the grid scale, or the post-front entropy jump) supports the distinction.

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The paper

The paper claims

The threshold for primordial black hole formation falls to about 0.45 when perturbations begin as pure growing modes.

The reading

The paper asks how large a density perturbation in the radiation-dominated early universe must be for it to collapse into a primordial black hole. Its central move is to start from a pure growing-mode perturbation while it is still larger than the horizon, rather than from data imposed at horizon crossing, and then follow the collapse numerically. The authors find that the threshold fractional mass excess drops from about $0.67$–$0.71$ in earlier work to about $0.43$ for a Mexican-hat profile and $0.47$ for a polynomial profile, while the power-law slope of the black-hole mass with distance above threshold stays near $0.36$–$0.37$. That matters because a lower threshold means more perturbations of a given amplitude make black holes, so abundance estimates tied to the older number need revisiting.

What carries the argument

The load-bearing machinery is a two-stage numerical scheme built around the initial-data choice. A super-horizon linear growing mode is specified with density perturbation $��e$ and velocity perturbation $��u = -��e/4$, then evolved with the Misner-Sharp cosmic-time equations until a null slice can be constructed; the collapse is continued with a Hernandez-Misner outgoing-null-slice code, where the lapse $f$ is fixed to unity at the outer FRW observer and evolved inward using $D_k[(��+U)/f] = -4��R(e+p)f$. The null slicing keeps singularity formation from halting the computation, and the paper reports that this alternative $f$ equation, rather than the more direct equation (24), is what preserves the synchronization $D_kR �� 0$ as $f �� 0$.

Extended reading notes

Core claim

The paper's central claim is that the threshold amplitude for primordial black-hole formation is substantially lower than previously reported when the initial perturbation is a pure growing mode. For Mexican-hat perturbations the critical fractional mass excess is $��_c �� 0.43$, and for polynomial perturbations $��_c �� 0.47$, compared with $��_c �� 0.67$ and $0.71$ obtained with horizon-crossing initial data. The authors explain the shift by noting that the earlier initial data contained a decaying-mode component which rapidly disappeared, so the effective growing-mode amplitude available for collapse was smaller than the nominal $��$. The scaling law $M_{��m BH} = K(��-��_c)^�� M_H(t_H)$ survives with $�� 0.36$–$0.37$, and adding a positive cosmological constant raises $��_c$ linearly while lowering $��$. For sub-critical perturbations close to threshold, the collapse rebounds, drives strong compression and rarefaction waves, and can produce a second, more violent bounce.

Load-bearing premise

The revised threshold depends on the assumption that a super-horizon perturbation from inflation is well represented by the linear growing mode with velocity perturbation $��u = -��e/4$; if that prescription misstates the actual perturbation, the lower $��_c$ is an artifact of the setup.

Editorial extensions

If this is right

For a fixed inflationary perturbation spectrum, using $��_c �� 0.43$–$0.47$ instead of $0.67$–$0.71$ raises the predicted number of primordial black holes, because more perturbations cross the collapse threshold.
The mass scaling $M_{��m BH} \propto (��-��_c)^��$ with $�� 0.36$–$0.37$ survives the change in initial data, so near-threshold black-hole masses are controlled mainly by $��-��_c$ rather than by the initial-data convention.
A positive cosmological constant makes collapse harder: it shifts $��_c$ upward by about $0.98\,y$ and lowers $��$ by about $8.3\,y$ for small vacuum-energy fraction $y$.
Near-threshold sub-critical perturbations do not fade quietly; they generate strong compression waves and a second, more violent bounce, so the surrounding medium can be dynamically disturbed even when no black hole forms.

Reading between the lines

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

If a decaying-mode component can inflate the apparent $��_c$ by roughly 0.2, then threshold values quoted across the literature are not directly comparable unless the initial-data slicing and mode content are specified.
The shift from $0.67$ to $0.43$ is large enough that primordial-black-hole abundance constraints on the inflationary spectral index should be recomputed with the lower threshold; a constraint derived with the older value may be too conservative.
The linear $��$ corrections suggest a possible consistency check: any measured primordial-black-hole abundance that requires a nonzero vacuum-energy fraction would also require a correlated change in the scaling exponent, not just in the threshold.
The second-bounce compression waves in near-critical sub-critical cases may be a source of secondary perturbations or gravitational radiation; simulating them with a shock-capturing code would test whether they leave an observable imprint.

Original abstract

Results are presented from general relativistic numerical computations of primordial black-hole formation during the radiation-dominated era of the universe. Growing-mode perturbations are specified within the linear regime and their subsequent evolution is followed as they become nonlinear. We use a spherically symmetric Lagrangian code and study both super-critical perturbations, which go on to produce black holes, and sub-critical perturbations, for which the overdensity eventually disperses into the background medium. For super-critical perturbations, we confirm the results of previous work concerning scaling-laws but note that the threshold amplitude for a perturbation to lead to black-hole formation is substantially reduced when the initial conditions are taken to represent purely growing modes. For sub-critical cases, where an initial collapse is followed by a subsequent re-expansion, strong compressions and rarefactions are seen for perturbation amplitudes near to the threshold. We have also investigated the effect of including a significant component of vacuum energy and have calculated the resulting changes in the threshold and in the slope of the scaling law.

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Results are presented from general relativistic numerical computations of primordial black-hole formation during the radiation-dominated era of the universe. Growing-mode perturbations are specified within the linear regime and their subsequent evolution is followed as they become nonlinear. We use a spherically symmetric Lagrangian code and study both super-critical perturbations, which go on to produce black holes, and sub-critical perturbations, for which the overdensity eventually disperses into the background medium. For super-critical perturbations, we confirm the results of previous work concerning scaling-laws but note that the threshold amplitude for a perturbation to lead to black-hole formation is substantially reduced when the initial conditions are taken to represent purely growing modes. For sub-critical cases, where an initial collapse is followed by a subsequent re-expansion, strong compressions and rarefactions are seen for perturbation amplitudes near to the threshold. We have also investigated the effect of including a significant component of vacuum energy and have calculated the resulting changes in the threshold and in the slope of the scaling law.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Results are presented from general relativistic numerical computations of primordial black-hole formation during the radiation-dominated era of the universe.	contribution 0.97	0.92	0.07	asserted 0.99
s1	Growing-mode perturbations are specified within the linear regime and their subsequent evolution is followed as they become nonlinear.	method 0.99	0.85	0.04	asserted 0.95
s2	We use a spherically symmetric Lagrangian code and study both super-critical perturbations, which go on to produce black holes, and sub-critical perturbations, for which the overdensity eventually disperses into the background medium.	method 0.89	0.95	0.08	asserted 0.97
s3	For super-critical perturbations, we confirm the results of previous work concerning scaling-laws but note that the threshold amplitude for a perturbation to lead to black-hole formation is substantially reduced when the initial conditions are taken to represent purely growing modes.	result 1.00	0.88	0.44	asserted 1.00
s4	For sub-critical cases, where an initial collapse is followed by a subsequent re-expansion, strong compressions and rarefactions are seen for perturbation amplitudes near to the threshold.	result 1.00	0.93	0.05	asserted 1.00
s5	We have also investigated the effect of including a significant component of vacuum energy and have calculated the resulting changes in the threshold and in the slope of the scaling law.	contribution 0.54	0.97	0.13	asserted 1.00

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Figures

**Figure 1.** Scaling behaviour for MBH as a function of (δ − δc) calculated for growing-mode Mexican-hat perturbations specified within the linear regime. The filled circles refer to the standard calculation discussed in section 3.2, while the open circles are for a calculation including a non-zero cosmological constant Λ, as discussed in section 3.3, giving y = 3.0 × 10−3 . Following these initial calculations to reproduce previous results, we then carried out further ones starting from growing-mode perturbations specified within the linear regime and with length-scales larger than the…

**Figure 2.** A typical evolution leading to black hole formation: the initial perturbation had a Mexican-hat profile and gave (δ − δc) = 2.37 × 10−3 at the horizon crossing time. The top left-hand panel shows the behaviour of the lapse function (the time sequence of the curves goes from bottom to top on the right hand side); the top right-hand panel shows the fluid-element worldlines (the time is measured in units of the horizon-crossing time tH). The bottom left-hand panel shows the profile of 2M/R at different times, with the inset showing the approach of the maximum value of 2M/R → 1; the bott

**Figure 3.** Worldlines for a Mexican-hat perturbation with (δ − δc) = −3.0 × 10−3 ). This plot shows alternating collapse and expansion of the perturbed region while the outer material continues to expand uniformly. The “cosmic” time is measured in units of the time at horizon crossing. the outer matter until eventually it starts to re-contract shortly after horizon crossing. The maximum infall velocity reached is, however, rather small [see row (a) of

**Figure 4.** Plots of local quantities as functions of R/RH(tH): the velocity U/c is shown in the left-hand column and the energy density e/eb in the right-hand column. The frames correspond to the following values of (t − t0)/tH: (a) 7.02; (b) 25.92, (c) 31.67; (d) 33.64; (e) 40.11 . Note that RH is increasing with time and so points with R/RH(tH) > 1 can be within the current horizon scale at times after horizon-crossing

**Figure 5.** Evolution of the energy density e and radial velocity U at three (comoving) locations: near to the centre of the perturbation, at an intermediate region and at the edge of the grid where the fluid is unperturbed. Each quantity is measured in units of its initial value at the same comoving location

Figures reproduced from arXiv:gr-qc/0412063 by the authors.

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See all 103 Pith citations

How the literature uses it

158 arXiv papers cite this in 216 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 · 128 sentences without a confident label

Uses its method or tool 7 papers

The numerical code used is the same of Refs. [ref] , which has been fully described previously and therefore we give only a brief outline of it here.

From Primordial Black Holes Abundance to Primordial Curvature Power Spectrum (and back) · Numerical simulations · 0.85
Assuming Gaussian primordial density fluctuations, the Press–Schechter formalism [ref] relates the initial PBH mass fraction to the curvature power spectrum through [equation] where we adopt the collapse threshold $δ_c≃0.45$ [ref] .

Fine-tuning in mixed Dark Matter models with Primordial Black Hole relics · The Fine-Tuning Definition and Measure of the Model · 0.94
The spherically symmetric numerical simulations used to compute the threshold $δ_c$, and the mass distribution discussed in the next section, have been performed with a numerical code developed in [ref] that has been widely used and tested [ref] , including also an adaptive mesh refinement (AMR) scheme, which makes the code very flexible.

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 · The threshold for PBHs · 1.00

Uses its data or values 38 papers

The mass of the resulting PBH depends on the horizon mass and the amplitude of the density perturbation $δ$ defined in the comoving gauge from which it formed, and is given by [ref] [equation] where $$ is the horizon mass at the time of horizon re-entry, and during radiation domination the constants have been numerically found to be given by $k=3.3, γ=0.36, δ_c=0.453$ (the values depend upon the radial profile of the perturbations being considered; we use the values given here in order to be concrete).

Primordial black holes with an accurate QCD equation of state · Extended mass function of primordial black hole formation · 0.91
where $t_PBH$ is the time of production of the PBH, which can be obtained when the density contrast between inside and outside of the Hubble volume exceeds the critical value $δ_c=0.45$ [ref] .

Primordial black holes as a probe of strongly first-order electroweak phase transition · PBH and GW productions from first-order phase transition · 0.86
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

Compares with it 1 paper

This corresponds to N = ln(a_m/a_hc) = 1/4ln(ρ_hc/ρ_m)∼0.6 e-folds passing from horizon crossing time up to the onset of the gravitational collapse process at $ρ=ρ_m$, thus being in agreement with the results from PBH numerical simulations [ref] .

Primordial black holes in loop quantum cosmology: The effect on the threshold · Results · 0.82

Cites it as support 2 papers

This assumption is supported by numerical simulations which show that maximum infall radial velocity remains rather small for a time considerably after horizon crossing, till the perturbation has become highly non-linear [ref] .

The Black Hole Formation -- Null Geodesic Correspondence · Introduction · 0.99
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 37 papers

PBHs will form by gravitational collapse at horizon reentry for the Hubble patches where the density contrast $δρ/ρ$ exceeds a threshold $δ_c$ [ref] , which depends on the EoS parameter $w$ at the re-entry moment [ref] .

NANOGrav Hints on Planet-Mass Primordial Black Holes · 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
As it is well known, in the standard scenario of a radiation dominated medium a critical collapse arises [ref] and the mass spectrum of PBHs is characterized by a scaling law [ref] given by [equation] where for $δ-δ_c ≲10^-2$ the critical exponent $γ$ depends only on the parameter of the equation of state, i.e the value of $w$, completely independent on the initial configuration of the initial conditions, given by the initial profile of $ζ(r)$, which affect instead the value of $𝒦$.

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 · The mass spectrum for PBHs · 0.92

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