Literature-claim verification report — PBH formation in matter domination

Sources: all eight papers were fetched as arXiv PDFs (latest versions: 2507.18312v2, 1801.03507v2, 2409.01934v3, 1609.01588v2, 2211.13950v3, 1810.03490v1, 2508.10070v1, 1805.03946v7) and converted locally with pdftotext; quotes below are verbatim from the text layer (equations re-typeset inline; equation numbers are those of the listed PDF version). Where the ar5iv/HTML rendering used a different equation numbering (Papers 4 and 8), this is flagged.

1. Ledger #

2. Verbatim quotes by paper #

Paper 1 — Ebrahimian, Abolhasani, Mirbabayi, arXiv:2507.18312v2 [gr-qc], 25 Mar 2026 (v1 24 Jul 2025; no journal reference) #

(a) Title / authors / abstract

Title: "Primordial black hole formation in matter domination" Authors (PDF): "E. Ebrahimian^{a,c}, A. Abolhasani^b, and M. Mirbabayi^a" — a: The Abdus Salam ICTP, Trieste; b: Sharif University of Technology, Tehran; c: IFPU, Trieste. (arXiv listing: Ehsan Ebrahimian, Ali Akbar Abolhasani, Mehrdad Mirbabayi.)

Abstract: "We study Primordial Black Holes (PBHs) formed by the collapse of rare primordial fluctuations during an early period of Matter Domination. The collapse threshold strongly depends on the shape of the peaks, decreasing as they become flatter and hence rarer. In the extreme limit of a top-hat perturbation, Harada, Kohri, Sasaki, Terada, and Yoo have argued that the growth of velocity dispersion prevents the formation of black holes unless the initial peak is larger than ζ_th ∼ ζ_rms^{2/5}. Including the shape distribution of the peaks, we find that for a realistic cosmic abundance of PBHs, the effective threshold is larger, ζ_th ∼ ζ_rms^{1/10}. And this model requires ζ_rms ∼ 10^{−1}, which is much larger than the observed value at the CMB scales. Hence, PBH formation during Matter Domination is barely more efficient than Radiation Domination. We estimate the dimensionless spin parameter to be a_rms ∼ ζ_rms^{7/4} ≪ 1, slightly larger than PBHs formed in Radiation Domination."

(b) Shell crossing, O(1) threshold, effective threshold, σ_v ↔ spectrum

Sec. 1: "A typical overdensity, even if perfectly spherically symmetric like a Gaussian, will experience shell-crossing and Virialization in a dust-dominated universe unless its initial amplitude is O(1)."

Sec. 1 (on [27]): "By studying the growth of perturbations in a dust-dominated universe, they find a black hole threshold ζ_th ∼ ζ_rms^{2/5}. … More explicitly log β ∼ −ζ_rms^{−6/5}."

Sec. 1, Eq. (1): "resulting in an effective threshold ζ_th ∼ ζ_rms^{1/10}, or log β ∼ −ζ_rms^{−1.8}. (1)"

Sec. 3, Eq. (29): "At t = t_c, the central density diverges, but black hole formation requires evaluating the apparent horizon condition, R < 2GM: 2GM/R|_{t_c} = 2K(r)r²/(1 − cos θ) ∼ (δ_m/A_2n^{2/3}) (r/r_m)^{(6−4n)/3}, (29) where we used K(0) ∼ δ_m/r_m², and 2π − θ(r, t_c) ∼ A_2n^{1/3}(r/r_m)^{2n/3}. This indicates that unless the curvature profile is sufficiently flat, i.e. either A_2 ≪ δ_m^{2/3} or n ≥ 2, an apparent horizon will not form by t_c."

Sec. 3: "Hence, Eq.(29) suggests that for a typical peak, namely one with n = 1 and A_2 = O(1), black hole formation threshold is δ_m = O(1). For a parametrically smaller δ_m, spherical shell-crossing will prevent black hole formation unless the profile is sufficiently flat, i.e., A_2 ≪ 1."

Sec. 3: "To summarize, the key difference between w = 0 and w → 0⁺ that underlies the discrepancy in the black hole formation thresholds, δ_th = O(1) vs. O(w), is the possibility of shell-crossing."

Sec. 4.1, Eq. (36): "Realistically, there are additional perturbations, represented by ζ_rms, that overlay this spherical perturbation. During the evolution of the top-hat background, fluctuations with wavelengths shorter than its size grow and induce random velocities, a velocity dispersion σ_v. If this grows large enough, then the centrifugal force can resist gravity: σ_v²/R(t) ∼ GM/R(t)². (36) If this equality is reached while R(t) > 2GM, the collapse will cease, and the sphere will begin to re-expand per the Virial theorem. Otherwise, a black hole forms. … In the following, we will find the parametric relation of δ_th on ζ_rms, ignoring coefficients of O(1)."

Sec. 4.1, Eqs. (37)–(40): "A typical perturbation with k > k_0, enters the horizon at a_k = k_0²/k² and grows as δ_k(t) ∼ ζ_k (k²/k_0²) a(t), t < t_max. (37) … δ_k(t) ∼ δ_k(t_max) b(t)^{−3/2} ∼ ζ_k (k²/k_0²) a(t_max) b(t)^{−3/2}, t > t_max. (38) Modes with wave-number k become nonlinear at the time t_k, when δ_k(t_k) = 1. The velocity dispersion induced at that moment can be found using the Virial theorem. They have an enclosed mass M_k ∼ M(k_0/k)³ and size R_k ∼ R(t)k_0/k: σ_{v,k}²(t_k) ∼ GM_k/R_k ∼ (GM/R(t)) k_0²/k². (39) Hence the condition (36) is satisfied at t_{k_0}, when δ_{k_0}(t_{k_0}) ∼ 1. … Setting k = k_0 in (38) and requiring it to be O(1) at b_BH given by (35) and using (33) results in the threshold δ_th ∼ ζ_{k_0}^{2/5}, (40) as originally derived in [27]."

Notation (Sec. 1): "when we write X_k for some field, we mean X_k = √(k³P(k)) where P(k) is the Fourier transformation of the two-point function of X(r)."

Sec. 4.1, Eq. (41): "Blueshifting (39), by a factor of b(t_k)²/b(t_{k_0})² gives σ_{v,k}(t_{k_0})/σ_{v,k_0}(t_{k_0}) ∼ (ζ_k/ζ_{k_0}) (k_0/k)^{2/3}, (41) which is less than 1, except for a very blue spectrum d(log(k³P_ζ(k)))/d log(k)|_{k_0} ≥ 2."

Sec. 4.2, Eqs. (45)–(50): "b^{3/2} ∼ A_2 (r_ed/r_m)² + A_4 (r_ed/r_m)⁴. (45) … When these two scales meet each other, the central region can't collapse further due to velocity dispersion. If no black hole has formed at this moment, the Virialized central region acts as a regulator for the spherical collapse described in section 3. Since the matter is non-interacting, the infalling outer layers just passes through the central region without any interruption and re-expand. According to the simulations discussed in Section 5.3, after this point a black hole can't form. … ζ_rms/δ_m ∼ A_2 (r_nl/r_m)⁴ + A_4 (r_nl/r_m)⁶, (46) … δ_m ≳ A_2^{1/3}, r_{2−4} > r_ed. (48) … δ_m ≳ ζ_rms^{1/10}, r_{2−4} < r_ed. (49) … δ_th = { A_2^{1/3}, ζ_rms^{3/10} < A_2 < 1 ; ζ_rms^{1/10}, A_2 < ζ_rms^{3/10} }. (50)" Also Eq. (51): "δ_th ∼ ζ_rms^{(4n−6)/10n}, (51) which in the n → ∞ limit reproduces the top-hat result (40)."

Sec. 6: "Hence, even though as σ_0 → 0 the PBH threshold asymptotes to ζ_rms^{2/5}, as derived in [27], for the realistic values of β, it is δ_th ∼ ζ_rms^{1/10}."

(c) Numerical method; post-caustic behaviour

Sec. 5: "In this section, we will use Newtonian N-body simulations to provide evidence not for the threshold of PBH formation, which would require relativistic simulations like those in [33], but for the halting mechanism based on the growth of velocity dispersion. We are going to carry out a series of simulations in which a spherical distribution of particles are going to collapse under Newtonian gravity. In Sections 5.1 and 5.2, we consider a top-hat background, where inhomogeneities are essential to prevent black hole formation. … In Section 5.3, we consider a collection of spherical shells that fallows a Gaussian density profile. The radial velocities for spherical shells are obtained from Misner-Sharp equations using the curvature profile which is equivalent with the Gaussian density profile. … In the first two simulations, we start with a sphere with GM = 1 and r = r_max = 1. Then, we divide the mass between N equal-mass particles and let it collapse until it starts to re-expand."

Sec. 5.1: "The initial positions of these particles are distributed with a uniform probability, so the power spectrum would be Poisson … we predict the minimum contraction: b_min ∼ 1/N^{1/3}. (55) … the solid line is 1.2 × N^{−1/3}." (Fig. 4 spans N ≈ 10–10³.)

Sec. 5.3: "The first simulation is the same N-body simulation code of Section 5.1, but … we use a Gaussian distribution with zero mean and unit variance to set the Cartesian coordinates of each particle. … We create 100 realizations with different particle numbers, N, from N = 100 to N = 2000, and let them evolve under gravity and measure the minimum compression factor. … The second simulation is a collection of spherical concentric shells with the same initial density profile as the first simulation. Each shell has a fixed mass and feels only the gravity of its inner shells. They evolve under Newtonian gravity with Plummer smoothing at r = 0, and the time evolution is carried out by the leapfrog method. We use a fixed number of shells (10000 shells) to ensure accuracy. … for the larger N the b_min shows a convergence to the value of second simulation b_min ≈ 0.41. … Note that in the second simulation, we only have perfect spherical shells that can pass through each other. Thus, we can deduce that the minimum value of b_min ≈ 0.4 in the N-body simulations comes from the shell-crossing. b_min generally depends on the shape of the profile, but for a typical profile with A_2 = O(1), it is an O(1) number."

Sec. 3 (after first caustic): "The divergence of density at the central means that we can't continue the evolution unless we somehow regulate the divergence. … Can a black hole form afterwards (assuming the evolution is continued beyond t_c with the help of a regulator)? In Section 5.3, we demonstrate that the numerical simulations do not support this idea. Heuristically, after regularizing the origin, if the matter can't interact with itself non-gravitationally, as we expect from any dust model, the central layers that were moving inward before t_c pass through the origin and move outward after t_c and start crossing the in-going outer layers. This is an example of the "shell-crossing" phenomenon, though one that starts from the origin after using a regulator. Afterwards, the out-flowing shells tend to reduce the density of the central region, and the infalling shells cannot compensate for this decrease."

Sec. 3 (only use of "cusp", pre-t_c): "Consequently, for a monotonically decreasing profile, the inner layers tend to shrink more rapidly than the outer layers. This results in the profile transforming into a cuspy shape over time. We will return to this point in Section 4.2."

Inner slope r^-2 / r^-9/4 / self-similar / Bertschinger / Fillmore–Goldreich: no occurrences in the paper (grep of "9/4", "r^-2", "power law", "Bertschinger", "Fillmore", "Goldreich", "self-similar" returns nothing).

(d) Abundance; comparison with Harada+2016

Sec. 6, Eqs. (63)–(68): "x ∼ A_2 ν, ν = δ_m/σ_0, σ_j² = 4π ∫ dk k² k^{2j} P_ζ(k). (63) Here we assumed r_m² σ_2/σ_0 ∼ 1. By definition, σ_0 = ζ_rms. … ν_th = { σ_0^{−1}, σ_0^{−1} < x ; x^{1/4}/σ_0^3 [sic, as extracted], σ_0^{−3/5} < x < σ_0^{−1} ; σ_0^{−0.9}, x < σ_0^{−3/5} } (64) … n(ν, x) ∝ x³ e^{−(ν²+x²−2νγx)/2(1−γ²)}, γ := σ_1²/(σ_0σ_2), (65) … β^{(4)} ∼ exp(−κ_4 ζ_rms^{−1.8}), (66) β^{(2,1)} ∼ ζ_rms^{0.3} exp(−κ_{2,1} ζ_rms^{−1.8}), (67) β^{(2,2)} ∼ ζ_rms^{−2} exp(−κ_{2,2} ζ_rms^{−2}), (68) where κ's are all O(1). The same approach applied to a radiation-dominated universe yields log β ∼ −ζ_rms^{−2}. … If dark matter is composed entirely of PBHs with mass m = 10^20 gr, then we must have log_10 β ∼ −35. This implies that ζ_rms should be around 0.1 for both the radiation-dominated and the matter-dominated cases. From Fig.7 it is evident that in log_10 β ∼ −35 the β_0^{(2,1)} is dominant and β^{(4)} is also close. This is because most PBHs form due to the flatness of their profile."

Comparison with Harada+2016 / 0.05556σ^5 / Khlopov–Polnarev: NOT FOUND. Reference list contains only [17] Harada, Yoo, Nakama, Koga; [24] Harada, Yoo, Kohri "Threshold of primordial black hole formation"; [27] Harada, Kohri, Sasaki, Terada, Yoo, JCAP 02 (2023) 038; [28] Harada, Yoo, Kohri, Nakao, PRD 96 (2017) 083517 (spins); [43] Saito, Harada, Koga, Yoo. No occurrence of "hoop", "Khlopov", "Polnarev", "0.055", "0.05556".

Sec. 7, Eq. (76): "a_rms ∼ ζ_rms^{7/4}. (76) … For ζ_rms ∼ 0.1, we have a_rms ∼ 0.01 in both MD and RD era."

Paper 2 — Mocz, Lancaster, Fialkov, Becerra, Chavanis, arXiv:1801.03507v2, Phys. Rev. D 97, 083519 (2018) #

Title: "On the Schrödinger-Poisson–Vlasov-Poisson correspondence"

Abstract: "We demonstrate that, while the density field of the superfluid always shows order unity oscillations as ħ/m → 0 due to interference and the uncertainty principle, the potential field converges to the classical answer as (ħ/m)². Thus, any dynamics coupled to the superfluid potential is expected to recover the classical collisionless limit as ħ/m → 0."

Sec. I (Introduction): "As ħ/m → 0, the period of the interference oscillations decreases as ħ/m but the envelope of the wave function remains constant. Therefore the density field always exhibits order unity differences from the classical solution. The potential (obtained from the density via the Poisson equation) and the force field (gradient of the potential) will also show oscillations on the scale of the de Broglie wavelength. However, fortunately the amplitude of the oscillations in the potential is not order unity, rather it is suppressed by a factor of (ħ/m)² by the ∇² operator in the Poisson equation, and likewise in the force field the amplitude is suppressed by a factor of (ħ/m). Therefore, these quantities are hypothesized here to converge to the classical solution without non-local manipulation/smoothing …" and "(non-convergence of density field, (ħ/m)² convergence of potential)."

Sec. I: "Of great interest is to test whether we can recover the classical potential V in a formal converged sense, with convergence rate faster than (ħ/m)¹ (so that the force field is also guaranteed to converge to the classical limit as ħ/m → 0)."

Sec. IV.A.3 (1D fixed potentials, harmonic + linear, "Discussion"): "The 1D examples highlight how the uncertainty principle regularizes caustics and how the classical solution is approached as ħ/m → 0. The norm of wave functions approaches the classical limit but retains order unity oscillations (with period the de Broglie wavelength) about the classical solution. Of interest, in the next section, is whether the self-potential from this oscillatory density field recovers the classical limit." (Eq. 31: caustic regulated to "a Gaussian of width σ = 2^{−1/2} ħ/m".)

Sec. IV.B.2 (2D cold ICs, resolution 2048², m = 1.25×10^-22–8×10^-21 eV): "Importantly, the potential is found to converge to the classical limit as m^{−2}, even though the density profile shows order unity oscillations on the scale of the local de Broglie wavelength." Fig. 4 caption: "The potential converges to the classical limit as m^{−2}."

Sec. IV.C.2 (2D multi-sheet, 13 phase sheets): "The SP equations again capture the classical solution as the boson mass increases, with an L1 norm error in the potential that goes as m^{−2} in the asymptotic limit. This is despite the fact that the density distribution has order unity errors at t = 0 due to the interference of the multiple phase sheets. This is of note because previously only cold initial conditions in 2D have been tested [32]."

Sec. IV.D.2 (2D warm ICs, v_disp = 0.1 km/s): "Even in the case of warm initial conditions, with random phases added to the different quantum wave modes, the solution recovers the classical answer as m → ∞, and the potential again converges as m^{−2}. This is good news, as it demonstrates that the Schrödinger equations can capture velocity dispersion in a meaningful way."

Sec. IV.E (3D cosmological, 250 h^-1 kpc box, 1024³ grid): no convergence rate is quoted; only "As the axion mass increases, more of the substructure of scale-free CDM is recovered" and the NFW-like profile discussion.

Sec. V (Concluding Remarks): "Our numerical experiments demonstrate that a correspondence between the SP and VP equations exists in the sense that for the wide range of complex test problems we simulated (caustics, multi phase-sheet, warm conditions, cosmological simulations), the potential converges to the classical solution as (ħ/m)². Hence the force field is also converged to the classical answer in the limit ħ/m → 0, despite the fact that the density field, riddled with order unity quantum interference patterns, does not converge to the classical limit."

Paper 3 — Harada, arXiv:2409.01934v3, "Primordial black holes: formation, spin and type II" (Universe, Special Issue "Primordial Black Holes from Inflation") #

Abstract: "Primordial black holes (PBHs) may have formed through the gravitational collapse of cosmological perturbations that were generated and stretched during the inflationary era, later entering the cosmological horizon during the decelerating phase, if their amplitudes were sufficiently large. In this review paper, we will briefly introduce the basic concept of PBHs and review the formation dynamics through this mechanism, the estimation of the initial spins of PBHs and the time evolution of type II fluctuations, with a focus on the radiation-dominated and (early) matter-dominated phases."

(a) Matter-era formulas (Sec. 3.6 "Matter domination")

"The PBH formation in matter domination has been pioneered by Khlopov and Polnarev (1980) [54–57], where the effects of anisotropy and inhomogeneity are studied as obstruction of PBH formation process. Harada, Yoo, Kohri, Nakao and Jhingan (2016) [58] revisited the anisotropic effects by combining the picture of pancake collapse of dark matter and the hoop conjecture by Thorne, which claims that black holes with horizons form when and only when a mass M gets compactified into a region whose circumference in every direction is C ≲ 4πM [59,60]. … They not only qualitatively reproduced the result of Ref. [54,56] but also updated the coefficient as β_aniso(M) ≃ 0.05556 σ_H^5(M), (19) where σ_H(M) is the standard deviation of δ_H in the mass scale of M and the Gaussian distribution for density perturbation is assumed."

Definition of δ_H (Sec. 3.3): "argument in terms of δ_H, the density perturbation at the horizon entry of the perturbation." and Sec. 3.9 Eq. (22): "where δ_max = 2/3 and σ_H(M) are the possible maximum value and the standard deviation of δ_H in the mass scale of M".

"Kokubu, Kyutoku, Kohri and Harada (2018) [63] revisited the inhomogeneity effects by utilising the LTB solution and not only qualitatively reproduced the result of Ref. [54,56] but also updated the coefficient so that the additional suppression factor is given by β_inhom(M) ≃ 3.70 σ_H^{3/2}(M) (20) with the caveat that its physical effect largely depends on the assumption that black hole formation is prevented by the appearance of an extremely high-density region before the black hole horizon formation surrounding it."

Velocity dispersion (only statement): "Harada, Kohri, Sasaki, Terada and Yoo (2022) [64] showed that the effects of velocity dispersion that may have been generated in possible nonlinear growth of perturbation in the earlier phase can suppress PBH formation. The effects of the angular momentum can play important roles and suppress PBH formation for smaller σ_H [65]."

Spin (Sec. 4.3): "a perturbative calculation under certain working assumptions gives ⟨a_²⟩ ∼ σ_H^{−1/2}, (23) where a_ is the nondimensional spin parameter of the region to collapse and σ_H is the standard deviation of δ_H at the the horizon entry. Although the nondimensional numerical factor of the order of the unity on the right-hand side should be determined yet, this implies that most of PBHs have spins a_* = O(1) if σ_H ∼ 0.1. The angular momentum effects will strongly suppress PBH formation if σ_H is even much smaller because of the Kerr bound |a_*| ≤ 1. These results have recently been updated based on peak theory [99]."

(b) Asteroid-mass window; Carr et al. reviews (Sec. 1)

"See Fig. 10 of Carr, Kohri, Sendouda and Yokoyama (2021) [23] for an overview of the constraints. Recent observational constraints indicate two intriguing windows for dark matter. One is M ∼ 10^17 − 10^23 g, where f(M) is virtually unconstrained. This implies that PBHs could account for all the CDM for this mass range. The other mass range is M ∼ 1 − 10³ M_⊙, where f(M) ≲ 0.1 … Although there is currently a mass window in which all the CDM might be explained by PBHs, a stricter constraint could be placed on this mass window in the near future."

"See Refs. [5,6] for the recent brief reviews of the history and the future of PBHs." — [5] Carr, B.J.; Green, A.M. The History of Primordial Black Holes 2024 [arXiv:2406.05736]; [6] Riotto, A.; Silk, J. The Future of Primordial Black Holes: Open Questions and Roadmap 2024 [arXiv:2403.02907]; [23] Carr, Kohri, Sendouda, Yokoyama, "Constraints on primordial black holes", Rept. Prog. Phys. (2021).

Sec. 6: "a very small value of β(M), as small as ∼ 10^{−17} for M ∼ 10^17 g, can yield f(M) = O(1), i.e., can explain all dark matter".

Paper 4 — Harada, Yoo, Kohri, Nakao, Jhingan, arXiv:1609.01588v2, ApJ 833, 61 (2016) #

Title: "Primordial black hole formation in the matter-dominated phase of the Universe". (Equation numbers below are from the v2 PDF; the ar5iv rendering numbers the same equations as (26), (29), (30), (32), (40) — an older version.)

Zel'dovich setup, Eqs. (4)–(7): "∂p_i/∂q_k = diag(−α, −β, −γ), (4) … D_ik = diag(a − αb, a − βb, a − γb). (5) … ρ = a³/[(a − αb)(a − βb)(a − γb)] ρ̄. (7)" Eq. (17): "χ := r_g/r_f = 2a(t_i)/a(t_f) = 4α (b/a)(t_i) = 4α δ_L(t_i)/(α+β+γ)."

Sec. 2.3 "Black hole formation criterion": "The hoop conjecture (Thorne 1972; Misner et al. 1973) states that black holes with horizons form when and only when a mass M gets compactified into a region whose circumference in every direction is approximately smaller than 4πGM/c². The hoop C of a region is defined as the maximum of its circumferences in all directions. For the pancake, it is given by the circumference of the ellipse. Since the eccentricity of the pancake is given by e² = 1 − (r_2(t_c)/r_3(t_c))² = 1 − ((α−β)/(α−γ))², (22) the hoop is calculated to C = 16(1 − γ/α) E(√(1 − ((α−β)/(α−γ))²)) r_f, (23) where E(e) is the complete elliptic integral of the second kind. Note that E(e) is a monotonically decreasing function of e ∈ [0, 1], where E(0) = π/2 in the circular limit and E(1) = 1 in the eccentric limit. According to the hoop conjecture, the condition for black hole formation is given by C ≲ 2πr_g. Thus, we find the criterion s(α, β, γ) ≲ χ, (24) where s(α, β, γ) := (8/π)(1 − γ/α) E(√(1 − ((α−β)/(α−γ))²)). (25) Equivalently, we can rewrite the criterion in the following form: h(α, β, γ) ≲ 1, (26) where we define h(α, β, γ) := C/(2πr_g) according to Yoshino (2008). Using Eq. (17), we can calculate the ratio to h(α, β, γ) = (2/π)((α−γ)/α²) E(√(1 − ((α−β)/(α−γ))²)), (27) where we have fixed the normalization of b so that (b/a)(t) = a(t)/a(t_i). (28)"

"If the above criterion is not satisfied, a sheet-like caustic occurs at t = t_c and matter particles should cross each other. These pancakes will then undergo violent relaxation by bouncing several times and eventually get virialized with a large velocity dispersion."

Terminology check: the strings "ellipticity", "prolateness" and a shape parameter "p" do not occur; the paper works in (α, β, γ) and the pancake eccentricity e.

Sec. 3.1, Eq. (29): "w(α, β, γ)dαdβdγ = −(27/(8√5 π σ_3^6)) exp[−(3/(5σ_3²)){(α²+β²+γ²) − ½(αβ+βγ+γα)}] · (α−β)(β−γ)(γ−α) dαdβdγ, (29) where ∞ > α ≥ β ≥ γ > −∞ is assumed and σ_3 is a positive constant."

Sec. 3.2, Eqs. (34)–(36): "σ² = ⟨δ_L²(t_i)⟩ = ⟨(α+β+γ)²⟩(b/a)²(t_i) = 5σ_3², (34) … β_0 = ∫_0^∞ dα ∫_{−∞}^α dβ ∫_{−∞}^β dγ θ(1 − h(α, β, γ)) w(α, β, γ), (35) … For small σ, β_0 tends to be proportional to σ^5 and is best fit by β_0 ≃ 0.056σ^5. (36)"

Sec. 3.3: "We here present the semi-analytic formula β_0 ≃ 0.05556σ^5. (37) This confirmes the best-fit curve (36) for small σ. In the same framework, we can analytically derive lower and upper bounds on β_0 so that 0.01338σ^5 ≲ β_0 ≲ 0.1280σ^5. (38)" Also Eq. (44) (circular limit e=0): "β_0 ≃ (7·5³/(2⁴·3⁶√(10π))) σ^5 ≃ 0.01338σ^5"; Eq. (45) (e=1): "≃ 0.1280σ^5".

Sec. 4.1 (Khlopov–Polnarev): "the new formula (37) for σ ≪ 1 agrees with the Khlopov-Polnarev one (48) only within a factor of 3 if we simply identify χ with σ." and "They finally obtained β_0 ≃ 0.02χ^{13/2}. This can be recast in the form β_0 ∼ σ^{13/2} in terms of σ in our formulation."

Sec. 4.3, Eqs. (57), (61): "σ_k² = 4((1+w)/(3w+5))² ⟨ζ_k²⟩{k=aH(t_i)}. (57) … In the matter-dominated phase, Eq. (57) with w = 0 reduces to σ_k² = (4/25)⟨ζ_k²⟩{k=aH(t_i)} (61) and σ_k can be identified with the standard deviation of the density perturbations at horizon entry in the linear regime in the Newtonian cosmology."

Paper 5 — Harada, Kohri, Sasaki, Terada, Yoo, arXiv:2211.13950v3, JCAP 02 (2023) 038 #

Title: "Threshold of Primordial Black Hole Formation against Velocity Dispersion in Matter-Dominated Era"

Abstract (last sentence): "We find that the threshold value of the density perturbation δ̃_th at the horizon entry for the PBH formation scales as δ̃_th ∝ σ_0^{2/5} for σ_0 ≪ 1."

Sec. 2: "the criterion for the PBH formation can be roughly evaluated by comparing the gravitational free-fall time Δt̃_ff ∝ ρ̃^{−1/2} ∝ R̃^{3/2} with the sound-wave crossing time Δt̃_cross ∼ R̃/σ_v, where ρ̃, R̃, and σ_v are the density, the physical size of the scale k̃, and the velocity dispersion, respectively."

Sec. 3, Eqs. (5)–(9): "Once the velocity dispersion is shared over the collapsing ball, the velocity dispersion simply increases as σ_v ∝ ρ̃^{1/3} as the system collapses adiabatically, according to Liouville's theorem. More explicitly, we have σ_v(t) = σ_v(t_) (ρ̃(t)/ρ̃(t_))^{1/3} = σ_v(t_) R̃(t_)/R̃(t), (5) … Δt̃_cross ≃ R̃(t)/σ_v(t) = (3/(2σ_v(t_))) (R̃(t)²/(R̃(t_)R̃_ent)) t̃_ent, (6) … Δt̃_ff ≃ (R̃(t)/R̃_ent)^{3/2} t̃_ent. (7) … the collapse halts when Δt̃_ff = Δt̃_cross if the radius is larger than the Schwarzschild radius at that time. … R̃_halt = (4/9) σ_v²(t_) (R̃²(t_)/R̃_ent²) R̃_ent. (8) If this is larger than 2M̃, i.e., R̃_halt > R̃_ent = 1/H̃_ent = 2M̃, the collapse halts before the PBH formation. Conversely, a PBH forms if R̃_halt < 2M̃. Thus, the condition for the PBH non-formation can be rewritten in the following simple form. σ_v(t_)R̃(t_) > (3/2)R̃_ent. (9)"

Eqs. (10)–(11): "δ_ent < k̃⁴/k⁴ for Case I : k²δ_ent > k̃²δ̃_ent, (10) δ_ent > δ̃_ent^{5/2} k/k̃ for Case II: k²δ_ent < k̃²δ̃_ent, (11) where we have ignored factors of order unity."

Velocity dispersion definition (Case I, virial; Appendix A.3, Eqs. (41)–(42)): "E = K + U = −K = −½Mσ_v² = −hM²/R_max. (41) From this equation, we find σ_v² = 2hM/R_max = h(2M/R_ent)(R_ent/R_max) ≃ hδ_ent, (42) where we used 2M = R_ent and Eq. (36)." Main text: "Since the gravitational potential Ψ is constant in time in the matter-dominated universe, and δ_ent ∼ Ψ at horizon crossing, the virial velocity dispersion is estimated as σ_v² ∼ Ψ ∼ δ_ent (see Eq. (42))."

Sec. 4, Eqs. (14)–(16): "hereafter, we replace δ_ent by its root-mean-square value σ_δ,ent(k). … σ_δ,ent(k) < (k/k̃)^{−4} for Case I …, (14) σ_δ,ent(k) > δ̃_ent^{5/2} (k/k̃) for Case II …, (15) … σ²_δ,ent(k) = σ_0² exp{−2μ[ln(k/k̃)]²}, (16) … where μ is the non-dimensional parameter characterizing the width of the log-normal distribution."

"Therefore, to avoid complications, we introduce the minimum applicable value for k, k ≥ k_min where k_min > k̃. Although we never assign an explicit value to k_min, we expect k_min/k̃ = O(10). We also mention that the introduction of k_min renders the condition derived below a sufficient condition for PBH non-formation or a necessary condition for PBH formation."

Eq. (23): "δ̃_ent > δ̃_th(σ_0, μ, k_min) ≡ { exp[−(2/5)μ ln²(k_min/k̃)] (σ_0(k̃/k_min))^{2/5} ; σ_0 < σ_cr(μ, k_min), exp[−(2/μ)(2 + √(4 + μ ln σ_0))] ; σ_0 > σ_cr(μ, k_min). (23)" with Eq. (18): "ln σ_0 < ln σ_cr(μ, k_min) ≡ μ ln²(k_min/k̃) − 4 ln(k_min/k̃)."

Sec. 4: "Thus we expect that σ_0 < σ_cr and the threshold in the first line of Eq. (23), δ̃_th ∝ σ_0^{2/5}, or the condition from Case II, is likely to be relevant in such cases. We mention that if the peak is very sharp so that the width of the spectrum is much less than k_min, namely, μ ln²(k_min/k̃) ≫ 1, the velocity dispersion becomes completely ineffective."

Sec. 5 (β): "β_0 ≃ ∫_0^∞ dα ∫_{−∞}^α dβ ∫_{−∞}^β dγ Θ(δ̃(α,β,γ) − δ_th) Θ(1 − h(α,β,γ)) w(α,β,γ), (27) … The result is that β_0 ∝ σ_0^5 with δ̃_th = 0. … The common slope can be approximated as β_0 ≃ 0.05556 σ_0^5 [86] (green dashed line). This is the effect of anisotropy. Different criteria on the density threshold result in different exponential falloffs. They can be understood by Carr's formula [94], according to which the PBH formation probability is roughly proportional to exp[−δ̃_th²/(2σ_0²)]. Applying this to our result (23) in the case σ_0 < σ_cr, δ̃_th² ∝ σ_0^{4/5}, the exponent is found to be proportional to σ_0^{−6/5}". On inhomogeneity: "However, as noted in [88], the threshold obtained there is a sufficient condition for the PBH formation. … Therefore, to be conservative, we do not take its effect into account in this paper."

Sec. 6 (Conclusions): "We obtained the simple criterion σ_v* R̃_* < 3M̃ for the formation of a black hole of mass M̃ … (see Eq. (9)). … We find that the effect of velocity dispersion can be a dominant factor to prevent PBH formation, although all these effects are subjected to various uncertainties … It should be also noted that, if the spectrum has a sufficiently sharp peak around the PBH scale, the velocity dispersion is likely to be ineffective. … since the effect of the velocity dispersion may act as an additional factor to prevent PBH formation, the allowed regions for the model parameters will get wider". Footnote 4: "the abundance produced in the RD epoch can be larger than the one in the MD at least for σ_0 ≳ O(0.1)."

No occurrence of "floor", "inevitable", "unavoidable" or "cannot be avoided" as applied to the threshold.

Paper 6 — Kokubu, Kyutoku, Kohri, Harada, arXiv:1810.03490v1, Phys. Rev. D 98, 123024 (2018) #

Title: "Effect of Inhomogeneity on Primordial Black Hole Formation in the Matter Dominated Era"

Abstract: "We investigate the effect of inhomogeneity on primordial black hole formation in the matter dominated era. In the gravitational collapse of an inhomogeneous density distribution, a black hole forms if apparent horizon prevents information of the central region of the configuration from leaking. Since information cannot propagate faster than the speed of light, we identify the threshold of the black hole formation by considering the finite speed for propagation of information. We show that the production probability β_inhom(σ) of primordial black holes, where σ is density fluctuation at horizon entry, is significantly enhanced from that derived in previous work in which the speed of propagation was effectively regarded as infinite. For σ ≪ 1, we obtain β_inhom ≃ 3.70σ^{3/2}, which is larger by about an order of magnitude than the probability derived in earlier work by assuming instantaneous propagation of information."

Sec. 1: "But whatever happens at the center, if that information is hidden behind the horizon before information reaches us, the collapsing matter will become a black hole. … Since any information propagates at most with the speed of light, the lower bound on the probability of PBH formation is obtained by tracking the null geodesic emanating from the central region to the characteristic radius."

Σ (Sec. 4, Eq. 46): "W(x) = (2/√(2πΣ²)) ∫_0^{u_max} du exp(−u²/(2Σ²)) = Erf(u_max/(√2 Σ)), (46) … In Ref. [12, 13], Σ ∼ 1 was assumed. Since there is no affirmative reason to take Σ ∼ 1, we leave Σ as it is in the formulation. … Only in carrying out calculations we assume Σ = 1 following Ref. [12]."

Sec. 4.1.1, Eqs. (53)–(54): "β_inhom^caus(σ) ≃ … ≃ (56/(3√(3π)Σ)) 2^{1/4} Γ(5/4) σ^{3/2} = 3.6979 σ^{3/2}/Σ (σ ≪ 1), (53) … β_inhom^inst(σ) ≃ 0.4484 σ^{3/2}/Σ (σ ≪ 1). (54) The power 3/2 in Eq. (53) and Eq. (54) was derived by Khlopov and Polnarev [12, 13]. The approximated function of Eq. (53) is valid up to σ ∼ 0.05, while Eq. (54) is valid up to σ ∼ 0.1."

Sec. 4.2, Eqs. (59)–(60): "For σ ≪ 1, β_inhom is given by Eq. (53) and β_aniso is semi-analytically given by [10] β_aniso ≃ 0.05556σ^5 (59) Ref. [10] showed that most of the collapses which result in PBH formation must be nearly spherically symmetric. In this case, we expect β_inhom+aniso may be given by the simple multiplication of β_inhom and β_aniso, β_inhom+aniso ≃ β_inhom × β_aniso = 0.2055σ^{13/2}. (60)"

Sec. 5 (Summary and Discussions): "Our new probability β_inhom^caus behaves as a power law of 3.6979σ^{3/2} for small σ … This approximation is valid up to σ ∼ 0.05." and "we emphasize that our formula for formation probability indeed evaluates the minimum rate for PBH formation because of the following reason: In this work, we derived our formula Eq. (44) from the condition Eq. (26) assuming conservatively that information of singularity formation propagates at the maximum possible velocity, i.e., the speed of light. However, it is expected that the smaller the propagation speed of information, the more PBHs tend to be formed."

Paper 7 — Ye, Gong, Harada, Kang, Kohri, Saito, Yoo, arXiv:2508.10070v1 [gr-qc], 13 Aug 2025 (no journal reference) #

Title: "Primordial Black Hole Formation and Spin in Matter Domination Revisited" Authors: Weitao Ye, Yungui Gong, Tomohiro Harada, Zhaofeng Kang, Kazunori Kohri, Daiki Saito, Chul-Moon Yoo

Abstract: "In this article, we calculate the mass distribution of primordial black holes (PBHs) formed in the matter-dominated (MD) era by the peak theory. We apply the Zel'dovich approximation to track the nonlinear evolution of overdensities and compute the PBH abundance and mass function by incorporating a PBH formation criterion based on the hoop conjecture. We find that the PBH abundance β follows the scaling law β ≃ A_γ σ_h*^5 for σ_h* ≪ 1. Here, σ_h* is the quantity that characterizes the variance of the density fluctuation at the horizon entry. We also find that, in contrast to the previous estimates, the PBH spin is very small for σ_h* ≪ 1 but could be larger for larger σ_h* and broader power spectra. Finally, specializing to a monochromatic power spectrum, we prove analytically that the PBH mass distribution becomes effectively monochromatic and reveal that the resultant PBH abundance is approximately 19 times the previous prediction."

Definitions: "q_0* := (σ_0/σ_2)^{1/2} the characteristic comoving radius of the power spectrum P(k)" (after Eq. 3.3); "σ_h := σ_0(t_h), the variance of density contrast at horizon cross … σ_h(q_0) = (q_0/q_0*)² σ_h* = (2ν/(λ_1λ_2λ_3)^{1/3}) σ_h*, (3.12)". Table I/Sec. III D: "Approximately, we have A_0.99 ≃ 9 × 10³, A_0.8 ≃ 7 × 10⁴, A_0.7 ≃ 3 × 10⁵." and "using either the coefficient A_γ or the ratio β/σ̄_h^5 to compare abundances across different values of γ is not very meaningful."

The 19× comparison (Sec. V): "Ref. [39] employs the Doroshkevich probability distribution, … and argues that the black hole abundance is β = ∫ dλ⃗ ω(λ⃗) Θ(σ_h − h(λ⃗)) ≃ 0.056σ_h^5. (5.5) In fact, Doroshkevich probability distribution represents the probability distribution of the eigenvalues λ_1, λ_2 and λ_3 of the rescaled tidal tensor within a localized overdense region. Therefore, this result actually represents the probability that an arbitrary peak collapses into a black hole, that is, the fraction of peaks forming PBHs. By contrast, the peak theory counts the distribution of peaks throughout the entire space, and therefore it provides a more appropriate way to compute the overall black hole abundance … σ̄_h := σ_h(M̄) = (M̄/M*)^{2/3} σ_h*. (5.6) In the limit σ_h* ≪ 1, for the monochromatic power spectrum we have σ̄_h = σ_h(M) = 6σ_h*; for γ = 0.99, σ̄_h = 6.1σ_h*. It can be seen that in the monochromatic case, we have β ≃ 1.08 σ̄_h^5, (5.7) which is 19 times the result reported in Ref. [39]." (HTML confirms the overbar on σ_h in Eq. 5.7. 1.08/0.056 = 19.3.) Also: "Since both our work and Ref. [39] adopt the Zel'dovich approximation and the hoop conjecture, the black hole formation condition in the σ_h ≪ 1 limit is the same, namely h(λ⃗) ≲ σ_h." [39] = "T. Harada, C.-M. Yoo, K. Kohri, K.-i. Nakao, and S. Jhingan, Primordial black hole formation in the matter-dominated phase of the Universe, Astrophys. J. 833, 61 (2016)."

Spin (Sec. I, IV, VI): "we demonstrate that for the vast majority of regions that eventually collapse into PBHs, the tidal torques are extremely small, implying ⟨ã⟩_BH ∼ σ_h^{1/2} ≪ 1. This conclusion is contrary to the previous studies [40, 41]." "Incorporating this constraint yields ⟨ã⟩_BH ∼ σ_h*^{1/2}, so that the PBH spins are typically very small for σ_h* ≪ 1." "However, for broader spectra and when σ_h* is not very small (e.g. γ = 0.3, σ_h* = 10^{−2}), the PBH spins can approach 1." Sec. VI: "only a spherical region of comoving radius q_0 = √6 q_0* can collapse to form a black hole. Consequently, the mass distribution of PBHs are monochromatic, with M = 6√6 M*, and their angular momentum vanishes exactly. Our calculation yields a PBH abundance β ≃ 1.08σ̄_h^5, which is 19 times the estimate given in Ref. [39]."

Paper 8 — Yoo, Harada, Garriga, Kohri, arXiv:1805.03946v7, PTEP 2018, 123E01 #

Title: "PBH abundance from random Gaussian curvature perturbations and a local density threshold". (Equation numbers from v7 PDF; ar5iv shows the monochromatic equations as (86)–(88).)

Sec. II, Eqs. (13)–(21): "⟨ζ̃*(k)ζ̃(k′)⟩ = (2π²/k³) P(k) (2π)³ δ(k − k′), (13) … Each gradient moment σ_n can be calculated by σ_n² := ∫ (dk/k) k^{2n} P(k). (14) Focusing on a high peak and taking it as the origin of the coordinates, we introduce the amplitude μ and the curvature scale 1/k_* of the peak as follows: μ = −ζ|{r=0}, (15) k² = Δζ|{r=0}/μ. (16) … According to the peak theory[45], for a high peak, we may expect the typical form of the profile ζ̄ can be described by using μ, k and the two point correlation function ψ as follows: ζ̄(r)/μ = g(r; k_) := g_0(r) + k_² g_1(r), (17) where g_0(r) = −(1/(1−γ²)) (ψ + (1/3) R_² Δψ), (18) g_1(r) = (1/(γ(1−γ²))) (σ_0/σ_2) (γ² ψ + (1/3) R_² Δψ), (19) with γ = σ_1²/(σ_0σ_2), R_ = √3 σ_1/σ_2 and ψ(r) = (1/σ_0²) ∫ (dk/k) (sin(kr)/(kr)) P(k). (20) It is worthy of note that, for k_ = k_c := σ_1/σ_0, we obtain g(r; k_c) = −ψ(r). (21) It will be shown that regarding k_ as a probability variable, we obtain k_c as the mean value of k_."

ν and k_* in peak-theory variables (Sec. III, before Eq. 58): "Let us change the variables from ν = −ζ_0/σ_0 = μ/σ_0 and ξ_1 = Δζ|{r=0}/σ_2 = μk²/σ_2 to variables μ and k_." Eqs. (49)–(50): "P_1(ν, ξ_1)dνdξ_1 = (μ/(2πσ_0σ_2√(1−γ²))) exp[−½ μ² (1/σ_0² + (1/(1−γ²)) (k_*² − k_c²)²/σ_2²)] dμdk_² … where k_c² = γσ_2/σ_0 = σ_1²/σ_0², and 1/σ̃(k_)² = 1/σ_0² + (k_*² − k_c²)²/((1−γ²)σ_2²). (50)" [45] = "J. M. Bardeen, J. R. Bond, N. Kaiser, and A. S. Szalay, Astrophys. J. 304, 15 (1986)".

Monochromatic case (Sec. V A, Eqs. 73–78): "P(k) = σ_0² k_0 δ(k − k_0). (73) Then, the moments are calculated as σ_n² = σ_0² k_0^{2n}. (74) It leads to k_c = k_0 and γ = 1. Since k_c = k_0, from Eq. (21), the functional form of g(r; k_0) is given by g(r; k_0) = −ψ(r) = −sin(k_0 r)/(k_0 r). (75) Then, we can find r̄_m(k_0) = ℓ/k_0 = 2.74/k_0, μ_c = 0.520 and g_c = −0.141. … lim_{γ→1} P_1(ν, ξ_1) = (1/√(2π)) δ(ξ_1 − γν) exp(−½ν²) = (σ_2/(2√(2π) μk_)) δ(k_ − k_0) exp(−μ²/(2σ_0²)). (76) … n_pk^{(μ)}(μ)dμ = 3^{−3/2}(2π)^{−2} (1/σ_0) k_0³ f(μ/σ_0) exp(−μ²/(2σ_0²)) dμ. (77)"

3. Points the brief should not state without qualification #

• Ebrahimian et al. do not compare their β with Harada et al. 2016 / 0.05556σ^5, do not cite that paper, and make no statement about the inner density slope after the caustic; their post-caustic statement is that the central density decreases and collapse halts at b_min ≈ 0.4.
• Harada et al. 2016 formulate the criterion in Zel'dovich eigenvalues (α,β,γ) with the pancake eccentricity e — not in the (ellipticity e, prolateness p) parameterisation.
• The Harada 2024 review contains no velocity-dispersion formula and no explicit spin-suppression formula beyond ⟨a_*²⟩ ~ σ_H^{-1/2}.
• Harada et al. 2023 never use the word "floor"; the defensible paraphrase is that their δ̃_th is "a necessary condition for PBH formation" and "can be the dominant factor," but is "likely to be ineffective" for a sufficiently sharp spectrum.
• Mocz et al.'s (ħ/m)² convergence is measured in 2D; the 1D and 3D sections support it qualitatively, and the conclusions generalise it.
• Ye et al.'s 19× is β ≃ 1.08 σ̄_h^5 (σ̄_h = σ_h at the mean PBH mass, = 6σ_h*) vs 0.056σ_h^5, and reflects peak number density vs Doroshkevich "fraction of peaks", with the same hoop criterion h(λ⃗) ≲ σ_h.

Local text extractions used: /tmp/claude-0/-root-PBH/a34e6be4-0ee0-4598-bfd0-88626255d779/scratchpad/p1.txt … p8.txt (from the corresponding *.pdf in the same directory).