PBH collapse: oscillating-scalar EKG → Vlasov limit — minimum decisive experiment

Sep 22, 2026 · @Someone

1. Executive answer #

Decision: USE VLASOV INSTEAD. At 10^19–10^20 g the scalar has q = m/H_k ≥ 3×10^9, where it behaves as collisionless (Vlasov) matter to ≲ 10^-5. The decisive open number is the converged GR collisionless threshold, which no one has computed.

• Regime [DERIVED]: q ≈ 7.5×10^4 γ^-1 (m/GeV)(M/10^19 g). Any modulus gives q ≥ 3×10^9 and the inflaton 10^18–10^19. Affordable EKG, at q ≈ 25–200, is soliton-dominated and has no physical counterpart.
• Literature spread [FACT]: at e = 0.2, p = 0, Yoo et al. find collisionless collapse from μ = 0.05 (unconverged), the analytic hoop estimate is 0.45, and dust runs crash below 0.95. That spread is worth ~6×10^4 in β and ~80× in P_ζ. The particle transition also sits where a top-hat estimate says collapse outlasts their 250 t_H runs (μ ≈ 0.048), so it may be a deadline, not a threshold [DERIVED].
• New causal bound [DERIVED]: for the Yoo profile, the spherical collisionless threshold cannot exceed the LTB value, μ_V,sph ≤ 0.33. A spherical GR-Vlasov run is therefore a cheap test of the 0.05 result.
• Abundance [DERIVED]: f = 1 needs β ≈ 5.8×10^-10 (GeV/T_RH). If the spherical threshold is a floor of ≈ 0.3 (H3, now roughly even odds), P_ζ ≈ 1.5×10^-3–1.2×10^-2, only ~1–7× below radiation-era needs. Without a floor, σ^5 scaling gives 10^-5–4×10^-2.
• f_PBH = 1 [DERIVED]: COMPATIBLE for a heavy modulus (T_RH 4 MeV–10^5 GeV) with gradual decay and a peaked spectrum; TENSION for an inflaton era or sudden reheating; EXCLUDED for broad spectra. Induced GWs at 10^-5–3×10^-2 Hz test it.
• Initial data [DERIVED]: flat-slice CTTK puts a density bump into the decaying mode. A separate-universe release at q_i ≈ 1, or an exact kinetic seed (φ = 0, Π = (2ρ)^(1/2)) for later starts, fixes this, pending validation.
• Compute [DERIVED]: 3D EKG cost scales as ≈ q^4.2 C^(−3/2), giving 2×10^4–2×10^6 core-hours per threshold point at q = 100 once the time step holds energy drift below 10^-3. All spherical work fits one server in weeks.
• Audit corrections [FACT]: the direct-perturbation difficulty is real but sits in de Jong, Aurrekoetxea & Lim 2022 (footnote 1), not in their 2023 paper; the Yoo et al. particle threshold is single-resolution with flagged constraint violations; Milligan et al. 2025 do not state m/H and resolve only κ ≥ 0.06.

First actions, both cheap: spherical GR-Vlasov with a deadline sweep (≲ 10^3 core-hours), and the Yoo et al. particle setup extended to 500–1000 t_H (10^3–10^4 core-hours). Together they show whether the 0.05 result is numerical, deadline-set or real, and whether a floor exists. That moves the P_ζ needed for f = 1 by up to 10^3.

2. Physical regime: q = m/H_k #

For 10^19–10^20 g PBHs, every conventional post-inflationary cosmology puts q at 10^9 or above; an inflaton-dominated era gives q ≈ 10^18–10^19. Nothing physical lives at the q ≈ 10–200 that direct EKG can reach. [DERIVED]

Tags used throughout: [FACT] = stated in a paper I opened; [DERIVED] = my calculation; [HYPOTHESIS] = working hypothesis; [SPECULATION] = unsupported guess.

Model-independent scaling [DERIVED] #

The last identity matters most: the scalar Compton wavelength divided by the Schwarzschild radius of the forming hole is 1/(γq). At q ≥ 10^9 the horizon-scale wave parameter is ≤ 10^-9.

γ is not a constant in a matter era. The apparent-horizon mass at formation can be ~10^-2 of M_H(t_k), followed by rapid accretion (de Jong thesis, [FACT]). I keep γ ∈ [0.01, 1]; q scales as 1/γ, so a small γ only raises q.

Timing and reheating window [DERIVED] #

In Yoo et al. 2026 the particle runs form horizons at t ≈ 100–250 t_H for μ ≈ 0.05 [FACT]. Linear growth δ ∝ a gives t_coll ∝ μ^(-3/2).

Representative q [DERIVED; model assumptions in column 1] #

So q ≈ 10^9 is not an assumption but the floor for the modulus case. The inflaton case sits nine decades higher.

A regime issue the proposal has not priced in [DERIVED] #

A long scalar era fragments the background before t_k. Sub-horizon modes grow ∝ a whenever k/a is below the Jeans scale 6^(1/4)(Hm)^(1/2), which exceeds H by q^(1/2). With P_ζ ≈ 2×10^-9 on small scales, those modes go nonlinear after ≈ 11 e-folds of matter domination.

The inflaton case needs ~27 e-folds before t_k, so the medium at t_k is a gas of inflaton halos and solitons, not a smooth condensate. Schrödinger–Poisson work on post-inflationary structure (Niemeyer & Easther 2020; Eggemeier et al. 2021, as cited in Milligan et al. 2025) shows exactly these objects form. [FACT for existence; DERIVED for timing]

A smooth oscillating background at t_k requires the scalar to dominate within ≲ 10 e-folds of t_k, or strongly suppressed small-scale power. At the PBH scale, a halo gas is a collisionless medium by construction. [HYPOTHESIS]

3. Expected effective theory: dust vs Vlasov vs EKG #

The q → ∞ limit of a classical real scalar should be collisionless (Einstein–Vlasov) matter with cold initial data, not single-stream dust — but only for coarse-grained, metric-level observables. This is demonstrated for Newtonian gravity and plausible, not proven, for strong-field GR collapse. [FACT for Newtonian; HYPOTHESIS for GR]

Which description holds where #

Dimensionless parameters that control the correspondence [DERIVED] #

m/H alone is not the control parameter. The collapse manufactures its own short scales, so the local ratios below are what matter.

The M_max coefficient (Seidel–Suen) is from memory, not re-verified here. The oscillaton row is the most consequential for design: at simulated q ≈ 10–30 with γ ≈ 0.1, γq ≈ 1–3, so a single soliton can hold much of the collapsing mass. That regime has no physical counterpart at q ≥ 10^9.

What is actually established #

• [FACT] Mocz et al. 2018: in 1D/2D/3D Newtonian tests, the Schrödinger–Poisson density keeps order-unity interference oscillations as ħ/m → 0, while the potential converges to the Vlasov answer as (ħ/m)². Dynamics coupled to the potential recover the collisionless limit.
• [FROM MEMORY, not re-opened] Rigorous results: Lions & Paul (1993, Wigner measures, weak limits); Zhang, Zheng & Mauser (2002, 1D pure states, global limit); Golse–Paul and Lafleche (quantitative, mixed states, conditions on the interaction). None covers 3D pure states through generic caustics, and none covers GR.
• [FROM MEMORY] Relativistic high-frequency limits give Vlasov-type effective stress tensors (Burnett's conjecture; Huneau–Luk under symmetry). Nothing covers massive-scalar strong-field collapse.
• [FACT] Milligan et al. 2025: spherical EKG with φ² departs from dust — central flattening, a soliton core matching the Schive profile to 2%, an NFW-like envelope. Every amplitude they could evolve (κ = 0.06–0.2) still formed a horizon. They bound C_th ≤ 0.2, δ_th ≤ 0.077, and could not go lower: constraint control failed even at N = 10,000. The m/H they used is not stated in the text I read.
• [FACT] Ebrahimian, Abolhasani & Mirbabayi 2025/26: Newtonian N-body and shell runs show shell crossing and velocity dispersion halt collapse. Typical (non-flat) peaks need δ_m = O(1). Their effective threshold for realistic abundances is δ_th ~ ζ_rms^(1/10).

Answer to the Part II questions #

The macroscopic strong-collapse dynamics should be reproduced by Einstein–Vlasov with cold initial data plus the physical small-scale velocity dispersion. The conditions: λ_dB/L ≪ 1 on every dynamically relevant scale including caustic widths; γq ≫ 1; metric-level observables only (horizon, mass), never pointwise density. At q ≥ 10^9 all hold by nine orders of magnitude. [DERIVED]

Failure routes, and why each is small at physical q:

• Local de Broglie length: 2π/(γqv) ≈ 10^-9 at r_s; failure needs near-critical fine-tuning.
• Dynamically generated gradients: caustic structure enters at order (λ_dB/L)^(2/3) at most. [HYPOTHESIS]
• Interference: order-unity in density, averaged out in the metric; a hazard only for max-density diagnostics.
• Oscillatons: cannot hold M ≫ M_max; only a core forms, as Milligan et al. saw.
• Fragmentation: real and physical (Section 2), but it is initial small-scale power. In the Vlasov limit it becomes velocity dispersion — the quantity Harada et al. 2023 and Ebrahimian et al. show controls the threshold. [HYPOTHESIS]
• Strong-field wave effects: 1/(γq), negligible.

The open physics sits inside the Vlasov problem (threshold and velocity dispersion), not in the EKG → Vlasov map.

4. Initial-data problem: can the dominant scalar be perturbed directly? #

Yes. The obstruction is a poor initial-data choice, not a fundamental one. But at the q that 3D EKG can afford, a real technical tension remains — the field cannot be both oscillating and cleanly super-horizon on the initial slice — and it must be solved and validated in spherical symmetry before any 3D run. [DERIVED]

What de Jong et al. actually did [FACT] #

In de Jong et al. 2023, the massive field is homogeneous and at rest (φ_0 = 7.8×10^-3 m_Pl, m ≈ 62 H_0). The perturbation is a Gaussian shell of a separate massless field ξ (R_0 = 1.2–1.5 H_0^-1, width 0.15 H_0^-1, amplitude 0.0825–0.09 m_Pl). Constraints are solved with CTTK on a flat conformal metric, solving for K and A_ij.

That paper contains no such statement. The warning is in their earlier paper, de Jong, Aurrekoetxea & Lim, JCAP 03 (2022) 029, footnote 1: large perturbations of the massive scalar injected enough potential energy to make the initial evolution non-matter-dominated [FACT; corrected 22 September]. It diagnoses that initialization, not an obstruction. Note also that their collapse is driven by the massless shell's own wave dynamics, so their thresholds are not growing-mode thresholds.

Diagnosis: flat-slice CTTK gives a decaying mode [DERIVED] #

With γ_ij = δ_ij and K from the Hamiltonian constraint, an overdense patch expands at its local Friedmann rate, H_loc² = 8πGρ_loc/3. At leading order in gradients that patch is the flat background at an earlier time: a local big-bang time shift, which is the decaying mode (the LTB t_B(r) freedom that Yoo et al. also classify as decaying).

Linear check: this data has δH/H = +δ/2, while the pure growing mode has δH/H = −δ/3. Decomposing gives δ_+ = 0. So a density bump in the dominant matter on a flat slice does not grow — a plausible reason the massless-field workaround was used.

Candidate growing-mode construction [DERIVED] #

1. The adiabatic growing mode is a separate-universe time shift. On uniform-density slices, at leading gradient order, the scalar is homogeneous (φ = φ̄(t_i), Π = Π̄(t_i)) and all perturbation lives in the metric, γ_ij = a²Ψ⁴δ_ij with Ψ = e^(ζ/2). Use exactly the Yoo et al. ζ profile with parameters (μ, e, p).
2. At next order, take the constant-mean-curvature long-wavelength data of Yoo et al. (K = −3H; ψ, γ̃_ij, Ã_ij from the functions q(x), p_ij(x)) and solve the constraints for E and J_i, as they did for dust.
3. Realize E and J_i with the scalar in WKB form φ = A(x) cos Θ(x): A = (2E)^(1/2)/m, and the cycle-averaged momentum density −(mA²/2)∂_iθ fixes the phase gradient as m × velocity potential.
4. The instantaneous J_i also carries a piece oscillating at 2m, suppressed by the Jeans ratio (k/a)²/(Hm). Absorb it by re-solving the momentum constraint with the actual scalar source; GRTresna supports scalar sources on periodic cosmological grids with CTTK and CTTK-Hybrid [FACT].
5. CMC and uniform-density slices differ at O(ε²), which becomes a phase shift ≈ q_i ε_i² (ε = k/aH). Keep cos Θ exact; never linearize the phase.

The tension at accessible q [DERIVED] #

In a matter era q_i = q_k (ε_i/ε_k)³, with ε_k = √6 for the Yoo profile. Yoo et al. start at ε_i = 0.2, which gives q_i = q_k/1840: at q_k = 100 the field is not yet oscillating (q_i ≈ 0.05). Demanding q_i ≥ 10 at that start needs q_k ≥ 1.8×10^4.

Requiring both an oscillating field and small phase corrections gives the window q_k^(-1/3) ≪ ε_i/ε_k ≪ (6q_k)^(-1/5). It is empty at q_k = 100 (0.22 vs 0.30) and only marginal at 10^4.

Resolution: start before the oscillations [DERIVED; must be validated] #

The separate-universe mode is exact at leading gradient order whatever the background equation of state, and ζ is conserved outside the horizon. So release the field at q_i ≈ 1 (de Jong et al. released theirs from rest at q_i = 62) while the peak is still super-horizon. Oscillations switch on within about one Hubble time. For later starts (q_i ≳ 10) an exact seed exists: on a comoving slice with J_i = 0, set φ = 0 and Π = (2ρ)^(1/2), keeping the dust metric and extrinsic curvature. Then E = ρ and J_i = 0, so both constraints hold exactly [DERIVED; from the external review, checked]. It is constraint-consistent but not yet the finite-q growing mode, so gates 2 and 5 below still decide.

The price is about one e-fold of non-dust background before entry; harmless for a super-horizon mode, but it must be checked.

Validation gates before any 3D run #

• [ ] Homogeneous limit: FLRW with the expected oscillating H, ΔH/H ≈ 1/q.
• [ ] Linear regime: fit δ(t) to a + a^(-3/2); decaying fraction below 1% at entry.
• [ ] Pre-shell-crossing agreement with the exact LTB solution built from the same Ψ(r) (Yoo et al. Sec. IV), with residuals scaling as 1/q².
• [ ] Hamiltonian and momentum constraints converge at three resolutions.
• [ ] μ_th unchanged, within the error budget, when q_i ∈ {0.5, 1, 2} and the initial oscillation phase are varied.

If gate 2 or gate 5 fails at the 1% level, 3D simulations should not proceed. A validated construction is publishable on its own as a short technical paper. [SPECULATION]

5. Stage 1: spherical convergence experiment #

Stage 1 measures μ_th in spherical symmetry for three matter models on one profile, and asks whether EKG converges to Vlasov, to dust, or to neither. It fits on a workstation; the binding risk is long-time constraint control, not flops (Milligan et al. failed there at N = 10,000). [DERIVED]

Fixed physics #

• Profile: the Yoo et al. spherical case, ln Ψ = (μ/2) exp(−k²r²/6), so ζ has peak μ and r_m = √6/k. Its exact LTB answer is known: locally naked (PBH) at μ = 0.33, globally naked at μ = 0.32 [FACT].
• Potential V = m²φ²/2. No reason to deviate: it is the case Milligan et al. studied and the one with a clean dust/Vlasov limit.
• Deadline: primary t_dead = 300 t_H (t_H = horizon-entry time; Yoo's particle runs needed up to 250 t_H). Report μ_th also at 100 and 1000 t_H to measure dμ_th/d ln t_dead, which later maps onto the reheating time. Compare each value with the top-hat deadline curve 1.91 D^(−2/3) of Section 7: a threshold that tracks it is delayed collapse, one that stays flat is a real floor.

Matter models #

1. A — Dust/LTB, semi-analytic (Yoo eqs. 4.10 and 4.12 map Ψ(r) to m(r) and k(r)) plus an NR dust run for code validation.
2. B — Spherical Einstein–Vlasov, cold start. Shells from the same LTB data, each given a small angular momentum L_reg so they pass the centre regularly. Converge in shell number (10^4 → 10^5) and in L_reg → 0. Validate against published Einstein–Vlasov critical-collapse results (Rein–Rendall–Schaeffer 1998; Olabarrieta–Choptuik 2002; Akbarian–Choptuik 2014 — cited from memory).
3. C — Spherical EKG with the Section 4 initial data. Run two independent codes: a Misner–Sharp scalar code (Milligan-type) and GRChombo in cartoon mode (the modified-cartoon scalar extension in de Jong's thesis), so the 1D numerics are the same as any later 3D numerics.

q ladder #

The 10–160 ladder in the brief is not adopted: q ≤ 20 is soliton-dominated with no physical counterpart, and q ≤ 50 cannot start from clean growing-mode data (Section 4).

Observables per run #

Primary: outcome class and, for collapse, apparent-horizon time t_AH and mass M_AH (area-based).

Secondary: max compaction C_max(t); central density; local shortest wavelength λ_min(t) and points per wavelength; gradient-energy fraction; kinetic energy and velocity dispersion (Vlasov); soliton-core radius and mass against the Schive profile and oscillaton mass limit; outgoing energy flux at 3 r_m; Hamiltonian and momentum constraint norms (L2 and max, normalized by 16πGρ).

Classification rule #

• COLLAPSE: an apparent horizon (Θ+ = 0, 2M/R ≥ 1) before t_dead; constraints under tolerance at that time; the horizon persists for ≥ 0.5 t_AH or is stably excised; same verdict at two resolutions.
• RESOLVED NON-COLLAPSE: no horizon by t_dead; C_max has peaked and fallen below 0.9 of its peak; bounce or virialization identified (virial ratio within 10% of 1, or a quasi-stationary soliton+envelope); constraints under tolerance throughout; same verdict at two resolutions.
• UNRESOLVED: everything else, including crashes, tolerance violations and resolution-dependent verdicts. Never counted as non-collapse.

Bracketing and fits #

Bisect μ to a relative bracket of 1% at each of three resolutions, then Richardson-extrapolate the bracket midpoint. Fit μ_th(q) with α free in μ_∞ + A q^(−α), and compare against (γq)^(−α), a log-corrected form, and a two-regime piecewise model by AIC/BIC. Require at least four q points with γq ≥ 10 in any accepted fit; do not assume the functional form.

Deliverables and pass criteria #

• μ_th for LTB, Vlasov and EKG(q), each with a full error budget (Section 9).
• The Vlasov-vs-LTB gap. This is itself new: I found no GR spherical collisionless threshold for this profile.
• The EKG convergence exponent α and asymptote μ_∞, compared with the Vlasov number.
• Pass: |μ_∞ − μ_Vlasov| below 2σ, and |μ_th(q = 400) − μ_Vlasov| below 1%. Then EKG at q ≥ 10^9 is Vlasov to far better than any other uncertainty, and 3D should be done with Vlasov (Section 13).

6. Stage 2: minimum novel 3D experiment #

One triaxial profile, e = 0.2 and p = 0, varying only μ and q. The choice is verified: it is exactly the configuration of the Yoo et al. particle runs [FACT]. But their particle threshold is unconverged, so Stage 2 is only meaningful with a matched, converged Vlasov benchmark run alongside. Without it there is nothing to converge to. [DERIVED]

Benchmarks at e = 0.2, p = 0 #

The analytic value uses E(k²=0.75) ≈ 1.211 in ĥ(e,0) = (15/π) e E/(1+3e)². For comparison, at e = 0.1 the analytic value is 0.34 and the dust runs first show a horizon at μ = 0.675 [FACT].

EKG runs #

• Code: GRChombo, CCZ4, Yoo-type cosmological slicing ((∂t − β^i∂i)α = −2α(K + 3H) plus their early e-fold term) or an equivalent tested gauge.
• Symmetry: the profile is even in X, Y and Z, so evolve one octant with reflection boundaries — an 8× saving. Octant symmetry suppresses symmetry-breaking fragmentation, so repeat one near-threshold pair on the full domain.
• Domain: L = 10/k as in Yoo et al., plus a domain test at 25/k with coarse outer levels; the comoving light-travel distance from entry to 250 t_H is ≈ 10.6 r_m ≈ 26/k.
• Profile: ln Ψ = (μ/2) exp(−k²r²/6)[1 + (k²/6)(p(2X²−Y²−Z²) + 3e(Y²−Z²))] with e = 0.2, p = 0.
• Initial data: Section 4 construction, released at q_i ≈ 1.
• q: 100 and 200. Add 400 only if the 100 → 200 shift exceeds 3σ.
• μ: bisect from [0.03, 1.0], which spans the particle and dust results, to a 2% relative bracket.
• AMR: refine where Δx > λ_loc/12 (local scalar wavelength), and on gradients of K and the conformal factor. Report the minimum points per wavelength over the whole run.
• Resolutions: octant base grids 64³, 96³ and 128³ with identical refinement criteria.

Matched Vlasov runs (required, same geometry) #

• Same curvature profile; E and J_i from the constraints as in Yoo et al.; cold particles.
• GR-PIC with 8, 27 and 64 particles per cell, a higher-order deposition kernel, and three grids.
• Acceptance test: the localized constraint violation at stream crossings must converge away with resolution and particle number. If it does not, reject the method and use the EKG code at the highest affordable q as a wave-regularized Vlasov solver instead. First, rerun the Yoo et al. particle setup at μ = 0.03–0.045 to 500 and 1000 t_H: Section 7 puts the 250 t_H deadline boundary at μ ≈ 0.048.

Diagnostics #

Outcome class (Section 5 rule); t_AH; M_AH and spin (should be ≈ 0 here); maximum density; shear and anisotropy from Ã_ij and the inertia tensor; λ_min(t); gradient-energy fraction; core/soliton detection; outgoing energy flux; quadrupole GW proxy; Hamiltonian and momentum constraints outside the horizon.

Output #

μ_th(q; 0.2, 0) ± σ at q = 100 and 200, and μ_th^Vlasov(0.2, 0) ± σ, each compared with the three benchmarks above. If only one number can be funded, fund the converged Vlasov threshold: it alone decides whether matter-era PBH formation is easier or harder than the analytic estimate.

7. Expected result: pre-registered hypotheses #

Recorded before any simulation. Each row gives the number I expect, the band that still counts as expected, and the basis. None of these values may enter the analysis (blinding rules at the end of this section).

Spin is exactly zero by the octant reflection symmetry of the e = 0.2, p = 0 profile. That is a code check, not a prediction.

Why H3 is a bound, not a guess [DERIVED] #

For the Yoo profile the enclosed density decreases outward, so in LTB no off-centre shell crossing precedes the first central focusing event p. Cold Vlasov data evolve exactly as dust everywhere outside the causal future of p.

• If μ ≥ 0.33 the singularity is not globally naked [FACT, Yoo et al.], so the outermost outgoing null ray from p is eventually trapped. Every shell that ray has not yet reached is already trapped when the ray arrives. Those trapped spheres therefore form outside the causal future of p and exist identically in the Vlasov solution. Cold Vlasov forms a horizon, so μ_V,sph ≤ 0.33.
• If μ ≤ 0.32 the first crossing precedes every trapped sphere, and the outcome is genuinely post-crossing dynamics.

Assumptions to check in the LTB solution: no off-centre crossing before trapping, and the L_reg → 0 limit.

This is exactly the assumption behind the inhomogeneity suppression of Kokubu et al. 2018: a black hole forms if the apparent horizon keeps information from the central region from leaking out [FACT]. Stage 1 model B tests that assumption directly.

A cheap test of the most consequential number in the literature #

If H3 holds and anisotropy only hinders collapse, then μ_V(0.2, 0) ≥ μ_V,sph ≥ 0.2, and the Yoo et al. particle threshold of 0.05 must be a numerical artifact. A spherical GR-Vlasov run on a workstation therefore tests it before any 3D run.

The monotonicity assumption is not guaranteed. Collisionless spindle collapse can end without a horizon (Shapiro–Teukolsky 1991, cited by Yoo et al.), which points the same way but is not a proof.

Deadline check on the particle result [added 22 September, before any run] #

The published particle transition may be a deadline, not a threshold. A spherical top-hat estimate gives the amplitude that collapses by D = t/t_k. For the Yoo profile, the linear density contrast averaged inside r_m = √6/k at entry is (12/5e) μ ≈ 0.883 μ; with δ ∝ t^(2/3) and δ_sc = 1.686:

That is 0.048 at D = 250, 0.030 at D = 500 and 0.019 at D = 1000 [DERIVED; raised by the external review, recomputed here]. Yoo et al. evolve their particles to about 250 t_H and place the transition between 0.045 and 0.05 [FACT]. The match is a coincidence until tested: anisotropy, nonlinearity and dispersion all shift collapse times.

The deadline sweep in V1 (H8) separates two readings:

• Delayed collapse: μ_th(D) tracks 1.91 D^(−2/3), falling 4.6× from 100 to 1000 t_H. No floor exists, F4 applies, and the σ^5 rows of Section 11 hold, because a physical matter era lasts D_RH ≈ H_k/Γ ≈ 10^3–10^18 entry times.
• Permanent support: μ_th(D) stays flat within 3% (H8). The floor rows apply, and the particle horizons at 0.05 must be numerical.

One more data point leans toward the first reading. In de Jong, Aurrekoetxea & Lim 2022, black holes still form by accretion of the ambient scalar when the initial shell fails the hoop test [FACT; massless-field trigger, so not a growing-mode case].

Revised credences: H4's claim that the 0.045–0.05 transition will not survive testing drops from ≈ 75% to ≈ 50%; H3 and H8 move to roughly even odds. [HYPOTHESIS]

Blinding rules #

• Fits keep exponents and asymptotes free; no value in this table enters any likelihood.
• Bisection starts from the full benchmark range, never from these predictions.
• Outcome classes come from the automated Section 5 rule; every manual reclassification is logged with its reason.

8. Falsifying outcomes #

Two outcomes would reverse the decision: spherical EKG failing to converge to Vlasov, or converging to dust instead. Ten others change priorities or the abundance, not the choice of Vlasov. Each row gives the observable, the level that counts as observed, and what follows. [DERIVED unless tagged]

F1 and F2 are the only rows that overturn Section 13. F3 halts work until fixed; the rest re-weight Sections 10–12.

9. Convergence tests and μ_th error budget #

Target accuracy is ≈ 0.8% on a spherical threshold and ≈ 3% on a 3D one. No single component may exceed half the total variance. Failed endpoints are reported as bounds, never as values. [DERIVED; component sizes are targets to be confirmed by the tests]

Tests every result must pass #

• Resolution: three resolutions per point (spherical N, 2N, 4N; 3D octant base grids 64³, 96³, 128³). The observed order must lie within ±1 of the scheme order, using the grid convergence index for non-uniform refinement ratios (Roache; from memory).
• Time step: m Δt ≤ 0.1 (≈ 63 samples per oscillation) for q ≤ 200 and ≤ 0.07 at q = 400, plus Courant factor 0.25 and 0.125. RK4 damps an oscillation by ≈ (ωΔt)^6/144 per step, a 4% amplitude loss over 10^5 steps at m Δt = 0.2. Require cumulative scalar-energy drift below 10^-3 in the homogeneous test, or use a higher-order integrator. [DERIVED; revised 22 September from 0.2/m]
• AMR: refinement at λ_loc/12, repeated at λ_loc/16 for one near-threshold pair.
• Domain: L = 10/k against 25/k in 3D with coarse outer levels, since light travels ≈ 26/k from entry to 250 t_H; spherical outer boundary moved out by 1.5–2×.
• Symmetry: octant runs suppress symmetry-breaking fragmentation, so one near-threshold pair is repeated on the full domain.
• Constraints: L2 and max norms of the Hamiltonian and momentum constraints, normalized by 16πGρ. Below 10^-3 (spherical) and 10^-2 (3D) outside the horizon. Violations at stream crossings must shrink with resolution at the expected order.
• Horizon finder: tolerance 10^-10 (spherical) and 10^-8 (3D); a search at least every 0.05 t_AH; the horizon must persist for 0.5 t_AH or more.
• Wave-resolution monitor: λ_min(t) = 2π divided by the largest phase-gradient magnitude on the grid. Target 12 or more points per λ_min; fewer than 8 anywhere outside the horizon makes the run UNRESOLVED.
• Fits: the highest q is withheld from each fit and used as an out-of-sample check.

Budget per threshold (1σ, relative) #

The largest single share is the spherical bisection bracket, at 38% of the variance. Deadline dependence is reported separately at 100, 300 and 1000 t_H: it defines the question rather than adding error.

When a gap counts as physical #

An EKG–Vlasov difference is a physical effect only if all four hold:

1. It exceeds 3σ at every resolution.
2. Its Richardson-extrapolated value lies within 1σ of the finest-grid value.
3. It survives the initial-data variations in the table.
4. Both spherical EKG codes reproduce it.

The Section 5 pass test compares two numbers of ≈ 0.8% each, so their difference carries ≈ 1.1%. The 1% criterion at q = 400 is therefore a point-estimate check; significance comes from the 2σ test on μ_∞.

An UNRESOLVED endpoint brackets nothing. If one side of a bisection is UNRESOLVED, the result is a one-sided bound.

10. Compute requirements #

The spherical work fits on one rented 32–64-core server in 2–4 weeks, at up to ≈ 1.1×10^4 core-hours in total. One converged 3D EKG threshold point at q = 100 costs 2×10^4–2×10^6 core-hours, and 3D EKG cost grows as ≈ q^4.2. Physical q ≥ 10^9 is 28 orders of magnitude out of reach, so it has to be done as Vlasov. [DERIVED; throughput is an assumption; revised 22 September for the Section 9 time step]

Summary by package #

Spherical EKG per run, single core, at m Δt = 0.1: under 1 hour at q ≤ 100; 0.2–6 hours at q = 200–400; 1–20 hours at q = 800; 4–70 hours at q = 1600. Runs are independent, so they parallelize trivially.

3D EKG cost model [DERIVED] #

• Δt ≈ 0.1/m on every level (≈ 0.07/m at q = 400), set by the oscillation and by RK4 amplitude error (Section 9), so AMR subcycling saves nothing.
• Steps to t_dead = 300 t_H = 200/H_k: 200 q/(m Δt), i.e. 2000 q at m Δt = 0.1.
• The pre-collapse halo is resolved at 12 points per local de Broglie wavelength: N_R ≈ 0.95 q C^(−1/2) points per halo radius, with C the halo compaction.
• Octant cells: ≈ 0.46 q³ C^(−3/2) in the halo plus ~10^6 base cells.
• Throughput 3–30 μs per cell per RK4 step per core for CCZ4 plus scalar [assumption: I found no published GRChombo core-hour figures].
• One threshold point ≈ 30 runs: bisection at mid resolution plus bracketing runs at the other two.

I take C ≈ 0.3 for compact thresholds (μ_th ≈ 0.4) and C ≈ 0.05 for dilute ones (μ_th ≈ 0.05). [HYPOTHESIS] A compact threshold with a short halo phase can cost up to ~10× less than listed.

Matched GR-PIC runs cost 300–3,000 core-hours each [assumption], so 10^4–10^5 per point.

Workstation or rented server versus HPC #

• One dedicated server with 32–64 cores and ≥ 128 GB RAM runs package A and the compact case of B. At 64 cores, 10^5 core-hours is about 9 weeks.
• Anything at q ≥ 200, or a dilute threshold at q = 100, needs an HPC allocation: in Austria the national VSC, otherwise a EuroHPC access call.
• Storage is dominated by checkpoints: 0.1–2 TB per point at q = 100 and 1–20 TB at q = 200.

Multiscale options for q ~ 10^6–10^9 #

• Brute force: going from q = 100 to q = 10^9 multiplies the cost by 10^28. No AMR scheme closes that.
• Schrödinger–Poisson or WKB phase methods remove the fast oscillation but assume weak fields and a single stream, so they fail near horizon formation.
• The scale separation is itself the multiscale method. At q ≥ 10^6, λ_dB/r_s ≤ 10^-6, so Vlasov is exact to that order. Use spherical shells and GR-PIC; if PIC crossings do not converge, use phase-space-sheet methods (Abel, Hahn & Kaehler 2012; Hahn & Angulo 2016; from memory) or EKG at q = 100–200 as a wave-regularized Vlasov solver.
• EKG at q = 100–400 then only measures the size of wave corrections, which extrapolate as (γq)^(−α) to physical q.

11. Abundance mapping: from P_ζ to f_PBH #

f_PBH = 1 at 10^19–10^20 g needs a formation fraction β ≈ 5.8×10^-10 (1 GeV/T_RH). At fixed T_RH, the collapse model moves the required P_ζ by 10^2–10^4, more than every other uncertainty combined. If the spherical collisionless threshold is a floor of ≈ 0.3 (H3), f = 1 needs P_ζ ≈ 1.5×10^-3–1.2×10^-2. [DERIVED]

The chain #

flowchart LR
  A[P_ζ at the PBH scale] --> B[Peak height and shape]
  B --> C[Collapse test against μ_th]
  C --> D[Formation fraction β]
  D --> E[Reheating at T_RH]
  E --> F[f_PBH today]

1. Spectrum to peaks. For a peaked spectrum the ζ peak height is μ = ν σ_ζ with σ_ζ ≈ √P_ζ. The density contrast at entry is σ_H ≈ 0.4 √P_ζ, since δ = (2/5)(k/aH)² ζ in a matter era. Shapes (e, p) follow the BBKS peak distribution at given ν [FROM MEMORY].
2. Peaks to collapse. A peak collapses if μ > μ_th(e, p; t_dead). This is the only step that needs simulation.
3. Collapse to β. β is the mass fraction in PBHs when the matter era ends, including accretion after horizon formation (H6: M_AH grows from ~10^-2 to ≥ 0.3 M_H).
4. β to today. During matter domination ρ_PBH/ρ_φ stays constant, so there is no radiation-era growth factor ∝ a. After reheating ρ_PBH/s is conserved:

with s_0 = 2891 cm^-3 and Ω_c h² = 0.12 [FROM MEMORY]. Hawking evaporation is irrelevant at these masses.

5. Mass function. M = γ M_H(t_k) with M_H ∝ k^-3 in a matter era. Lighter PBHs enter earlier and get more of their own Hubble times before reheating, so μ_th(t_dead) falls toward low masses and rises near M_H(t_RH).

Why σ^5, and why β ∝ μ_th^-5 is not general [DERIVED] #

For dust with the hoop criterion, the threshold vanishes for a spherical peak and grows linearly with ellipticity near e = 0. This is the small-e limit of the Yoo et al. analytic threshold:

For high peaks the BBKS weight is e(e² − p²) de dp = e^4 (1 − t²) de dt. Collapse requires e < μ/S(t), which gives

The fifth power comes from the five powers of e in the measure. Integrated over peaks this is β_aniso ≈ 0.05556 σ_H^5 (Harada et al. 2016, via the Harada 2024 review) [FACT]. Ye et al. 2025 find ~19× more for a monochromatic spectrum [FACT]. The collapsing peaks have ν ≈ 3–4 and are unusually spherical.

β ∝ μ_th^-5 holds only for a uniform rescaling μ_th → λ e S(t), which gives β → λ^-5 β. It fails if the threshold has a floor at e = 0, changes shape in t, or is not linear in e. Simulations must therefore deliver μ_th along rays of fixed t at small e, not one number at e = 0.2.

Three regimes #

• (a) No floor: μ_th → 0 as e → 0, giving the σ^5 law. It assumes the matter era outlasts the collapse; the physical deadline is D_RH ≈ H_k/Γ ≈ 10^3–10^18 entry times. Kokubu et al. 2018 add a causal inhomogeneity factor 3.70 σ^(3/2), valid to σ ≈ 0.05 with inhomogeneity width Σ = 1 assumed, and propose the product β ≈ 0.2055 σ^(13/2) [FACT]. They call their rate a minimum, because information is assumed to travel at light speed.
• (b) Floor: a spherical threshold μ0 > 0 from post-crossing dynamics, velocity dispersion or wave pressure. β becomes a Gaussian tail, ~exp(−μ0²/2σ_ζ²), and collapsing peaks have ν ≈ 5–8. H3 puts this regime at μ0 ≈ 0.2–0.33. A floor must be deadline-independent: a threshold that falls as 1.91 D^(−2/3) is delayed collapse, which is regime (a) (Section 7).
• (c) Ebrahimian et al.: shell crossing and dispersion halt collapse unless δ_m = O(1), with δ_th ~ ζ_rms^(1/10) [FACT, Newtonian]. This is (b) with a higher floor.

P_ζ required for f_PBH = 1 [DERIVED] #

• Yoo particle slope: the Ye column divided by 81, i.e. a small-e threshold 9× below the hoop value, as 0.05 against 0.45 at e = 0.2.
• Floor column: β = γ C ν³ exp(−ν²/2), with γ ∈ [0.1, 1] and peak-counting prefactor C ∈ [0.004, 0.02]; that range moves P_ζ by 15–30%. With μ0 = 0.20, the low end of H3, divide by 2.6.
• Floor plus anisotropy: if the hoop slope adds to the floor (μ_th = μ0 + e S), P_ζ rises 1.8–3.5×, to 1.2×10^-2 at 4 MeV, 4.3×10^-3 at 100 GeV and 2.9×10^-3 at 10^5 GeV. If the larger of the two applies (μ_th = max(μ0, e S)), it rises only 4–22%.

The required enhancement over the CMB-scale 2.1×10^-9 is therefore 10^2–10^7. The decisive inputs are μ_V,sph for the Yoo profile and two peak-theory mean profiles (is there a floor, and where), then the small-e slope along t = −1, 0, 1 (additive or max). Both are Vlasov computations (Section 13).

12. Is f_PBH = 1 at 10^19–10^20 g viable? #

The verdict depends on the scenario: COMPATIBLE for a heavy modulus with gradual decay and a peaked spectrum; TENSION for an inflaton-dominated era or sudden reheating; EXCLUDED for broad spectra or T_RH below BBN. No current observation constrains P_ζ at k ≈ 10^11–10^13 Mpc^-1, but LISA will. [DERIVED unless tagged]

The asteroid-mass window, 10^17–10^22 g, is treated as open in the Carr et al. 2026 review as cited by Yoo et al. [FACT, as cited]. Gorton & Green 2024 (arXiv:2403.03839) reach the same conclusion; I saw only the abstract. A 2025 star-survival limit from ultra-faint dwarfs (Esser et al.) is set aside: Gottlieb et al. 2026 model stars after PBH capture and find that slow, steady accretion lets the star survive, which undercuts the prompt-destruction assumption such limits rely on [search results; the paper's text was not machine-readable].

Verdicts #

Spectral shape #

From 10^19 g down to 10^15 g, β must fall by ~10^8 to respect evaporation bounds. That is 1.3 decades in k, since M ∝ k^-3. In the σ^5 regime P_ζ must fall ~10^3 over that range, i.e. faster than k^-2.4 on the small-scale side. In the floor regime a fall of ~2× suffices, because the tail is Gaussian.

The matter era also cuts the heavy side automatically: modes entering close to reheating run out of time before collapse (Section 11, step 5).

Induced gravitational waves #

For γ ∈ [0.01, 1] and T_RH from 4 MeV to 2×10^5 GeV, this spans ≈ 10^-5–3×10^-2 Hz. Most of that range lies in the LISA band; the lowest-T_RH cases fall below 10^-4 Hz, the target of μAres-type proposals [FROM MEMORY].

On radiation-era scaling, Ω_GW h² ≈ 10^-5 P_ζ² [FROM MEMORY]. The floor regime then gives ~10^-11–10^-9, above LISA's ~10^-12 sensitivity. The high-T_RH σ^5 case (P_ζ ~ 10^-5) gives ~10^-15, which is invisible. At known T_RH, LISA would therefore discriminate between collapse regimes, subject to the matter-to-radiation transition, which suppresses the signal if gradual and enhances it if sudden.

13. GO/NO-GO decision #

The open physics is the converged GR collisionless threshold, not the EKG → Vlasov map, so the program should run as Vlasov. EKG survives only as a spherical side-check that sizes wave corrections. The decision is the last line of this section.

Stop conditions from the brief (Part XII) #

The condition labels below are my paraphrases of the brief.

Conditions 2 and 5 decide it. Direct EKG at physical q is irrelevant, and the number worth computing, a converged GR Vlasov threshold, does not exist yet.

Program #

1. V0: rerun the Yoo et al. particle setup at μ = 0.03–0.045 to 500 and 1000 t_H, 10^3–10^4 core-hours. It shows whether the published 0.045–0.05 transition is numerical, deadline-set or real (Section 7).
2. V1: spherical GR-Vlasov. Yoo profile plus two peak-theory mean profiles; deadlines 100, 300 and 1000 t_H, each compared with the top-hat curve 1.91 D^(−2/3) to separate a floor from delayed collapse; a small velocity-dispersion scan. Workstation, ≲ 10^3 core-hours, weeks. Gate: μ_V,sph ≤ 0.33, the causal bound. This alone moves the f = 1 requirement by 10^2–10^3.
3. V1b: spherical EKG side-check at q = 50–1600 with two codes, 4×10^3–10^4 core-hours at the corrected time step. It measures α and tests H1, clearing Section 4 gates 2 and 5 along the way. A failure (F1 or F2) reverses this decision.
4. V2: converged 3D GR-PIC at e = 0.2, p = 0, on a large server or Tier-1, 10^4–10^5 core-hours. It must pass the Section 6 crossing test; otherwise fall back to EKG at q = 100 as wave-regularized Vlasov, 2×10^4–2×10^6 core-hours.
5. V3: small-e rays at e = 0.05 and 0.1 along t = −1, 0, 1, plus a velocity-dispersion floor, ~10^5 core-hours. It fixes the slope and the floor-plus-anisotropy combination that Section 11 needs.

Why not the other three options #

• GO TO STAGE 1, the external review's verdict: its Stage 1, a matched spherical EKG–Vlasov comparison with deadline tests, is V0 + V1 + V1b here, so the first months of work coincide. The disagreement is what counts as the result. The review targets the EKG → Vlasov boundary map, citing limits that may not commute (q → ∞, μ → μ_th, t_D → ∞). That risk is real but narrow: wave effects need γ_AH q ≲ 1, which at q ≥ 10^9 means near-critical holes below ~10^-9 M_H, a sliver of amplitude space with no weight in the abundance [SPECULATION]. F1 and F2 test it anyway.
• DO MORE ANALYTIC WORK: the causal bound, the σ^5-versus-floor analysis and the deadline estimate are done. The remaining numbers need simulation.
• STOP — QUESTION ALREADY ANSWERED: conditions 3 and 5 are unmet. No converged threshold exists.

USE VLASOV INSTEAD

14. Appendix: verification ledger and sources #

Every literature claim in this memo is listed with its status. VERIFIED means I read it in the source. NOT FOUND means I looked and it was absent. NOT CONVERGED means it is reported but flagged unconverged by its authors. FROM MEMORY means it was not re-checked in this session.

Added on 22 September, after the external review:

Cited but not opened: Harada et al. 2016 (ApJ 833, 61; arXiv:1609.01588); Harada et al. 2023 (JCAP 02, 038; arXiv:2211.13950); Padilla et al. 2026 (JCAP 04, 049; arXiv:2509.10431); Carr et al. 2026 (arXiv:2601.06024); Escrivà et al. 2026 (arXiv:2605.04487); Gorton & Green 2024 (arXiv:2403.03839); the Einstein–Vlasov critical-collapse papers in Section 5. From the external review, also not opened: Tanaka & Sasaki on gradient-expansion growing modes; Widrow's relativistic Vlasov limit; Salvi's Klein–Gordon–Maxwell limit; the revised Inomata et al. induced-GW papers; De la Torre Luque et al. (March 2026 erratum on evaporation limits).