# 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 \[DERIVED\]

```latex
M_H(t)=\frac{4\pi M_p^2}{H}=\frac{1}{2GH}\approx 1.33\times10^{14}\,\mathrm{g}\left(\frac{1\,\mathrm{GeV}}{H}\right)
```

```latex
M_{\rm PBH}=\gamma\,M_H(t_k)\;\Rightarrow\;H_k\approx1.33\times10^{-5}\,\gamma\,\mathrm{GeV}\left(\frac{10^{19}\,\mathrm{g}}{M_{\rm PBH}}\right)
```

```latex
q\equiv\frac{m}{H_k}\approx7.5\times10^{4}\,\gamma^{-1}\left(\frac{m}{1\,\mathrm{GeV}}\right)\left(\frac{M_{\rm PBH}}{10^{19}\,\mathrm{g}}\right),\qquad r_s\,m=2GM_{\rm PBH}\,m=\gamma\,q
```

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](https://arxiv.org/html/2403.02878v1), \[FACT\]). 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](https://arxiv.org/pdf/2609.14218v1) 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).

| Requirement | Condition | Value for 10^19–10^20 g, γ = 1 |
| --- | --- | --- |
| Matter era survives until collapse | Γ ≲ H\_coll ≈ H\_k/200 (μ ≈ 0.05) | Γ ≲ 7×10^-9 – 7×10^-8 GeV |
| Reheating temperature ceiling | T\_RH ≈ 0.54 (Γ M\_p)^1/2 | T\_RH ≲ 7×10^4 – 2×10^5 GeV |
| BBN floor | T\_RH ≳ few MeV | Γ ≳ 10^-23 GeV |
| Planck-suppressed decay, Γ \~ m³/M\_p² | BBN floor on Γ | m ≳ 30–100 TeV |

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

| Scenario (assumption) | m | q at 10^19 g | q at 10^20 g | Comment |
| --- | --- | --- | --- | --- |
| Inflaton, Starobinsky-like | \~3×10^13 GeV | \~2×10^18 | \~2×10^19 | needs \~27 e-folds of matter era before t\_k |
| Heavy modulus | 10^5–10^10 GeV | 10^10–10^15 | 10^11–10^16 | standard moduli range |
| Lightest gravitationally decaying modulus | \~35 TeV | \~3×10^9 | \~3×10^10 | set by BBN |
| Scalar with unsuppressed couplings | 1 GeV | 7.5×10^4 | 7.5×10^5 | exotic; must still dominate at H\_k |
| Numerically accessible EKG | — | 10–200 | — | de Jong et al. ran m ≈ 62 H\_0 |

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](https://arxiv.org/pdf/2504.02600)) 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

| Description | Valid while | Breaks at |
| --- | --- | --- |
| Dust (single stream) | before the first caustic | shell crossing; Yoo et al. dust runs crash unless the singularity is already inside a horizon \[FACT\] |
| Einstein–Vlasov, cold start | after shell crossing, if λ\_dB is far below every scale carrying the dynamics | cannot represent interference or soliton cores; caustics need regularization |
| Schrödinger–Poisson / nonrelativistic EFT | H ≪ m, k/(am) ≪ 1, weak field | near horizon formation |
| Full EKG | always (classical field; occupation numbers are enormous) | only numerically |

### 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.

| Parameter | Definition | Value at horizon entry | Value near horizon formation |
| --- | --- | --- | --- |
| Expansion/oscillation | H/m | 1/q | smaller still |
| Gradient/mass | k/(am) | (k/aH)/q ≈ 2.4/q | grows as the region compresses |
| Jeans ratio | (k/a)²/(Hm) | ≈ 6/q; linear-growth correction ≈ 6/q² | — |
| de Broglie ratio | λ\_dB/L = 2π/(m v L) | — | 2π/(γ q v) at L ≈ r\_s |
| Compton/curvature | 1/(m r\_s) | — | 1/(γq) |
| Caustic width (fold, Airy scaling) | (λ\_dB/L)^(2/3) | — | (2π/γqv)^(2/3) |
| Collapsing mass / max oscillaton mass | M/M\_max, M\_max ≈ 0.6 M\_Pl²/m | — | ≈ 0.8 γq |
| Oscillations before collapse | m t\_coll/2π | — | ≈ 20q for μ ≈ 0.05 |

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](https://arxiv.org/abs/1801.03507): 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](https://arxiv.org/pdf/2504.02600): 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](https://arxiv.org/pdf/2507.18312): 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](https://arxiv.org/html/2306.11810), 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.](https://arxiv.org/pdf/2609.14218v1) (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](https://arxiv.org/abs/2501.13046) 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.

| q\_k | ε\_i/ε\_k at q\_i = 1 | gradient parameter ε\_i² (r\_m-based) | e-folds from start to entry |
| --- | --- | --- | --- |
| 20 | 0.37 | 0.14 (marginal) | 2.0 |
| 50 | 0.27 | 0.07 | 2.6 |
| 100 | 0.22 | 0.05 | 3.1 |
| 200 | 0.17 | 0.03 | 3.5 |
| 1000 | 0.10 | 0.01 | 4.6 |

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

| q\_k | Role | Why this value |
| --- | --- | --- |
| 25 | soliton-regime diagnostic | γq ≈ 2–10: a single oscillaton can hold much of the mass; expected to deviate; excluded from the asymptotic fit |
| 50 | lower end of fit | initial data marginal (ε\_i² ≈ 0.07) |
| 100 | fit | clean initial data (ε\_i² ≈ 0.05) |
| 200, 400, 800 | fit | ≈ 1.1q–21q oscillations before collapse; minutes to hours per run |
| 1600 | asymptote check | ≈ 2 days per run single-core; include only if constraint control holds |

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

| Description | Threshold μ | Status |
| --- | --- | --- |
| Dust, 3D NR | runs crash before horizon formation for μ ≤ 0.95; the value is a sufficient condition only | \[FACT\] Yoo et al. 2026, 80³ grid |
| Analytic hoop + Zel'dovich | ĥ(0.2, 0) ≈ 0.45 (no inhomogeneity effect) | \[DERIVED\] from Yoo et al. eq. 5.15 |
| Collisionless particles, GR-PIC | horizon for μ ≥ 0.050; none for μ ≤ 0.045 | \[FACT\] single resolution (80³); large localized Hamiltonian-constraint violation at particle crossings, flagged by the authors |

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).

| ID | Quantity | Expectation | Still counts as expected | Basis |
| --- | --- | --- | --- | --- |
| H1 | Spherical EKG vs Vlasov: Δ(q) = μ\_EKG(q) − μ\_V,sph | Δ ∝ q^(−α) with α ≈ 1 | α ∈ \[2/3, 2\]; Δ within ±1% by q = 400 | α = 2/3 caustic width; α = 1 soliton-core mass fraction; α = 2 potential-level convergence as in Mocz et al. \[HYPOTHESIS\] |
| H2 | Small-q deviation at q = 25 | Δ of 5% or more; soliton core holding 1–5% of M\_H(t\_k) | either sign | core mass \~ C^(1/2)/(Gm), comparable to the first-trapped mass at q = 25 \[DERIVED scaling\] |
| H3 | Spherical Vlasov threshold μ\_V,sph, Yoo profile | ≈ 0.30 | uniform on \[0.20, 0.33\]; above 0.33 means a bug | causal bound below \[DERIVED\]; value below 0.32 set by post-crossing dynamics \[HYPOTHESIS\] |
| H4 | 3D Vlasov threshold μ\_V(0.2, 0) | above the hoop value 0.45, more likely than not | \[max(μ\_V,sph, 0.3), 0.95\] | anisotropy only hinders collapse; Yoo's 0.045–0.05 does not survive convergence testing (credence ≈ 75%) \[SPECULATION\] |
| H5 | 3D EKG vs Vlasov at e = 0.2 | Δ within ±2% at q = 200 | within ±5% at q = 100 | H1 carried over to 3D \[HYPOTHESIS\] |
| H6 | Horizon at formation | M\_AH ≈ 10^-2–10^-1 M\_H(t\_k), growing to 0.3 M\_H(t\_k) or more by t\_dead | factor 3 either way | de Jong thesis, \~10^-2 at formation \[FACT\]; growth \[HYPOTHESIS\] |
| H7 | Dust failure is an artifact | μ\_V(0.2, 0) < 0.95 | — | shell crossing, not absence of collapse, stops the dust runs \[HYPOTHESIS\] |
| H8 | Deadline dependence | μ\_V,sph shifts by less than 3% between t\_dead = 100 and 1000 t\_H | less than 5% | near μ ≈ 0.3, turnaround comes within a few t\_H of entry \[HYPOTHESIS\] |

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](https://arxiv.org/abs/1810.03490): 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:

```latex
\mu_{\rm deadline}(D)\simeq1.91\,D^{-2/3}
```

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. \[DERIVED unless tagged\]

| # | Outcome | Counts as observed when | Meaning | Consequence |
| --- | --- | --- | --- | --- |
| F1 | EKG does not approach Vlasov | fitted α < 0.3, or Δ(q) flat over q = 100–1600 at more than 3σ | wave effects survive q ≫ 1 in strong-field collapse | decision reverses: Vlasov is not the effective theory; 3D EKG at q = 100–200 becomes the only handle |
| F2 | EKG converges to LTB, not Vlasov | μ\_∞ within 1σ of 0.32–0.33 while μ\_V,sph differs by more than 3σ | the scalar stays single-stream through shell crossing | decision reverses: dust with wave regularization replaces Vlasov; Section 3 argument fails |
| F3 | Spherical Vlasov above the causal bound | μ\_V,sph > 0.33 at more than 3σ | contradicts the Section 7 bound | code bug; stop and debug before any 3D run |
| F4 | Spherical Vlasov far below LTB | μ\_V,sph ≤ 0.1 | collisionless matter collapses where LTB is globally naked; the Kokubu criterion is not physical here | no floor: abundance moves from the floor rows to the σ^5 rows of Section 11 |
| F5 | Yoo et al. particle threshold confirmed | converged μ\_V(0.2, 0) ≤ 0.06 | collapse far easier than the hoop estimate | β up \~6×10^4 at fixed σ if the small-e slope scales the same way; required P\_ζ down \~80× |
| F6 | Almost no collisionless collapse at e = 0.2 | no horizon for μ ≤ 0.95, or μ\_V(0.2, 0) ≥ 0.9 | anisotropy blocks collapse | matter-era advantage gone; f = 1 needs P\_ζ \~ 10^-2, as in radiation |
| F7 | Anisotropy helps collapse | μ\_V(0.2, 0) < μ\_V,sph by more than 3σ | spindle focusing survives in GR | the cheap spherical test of Section 7 no longer bounds 3D; V3 becomes mandatory |
| F8 | Soliton core persists | core above 10% of collapsing mass at q ≥ 400 (expected ≈ 0.3%) | core mass does not scale as C^(1/2)/(Gm) | the α ≈ 1 argument fails; extend the ladder to q = 1600 before concluding |
| F9 | EKG-only fragmentation | two or more collapsing clumps in EKG at q ≥ 200, absent in Vlasov, at two resolutions | a wave-specific instability | physical-q collapse may differ; model fragmentation explicitly |
| F10 | Horizon properties diverge | t\_AH or M\_AH differ by more than 30% between EKG (q ≥ 400) and Vlasov while μ\_th agrees | threshold is Vlasov-like, mass function is not | thresholds from Vlasov, γ from EKG; extra systematic in the mass function |
| F11 | GR-PIC fails its acceptance test | crossing-localized constraint violation does not shrink with grid and particle number | the particle method is unconverged in GR | fall back to EKG at q = 100 as a wave-regularized Vlasov solver, at 10–100× the cost |
| F12 | Strong deadline dependence | μ\_th falls roughly as D^(−2/3) (Section 7), or d ln μ\_th / d ln t\_dead exceeds 0.3 in magnitude | the reheating time, not geometry, sets the threshold | no permanent floor: the σ^5 rows of Section 11 apply at the physical deadline H\_k/Γ; tabulate μ\_th against t\_dead |

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)

| Component | Test that sizes it | Spherical | 3D |
| --- | --- | --- | --- |
| Bisection bracket | 1% (spherical) or 2% (3D) relative bracket | 0.5% | 1.0% |
| Spatial resolution | Richardson residual across three grids | 0.3% | 1.2% |
| Time step | m Δt 0.1 against 0.05; Courant 0.25 against 0.125 | 0.1% | 0.5% |
| AMR criterion | λ\_loc/12 against λ\_loc/16 | 0.2% | 1.0% |
| Domain and boundary | L = 10/k against 25/k; outer radius × 1.5–2 | 0.2% | 1.0% |
| Initial data | q\_i ∈ {0.5, 1, 2}; four initial phases; ε\_i × 1.5 | 0.3% | 1.0% |
| Gauge and damping | κ1 × 2 and ÷ 2; a second slicing condition | 0.2% | 1.0% |
| Horizon finder | tolerance and search cadence | 0.1% | 0.5% |
| Vlasov sampling | shells 10^4 → 10^5 and L\_reg → 0; 8, 27, 64 particles per cell | 0.3% | 1.2% |
| Total, in quadrature | — | ≈ 0.8% | ≈ 2.9% |

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

| Package | Content | Core-hours | RAM per run | Storage | Hardware | Wall-clock |
| --- | --- | --- | --- | --- | --- | --- |
| A | Stage 1 spherical: LTB, Vlasov, EKG at q = 25–1600 with two codes | 4×10^3–10^4 EKG + ≲ 10^3 Vlasov | < 1 GB | < 0.1 TB | one 32–64-core server or workstation | 2–4 weeks |
| B | one 3D EKG point, q = 100, e = 0.2; plus matched GR-PIC | EKG 2×10^4–2×10^6; PIC 10^4–10^5 | 5–50 GB | 0.1–2 TB | compact case: one large node; dilute case: HPC | weeks (HPC) to months (one node) |
| C | q scan {100, 200} at e = 0.2 | 2.6×10^5–3.5×10^7; ≈ 20× more with q = 400 | 30–400 GB | 1–20 TB | Tier-1 national or EuroHPC | months |
| D | e scan, five values at q = 100 | EKG 10^5–10^7; GR-PIC 5×10^4–5×10^5 | 5–50 GB | 0.5–10 TB | HPC for EKG; large server for PIC | months |

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.

```latex
\mathrm{cost_{run}}\approx\left[10^{6}+0.46\,q^{3}C^{-3/2}\right]\times\frac{200\,q}{m\Delta t}\times(3\text{--}30)\,\mu\mathrm{s}\;\propto\;q^{4.2}C^{-3/2}
```

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.

| q | C | Octant cells | Core-h per run | Core-h per point | RAM per run | Wall-clock per run |
| --- | --- | --- | --- | --- | --- | --- |
| 100 | 0.3 | 3.8×10^6 | 6×10^2–6×10^3 | 2×10^4–2×10^5 | \~5 GB | 5–50 h on 128 cores |
| 100 | 0.05 | 4.2×10^7 | 7×10^3–7×10^4 | 2×10^5–2×10^6 | \~50 GB | 14–140 h on 512 cores |
| 200 | 0.3 | 2.3×10^7 | 8×10^3–8×10^4 | 2.4×10^5–2.4×10^6 | \~30 GB | 16–160 h on 512 cores |
| 200 | 0.05 | 3.3×10^8 | 1.1×10^5–1.1×10^6 | 3.3×10^6–3.3×10^7 | \~400 GB | 2–22 days on 2048 cores |
| 400 | 0.3 | 1.8×10^8 | 1.7×10^5–1.7×10^6 | 5×10^6–5×10^7 | \~200 GB | not planned |
| 400 | 0.05 | 2.6×10^9 | 2.5×10^6–2.5×10^7 | 7.5×10^7–7.5×10^8 | \~3 TB | not planned |
| 1000 | 0.3 | 2.8×10^9 | ≥ 8×10^6 | ≥ 2×10^8 | \~3 TB | infeasible |

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

```mermaid
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:

```latex
f_{\rm PBH}=\frac{3}{4}\,\beta\,\frac{T_{\rm RH}\,s_0}{\rho_{\rm DM,0}}\approx1.7\times10^{9}\,\beta\left(\frac{T_{\rm RH}}{1\,\mathrm{GeV}}\right)
```

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:

```latex
\mu_{\rm th}(e,p)\approx e\,S(t),\qquad t=\frac{p}{e}\in[-1,1],\qquad S(t)=\frac{15}{\pi}\,E\!\left(\sqrt{1-\tfrac{(1+t)^2}{4}}\right)\in[4.8,\,7.5]
```

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

```latex
P(\mathrm{collapse}\mid\nu)\;\propto\;(\nu^{2}\sigma_\mu)^{5}\int_{-1}^{1}(1-t^{2})\,S(t)^{-5}\,dt
```

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](https://arxiv.org/pdf/2409.01934)) \[FACT\]. [Ye et al. 2025](https://arxiv.org/pdf/2508.10070) 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](https://arxiv.org/pdf/1810.03490) 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.](https://arxiv.org/pdf/2507.18312): 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 \[DERIVED\]

| T\_RH | β for f = 1 | Yoo particle slope | σ^5, Ye ×19 | σ^5, Harada | σ^(13/2), Kokubu | Floor μ0 = 0.32 |
| --- | --- | --- | --- | --- | --- | --- |
| 4 MeV | 1.5×10^-7 | 1.4×10^-4 | 1.1×10^-2 | 3.7×10^-2 | outside validity (σ > 0.05) | 3.0–4.0×10^-3 |
| 1 GeV | 5.8×10^-10 | 1.5×10^-5 | 1.2×10^-3 | 4.0×10^-3 | 1.5×10^-2 | 2.2–2.7×10^-3 |
| 100 GeV | 5.8×10^-12 | 2.4×10^-6 | 2.0×10^-4 | 6.4×10^-4 | 3.6×10^-3 | 1.8–2.2×10^-3 |
| 10^4 GeV | 5.8×10^-14 | 3.8×10^-7 | 3.1×10^-5 | 1.0×10^-4 | 8.6×10^-4 | 1.6–1.8×10^-3 |
| 10^5 GeV | 5.8×10^-15 | 1.5×10^-7 | 1.2×10^-5 | 4.0×10^-5 | 4.2×10^-4 | 1.5–1.7×10^-3 |

- 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](https://arxiv.org/pdf/2606.02700) 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

| Scenario | Requirement | Verdict | Main reason |
| --- | --- | --- | --- |
| Heavy modulus, m ≈ 50 TeV–5×10^9 GeV (T\_RH 4 MeV–10^5 GeV), gradual decay, peaked P\_ζ, no floor | P\_ζ ≈ 10^-5–4×10^-2 (Section 11) | COMPATIBLE | no bound on P\_ζ at these k; needs domination within \~10 e-folds of t\_k or suppressed small-scale power (Section 2) |
| Same, with the H3 floor | P\_ζ ≈ 1.5×10^-3–1.2×10^-2 | COMPATIBLE, testable | induced GWs of Ω\_GW h² \~ 10^-11–10^-9 in the LISA band on radiation-era scaling \[FROM MEMORY\] |
| Sudden reheating | P\_ζ ≳ 10^-3 | TENSION | the poltergeist enhancement of induced GWs at the transition can exceed the ΔN\_eff bound (Inomata et al., revised versions) \[FROM MEMORY\] |
| Inflaton-dominated era | \~27 e-folds of matter era before t\_k | TENSION | plateau models lose ΔN\_\* ≈ 8, moving n\_s from ≈ 0.965 to ≈ 0.959 against ACT DR6 0.974 ± 0.003 \[FROM MEMORY\]; the medium is already fragmented into halos |
| Broad, flat enhancement | flat P\_ζ over 10^15–10^20 g | EXCLUDED | overproduces evaporating 10^15–10^17 g PBHs; gamma-ray and e± bounds reach f ≲ 10^-8 near 10^15 g \[FROM MEMORY\]; use the March 2026 corrected limits of De la Torre Luque et al. (per the review; not opened) |
| Very late reheating | T\_RH below a few MeV | EXCLUDED | BBN |

### 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

```latex
f_{\rm GW}\approx2.6\times10^{-2}\,\mathrm{Hz}\left(\gamma\,\frac{10^{19}\,\mathrm{g}}{M_{\rm PBH}}\,\frac{T_{\rm RH}}{10^{5}\,\mathrm{GeV}}\right)^{1/3}
```

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.

| # | Condition | Status | Evidence |
| --- | --- | --- | --- |
| 1 | EKG → Vlasov correspondence already established | Partly met | Newtonian only (Mocz et al. 2018; rigorous 1D results); nothing in strong-field GR (Section 3) |
| 2 | Physical q so large that direct EKG is irrelevant | Met in substance | q ≥ 3×10^9 for any modulus, 10^18–10^19 for the inflaton; wave corrections at the horizon ≲ 10^-5 (Section 2) |
| 3 | μ\_th(e, p) for a directly perturbed oscillating scalar already published | Not met | the audit gap is confirmed: Yoo et al. 2026 use dust and particles, Milligan et al. 2025 are spherical, de Jong et al. 2023 perturb a massless field |
| 4 | Growing-mode initial data for the scalar impossible | Not met | separate-universe construction released at q\_i ≈ 1 (Section 4), pending its validation gates |
| 5 | Vlasov is the limit and has been computed in a controlled way | First half met, second not | Vlasov is the expected limit; the only GR collisionless threshold (0.05) is flagged unconverged by its authors |
| 6 | Abundance insensitive to the collapse physics | Not met | collapse physics moves the required P\_ζ by 10^2–10^4 at fixed T\_RH (Section 11), and that physics is Vlasov physics |

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.

| Claim | Status | Source |
| --- | --- | --- |
| Yoo et al. profile; entry at k/aH = √6; CMC start at H\_i = 5k | VERIFIED | [Yoo et al. 2026](https://arxiv.org/pdf/2609.14218v1) |
| Spherical LTB: PBH (locally naked) at μ = 0.33, globally naked at 0.32 | VERIFIED | [Yoo et al. 2026](https://arxiv.org/pdf/2609.14218v1) |
| Analytic threshold ĥ(e, p); ĥ(0.2, 0) ≈ 0.45 | VERIFIED formula; value DERIVED | [Yoo et al. 2026](https://arxiv.org/pdf/2609.14218v1) |
| 3D dust at e = 0.2 crashes without a horizon for μ ≤ 0.95 | VERIFIED | [Yoo et al. 2026](https://arxiv.org/pdf/2609.14218v1) |
| GR-PIC at e = 0.2: horizon for μ ≥ 0.050, none for μ ≤ 0.045 | NOT CONVERGED | [Yoo et al. 2026](https://arxiv.org/pdf/2609.14218v1): 80³ only; constraint violations flagged |
| Soliton core within 2% of Schive; all κ ∈ \[0.06, 0.2\] collapse; C\_th ≤ 0.2 | VERIFIED | [Milligan et al. 2025](https://arxiv.org/pdf/2504.02600) |
| m/H used by Milligan et al. | NOT FOUND in the text read | [Milligan et al. 2025](https://arxiv.org/pdf/2504.02600) |
| de Jong et al.: massive field at rest, m ≈ 62 H\_0, massless-shell trigger, CTTK | VERIFIED | [de Jong et al. 2023](https://arxiv.org/html/2306.11810) |
| de Jong et al. 2023: difficulty when the massive scalar is perturbed directly (the warning is in their 2022 paper, below) | NOT FOUND | [de Jong et al. 2023](https://arxiv.org/html/2306.11810) |
| Horizon mass at formation \~10^-2 M\_H, then accretion | VERIFIED | [de Jong thesis](https://arxiv.org/html/2403.02878v1) |
| Shell crossing and dispersion halt collapse; δ\_th \~ ζ\_rms^(1/10) | VERIFIED (Newtonian) | [Ebrahimian et al.](https://arxiv.org/pdf/2507.18312) |
| Schrödinger–Poisson potential converges to Vlasov as (ħ/m)² | VERIFIED (Newtonian) | [Mocz et al. 2018](https://arxiv.org/abs/1801.03507) |
| β\_aniso ≈ 0.05556 σ\_H^5 | VERIFIED | [Harada 2024 review](https://arxiv.org/pdf/2409.01934) |
| \~19× larger β for a monochromatic spectrum; small spin | VERIFIED | [Ye et al. 2025](https://arxiv.org/pdf/2508.10070) |
| β\_inhom ≈ 3.70 σ^(3/2) (Σ = 1, σ ≲ 0.05); product 0.2055 σ^(13/2); rate is a minimum | VERIFIED | [Kokubu et al. 2018](https://arxiv.org/pdf/1810.03490) |
| CTTK and CTTK-Hybrid with scalar matter on periodic grids | VERIFIED | [GRTresna](https://arxiv.org/abs/2501.13046) |
| GRChombo core-hour costs | NOT FOUND | Section 10 uses assumed throughput |
| Asteroid-mass window 10^17–10^22 g open | FROM MEMORY and search results | Carr et al. 2026 as cited by Yoo et al.; Gorton & Green 2024 abstract |
| Oscillaton M\_max ≈ 0.6 M\_Pl²/m | FROM MEMORY | Seidel–Suen |
| BBKS peak-shape distribution | FROM MEMORY | Bardeen, Bond, Kaiser & Szalay 1986 |
| Wigner-limit theorems; relativistic high-frequency limits | FROM MEMORY | Section 3 |
| ACT DR6 n\_s ≈ 0.974 ± 0.003 | FROM MEMORY | not re-checked |
| Evaporation bounds near 10^15 g; poltergeist GWs; Ω\_GW scaling; LISA sensitivity | FROM MEMORY | not re-checked |

Added on 22 September, after the external review:

| Claim | Status | Source |
| --- | --- | --- |
| Large massive-field perturbations made the initial evolution non-matter-dominated (footnote 1) | VERIFIED | [de Jong, Aurrekoetxea & Lim 2022](https://ar5iv.labs.arxiv.org/html/2109.04896) |
| Black holes form by accretion even when the initial shell fails the hoop test | VERIFIED | [de Jong, Aurrekoetxea & Lim 2022](https://ar5iv.labs.arxiv.org/html/2109.04896) |
| Triangulum II star-survival limit near 10^19 g (f = 1 excluded at 3.7σ) | CHALLENGED; not used | [Esser et al. 2025](https://arxiv.org/abs/2503.03352), challenged by [Gottlieb et al. 2026](https://arxiv.org/pdf/2606.02700) |
| μ\_deadline(D) ≈ 1.91 D^(−2/3) for the Yoo profile | DERIVED, recomputed | Section 7 |
| Exact kinetic seed φ = 0, Π = (2ρ)^(1/2) satisfies both constraints | DERIVED, checked | Section 4 |
| RK4 amplitude loss ≈ (ωΔt)^6/144 per step | DERIVED | Section 9 |

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).