# PBH collapse: independent verification and V0 / V1 / V1b implementation plan **Audit date:** 23 September 2026. **Specification:** the supplied `preview.md`, not an established result. **Status:** literature audit, independently evaluated analytic/LTB diagnostics, and an implementation specification. No Einstein–Vlasov (EV), Einstein–Klein–Gordon (EKG), or production PIC collapse simulations were run. No production-code performance measurements were made. ## Executive decision Proceed with a **collisionless-first investigation, accompanied by a necessary scalar validation track**. Do not adopt the brief's stronger conclusion that the EKG-to-EV collapse boundary is already known to an error below 10^-5. Nor should a spherical threshold above 0.33 automatically be labelled a code bug. The three packages have distinct purposes: | Package | Question | Minimum useful result | |---|---|---| | V0 | Does the published triaxial particle transition survive reproduction, longer deadlines, and numerical controls? | An outcome map with retained unresolved cases; a converged threshold only if the controls actually pass | | V1 | What is the spherical relativistic collisionless boundary, including its dependence on deadline, profile, and physical/numerical velocity dispersion? | A regulator-controlled family of finite-deadline brackets | | V1b | Do directly perturbed massive-scalar initial data and collapse outcomes approach the matched collisionless system over a controlled overlap regime? | Validated growing-mode initialization and measured finite-q differences, or a documented failure of that comparison | The expensive nonspherical EKG campaign remains gated. If the original PIC branch is unavailable, V0 is a reconstruction project, not a cheap rerun. V1 can start independently with exact LTB and cosmological-geometry tests. ### Evidence labels - **VERIFIED:** directly supported by a primary source checked for this audit. - **DERIVED:** calculation reproduced here; assumptions are stated. - **WORKING HYPOTHESIS:** scientifically plausible but not an acceptance criterion. - **UNVERIFIED / OVERSTATED:** not established by the cited evidence. - **IMPLEMENTATION CHOICE:** a proposed design, not a result or measured cost. ## 1. Verification ledger ### 1.1 Relevant versions and scope | Source | Version checked | What it establishes for this project | |---|---|---| | Yoo, Escrivà, Harada & Kohri, *Simulation of PBH formation in a matter-dominated universe* [R01] | 2609.14218v1, 13 September 2026 | The profile, dust/particle comparison and LTB causal diagrams; the particle result is explicitly quantitatively limited by constraint violations | | Milligan et al., *Primordial Black Hole Formation in a Scalar Field Dominated Universe* [R02] | 2504.02600v2, 26 August 2025 | Spherical directly perturbed scalar calculations; not an EKG-to-EV threshold-convergence sequence | | de Jong et al., *Spinning primordial black holes formed during a matter-dominated era* [R03] | 2306.11810v2, 7 July 2023 | Relevant 3D EKG machinery, but a separate massless perturbing field | | de Jong, Aurrekoetxea & Lim, *Primordial black hole formation with full numerical relativity* [R04] | 2109.04896v2; JCAP 03 (2022) 029 | Footnote 1 contains the warning about a large massive-field perturbation injecting potential energy | | Ebrahimian, Abolhasani & Mirbabayi, *Primordial black hole formation in matter domination* [R05] | 2507.18312v2, 25 March 2026 | Analytic arguments and Newtonian numerical evidence for the importance of profile and multistream dispersion, not a calibrated cosmological EV horizon boundary | | Mocz et al., *On the Schrödinger–Poisson–Vlasov–Poisson correspondence* [R06] | 1801.03507v2 | Newtonian tests of coarse gravitational convergence despite density interference | | Ye et al., *Primordial Black Hole Formation and Spin in Matter Domination Revisited* [R07] | 2508.10070v1 | A peak-theory calculation within its assumed collapse prescription, not a measured EV threshold | | Padilla et al. [R08] | 2509.10431v1 | Scalar critical-collapse follow-up focused on the quartic/radiation-like case; it does not close the quadratic EKG–EV gap | | Escrivà et al. [R09] | 2605.04487v1 | Semirelativistic/Newtonian-force collapse and GW modelling; not a full Einstein–Vlasov apparent-horizon calibration | The search did not identify a published calculation meeting all of the project's requirements: matched growing-mode data, directly perturbed real quadratic scalar, controlled collisionless comparison, post-caustic evolution, and converged finite-deadline PBH boundaries. This is a bounded literature-search conclusion, not proof that no such unpublished or differently indexed result exists. ### 1.2 Claims to retain, qualify, or remove | Brief claim | Audit result | Required action | |---|---|---| | q = 7.5×10^4 (m/GeV)(M/10^19 g)/gamma | **DERIVED; correct** with the stated entry-mass convention | Retain; distinguish seed and later PBH masses | | Every asteroid-mass model has q >= 3×10^9 | **OVERSTATED** | Restrict to an explicitly specified gravitationally decaying heavy-modulus scenario | | Large q proves EV threshold errors below 10^-5 | **NOT ESTABLISHED** | Remove the numerical error guarantee | | The particle bracket is 0.045–0.050 at e=0.2,p=0 | **VERIFIED AS A REPORTED RESULT, NOT A CONVERGED THRESHOLD** [R01] | Reproduce it with horizon and constraint diagnostics | | Dust's 0.95 boundary is a physical non-collapse threshold | **INCORRECT** | It is a breakdown boundary of those runs | | The hoop comparator is about 0.45 | **DERIVED; 0.45175** for the stated convention | Keep as a model comparator, not ground truth | | The spherical LTB cases are globally naked at 0.32 and only locally naked at 0.33 | **VERIFIED** [R01] | Do not relabel them as measured EV outcomes | | The bound mu_EV,sph <= 0.33 is already an unconditional theorem | **NOT ESTABLISHED** | A conditional causal-protection argument is promising; see the independent check below | | A spherical result can lower-bound the triaxial threshold because anisotropy only hinders collapse | **UNPROVEN** | Remove the monotonicity assumption | | Milligan establishes a q-dependent threshold sequence | **INCORRECT** | Its reliable quadratic cases give upper bounds and demonstrate intermediate core dynamics [R02] | | The de Jong initialization difficulty belongs to the 2023 spinning paper | **MISATTRIBUTED** | Cite the 2022 paper, footnote 1 [R04] | | Any flat-conformal-metric CTTK solve selects the decaying mode | **OVERGENERALIZED** | Distinguish a flat physical slice from conformal flatness; test the realized mode | | phi=0, Pi=sqrt(2 rho) on a comoving dust slice satisfies both constraints | **DERIVED; correct** | Keep as a constraint-consistent seed, not a pure finite-q growing-mode proof | | Schrödinger–Poisson assumes single-stream matter | **INCORRECT** | It supports wave-encoded multistreaming; its limiting issue near a horizon is the weak-field/nonrelativistic approximation [R06] | | No apparent horizon by D=1000 proves permanent support | **INCORRECT** | Report a finite-time statement | | A falling deadline boundary establishes beta proportional to sigma^5 | **INCORRECT** | The shape/profile statistical integral remains necessary | | EKG at q=100 is automatically a replacement Vlasov solver when PIC fails | **CIRCULAR** | Use an independently validated kinetic method, or call the comparison unresolved | | A factor-nine shift at one ellipticity determines beta×9^5 and P_zeta/81 | **UNJUSTIFIED** | Remove the “Yoo particle slope” abundance column | | The heavy-modulus / inflaton / broad-spectrum compatibility labels are established | **NOT ESTABLISHED** | Evaluate specific models and full mass functions | | All spherical work costs <=10^3–10^4 core-hours | **UNMEASURED** | Replace with a profiling-based allocation gate | ## 2. Physical regime and effective-theory assumptions ### 2.1 Correct mass scaling Use c=hbar=1 and the reduced Planck mass Mbar_P: \[ M_H=\frac{4\pi\bar M_P^2}{H_k},\qquad M_{\rm PBH}=\gamma M_H, \] \[ H_k\simeq1.33\times10^{-5}\,\gamma\, \left(\frac{10^{19}\mathrm g}{M_{\rm PBH}}\right)\mathrm{GeV}, \] \[ q\simeq7.53\times10^4\gamma^{-1} \left(\frac{m}{\mathrm{GeV}}\right) \left(\frac{M_{\rm PBH}}{10^{19}\mathrm g}\right). \] For M=10^19 g and gamma=1, m=1 GeV, 200 TeV and 10^13 GeV give q about 7.5×10^4, 1.5×10^10 and 7.5×10^17. The first is an illustrative weakly decaying scalar scale, not a complete model. At M=10^20 g multiply by ten. The lifetime and energy density needed for scalar domination remain independent assumptions. If Gamma=c m^3/Mbar_P^2, the BBN floor produces a heavy-mass floor only after specifying c and the decay/thermalization history. For c=1/(8 pi), m=200 TeV and g*=10.75, the usual instantaneous-reheating estimate gives T_R about 11 MeV. Neither this decay law nor its coefficient follows from M_PBH. In a matter-like era, collapse after D=t_c/t_k requires approximately Gamma < H_k/D. At D=200 and gamma=1, the corresponding temperature ceilings are about 2.2×10^5 GeV for 10^19 g and 6.9×10^4 GeV for 10^20 g, taking g*=106.75. Near reheating, evolve radiation production rather than replacing it with an arbitrary hard deadline. ### 2.2 Local validity, not background validity The proposed correspondence needs a hierarchy involving H/m, K/(am), local dynamical time, de Broglie wavelength, envelope gradients, sheet thickness, curvature, and self-interaction/decay scales. In particular, \[ m r_{s,\rm AH}=q\gamma_{\rm AH},\qquad \gamma_{\rm AH}=M_{\rm AH}/M_H(t_k). \] Use the first-horizon mass here, not an accreted final mass. A tiny first seed can influence the ultimate PBH outcome by later accretion; its importance is not disproved by a small instantaneous mass fraction. At a minimum record \[ \epsilon_{\rm WKB}=1/(|p_{\rm local}|L_{\rm local}),\quad \epsilon_{\rm dyn}=1/(m t_{\rm dyn}),\quad \epsilon_{\rm curv}=\sqrt{\max|R_{\hat a\hat b\hat c\hat d}|}/m, \] and separately the local gradient/rest-energy ratio, the wavelength/sheet-width ratio, and the amplitude of anharmonic corrections. Relativistic streaming energy is not itself a failure of relativistic Vlasov. No universal Airy-caustic-width estimate translates directly into a PBH-boundary error. The free real-scalar oscillaton maximum is about 0.607/(Gm) [R17]. Consequently M/Mmax≈0.824 gamma q. This identifies potentially core-dominated small-q runs, but is not an upper bound on every extended configuration or a theorem about collapse probabilities. ### 2.3 What the mathematical literature does and does not establish The relevant distinctions are: - A single WKB phase gives a monokinetic dust description before caustics. - A coarse phase-space description can admit multiple streams. Mocz et al. establish numerical Newtonian potential convergence in their tested configurations, not convergence of a black-hole indicator [R06]. - Widrow develops a relativistic scalar/Vlasov representation with a smoothing hierarchy; it is not a self-consistent cosmological horizon-convergence proof [R14]. - The one-dimensional pure-state semiclassical problem has delicate weak-solution/selection issues; the later selection work [R15] is a reason to control the cold regulator, not a general 3D strong-field theorem. - Huneau–Luk's Burnett result concerns high-frequency vacuum metrics and an **Einstein–massless** Vlasov limit [R16]. It is not the massive matter limit in this project. - Salvi's result is massive Klein–Gordon–Maxwell to relativistic Euler–Maxwell under its hypotheses [R18], not Einstein–Vlasov through cosmological caustics. The working target remains EV, but the limits q→infinity, regulator width→0, amplitude→threshold, and deadline→infinity need not commute. A finite-q discrepancy over a limited scan can falsify a proposed convergence fit without falsifying all possible asymptotic correspondence. ### 2.4 Fragmentation before the target mode enters With an assumed small-scale density amplitude around 2×10^-5, linear matter-era growth needs about ln(1/delta)≈11 e-folds to reach unity. This conditional estimate is consistent with the mechanism investigated in inflaton-cluster and post-inflationary structure studies [R19,R20]. It does not establish that every model's target background has fragmented. A fragmented medium is not automatically equivalent to smooth cold particles. Its clump masses, internal structure, tidal evolution, velocity dispersion, and relaxation times must justify that approximation. V1 should distinguish physical initial dispersion from an artificial numerical regulator. ## 3. Independent LTB and deadline checks ### 3.1 Fixed conventions Let k_p denote the Gaussian profile parameter, not the characteristic horizon-entry wavenumber: \[ \zeta(r)=\mu e^{-k_p^2r^2/6},\quad r_*=\sqrt6/k_p,\quad K_*=1/r_*,\quad a_kH_k=K_*. \] For reproducing the paper use a_i=1, H_i=5k_p and its constant-bang-time matter solution. Then t_i=2/(15 k_p) and t_k=100 sqrt(6)/k_p. All outputs must carry the convention explicitly. The spatial q(x) used in the paper's gradient expansion must be renamed in software to avoid confusion with m/H_k. ### 3.2 The proposed causal upper bound is promising, but conditional I reconstructed the exact LTB functions and integrated an approximation to the earliest outgoing central null ray, taking its launch radius successively smaller. The accompanying Python code reproduces these diagnostics. For k_p=1, mu=0.33, a representative regular sphere has: | Quantity | Value, rounded | |---|---:| | k_p r | 2.036771 | | Collapsing apparent-horizon time / t_k | 16.173458 | | Earliest central outgoing-ray time / t_k | 16.183458 | | Probe time / t_k | 16.178458 | | 2M_MS/R at the probe | 1.002735 | | R' at the probe | positive | | Both future null expansions | negative | The ray endpoint changed by less than 10^-7 in k_p r between the two finest launch-radius/step choices. The mu=0.32 ray did not enter a trapped region on the computed interval to k_p r=4. This is consistent with the paper's LTB classification [R01]. Thus the **geometrical ingredient** of causal protection is supported: at mu=0.33, there is a strictly trapped sphere earlier than the central signal reaches it. Matter evolution modified only inside the central event's causal future cannot remove a trapped sphere outside that future. What is not established here is the full transfer to a family of regularized EV solutions. That requires demonstrating convergence on the protected exterior region, controlling any earlier caustic or boundary influence, matching the initial geometry, and retaining a strict trapping margin. The launch calculation uses an asymptotic expansion and floating-point ODE integration, not interval arithmetic or a proof identifying every null generator. **Use mu=0.33 as a conditional cold-limit consistency check, not an unconditional bug detector.** A finite-dispersion family has different stresses everywhere from the initial slice. The argument gives no lower bound near 0.2 and no theorem ordering spherical and triaxial thresholds. ### 3.3 Deadline estimate: correct diagnostic, not a collapse law For the linear matter growing mode, \[ \bar\delta_k(r_*)=\frac{12}{5e}\mu\simeq0.883\mu. \] Combining delta∝t^(2/3) with the spherical top-hat linear collapse value 1.686 yields \[ \mu_{\rm deadline}(D)\simeq1.91013D^{-2/3}. \] | D | Approximate amplitude | |---|---:| | 100 | 0.08866 | | 250 | 0.04813 | | 300 | 0.04262 | | 500 | 0.03032 | | 1000 | 0.01910 | The proximity to the published particle transition motivates V0. It does not validate top-hat dynamics for the triaxial Gaussian peak. A finite-time plateau does not prove permanent support. A decreasing boundary does not prove that no nonzero asymptotic floor exists. A floor plus a delay, critical slowing near an unstable state, and numerical artifacts can all produce intermediate behavior. Fit and publish the finite-deadline surface first; do not force a binary interpretation. ## 4. Common implementation contract ### 4.1 Three outcomes, with finite-time meaning **COLLAPSE:** a future black-hole marginally outer-trapped surface is found, with outgoing expansion zero and ingoing expansion negative, and detection is stable under numerical checks. A falling lapse or 2M/R>=1 without checking the expansion signs is insufficient in an expanding cosmology. **RESOLVED NON-COLLAPSE BY D:** no such surface forms by the specified exterior cosmological deadline, and the evolution to that deadline is resolved. It may still be infalling. A bounce, virial ratio near unity, or falling compaction is not a prerequisite. **UNRESOLVED:** the calculation cannot support either classification because of numerical failure, insufficient resolution, uncontrolled constraints, an unvalidated regulator, or ambiguous horizon identification. Give separate morphology tags: infall, rebound, multistream halo, long-lived core, dispersion. These tags are not substituted for horizon outcomes. Do not require a newly formed horizon to survive for half the full collapse time as a generic detection rule. Use resolution-stable expansion signs, a short resolved continuation, or validated excision. Historical detection before D remains recorded even if a later optional continuation fails. ### 4.2 Shared geometry and matter interface Use a cosmological, horizon-capable spherical geometry formulation for V1 and V1b, for example reference-metric BSSN with regular origin treatment. A useful general line element is \[ ds^2=-\alpha^2dt^2+A^2(dr+\beta dt)^2+R^2d\Omega^2. \] An asymptotically flat polar-areal solver cannot simply be copied onto the entire initial superhorizon cosmological domain. Likewise, a dust-comoving coordinate system need not survive multistreaming. An independently formulated check can use another admissible gauge on overlapping regular regions. Define a common matter-source interface returning E, J_i and S_ij, and independent diagnostic routines for the Hamiltonian/momentum constraints, Misner–Sharp mass, and both null expansions. Match scalars at equal exterior cosmological time and compare invariant profiles or proper/areal radii, not unmatched coordinate radii. COSMOS is a useful public cosmological code base, but its release is advertised for a linear-EOS fluid plus a massless scalar; the code paper explicitly notes that publication-specific additions can be absent from the public release [R10]. COSMOS-S supplies a spherical starting point, not a verified ready-made EV or massive-EKG module. GRChombo/GRTresna provide alternatives, but a validated spherical/cartoon implementation must be demonstrated rather than assumed [R11,R12]. ### 4.3 Record every run reproducibly Required metadata: source commit and patch digest; compiler/build flags; initial-data digest; amplitude/profile definition; scalar q and phase or kinetic regulator; metric gauge; spatial/phase-space grid; refinement/deposition rules; time-step history; domain and boundary treatment; exterior time mapping; resource usage; outcome by each deadline; and unresolved reason. Required products: initial constraint residuals, run-time local and global constraints, horizon diagnostics, mass/flux accounting, density/stress profiles, field wavelength or particle sampling diagnostics, restart-validation logs, and threshold endpoint records. Test monotonicity in amplitude before bisection. Keep interval-valued boundaries. An unresolved interior trial does not erase a valid outer bracket; an unresolved endpoint cannot establish that side of a bracket. ## 5. V0: reproduce and extend the published triaxial particle calculation ### Goal Separate a reproducible physical finite-deadline transition from a plotting deadline, numerical breakdown, and particle/discretization artifacts. V0 is not automatically a publication-quality EV continuum calculation. ### V0.0 — provenance gate Obtain the original PIC module or an author-identified equivalent, its commit, parameter files, coordinate mapping, particle loading and deposition, gauge settings, horizon finder, domain, and actual termination reasons for the low-amplitude runs. Archive the exact paper version. If the original branch is unavailable, label the work an **independent reconstruction**. Do not budget it as extending a checkpoint. No contact with the authors or acquisition of an unpublished branch has occurred in this audit. ### V0.1 — reproduce controls before extending weak peaks Use the paper-matched e=0.2,p=0 profile and normalization, with \[ \zeta(\mathbf X)=\mu e^{-k_p^2r^2/6} \left[1+\frac{k_p^2}{6}\left\{p(2X^2-Y^2-Z^2)+3e(Y^2-Z^2)\right\}\right], \qquad \Psi=e^{\zeta/2}. \] First run mu=0.045, 0.050 and 0.060 to D=250, keeping the published baseline numerical settings. Include a homogeneous cosmological particle test and a spherical pre-caustic dust/PIC comparison. Validate the horizon itself, the timing convention, and the constraint histories; matching only a central lapse curve is insufficient. Differences between the reconstruction and the original setup must be recorded before claiming reproduction. ### V0.2 — deadline screen Use mu={0.030,0.035,0.040,0.045,0.050,0.060}; evolve suitable cases to D=1000. Record outcomes at D={100,250,300,500,1000} from each trajectory. Include collapsing controls rather than testing only the previously non-collapsing interval. Stop an individual run when its output becomes unresolved; never propagate its non-collapse label to a later deadline. A restart from D=250 must agree with an uninterrupted control evolution within the measured tolerance. ### V0.3 — targeted controls, not an indiscriminate full factorial After screening, choose near-boundary pairs and test spatial grids N={80,120,160} per octant, physical central resolution held meaningful under the coordinate mapping. Vary particle sampling at fixed grid, then grid at controlled sampling. Candidate checks include 8,27,64 particles per cell with a quiet initial load; retain the original loading as a reproduction control. Change deposition order and check the resulting source conservation and force interpolation consistently. These values are pilot choices. Increasing particles alone cannot cure a metric/deposition error. Increasing the grid while holding particles fixed can worsen sampling. Test both Hamiltonian and momentum constraints around crossings, not just a whole-domain mean. ### V0.4 — boundary control For an ideal matter background the light-travel distance from entry to D is \[ \Delta\chi=2r_*\left(D^{1/3}-1\right). \] At D=250 this is 10.60 r_*=25.96/k_p, and at D=1000 it is 18 r_*=44.09/k_p. The original initial slice adds an earlier travel distance of about 1.84 r_* before entry. Therefore simply increasing an outer face from 10/k_p to 25/k_p is not a general causal-isolation certificate. Choose the boundary from the region being diagnosed, full start-to-deadline characteristic travel, and the actual boundary/gauge characteristics. Then demonstrate domain convergence; this may require a much larger coarse exterior. An absorbing prescription is not assumed perfect. ### V0 acceptance and interpretation - Reproduction succeeds only if the original controls are recovered with their caveats. - Evidence for delayed collapse requires newly formed **resolved** horizons at larger D, with the effect surviving numerical controls. - Evidence for a persistent finite-time plateau requires converged neighboring brackets; it is not a proof about D→infinity. - If particle constraints or outcomes do not converge, report V0 as unresolved and use V1 to distinguish physics from the 3D method. Do not relabel an affordable EKG run as a validated Vlasov replacement. **Deliverables:** provenance report, an outcome-by-deadline table, convergence panels, and either controlled brackets or explicit one-sided/unresolved intervals. Allocation for subsequent V2 depends on these deliverables, not on a preferred threshold value. ## 6. V1: spherical Einstein–Vlasov implementation ### V1.0 — exact geometry tests Implement the constant-bang-time LTB solution from the supplied Gaussian curvature profile. Reproduce the small-radius limit, FLRW exterior, mass function, density, null expansions and causal diagnostic above. Compute R' and identify shell crossing independently of central focusing. Add automatic tests of positive density and coordinate regularity. Use mu=1.3 only as a separate published-style strong spherical control when the code/data support its geometry; the small-radius diagnostic script supplied here is deliberately restricted to 0