Results

Key findings #

Spherical collapse (1D) #

Code: pbhgr/cosmo_ev.py — spherical Einstein–Vlasov in the areal gauge with CMC slicing K = −3H(t) and a shift, horizon-penetrating and fully constrained (equations in Appendix B of the plan). Initial data: the long-wavelength CMC data of Yoo et al. (2026), pbhgr/cosmo_id.py; reference: the exact LTB solution, pbhgr/ltb.py.

Code checks #

• A homogeneous Universe (FLRW) is reproduced to 10⁻⁹; the LTB cycloid to 1.7·10⁻⁴ at 0.8 t_C, with second-order convergence.
• Causal-bound test at μ = 0.36: every shell is trapped at its LTB t_AH to 10⁻⁴.
• A static Einstein–Vlasov polytrope stays static to ~10⁻³.
• Convergence in the grid (K = 2000…8000, re-stretched spacing dR = 0.3/k → 0.15/k) and in the number of shells per cell (4…64): at small μ the multi-stream core suffers discreteness relaxation, which saturates at ≥ 32 shells per cell (≈ 1000 shells in the core).
• Independent check of the horizon diagnostic with the sphdiag module (the PBH-VERIFY-01 pilot): the resolved horizon appears 0.29 t_H after the formal criterion at μ = 0.10 — a ~1 % effect.

Threshold map (cold case) #

Thresholds where the curve crosses the deadlines: μ_th(100) ≈ 0.039, μ_th(300) ≈ 0.019, μ_th(1000) ≈ 0.0086. The first horizon is tiny (0.001–0.02 M_H) and grows: at μ = 0.1, 0.1 M_H is trapped by 1.43 t_C(0) and 0.3 M_H by 1.92.

For comparison: in the 3D particle runs of Yoo et al. at e = 0.2 there is a horizon for μ ≥ 0.050 and none for μ ≤ 0.045 within their ≈ 250 t_H. The spherical map gives a horizon by ~80 t_H at μ = 0.045, so the absence of a horizon in Yoo et al. at that amplitude is an effect of non-sphericity (or run length), not of the spherical threshold. The μ = 0.1 and 0.05 runs at e = 0.2 test this directly.

Velocity dispersion #

A real field is not perfectly cold dust: it has a velocity dispersion (de Broglie, σ_ent = √6/q at horizon entry) and kicks from small-scale structure that switch on during the collapse.

Conclusion: a threshold floor is set by the kick amplitude times the time it switches on. A later release (b_nl = 0.6 and 0.3) shortens the delay. The cold "no floor" result is fragile against realistic small-scale power — the decisive test is 3D with a real spectrum.

Profile shape #

The profile shape is a first-order effect: it controls how synchronously the shells fall. A flat core collapses as a whole; a steep peak assembles slowly. With σ = 0.03 the lognormal Δ = 0.1 is unchanged (1.110) and Δ = 0.5 is delayed by another 12 %.

Non-sphericity: semi-analytics and Newtonian 3D #

The Bond–Myers ellipsoid model for ν = 4 peaks: the median ellipticity e = 0.13 gives the pancake at 0.91 and the last axis at 1.05 of the spherical collapse time; the 90th percentile e = 0.19 gives 0.85 and 1.09. In this model non-sphericity shifts the threshold by at most 3 %.

A Newtonian particle-mesh code (pbhgr/pm3d.py, N = 192³, 7M particles) showed that this is a strong underestimate:

The third axis collapses 1.7–2.7× later than the sphere. The horizon must appear somewhere between the filament stage (~0.95) and the third-axis collapse, hence μ_th(100) ∈ [0.039, ~0.06–0.08]; the 0.045–0.05 of Yoo et al. falls inside. With a small-scale tail in the spectrum (ζ = 0.05, k = 2–10), one of three random fragments breaks up and the other two build one dominant clump. Only 3D GR can narrow the bracket.

3D GR with a scalar field (GRChombo) #

Stage 1: does a coherent scalar field in 3D give the same horizon as the 1D map at e = 0? A PBHCosmo example for GRChombo (adaptive meshes, 5 levels, box 8 r_m), μ = 0.1, m t_H = 30.

The result is a bounce: the central density reached 1739 times the background at the caustic and fell to 0.3 by 1.9 t_C(0) — the core emptied and no horizon formed. The reason: the Kaup mass 0.633/m = 0.028 M_H exceeds the first-horizon mass in 1D (0.016 M_H warm, 0.001 cold). At affordable m t_H the field behaves like a boson star, not like dust. A scalar 3D calculation would need m t_H ≥ 100–300 (3–10× the cost) and still could not see the tiny first horizon. Hence stage 2 — Vlasov particles in 3D.

3D GR with particles (PBHVlasov) #

Stage 2: our own Vlasov particle module PBHVlasov for the GRTeclyn code (AMReX, CCZ4): deposition on all AMR levels, geodesic motion with each particle's proper time, a gauge referenced to the K profile, ray and sphere horizon diagnostics.

Stage 1b: early start and non-spherical peaks #

To compute non-spherical peaks, the run starts early, at t_i, from Yoo's long-wavelength data for any ζ(x): the matter is built from the constraints; geodesic slicing is used until 0.5 t_C(0), then 1+log slicing (coefficient 2) with a Γ-driver; the time step follows the expansion. Two systematics with an O(dt/t) lag were found by the FLRW test and fixed; the particle density is matched to the constraint density to ~10⁻⁵, because the binding energy 2E at a super-horizon start is a small difference of large numbers.

Check at e = 0 (7 levels, 20 h on the CCX33): the early start matched the late start to 0.007 t_C(0) and LTB to 0.010 in the proper time of the matter at the horizon, for M = 0.11–0.6 M_H.

e = 0.2, p = 0, μ = 0.3:

The non-spherical peak collapses in turn — pancake, filament, third axis — and the horizon appears only at τ = 2.09 t_C(0), but straight away with ≈ 1 M_H, about the mass a spherical horizon would have grown to by then. At first it is clearly non-spherical (R_min/R_max = 0.8) and rounds off to 2–6 % within 0.5 t_C(0). At e = 0 the horizon was born tiny (0.105 M_H at τ = 1.13).

In Yoo et al., at the same μ and e, the lapse collapses at ≈ 20 t_H = 2.27 t_C(0). Our far-field times (2.73 and 2.97) refer to a different slicing, and their criterion is the lapse, not a trapped surface; the invariant statement of our run is τ = 2.09 t_C(0) for the matter of the first trapped sphere.

What next #

• The warm first horizon and kicks from small-scale power in 3D (a real spectrum instead of a single mode).
• Peak statistics: from μ_th(D; e, p) to the fraction β and f_PBH, with the distribution of e and p at ν ≈ 4.
• A higher-resolution check; a request to the Yoo group (V0) to extend their particle runs to 500–1000 t_H.