Research programme

Goal #

To test whether there is a physically realisable region of parameters in which primordial black holes of 10¹⁹–10²⁰ g make up all of the dark matter (f_PBH = 1), given that before reheating the Universe went through a dust era — an era of an oscillating heavy scalar field (the inflaton or a modulus). The original problem statement (12 September 2026) also included the field's decay and baryogenesis; see the full problem statement and the programme changes (both in Russian).

The key quantity is the threshold μ_th(D): the smallest curvature-peak amplitude for which an apparent horizon (a closed surface with θ₊ = 0, θ₋ < 0) forms before the deadline D, the end of the dust era. Through peak statistics the threshold sets the fraction β of collapsing regions and hence the required spectral amplitude P_ζ. Black-hole formation is not decided by δ = 1, δ_lin = 1.686 or a density maximum — only by a trapped surface.

Why Vlasov particles, not the field #

The 22 September brief showed that for 10¹⁹–10²⁰ g holes the ratio of the field mass to the Hubble rate q = m/H_k ≥ 3·10⁹ (10¹⁸–10¹⁹ for an inflaton). At such q the scalar field behaves as collisionless matter to ≲ 10⁻⁵, and the right model is the Einstein–Vlasov system. A direct Einstein–Klein–Gordon calculation is affordable only at q ≈ 25–200, where the field forms solitons with no physical counterpart. Our stage 1 confirmed this: the scalar core in 3D bounced off the centre (Kaup limit).

The literature spread is huge: at e = 0.2 the particle runs of Yoo et al. (2026) collapse from μ = 0.05, the analytic hoop estimate gives 0.45, and dust runs crash below 0.95. That spread is worth ~6·10⁴ in β and ~80× in P_ζ. For the Yoo profile, the spherical collisionless threshold cannot exceed the LTB value: μ_V,sph ≤ 0.33 (the causal bound).

Plan and stages #

The programme (verification and plan of 23 September, independently re-checked by a second audit):

• V0 — extend the 3D particle runs of Yoo et al. to 500–1000 t_H (a request to the Yoo group, or our own CosmoGRaPH build).
• V1 — spherical GR Vlasov: the Yoo profile and peak profiles, deadlines 100/300/1000 t_H, a velocity-dispersion scan.
• V1b — spherical EKG at q = 50–1600 with two codes (connecting field and particles).
• V2/V3 — three-dimensional runs with non-spherical peaks.

Methods #

Every result climbs a ladder of checks before it enters the map: a homogeneous Universe → the exact LTB solution → static solutions → convergence in grid, particle number and gauge → comparison with an independent code. Horizon formation times are compared in invariant form — by the proper time of the matter at the horizon, not by the coordinate time, which depends on the slicing.

• 1D: pbhgr (Python + C) — spherical Einstein–Vlasov in the areal–CMC gauge; exact LTB; profiles; Newtonian shells for comparison.
• Newtonian 3D: the particle-mesh code pbhgr/pm3d.py for triaxial peaks.
• 3D GR: GRChombo (stage 1, scalar field) and GRTeclyn/AMReX with our particle module PBHVlasov (stages 2 and 1b), up to 7 adaptive mesh levels and 18 million particles.
• Diagnostics: ray and sphere apparent-horizon searches, the Misner–Sharp mass, constraint residuals; the sphdiag module.

Details are in Code and in the log.

Resources #

How we work #

• An open notebook. The computation log records every run, every convergence check and every bug found. Withdrawn results are not deleted; they are marked with the reason.
• Independent checks. The plan was checked by two independent audits; the literature by five reports; the horizon diagnostic by a pilot with separated roles (derivation, implementation, verification, audit).
• Comparing invariants. 3D and 1D are compared by the proper time of the matter and the horizon mass, not by slicing-dependent coordinate time.