Inflaton fragment

A random curvature field ζ with a peaked spectrum. High peaks collapse into PBHs if they make it before reheating.

PBHs: 0

peak dust, expanding peak dust, falling background: pancakes and filaments PBH (ring is a marker, not to scale)

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comoving coordinates

A PBH seen from a particle

Exact LTB solution for the Yoo et al. (2026) profile. The camera sits on a single dust particle — at the centre, at the edge of the perturbation or far out — and looks towards the centre of the collapse.

particle

expanding

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field of view 60°

τ = 0.113 t_C(0)

= 1.00 t_H · observer's clock

Speed

×1.0

Observer expanding

shell label: 1.00 r_m
mass inside, m(r): 1.39 M_H
distance to centre, R: —
dR/dτ: —
density at particle ρ/ρ̄: —
mean contrast δ(<r): —
cycloid phase η: —

compactness 2m/R—

tick: 2m/R = 1 — the horizon during contraction, the Hubble radius during expansion

Black hole not yet

horizon mass M_AH: —
radius R_AH = 2M_AH: —
angular radius from particle: —
labels trapped up to: —

Shells R(τ)

Comoving radius R/(a·r_m) of shells from 0.2 to 2 observer radii. Yellow: the observer; red segment: shell inside the horizon; dashed: the horizon.

Mass growth M_AH(τ)

Dashed line at the top: the mass inside the observer. When M_AH grows to it, the horizon passes through the camera.

Parameters

Curvature amplitude μ0.3

Observer label r / r_m1.00

Particle colour

Reference shell rings (labelled with mass inside) Slow down before the observer's R → 0 Look inside the horizon Particles beyond the observer

How this is computed

Curvature profile of Yoo et al. (2026): ζ(r) = μ·exp(−r²/6) with k = 1. The scale r_m = √6 is where the compactness is maximal; it enters the Hubble horizon at time t_H. M_H = 0.75 t_H is the mass within the Hubble radius at that moment.

Each label r is a spherical dust shell with its own Misner–Sharp mass and energy:

m(r) = ½ H_i² (e^ζ r)³, E(r) = rζ′ (1 + rζ′/2) < 0

A shell moves along a cycloid: turnaround at η = π, R = 0 at η = 2π.

R = m/(−2E) · (1 − cos η), τ = m/(−2E)^3/2 · (η − sin η)

A shell falls inside the horizon when 2m/R = 1 during contraction, i.e. cos η_AH = 1 + 4E. The horizon mass is the mass of the last trapped shell: M_AH(τ) = m(r_AH(τ)).

The “by density” colour is the local contrast from the same solution: ρ = m′/(4πR²R′), background ρ̄ = 1/(6πτ²). The centre turns around at ρ/ρ̄ = 9π²/16 ≈ 5.55, as in a closed Friedmann universe.

From the particle, uniform contraction is invisible: if everything around contracts equally, the angles between neighbours do not change. What you see is the lead: inner shells fall earlier, and the core pulls together into a clump in front of the camera. The early convergence is easier to follow with the “by density” colour or in the outside view: there everything is divided by the scale factor a(τ), the background stays still and the clump shrinks.

τ is the proper time of the particles. In the LTB solution it is the same for all of them, so each frame is a slice τ = const. In this same time the 1D and 3D Vlasov runs fall on the trapping curve: at μ = 0.3, masses of 0.1 and 0.3 M_H are trapped at τ = 1.13 and 1.32 t_C(0), and the 3D PBHVlasov run matched LTB to within ±0.002 t_C(0).

What is left out. The radius is drawn as the areal radius R in Euclidean space. Light is not modelled: the real view from a particle would be delayed by the light travel time and distorted by lensing. In the Vlasov runs particles pass through the centre rather than stopping at R = 0, but outside the horizon the picture is the same. Inside the observer's label (but not beyond r_m) the density of points is proportional to mass; outside it the points are thinned out and dimmed.

Source: pbhgr/ltb.py (LTBGaussian), devlog V1 and the stage 2 comparison of 3 October 2026.

τ = 2.00 t_H

until reheating: ×50 in time

Speed

×1.0

Fragment seed 7

PBHs now: 0
mass in PBHs now: —
at reheating: —
fraction of fragment mass: —
heaviest: —

Threshold: amplitude and collapse time

Points: peaks with ν > 1.5. Horizontal: μ = ζ at the peak; vertical: t_C(0) of the peak centre (LTB from the height and Laplacian of ζ). Band: the same calculation for a Gaussian peak with k = 1…1.35 k_p. Everything below the reheating line has time to become a PBH.

Peaks that collapse

t_C(0) and M in units of t_H, M_H for k_p. M(D) is the horizon mass at reheating, capped at the mass within 0.9 r_m of the peak.

Parameters

Amplitude σ_ζ0.0080

Reheating D (end of dust stage)100 t_H

Spectral width Δ0.3

Labels for peaks and PBHs Background dust

How the fragment works

The curvature field ζ(x) is a Gaussian random field in a periodic box L = 24/k_p (≈ 10 Hubble radii at time t_H) on a 64³ grid. The spectrum is lognormal, as in pbhgr/profiles.py:

𝒫_ζ(k) ∝ exp(−ln²(k/k_p) / 2Δ²), σ_ζ = rms ζ

At each peak we take the height μ = ζ and the Laplacian ∇²ζ. This gives a Yoo Gaussian profile with k² = −∇²ζ/μ, for which the exact LTB solution yields the collapse time of the centre and the growth of the horizon:

t_C(0) = π H_i² e^3μ / (2μk²/3)^3/2, M_AH(τ) = m(r_AH(τ))

Dust within 0.8 r_m of a peak moves along the cycloids of this solution. The rest of the dust follows the Zel'dovich approximation, x = q + a·(2/5)∇ζ/(aH)², and stops on a pancake once its neighbourhood has collapsed along the first axis. In the 0.8–1.15 r_m zone the two motions are blended.

A PBH is counted if the peak centre has collapsed before reheating D. After D the dust stage ends: the dust decays into radiation and the PBH masses freeze.

Limitations. This is a cold spherical model, i.e. an upper estimate. In 3D, non-sphericity and velocity dispersion delay the horizon: PM-3D gives collapse of the third axis 1.7–2.7 times later than the spherical model, so many low peaks will not produce PBHs. The amplitude σ_ζ is inflated so that one box holds several PBHs: in a realistic scenario such peaks are extremely rare. In LTB the horizon appears at t_C(0) with zero mass; in the Vlasov runs the first resolved horizon appears later, at 1.12–1.3 t_C(0). Interactions between neighbouring peaks and accretion of the background onto PBHs are not included.