Numerical simulations of primordial black hole formation via delayed first-order phase transitions

We perform fully nonlinear, spherically symmetric numerical simulations of superhorizon false-vacuum-domain (FVD) collapse in a coupled gravity-scalar-fluid system to study primordial black hole (PBH) formation during delayed first-order phase transitions (FOPTs). Using adaptive mesh refinement to resolve the bubble wall, we identify three dynamical outcomes: type B (supercritical) PBHs with an interior baby universe and a bifurcating trapping horizon, type A (subcritical) PBHs with an apparent horizon formed by direct wall collapse, and dispersal with no PBH formation. To separate these three cases, we evaluate two commonly used PBH-formation criteria: the time scale ratio $t_\mathrm{H}/t_\mathrm{V}$ (horizon crossing time versus vacuum-energy domination time) and the local density contrast $δ(t_\mathrm{H})$ at horizon crossing. For the parameter space explored, we find that $t_\mathrm{H}/t_\mathrm{V}$ is a more robust predictor of outcome: type B PBHs form when $t_\mathrm{H}/t_\mathrm{V} \gtrsim 1$ (critical range $\sim 1.1 - 1.6$ in our survey), type A PBHs arise when $t_\mathrm{H}/t_\mathrm{V}$ is below this threshold but remains above a lower bound (typical range $\sim 0.35 - 0.7$), and no-PBH dispersal occurs when $t_\mathrm{H}/t_\mathrm{V}$ falls below this lower bound. When a clear thin-wall FVD boundary exists, $δ(t_\mathrm{H})$ can correspondingly distinguish different outcomes (roughly $δ_c(t_\mathrm{H}) \sim 1 - 1.7$ for type B and $δ_c(t_\mathrm{H}) \sim 0.35 - 0.5$ for type A), but is highly sensitive to wall structure and model details and thus less universal. These results offer new insights into the dynamics of FVD collapse, quantify practical PBH-formation thresholds, and pave the way for precise predictions of PBH abundance from delayed FOPTs.

💡 Research Summary

This paper presents fully nonlinear, spherically symmetric numerical simulations of the collapse of super‑horizon false‑vacuum domains (FVDs) that arise during delayed first‑order phase transitions (FOPTs) in the early universe. The authors model a coupled gravity‑scalar‑fluid system: Einstein gravity, a scalar field with a temperature‑independent quartic‑type potential that can be tuned between thin‑wall (large λ) and thick‑wall (small λ) regimes, and a radiation fluid (p = ρ/3) that interacts only gravitationally with the scalar. The metric is written in the form ds² = −A²dt² + B²dr² + R²dΩ², and the gauge choice A = 1 (geodesic slicing) is adopted for numerical stability. Adaptive mesh refinement resolves the bubble wall, allowing the authors to follow the dynamics from the initial FLRW radiation background with an embedded FVD to the final outcome.

Three distinct dynamical outcomes are identified:

1. Type‑B (supercritical) PBH – occurs when the ratio of the horizon‑crossing time t_H to the vacuum‑energy‑domination time t_V satisfies t_H/t_V ≳ 1 (critical range ≈ 1.1–1.6 in the explored parameter space). The FVD collapses while its interior inflates, producing a baby universe connected to the exterior by a wormhole. A bifurcating trapping horizon (Θ⁺ = Θ⁻ = 0) forms, signalling the presence of a wormhole throat. An external observer perceives a black hole whose mass is roughly the volume of the original domain times the vacuum‑energy density.

2. Type‑A (subcritical) PBH – appears for intermediate ratios 0.35 ≲ t_H/t_V ≲ 0.7. The bubble wall shrinks directly, forming an apparent horizon without generating an interior baby universe. No bifurcating trapping horizon is observed; the collapse is more akin to the standard radiation‑perturbation PBH formation.

3. No‑PBH (dispersal) – for t_H/t_V < 0.35 the wall collapses to zero radius and the overdensity is radiated away; no horizon forms.

To assess the predictive power of commonly used PBH‑formation criteria, the authors compare (i) the time‑scale ratio t_H/t_V and (ii) the local density contrast δ(t_H) evaluated at horizon crossing. Their extensive parameter scan shows that t_H/t_V is a robust discriminator across thin‑ and thick‑wall regimes, while δ(t_H) can separate outcomes only when a clear thin‑wall boundary exists. In that case, critical density contrasts are roughly δ_c ≈ 1–1.7 for type‑B and δ_c ≈ 0.35–0.5 for type‑A, but these values shift dramatically with wall thickness, λ, and initial profile, making δ a model‑dependent indicator.

The paper also discusses the limitations of previous approaches based on Israel junction conditions, which cannot capture subcritical PBH formation because they treat the wall as an infinitesimally thin, impermeable membrane and neglect the radiation fluid inside the bubble. By solving the full set of Einstein‑scalar‑fluid equations, the authors resolve the wall dynamics, radiation inflow, and the evolution of the Misner‑Sharp mass and expansion scalars, thereby providing a more complete picture.

Implications for cosmology are highlighted. The time‑scale ratio t_H/t_V depends on the phase‑transition parameters: the vacuum‑energy difference ρ_V, the bubble radius at nucleation, and the background expansion rate. Consequently, the PBH mass function can be expressed as M_PBH ∼ (4π/3) R_FVD³ ρ_V for type‑B, with a reduced mass for type‑A due to radiation accretion and wall thickness effects. This dual‑population picture affects predictions for PBH contributions to dark matter, LIGO/Virgo black‑hole merger rates, and microlensing constraints. Moreover, the identification of a universal t_H/t_V threshold offers a practical tool for translating particle‑physics models of delayed FOPTs into observable PBH abundances.

In conclusion, the study establishes t_H/t_V as the most reliable, model‑independent criterion for PBH formation from delayed first‑order phase transitions, while confirming that density‑contrast criteria are highly sensitive to microphysical details. The work paves the way for precise calculations of PBH abundances in a broad class of early‑Universe phase‑transition scenarios and suggests future extensions to non‑spherical geometries, multiple‑bubble interactions, and reheating dynamics.