[Paper Review] Detailed Calculation of Primordial Black Hole Formation During First-Order Cosmological Phase Transitions

Michael J. Baker, Moritz Breitbach+2 more|arXiv (Cornell University)|Sep 30, 2021

Cosmology and Gravitation Theories22 citations

TL;DR

This paper presents a detailed numerical study of primordial black hole (PBH) formation during first-order cosmological phase transitions, where particles (χ) with a large mass jump in the true vacuum are reflected by advancing bubble walls, creating localized energy overdensities. The key result is that PBHs form for a wide range of parameters—particularly large mass gains (mχ^∞ ≳ 10Tₙ) and initial bubble radii (r₀^w ≳ 1.5rₕ⁰)—when the overdensity collapses under its own gravity, satisfying the Schwarzschild criterion.

ABSTRACT

Primordial black holes could potentially form during a first-order cosmological phase transition due to a build-up of particles which are predominantly reflected from the advancing bubble walls. After discussing the general mechanism, we examine the criteria that need to be satisfied for a black hole to form. We then set out the Boltzmann equation that describes the evolution of the relevant phase space distribution function, carefully describing our treatment of the Liouville operator and the collision term. Assuming a spherical false vacuum pocket of sufficient size and a constant wall velocity, we find that black holes can form in a range of different scenarios.

Motivation & Objective

• To develop a comprehensive numerical framework for modeling primordial black hole (PBH) formation during first-order cosmological phase transitions.
• To investigate the conditions under which particle energy overdensities from reflected particles can collapse into PBHs, focusing on the Schwarzschild and Jeans instability criteria.
• To validate the mechanism using the Boltzmann equation with a simplified phase space treatment and collision terms, enabling tractable numerical simulations.
• To determine the parameter space—particularly wall velocity, mass gain, and initial bubble size—where PBH formation is viable.
• To demonstrate that PBH formation is robust to variations in wall velocity and thickness, provided the wall's relativistic gamma factor remains below mχ^∞/Tₙ.

Proposed method

• Formulates the Boltzmann equation for the phase space distribution function f(x,p,t) of χ particles, incorporating the Liouville operator to describe free streaming in the bulk and near-wall regimes.
• Applies a spherical symmetry reduction to eliminate two spatial and one momentum dimension, simplifying the Liouville operator to a 1+1 dimensional problem.
• Models collision terms via inverse decay (χχ̄ ↔ φ), annihilation (χχ̄ ↔ φφ), and momentum redistribution from χ scattering, with dominant contributions from inverse decay and annihilation.
• Solves the Boltzmann equation numerically using the method of characteristics, which tracks particle trajectories and enables intuitive visualization of reflection and accumulation.
• Imposes consistency checks on energy and particle number conservation to validate the numerical solution.
• Assumes a radiation-dominated universe, constant wall velocity and thickness, spherically symmetric false vacuum regions, and thermal contact between χ and the SM bath.

Experimental results

Research questions

• RQ1 Under what conditions does the energy overdensity from reflected χ particles collapse into a primordial black hole?
• RQ2 How do the Schwarzschild and Jeans instability criteria compare in predicting PBH formation in this scenario?
• RQ3 What is the role of wall velocity and thickness in determining the efficiency of particle reflection and energy accumulation?
• RQ4 How does the Yukawa coupling strength and mass gain of χ particles affect the formation of PBHs?
• RQ5 To what extent is PBH formation independent of the wall profile, and how robust is the mechanism under varying assumptions?

Key findings

• PBH formation is primarily governed by the Schwarzschild criterion: when the radius of the energy overdensity becomes smaller than its Schwarzschild radius, collapse is inevitable.
• The mechanism produces PBHs for a wide range of parameters, particularly when the final χ mass exceeds 10Tₙ and the initial bubble radius is greater than 1.5 times the Hubble radius at the transition.
• Numerical simulations confirm that energy overdensities build up inside shrinking bubbles due to particle reflection, with peak energy densities exceeding the critical threshold for black hole formation.
• The Schwarzschild criterion is more restrictive and predictive than the Jeans instability criterion, which rarely leads to earlier collapse and is thus less relevant.
• Black hole formation is largely insensitive to wall velocity and thickness as long as the wall’s Lorentz factor γ < mχ^∞ / Tₙ, indicating robustness to dynamical assumptions.
• The method of characteristics successfully captures particle trajectories and reflection dynamics, enabling accurate simulation of the phase space evolution without requiring full 6D resolution.