APS : Detailed Calculation of Primordial Black Hole Formation During First-Order Cosmological Phase Transitions

Baker, Michael J.; Breitbach, Moritz; Mittnacht, Lukas; Kopp, Joachim

doi:10.1103/PhysRevD.111.063544

Article
Report number	arXiv:2110.00005 ; CERN-TH-2021-113 ; MITP-21-036
Title	Detailed Calculation of Primordial Black Hole Formation During First-Order Cosmological Phase Transitions
Author(s)	Baker, Michael J. (Melbourne U.) ; Breitbach, Moritz (Mainz U., Inst. Phys. ; U. Mainz, PRISMA) ; Kopp, Joachim (Mainz U., Inst. Phys. ; U. Mainz, PRISMA ; CERN) ; Mittnacht, Lukas (Mainz U., Inst. Phys. ; U. Mainz, PRISMA)
Publication	2025-03-15
Imprint	2021-09-30
Number of pages	33
Note	33 pages, 6 figures; v2: improved discussion of phase transition dynamics; matches published version
In:	Phys. Rev. D 111 (2025) 063544
DOI	10.1103/PhysRevD.111.063544 (publication)
Subject category	Particle Physics - Phenomenology ; Astrophysics and Astronomy
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.
Copyright/License	publication: © 2025 The author(s) (License: CC BY 4.0) preprint: © 2021-2025 The author(s) (License: arXiv nonexclusive-distrib 1.0)

$A schematic picture of the first-order cosmological phase transition: regions of true vacuum (blue) are expanding with speed $v_w$ and have coalesced, resulting in a shrinking, approximately-spherical bubble of false vacuum (light red). Very high momentum $\chi$ particles pass through the bubble wall and convert kinetic energy into a large mass, while lower momentum $\chi$ particles are reflected due to energy conservation. The reflected $\chi$ particles build up inside the bubble, creating an energy overdensity which may lead to a black hole. The local coordinate system and the bubble wall thickness, $l_w$, are also shown.$ $Two example trajectories in position space and phase space. Panels (i) and (ii) both show the same type (A) trajectory, which is reflected by the bubble wall several times and remains inside the bubble until a black hole forms (grey disc). Panels (iii) and (iv) show a type (B) trajectory, which passes through the bubble wall before the black hole forms. We show the trajectories in position space on the left, in (i) and (iii), while on the right, in (ii) and (iv), we illustrate the same trajectories in $r$--$p_r$ phase space. The top two sub-panels in (ii) and (iv) show the $r$--$p_r$ plane far away from (left) and near to (right) the wall. The small coloured arrowheads show where the trajectories pass between the bulk regime and the near-wall regime. In the lower sub-panels of (ii) and (iv) we show the wall profile at the time of each reflection. The colour of the trajectories represents the phase space enhancement factor $\A$. The starting position of the trajectories is indicated with a large blue arrowhead, while a black dot or a large red arrowhead indicates the moment of black hole formation or when the trajectory leaves the simulation, respectively.$ $Two example trajectories in position space and phase space. Panels (i) and (ii) both show the same type (A) trajectory, which is reflected by the bubble wall several times and remains inside the bubble until a black hole forms (grey disc). Panels (iii) and (iv) show a type (B) trajectory, which passes through the bubble wall before the black hole forms. We show the trajectories in position space on the left, in (i) and (iii), while on the right, in (ii) and (iv), we illustrate the same trajectories in $r$--$p_r$ phase space. The top two sub-panels in (ii) and (iv) show the $r$--$p_r$ plane far away from (left) and near to (right) the wall. The small coloured arrowheads show where the trajectories pass between the bulk regime and the near-wall regime. In the lower sub-panels of (ii) and (iv) we show the wall profile at the time of each reflection. The colour of the trajectories represents the phase space enhancement factor $\A$. The starting position of the trajectories is indicated with a large blue arrowhead, while a black dot or a large red arrowhead indicates the moment of black hole formation or when the trajectory leaves the simulation, respectively.$ Show more plots

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Record created 2021-10-05, last modified 2025-08-22

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