[Paper Review] Primordial Black Holes from First-Order Cosmological Phase Transitions

Michael J. Baker, Moritz Breitbach+2 more|arXiv (Cornell University)|May 16, 2021

Cosmology and Gravitation TheoriesPhysics and Astronomy63 references42 citations

TL;DR

The paper presents a mechanism for primordial black hole production during a first-order cosmological phase transition, where reflected particles off advancing bubble walls create overdensities that collapse into PBHs. It provides a Boltzmann-equation-based analysis to compute PBH abundance and mass distribution as functions of transition parameters.

ABSTRACT

We discuss the possibility of forming primordial black holes during a first-order phase transition in the early Universe. As is well known, such a phase transition proceeds through the formation of true-vacuum bubbles in a Universe that is still in a false vacuum. When there is a particle species whose mass increases significantly during the phase transition, transmission of the corresponding particles through the advancing bubble walls is suppressed. Consequently, an overdensity can build up in front of the walls and become sufficiently large to trigger primordial black hole formation. We track this process quantitatively by solving a Boltzmann equation, and we delineate the phase transition properties required for our mechanism to yield an appreciable abundance of primordial black holes.

Motivation & Objective

• Motivate the study of PBH formation during first-order phase transitions as an alternative to inflation-generated perturbations.
• Propose a mechanism where particle masses increase across the bubble wall, causing overdensities via reflected particles.
• Quantitatively track the evolution of the reflected-particle population with a Boltzmann equation to assess PBH formation conditions.
• Compute the resulting PBH mass spectrum and abundance as functions of nucleation temperature and phase-transition parameters.
• Discuss implications for dark matter, seeds of structure, and observational constraints.

Proposed method

• Introduce a toy model with a scalar field phi and a Dirac fermion chi interacting via a Yukawa coupling y_chi, giving m_chi = m_chi^0 + y_chi <phi> when phi gains a vev.
• Assume a first-order phase transition with bubble walls separating false and true vacuum regions.
• Model chi as relativistic inside the bubble and analyze transmission vs reflection at the wall due to mass gain, leading to overdensities in the false vacuum.
• Solve the Boltzmann equation L[f_chi] = C[f_chi] for f_chi under spherical symmetry to track f_chi(r, p_r, p_sigma, t).
• Include collision terms from chi chibar <-> phi and chi chibar <-> phi phi in the Boltzmann equation to account for annihilation and production.
• Derive a Schwarzschild-formation criterion r_w(t) < r_s to determine black hole formation, with r_w(t)/r_H depending on g_star, g_chi, and bubble initial radius r_w^0.
• Use numerical simulations (method of characteristics) to evolve the phase-space distribution and evaluate the conditions under which PBHs form.

Experimental results

Research questions

• RQ1 Under what phase-transition conditions (nucleation temperature T_n, wall properties, and particle masses) does a PBH form from reflected chi particles?
• RQ2 How does the PBH mass and abundance depend on the bubble initial size, bubble-wall dynamics, and the phase-transition parameters (phi vev, y_chi, and g_star)?
• RQ3 What is the resulting PBH mass spectrum and its potential to account for dark matter, seeds for SMBHs, or other cosmological phenomena?
• RQ4 How do annihilations, wall transmission, and Hubble effects influence the viability of PBH formation in this mechanism?

Key findings

• PBH formation is possible if large bubbles (r_w^0 ~ r_H) shrink by a factor of a few while capturing most reflected chi particles.
• The overdensity grows approximately as rho_chi ~ (r_w^0/r_w(t))^4 due to number-density increase and energy boosts from wall reflections.
• Black holes form when the shrinking bubble’s radius falls below the Schwarzschild radius corresponding to the enclosed chi energy, with the criterion depending mainly on g_star(T_n) and the ratio r_w^0/r_H.
• Parameter scans show successful PBH formation for sufficiently large m_chi/T_n (to suppress transmission) and small y_chi (to avoid excessive annihilation).
• The resulting PBH mass can span from approximately the Planck scale up to 10^18 solar masses, with the abundance quantified by f ≡ Omega_PBH/Omega_DM and depending on the phase transition temperature and the probability p of PBH formation per Hubble volume.
• Supercooled, strongly first-order transitions are especially conducive to PBH production due to favorable beta/H and extended bubbles.
• Observationally relevant mass windows emerge, including possibilities for PBHs as dark matter, SMBH seeds, and OGLE hints, while respecting current constraints.