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arxiv: 2605.11332 · v2 · pith:32KD3RDKnew · submitted 2026-05-11 · ✦ hep-ph · astro-ph.CO· gr-qc

Reviving primordial black hole formation in slow first-order phase transitions

Wen-Yuan Ai , Ke-Pan Xie This is my paper

Pith reviewed 2026-05-22 10:10 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COgr-qc
keywords primordial black holesfirst-order phase transitionssupercooled transitionsearly matter dominationreheatingcurvature perturbationsdensity contrast
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The pith

Slow reheating after a supercooled first-order phase transition lets small overdensities grow into primordial black holes during an early matter-dominated era.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that large curvature perturbations from slow first-order phase transitions can still produce primordial black holes. Recent work had suggested the mechanism fails when density contrast and collapse threshold are computed consistently in one gauge. Here the authors show that a supercooled transition followed by sufficiently slow reheating drives the universe into a prolonged early matter-dominated era. In that era even modest overdensities grow linearly with the scale factor and can collapse once they exceed the formation threshold. This revives the scenario as a viable source of primordial black holes without needing large initial contrasts.

Core claim

Large curvature perturbations generated during slow first-order phase transitions remain a viable source of primordial black holes because, after a supercooled transition, reheating can be slow enough for the Universe to enter an early matter-dominated era during which even small overdensities grow and collapse into black holes.

What carries the argument

The early matter-dominated era triggered by slow reheating after a supercooled first-order phase transition, during which density perturbations grow linearly with the scale factor until they reach the collapse threshold.

If this is right

  • Primordial black holes can form from smaller initial perturbations than previously required in radiation-dominated scenarios.
  • The parameter space of first-order phase transitions that can source observable black-hole abundances is enlarged.
  • Gravitational-wave signals from the same phase transition must be re-evaluated in the presence of the intervening matter era.
  • Constraints on the duration and strength of supercooling from black-hole abundance limits become weaker.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the mechanism operates, the mass distribution of primordial black holes would be set by the horizon size at the end of the matter era rather than at the phase transition itself.
  • This channel could contribute to dark-matter candidates in mass windows where other production mechanisms are suppressed.
  • Future pulsar-timing or gravitational-wave observatories might detect a stochastic background whose spectrum encodes the length of the matter-dominated phase.

Load-bearing premise

Reheating after the supercooled first-order phase transition must be slow enough that the universe spends a long period in a matter-dominated phase before radiation domination resumes.

What would settle it

A direct calculation or simulation showing that the linear growth of overdensities during the matter era fails to push the density contrast above the collapse threshold for the curvature perturbations produced by the transition.

Figures

Figures reproduced from arXiv: 2605.11332 by Ke-Pan Xie, Wen-Yuan Ai.

Figure 1
Figure 1. Figure 1: FIG. 1. PDF of the density contrasts in the comoving volume [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: illustrates the PBH profile for k = 0.9 kmax and TV = 106 GeV. The blue and orange bands show the ranges of Mpbh and fpbh, respectively. In the upper panel, we fix Treh = T max reh while vary β/Hn from 5 to 8. Mpbh lies within 1020 g – 1022 g, an “asteroid-mass” window that can explain all of the dark matter while satisfying current constraints [51– 53], and that could be probed in the future [54–59]. fpbh… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
read the original abstract

Large curvature perturbations generated during slow first-order phase transitions are a promising source of primordial black holes. However, recent analyses suggested that the mechanism is ruled out once the density contrast and the formation threshold are evaluated in the same gauge. In this work, we show that this mechanism remains viable: after a supercooled transition, reheating can be sufficiently slow that the Universe enters an early matter-dominated era, during which even small overdensities grow and collapse into primordial black holes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript argues that large curvature perturbations generated during slow first-order phase transitions remain a viable source of primordial black holes. It claims that after a supercooled transition, sufficiently slow reheating allows the Universe to enter a prolonged early matter-dominated era in which even small overdensities grow linearly and collapse, circumventing recent gauge-dependent objections to the mechanism.

Significance. If substantiated with quantitative support for the reheating dynamics, the result would revive a promising channel for PBH formation tied to observable phase-transition gravitational waves and potentially dark matter. The approach directly engages prior no-go results by incorporating post-transition cosmological evolution rather than assuming immediate radiation domination.

major comments (2)
  1. [Reheating and early MD era discussion (near Eq. for equation-of-state evolution)] The central claim requires a matter-dominated era lasting multiple Hubble times after the transition, but no explicit computation of the scalar decay width Γ relative to the Hubble rate H at the end of supercooling is provided to demonstrate that the required duration is achieved. This leaves the viability dependent on an unquantified assumption about reheating speed.
  2. [Abstract and Section on post-transition cosmology] The abstract and main text assert that reheating 'can be sufficiently slow' without an error analysis, parameter scan, or falsifiable range of couplings that would produce a prolonged MD phase. This is load-bearing because if Γ is set by the same interactions enabling supercooling, the MD era may be too brief for linear growth to the collapse threshold.
minor comments (2)
  1. [Introduction] Notation for the curvature perturbation and density contrast should be clarified when switching between gauges to avoid reader confusion with prior literature.
  2. [Figures] Figure captions describing the scale factor or equation-of-state evolution during reheating could include explicit labels for the onset and end of the MD phase.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and the recommendation for major revision. The comments highlight important aspects of the reheating dynamics that warrant clarification. We address each point below and describe the planned revisions.

read point-by-point responses

  1. Referee: [Reheating and early MD era discussion (near Eq. for equation-of-state evolution)] The central claim requires a matter-dominated era lasting multiple Hubble times after the transition, but no explicit computation of the scalar decay width Γ relative to the Hubble rate H at the end of supercooling is provided to demonstrate that the required duration is achieved. This leaves the viability dependent on an unquantified assumption about reheating speed.

    Authors: We agree that an explicit estimate of the reheating duration strengthens the central claim. In the revised manuscript we will add a calculation of Γ/H at the termination of supercooling for a representative scalar potential. This will show that, for couplings and masses that permit supercooling, Γ/H can remain ≪ 1 for several Hubble times, allowing the early matter-dominated era to persist long enough for linear growth of overdensities to reach the collapse threshold. The addition will be placed near the equation-of-state evolution discussion. revision: yes

  2. Referee: [Abstract and Section on post-transition cosmology] The abstract and main text assert that reheating 'can be sufficiently slow' without an error analysis, parameter scan, or falsifiable range of couplings that would produce a prolonged MD phase. This is load-bearing because if Γ is set by the same interactions enabling supercooling, the MD era may be too brief for linear growth to the collapse threshold.

    Authors: The manuscript presents the mechanism under the assumption of sufficiently slow reheating, but we acknowledge that the conditions should be stated more precisely. We will revise the abstract and the post-transition cosmology section to include a qualitative discussion of the relevant parameter space. In particular, we will note that Γ can be suppressed relative to the supercooling scale through small additional couplings or symmetries that do not alter the phase-transition dynamics, thereby defining a falsifiable window of model parameters where the matter-dominated era lasts multiple Hubble times. A full numerical scan lies beyond the scope of the present work, which focuses on the cosmological evolution, but the revisions will clarify the viable range and the conditions under which the same interactions do not preclude a prolonged MD phase. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation relies on independent physical assumptions about reheating dynamics

full rationale

The paper argues that after a supercooled first-order phase transition, sufficiently slow reheating allows an early matter-dominated era in which small overdensities can grow to form primordial black holes. This is presented as a resolution to prior gauge-dependent objections, based on the equation-of-state evolution during reheating. No quoted equations or steps reduce the central result to a fitted parameter, self-citation, or input by construction. The reheating duration is treated as a viable parameter regime rather than derived tautologically from the PBH formation claim itself. The derivation chain remains self-contained against external benchmarks of phase transition and cosmological perturbation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; free parameters, axioms and invented entities cannot be audited from the provided text.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Gravitational Waves from Black Hole Reheating: The Scalar-Induced Component

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    Accounting for the minimal mass spread of primordial black holes from gravitational collapse suppresses the Poltergeist GW background to the level of generic scalar-induced signals and reopens ultra-light PBH parameter space.

  2. Primordial Black Hole from Tensor-induced Density Fluctuation: First-order Phase Transitions and Domain Walls

    astro-ph.CO 2026-05 unverdicted novelty 6.0

    Tensor perturbations from first-order phase transitions and domain wall annihilation induce curvature fluctuations at second order that form primordial black holes, allowing asteroid-mass PBHs to comprise all dark mat...

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