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arxiv: 2503.01962 · v2 · submitted 2025-03-03 · ✦ hep-ph · astro-ph.CO· gr-qc

Curvature Perturbations from First-Order Phase Transitions: Implications to Black Holes and Gravitational Waves

Gabriele Franciolini , Yann Gouttenoire , Ryusuke Jinno This is my paper

Pith reviewed 2026-05-23 01:16 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COgr-qc
keywords first-order phase transitionsprimordial black holesgravitational wavesgauge dependencepulsar timing arraycosmological perturbationscurvature perturbations
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0 comments X

The pith

Covariant treatment shows strong suppression of black hole and gravitational wave signals from first-order phase transitions.

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

The paper studies curvature perturbations generated during first-order phase transitions in the early universe. It applies a fully covariant formalism that exposes gauge dependencies missed by earlier non-covariant calculations. This reveals that previous estimates overstated both the abundance of primordial black holes and the amplitude of scalar-induced gravitational waves. Accounting for the gauge issues produces strong suppression of both signals. The result directly affects whether first-order phase transitions can explain the stochastic background reported by pulsar timing arrays.

Core claim

A fully covariant formalism applied to cosmological perturbations during strong first-order phase transitions demonstrates that non-covariant treatments overestimate both primordial black hole formation and scalar-induced gravitational waves; once gauge dependencies are removed, both signals are strongly suppressed, carrying direct implications for the first-order phase transition interpretation of the pulsar timing array signal.

What carries the argument

The fully covariant formalism for cosmological perturbations, which removes gauge artifacts from curvature calculations during first-order phase transitions.

If this is right

  • Primordial black holes form at much lower rates during first-order phase transitions than non-covariant models predicted.
  • Scalar-induced gravitational waves from these transitions have substantially lower amplitudes.
  • Any first-order phase transition explanation of the pulsar timing array signal must incorporate the reduced strength after gauge correction.

Where Pith is reading between the lines

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

  • Similar gauge issues may affect other early-universe observables calculated without a covariant treatment.
  • Future numerical simulations of phase transitions should adopt covariant perturbation methods to produce reliable signal forecasts.
  • Observational upper limits on primordial black holes or gravitational waves can now be compared against the lower expected rates from phase transitions.

Load-bearing premise

The covariant formalism removes every gauge artifact without adding new modeling uncertainties or extra assumptions about bubble dynamics.

What would settle it

A confirmed pulsar timing array signal whose amplitude matches non-covariant first-order phase transition predictions but exceeds the suppressed covariant prediction would falsify the suppression result.

Figures

Figures reproduced from arXiv: 2503.01962 by Gabriele Franciolini, Ryusuke Jinno, Yann Gouttenoire.

Figure 1
Figure 1. Figure 1: Time evolution of the GI curvature perturbation [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Probability distribution of the density contrast in the co [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: SIGW contribution to the GWs from FOPT. The solid line [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

Understanding whether primordial black holes form during strong first-order phase transition (FOPT) is a crucial open question in cosmology. We address this using a fully covariant formalism to study cosmological perturbations, highlighting previously overlooked gauge dependencies. We show that non-covariant treatments can overestimate primordial black holes and scalar-induced gravitational waves. Once gauge dependencies are accounted for, both signals are strongly suppressed, with direct implications for the FOPT interpretation of the Pulsar Timing Array signal.

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

0 major / 1 minor

Summary. The paper claims that a fully covariant formalism for cosmological perturbations during first-order phase transitions reveals gauge dependencies overlooked in prior non-covariant treatments. These dependencies cause overestimation of primordial black hole (PBH) formation and scalar-induced gravitational waves (SIGW). Once accounted for, both signals are strongly suppressed, with direct implications for ruling out a first-order phase transition (FOPT) origin for the Pulsar Timing Array (PTA) signal.

Significance. If the result holds, the work would be significant for early-universe cosmology and gravitational-wave phenomenology. It would substantially weaken the case for FOPT as a source of observable PBH or the PTA background, redirecting attention to alternative mechanisms. The covariant approach, if rigorously implemented, would represent a methodological advance in handling gauge artifacts in phase-transition perturbation calculations.

minor comments (1)
  1. [Abstract] The abstract states the suppression result but provides no indication of the specific phase-transition parameters, bubble-wall velocity, or transition strength used to reach the 'strongly suppressed' conclusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and summary of our manuscript. The referee correctly identifies the central claim: a covariant treatment of curvature perturbations during first-order phase transitions exposes gauge artifacts in earlier non-covariant calculations, resulting in strong suppression of both primordial black hole formation and scalar-induced gravitational waves. We agree that this has significant implications for the interpretation of the PTA signal. Below we respond to the major comments; none were listed in the report provided, so we have no point-by-point replies. We are happy to address any additional questions the referee may raise.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract contains no equations, derivations, or explicit modeling steps that could be inspected for self-definition, fitted inputs presented as predictions, or load-bearing self-citations. Without access to the manuscript's specific equations or cited prior results in the derivation chain, no reduction of outputs to inputs by construction can be exhibited. The central claim rests on the application of a covariant formalism, but absent any quoted technical steps, the derivation chain cannot be shown to collapse into its own assumptions. This is the expected outcome when source material supplies no load-bearing technical content to analyze.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5609 in / 940 out tokens · 42675 ms · 2026-05-23T01:16:29.583268+00:00 · methodology

discussion (0)

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Forward citations

Cited by 7 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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  2. Reviving primordial black hole formation in slow first-order phase transitions

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    Primordial black hole formation from slow first-order phase transitions remains viable if slow reheating after supercooling creates an early matter-dominated era allowing small overdensities to grow, producing black h...

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

    astro-ph.CO 2026-05 unverdicted novelty 6.0

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

    hep-ph 2026-05 unverdicted novelty 5.0

    Slow reheating after a supercooled first-order phase transition allows an early matter-dominated era in which small curvature perturbations grow sufficiently to form primordial black holes.

  5. Phenomenology of Vector Dark Matter produced by a First Order Phase Transition

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    Dark sector first-order phase transitions near 10 MeV can substantially modify vector dark matter relic densities away from standard thermal freeze-out predictions, with distinct mass windows and calculable gravitatio...

  6. Thermodynamical uncertainties for primordial black holes from cosmological phase transitions

    hep-ph 2025-06 unverdicted novelty 5.0

    A state-of-the-art thermodynamic analysis of supercooled phase transitions yields a universal lower bound β/H_* ≃ 5 and shows that viable PBH dark-matter parameter space in classically conformal gauge-Higgs theories i...

  7. Primordial black holes and magnetic fields in conformal neutrino mass models

    hep-ph 2025-05 unverdicted novelty 5.0

    Conformal U(1)' seesaw models produce PBHs contributing to dark matter and helical magnetic fields at seesaw scales of 10^4-10^11 GeV, with observable GW, microlensing, and Hawking signals at LISA, Roman, and future g...

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