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arxiv: 2505.09202 · v3 · submitted 2025-05-14 · ✦ hep-ph · astro-ph.CO· hep-ex

Angular momentum of vacuum bubbles in a first-order phase transition

Jan Tristram Acu\~na , Danny Marfatia , Po-Yan Tseng This is my paper

Pith reviewed 2026-05-22 15:58 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COhep-ex
keywords first-order phase transitionfalse vacuum bubblesangular momentumprimordial black holescosmological perturbationsdark sectordimensionless spinscaling relation
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The pith

False vacuum bubbles acquire dimensionless spins from 10^{-5} to 10 during dark-sector first-order phase transitions

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

This paper calculates the angular momentum of spherical false vacuum bubbles that form during a first-order phase transition in a dark sector, induced by cosmological perturbations. The angular momentum is obtained from the product of density and velocity perturbations, with the background quantities and transfer functions evolved through the transition. For phase transitions with temperatures from 10 keV to 100 GeV and a dark sector 0.1 to 0.4 times cooler than the visible sector, the dimensionless spin parameter s = J/(G_N M^2) spans values from order 10^{-5} to order 10. A scaling relation is found that connects the root-mean-square spin to the transition timescale, bubble wall velocity, and temperature ratio. This calculation is presented as the first step toward determining the spins of any primordial black holes that might form from the collapse of such bubbles.

Core claim

False vacuum bubbles induced by cosmological perturbations during a first-order phase transition acquire angular momentum equal to the product of density and velocity perturbations. By tracking the background evolution and transfer functions through the transition, the dimensionless spin parameter s = J/(G_N M^2) of bubbles of mass M ranges from O(10^{-5}) to O(10) for FOPTs between 10 keV and 100 GeV when the dark sector is 0.1 to 0.4 times cooler than the visible sector. A scaling relation exists between the root-mean-square spin value, the FOPT timescale, the bubble wall velocity, and the dark-to-visible temperature ratio.

What carries the argument

Angular momentum of spherical false vacuum bubbles computed as the product of density and velocity perturbations while evolving background quantities and transfer functions through the first-order phase transition

If this is right

  • Primordial black holes formed from collapse of these bubbles would carry initial spins distributed across the calculated wide range.
  • The scaling relation allows direct estimation of typical spins for varying FOPT timescales, wall velocities, and temperature ratios without repeating the full transfer-function calculation.
  • Spin values in this range would modify the expected Hawking evaporation rates and gravitational-wave emission patterns from any resulting black holes.
  • This result supplies the starting spin distribution needed to model the cosmological evolution and observational signatures of dark-sector primordial black holes.

Where Pith is reading between the lines

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

  • The spin range connects hidden-sector phase-transition parameters directly to macroscopic properties of potential dark-matter candidates in the form of spinning black holes.
  • Observations of spin distributions in black holes within the relevant mass window could constrain the temperature ratio and duration of an early dark-sector transition.
  • Relaxing the sphericity assumption in future calculations would test whether bubble wall collisions or mergers systematically shift the spin values upward or downward.

Load-bearing premise

Angular momentum of the bubbles is given exactly by the product of density and velocity perturbations, with bubbles remaining spherical while background quantities and transfer functions are tracked through the FOPT.

What would settle it

A full numerical simulation of bubble formation that includes non-spherical deformations and nonlinear perturbation growth during the phase transition would produce spin values outside the reported O(10^{-5}) to O(10) range for the same transition temperatures and temperature ratios.

read the original abstract

The formation of primordial black holes (PBHs) during a first-order phase transition (FOPT) in a dark sector has been of recent interest. A quantity that characterizes a black hole is its spin. We carry out the first step towards determining the spin of such PBHs, by calculating the spin of spherical false vacuum bubbles induced by cosmological perturbations. The angular momentum is given by the product of density and velocity perturbations. We carefully track the evolution of background quantities and calculate the transfer functions during the FOPT. We find that the dimensionless spin parameter $s = J/(G_{\rm N} M^2)$ of false vacuum bubbles of mass $M$ and angular momentum $J$, take a wide range of values from ${\cal{O}}(10^{-5})$ to ${\cal{O}}(10)$ for FOPTs between 10 keV and 100 GeV and a dark sector that is 0.1 to 0.4 times cooler than the visible sector. We also find a scaling relation between the root-mean-square value of the spin, the FOPT time scale, the bubble wall velocity, and the dark sector-to-visible sector temperature ratio.

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 / 1 minor

Summary. The paper calculates the angular momentum of spherical false vacuum bubbles induced by cosmological perturbations during a first-order phase transition (FOPT) in a dark sector. It reports that the dimensionless spin parameter s = J/(G_N M^2) for bubbles of mass M takes values from O(10^{-5}) to O(10) for FOPTs between 10 keV and 100 GeV with dark-to-visible sector temperature ratios of 0.1–0.4, and identifies a scaling relation for the root-mean-square spin involving the FOPT timescale, bubble wall velocity, and temperature ratio. Angular momentum is obtained from the product of density and velocity perturbations while evolving background quantities and transfer functions under maintained spherical symmetry.

Significance. If the central modeling assumptions hold, the work supplies the first quantitative estimate of spins for PBHs potentially formed from FOPT bubbles. This could inform PBH spin distributions, merger rates, and associated gravitational-wave or electromagnetic signatures. The explicit tracking of transfer functions through the FOPT is a methodological strength.

major comments (2)
  1. [Abstract] Abstract: The claim that angular momentum J is given exactly by the product of density and velocity perturbations while bubbles remain spherical is load-bearing for both the reported range up to O(10) and the scaling relation. Non-zero net angular momentum necessarily sources quadrupole or higher moments that can deform the bubble wall or alter the local expansion rate, potentially invalidating the spherical-symmetry premise and the linear transfer functions once s approaches or exceeds O(1).
  2. [Abstract] Abstract: No error bars, sensitivity tests to modeling choices, or validation of the transfer functions against the spherical-symmetry assumption are provided, leaving the upper end of the O(10) range and its dependence on the dark-sector temperature ratio without quantified robustness.
minor comments (1)
  1. The manuscript would benefit from an explicit discussion of the regime of validity of the linear perturbation treatment when the resulting spin parameter reaches O(1) or larger.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address the two major comments point by point below, clarifying the scope of our linear-perturbation calculation and outlining planned revisions to improve robustness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that angular momentum J is given exactly by the product of density and velocity perturbations while bubbles remain spherical is load-bearing for both the reported range up to O(10) and the scaling relation. Non-zero net angular momentum necessarily sources quadrupole or higher moments that can deform the bubble wall or alter the local expansion rate, potentially invalidating the spherical-symmetry premise and the linear transfer functions once s approaches or exceeds O(1).

    Authors: We agree that a non-zero net angular momentum will source higher multipoles and can induce deviations from perfect sphericity at second order. Our calculation is performed entirely within linear cosmological perturbation theory: we evolve the background quantities and first-order transfer functions while imposing spherical symmetry on the unperturbed bubble, then extract the angular momentum from the product of the linear density and velocity perturbations. This yields the leading-order estimate of the spin parameter s for the bubble at the end of the FOPT. For s ≳ O(1) the linear approximation necessarily breaks down and nonlinear evolution (including possible wall deformation) must be studied separately; we will add an explicit statement of this limitation and note that the upper end of our reported range should be viewed as an indication that significant spins are possible rather than a precise prediction. revision: partial

  2. Referee: [Abstract] Abstract: No error bars, sensitivity tests to modeling choices, or validation of the transfer functions against the spherical-symmetry assumption are provided, leaving the upper end of the O(10) range and its dependence on the dark-sector temperature ratio without quantified robustness.

    Authors: We accept that the present manuscript lacks quantitative error estimates and sensitivity tests. In the revised version we will (i) vary the FOPT duration, wall velocity, and temperature ratio over the ranges already considered and report the resulting spread in the rms spin, (ii) propagate the uncertainty in the amplitude of the primordial curvature perturbations into error bands on s, and (iii) add a dedicated paragraph discussing the regime of validity of the spherical-symmetry assumption within linear theory. A full numerical validation against three-dimensional effects lies beyond the analytic scope of this work but will be flagged as a direction for future study. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation evolves standard perturbations to outputs

full rationale

The paper begins from standard cosmological perturbation equations, states that angular momentum is given by the product of density and velocity perturbations, and evolves background quantities plus transfer functions through the FOPT while maintaining spherical symmetry. The reported range O(10^{-5}) to O(10) for s and the scaling relation with FOPT timescale, wall velocity, and temperature ratio are computed outputs obtained by varying input parameters such as the dark-to-visible temperature ratio; they are not fitted to the target spin values or defined in terms of the final result. No self-citations, uniqueness theorems, or ansatzes from prior author work are invoked to force the central claim by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard linear perturbation theory in an expanding universe, the assumption that bubbles remain spherical, and the choice of temperature-ratio and wall-velocity ranges as representative inputs rather than derived quantities.

free parameters (2)
  • dark-sector to visible-sector temperature ratio
    Input range 0.1–0.4 used to scan possible dark-sector models; not derived from first principles within the paper.
  • FOPT timescale and bubble wall velocity
    Parameters that enter the scaling relation; treated as free inputs characterizing the transition.
axioms (2)
  • domain assumption Linear cosmological perturbations evolve according to standard transfer functions through the FOPT.
    Invoked when tracking background quantities and calculating transfer functions during the transition.
  • domain assumption Angular momentum of a spherical bubble is the product of its density and velocity perturbations.
    Stated as the starting point for the angular-momentum calculation.

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