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How large are curvature perturbations from slow first-order phase transitions? A gauge-invariant analysis

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A gauge-invariant calculation shows that curvature perturbations from slow first-order phase transitions are far smaller than earlier separate-universe estimates, making primordial black hole formation from this mechanism very unlikely.

desk verdict Solid confirmation that slow-FOPT super-horizon inhomogeneities don't form PBHs once gauge issues are handled; the new fitting template and PTA analysis are useful, but the linearization assumption needs a check. read the letter →

arxiv 2601.14412 v2 pith:P5OK6JWT submitted 2026-01-20 hep-ph

Xiao Wang , Csaba Balázs , Ran Ding , Chi Tian This is my paper · ORCID
classification hep-ph
keywords first-orderphasetransitionscurvatureperturbationsprimordialblackholesgaugeinvarianceseparateuniversesupercooledscalar-inducedgravitationalwavesmulti-fluidperturbationtheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the super-horizon density contrasts produced when a slow, strongly supercooled cosmological first-order phase transition nucleates bubbles are much smaller than previously thought. Working with a two-fluid system of radiation and vacuum, the authors compute the gauge-invariant comoving curvature perturbation and find its amplitude is suppressed relative to earlier separate-universe results. Consequently, the density contrast distribution lies below the typical black-hole formation threshold, so primordial black holes are unlikely to form through this mechanism. The paper also provides a fitting formula for the curvature power spectrum and shows that scalar-induced gravitational waves from these perturbations are subdominant.

What carries the argument

The central object is the gauge-invariant comoving curvature perturbation R (equivalently zeta), evolved on super-Hubble scales via d(zeta)/dt = -H/(rho+P) deltaP_nad, where deltaP_nad is the total non-adiabatic pressure perturbation. For a two-fluid system of radiation and vacuum, the intrinsic non-adiabatic pressure vanishes because each fluid has a definite barotropic equation of state, leaving only the relative entropy perturbation between the fluids. The authors avoid gauge ambiguities by treating the density contrasts from separate-universe-style nucleation simulations as linear-order perturbations on a specific but unspecified time slicing, then use the gauge-invariant equations to co

What would settle it

A fully nonlinear simulation of bubble nucleation in an expanding universe that computes the comoving curvature perturbation without invoking the linear-slicing assumption would settle the issue; if it produced a density contrast distribution with a tail above about 0.45 for some parameters, the central suppression claim would be refuted.

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Extended reading notes

Core claim

The central claim is that the comoving curvature perturbation generated by super-horizon inhomogeneities from slow first-order phase transitions is much smaller than implied by previous separate-universe studies, and the resulting density contrast does not exceed the conventional primordial black hole formation threshold. Using gauge-invariant multi-fluid perturbation theory, the authors evolve the curvature perturbation on super-Hubble scales, accounting for the relative entropy perturbation between radiation and vacuum. They derive a fitting formula for the curvature power spectrum that scales as k^3 at small k and beta^-5 at large beta, but with an amplitude suppressed compared to earlier

Load-bearing premise

The calculation assumes that the density contrasts from separate-universe simulations can be approximated, at linear order, as perturbations on a specific but unspecified time slicing, so that the gauge-invariant multi-fluid formalism applies.

Editorial extensions

If this is right

  • Primordial black hole formation from super-horizon inhomogeneities at slow first-order phase transitions is strongly disfavored; the density contrast distribution stays below the critical threshold.
  • Scalar-induced gravitational waves from these curvature perturbations are subdominant compared to primary bubble-collision or sound-wave signals.
  • The derived fitting formula for the curvature power spectrum can be used to translate current and future constraints on primordial perturbations into constraints on phase transition parameters.
  • Future radio and dark-matter-halo observations may constrain both slow and fast phase transitions, with the suppressed spectrum shifting the reachable parameter space.
  • The gauge-invariant treatment provides a template for evaluating other cosmological observables arising from large-scale phase-transition inhomogeneities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the suppression holds, the window for primordial black hole dark matter from this specific mechanism closes, though other formation channels such as false vacuum islands remain unaffected.
  • The linear-slicing assumption could be tested by comparing against a fully nonlinear simulation of bubble nucleation; if nonlinear corrections are large, the suppression may change.
  • The fitting formula offers a ready way to forecast LISA-era constraints on supercooled transitions, which the paper only sketches via existing datasets.
  • The result underscores that any claim of black hole production from phase transitions must specify the gauge in which the density contrast is defined.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper claims that super-horizon curvature perturbations generated by slow, strongly supercooled first-order phase transitions are much smaller than earlier separate-universe estimates suggested, so primordial black hole production from these perturbations is unlikely. The authors use the multi-fluid gauge-invariant perturbation formalism of Refs. [32,33], feeding in density contrasts δ_NG obtained from the DeltaPT separate-universe simulation of Ref. [24]. They compute the comoving curvature perturbation R via the relative entropy perturbation S_Vr, present its variance for several β/H_n and k/k_max, derive a fitting formula for P_R(k) (Eq. 20), and use this template to discuss constraints from CMB, Lyman-α, FIRAS, PTA, SKA, LISA, and UCMHs, as well as scalar-induced gravitational waves and a NANOGrav 15-year fit. The central conclusion is that the δ_C distribution lies far below the adopted PBH threshold δ_crit ≈ 0.45.

Significance. If the main claim is correct, the paper resolves a gauge ambiguity that has affected recent estimates of PBHs and SIGWs from slow FOPTs, and it provides a useful fitting formula for phenomenological studies. The use of the standard multi-fluid gauge-invariant formalism is appropriate, and the paper is explicit about its key assumption. The numerical analysis with 50,000 realizations per parameter point is a strength, as is the inclusion of current and projected observational constraints and a PTArcade-based posterior analysis. However, the central result is load-bearing on a linearization assumption that is not quantitatively validated, and the simulation data/code are not shipped, so the core suppression claim is not yet fully demonstrable to the standards of a journal publication.

major comments (3)
  1. [Sec. IV.A; Eq. (18); Fig. 1] The central result depends on the assumption, stated in Sec. IV.A, that the separate-universe density contrasts δ_NG can be approximated at linear order as density contrasts on a specific but unknown time-slicing. While a linear time shift cancels in S_Vr, the linearization itself is not justified: the manuscript nowhere shows δ_NG ≪ 1. Figure 1 plots |δ_NG| and |9/4 δ_NG| on an axis extending to order unity, and no convergence test in δ is reported. Since the final |R| values are orders of magnitude smaller (~10^-4 or less), second-order corrections to the gauge transformation and to the mapping from a finite patch to a global perturbation enter S_Vr and δP_rel at O(δ^2) and could change the sign or magnitude of R. The conclusion that PBHs are unlikely is therefore not robust against this most obvious failure mode. A quantitative bound on δ_NG in the relevant (β/H_n, k/k_max) regime, or
  2. [Sec. IV.B; Fig. 2; Eq. (20)] No statistical uncertainties are given for σ_R^2. The claim that deviations from k^3 scaling at large β are 'likely attributable to the limited number of bubbles' is not supported by any convergence test or error estimate. Since Eq. (20) is fitted to these same σ_R^2 values, the fitting coefficients A=0.0038, a=0.043, b=1.77 inherit these unquantified uncertainties. The observational constraints in Sec. V and Table I are computed using this template, so the presented exclusion regions are not accompanied by a systematic error budget. Please provide error bars (e.g., jackknife or bootstrap over histories) and a convergence test in the number of bubbles/realizations, or state clearly that the template is a central-value fit.
  3. [Sec. V; Eq. (20); Table I] The fitting parameters in Eq. (20) are fitted to the same DeltaPT simulations that produce the central quantities, and the same template is then used to derive all observational constraints and the SIGW spectra in Sec. V. This is not circular for the direct PBH conclusion, which is computed from the simulated R distribution, but it limits the independent validation of the template. The manuscript would be stronger if the simulation input (or at least the σ_R^2 data points) were provided as supplementary material, and if the template fit were validated against out-of-sample parameter points. Without this, the Table I constraints should be seen as dependent on an unvalidated functional form.
minor comments (5)
  1. [Sec. IV.B; Fig. 2] The horizontal axis label in the right panel appears garbled: '1016 × 100 2 × 101' should read '10^0, 2×10^1' (or similar). Please fix this and ensure all axis labels are clear.
  2. [Sec. IV.A] The text states 'For each super-horizon wave number k > kmax', but the figures and earlier discussion concern modes k < k_max that re-enter after percolation. Please correct this apparent typo.
  3. [Eq. (18) and surrounding text] The cancellation of the divergence in S_Vr when ρ̇_r = 0 is described only in words. Since the prefactor in Eq. (17) also contains 1/(6H ρ̇), it would be helpful to show the explicit cancellation or state the limiting behavior when both ρ̇_r and ρ̇_V approach zero.
  4. [Eq. (20)] Typo: 'heaviside' should be 'Heaviside'. Also, the phrase 'Θ(k_max − k)' should be written with a space or consistent notation.
  5. [Table I] The ranges for β/H_n (e.g., '>~ 35−80') are not explained. Are these ranges due to different constraints inside a column, or to the adopted reference values? Please clarify in the text or caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central PBH-suppression claim is computed directly from the simulated density contrasts via standard gauge-invariant multi-fluid perturbation theory, not from the fitted template.

full rationale

The derivation chain is self-contained. The paper takes DeltaPT separate-universe density contrasts δ_NG (Ref. [24]) and, using the standard gauge-invariant multi-fluid formalism (Eqs. 9–18, citing Refs. [31–33]), computes the relative entropy perturbation S_Vr and integrates Eq. (13) to obtain ζ and R ≈ −ζ. The key assumption—that δ_NG can be treated, at linear order, as density contrasts on some unknown slicing—is an explicit physical premise (Sec. IV.A), not a reduction of the output to the input; the linear-order gauge invariance of S_Vr is what makes the computation well-defined. The PBH-unlikely conclusion is read directly off the simulated δ_C distribution in Fig. 3, not off the fitted template. Equation (20)'s coefficients A, a, b are fitted to the same simulations, but the paper labels it a fitting formula and uses it only for observational projections (Sec. V); it is not used to derive the central suppression claim. No load-bearing self-citation or imported uniqueness theorem appears; the authors' own previous work (Refs. [9–12]) is cited only for background on SGWB and phase-transition dynamics. The main risk is the unvalidated linearization premise, which is a correctness/physics concern, not circularity.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central calculation relies on standard multi-fluid perturbation theory plus a stated but unvalidated mapping from separate-universe simulation outputs to a time slicing. The only free parameters introduced for the paper's own purposes are the three fitting coefficients in Eq. (20).

free parameters (3)
  • A = 0.0038
    Amplitude of the fitting formula for P_R in Eq. (20), fitted to the DeltaPT simulation variances in Sec. IV.B.
  • a = 0.043
    Shape parameter in Eq. (20), fitted to the simulated σ_R^2(k, β/H_n) data.
  • b = 1.77
    Shape parameter in Eq. (20), fitted to the simulated σ_R^2(k, β/H_n) data.
assumptions (7)
  • ad hoc to paper Separate-universe density contrasts δ_NG are approximated, at linear order, as density contrasts on some specific but unknown time-slicing.
    Sec. IV.A explicitly states this assumption. It is necessary to map simulation outputs into the gauge-invariant multi-fluid formalism, and it is the main unresolved premise.
  • domain assumption The bubble nucleation rate follows Γ = H_n^4 exp(β t).
    Eq. (1), standard exponential nucleation model used throughout FOPT phenomenology.
  • domain assumption The false-vacuum fraction F(t) obeys the standard expression involving the integral of the nucleation rate.
    Eq. (3), taken from Refs. [28,29]; standard for radiation-filled backgrounds with vacuum domination.
  • domain assumption Vacuum energy behaves as a barotropic fluid with P_V = -ρ_V and c_V^2 = -1.
    Used to set the vacuum sound speed and to cancel the divergence in S_Vr via background quantities; standard in multi-fluid treatments of vacuum.
  • standard math Linear perturbation theory and the super-horizon relation R ≈ -ζ are valid for the computed density contrasts.
    Invoked in Sec. III and IV; the paper notes the assumption is 'at linear order', but does not quantify the size of neglected nonlinear terms.
  • domain assumption The PBH formation threshold is δ_crit ≈ 0.45.
    Adopted in Sec. IV.B from Refs. [44-46]; the conclusion that PBHs are unlikely depends on this threshold value.
  • domain assumption UCMH-based constraints assume WIMP dark matter annihilating into b bbar with m_DM = 1 TeV and s-wave thermal relic cross section.
    Stated in Sec. V.A and Fig. 4 caption; the authors note other assumptions would relax these limits.

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Pith. "Pith review of How large are curvature perturbations from slow first-order phase transitions? A gauge-invariant analysis." pith.science (2026). https://pith.science/paper/P5OK6JWT

@misc{pith2026260114412,
  author       = {Pith},
  title        = {Pith review of: How large are curvature perturbations from slow first-order phase transitions? A gauge-invariant analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P5OK6JWT}},
  note         = {Machine review of arXiv:2601.14412}
}
read the original abstract

When strongly supercooled cosmological first-order phase transitions (FOPTs) are sufficiently slow, super-horizon inhomogeneities can be generated. We compute these super-horizon curvature perturbations by employing a gauge-invariant, multi-fluid formalism. By resolving the gauge ambiguities inherent in conventional separate-universe simulations, we demonstrate that Primordial Black Holes are unlikely to be produced by these super-horizon inhomogeneities. We also derive a fitting formula for the resulting curvature perturbations and discuss potential observational constraints on FOPTs imposed by limits on primordial curvature perturbations and associated scalar-induced gravitational waves.

Figures

Figures reproduced from arXiv: 2601.14412 by the authors.

Figure 1
Figure 1. FIG. 1. Time evolution of the absolute value of the false vac [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Variance of the comoving curvature perturbations [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Distributions of density contrast of independently [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Various benchmark spectra [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. GWs spectra from [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Posterior distributions of phase transition parameters from NANOGrav-15 data, based on [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Can the universe be matter-dominated after a supercooled first-order phase transition?

    hep-ph 2026-07 conditional novelty 7.0 of 10

    After a supercooled first-order phase transition, the scalar field's equation of state is set by the bubble-wall Lorentz factor γ*, and matter domination is delayed until a/a* ≃ γ* in the free-streaming limit.

  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 of 10

    Tensor perturbations from FOPT and domain-wall sources are claimed to induce second-order scalar perturbations large enough to form primordial black holes, potentially all of the dark matter.

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