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This paper claims current CMB temperature data cap a completed slow dark-sector phase transition at about 1% of the dark energy via anisotropic photon redshifting at a stochastic first-encounter surface.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-04 22:51 UTC pith:EZXXKBN5

load-bearing objection Novel constraints on late-time phase transitions via a first-encounter-surface mechanism that are orders of magnitude stronger than the Hubble-budget bound, but the central quadratic scaling rests on an asserted distance-matching condition; send to a referee who will push on it. the 3 major comments →

arxiv 2509.07076 v1 pith:EZXXKBN5 submitted 2025-09-08 hep-ph astro-ph.CO

Boiling After the Dust Settles: Constraining First-Order Phase Transitions During Dark Energy Domination

Seth Koren , Yuhsin Tsai , Runqing Wang This is my paper
classification hep-ph astro-ph.CO
keywords first-order phase transitiondark energyCMB temperature anisotropyintegrated Sachs-Wolfe effectbubble nucleationfirst-encounter surfacevacuum energylate-time cosmology
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that a first-order phase transition in the dark sector during the last few billion years—when vacuum energy already dominates—is far more constrained by CMB temperature anisotropies than by measurements of the Hubble expansion rate. The mechanism is that bubble nucleation is stochastic, so different lines of sight enter the true vacuum at slightly different redshifts; the resulting corrugated "first-encounter surface" makes CMB photons redshift by different amounts across the sky. In the minimal case where the released latent heat becomes free-streaming dark radiation, the authors derive a quadratic relation δz0 ∝ δz_pt² and project the induced redshift power spectrum onto low multipoles. Their central bound is r ≲ 10⁻⁵ (β/H⋆)², where β/H⋆ measures the transition's inverse duration in units of the Hubble rate at completion: a transition with β/H⋆ ≲ 25 can release no more than about 1% of the dark energy. This matters because Hubble-expansion measurements alone miss everything below roughly ten percent of the dark energy, so the CMB turns an otherwise nearly invisible late-time event into a testable signal.

Core claim

Late-time first-order phase transitions leave a CMB anisotropy even if the dark radiation they release is homogeneous afterwards. Fluctuations in local transition time enter the observed photon redshift only at second order, δz0 ≈ δz_pt² rΩΛ /[(1+z̄_pt)(ΩΛ+Ωm(1+z̄_pt)³)^{3/2}], because the equality of comoving distances before and after the transition cancels the linear term. Squaring this with the finite-bubble-statistics spectrum Pδt(k) gives an induced CMB power spectrum peaked at ℓ<20 with amplitude ∝ r²(β/H⋆)⁻⁴. Equating that to the observed scalar-perturbation amplitude yields r ≲ 10⁻⁵ (β/H⋆)²: slow completed transitions are capped near 1% of dark energy, negative-vacuum endpoints requ

What carries the argument

The load-bearing object is the "first-encounter surface": the corrugated surface at z̄_pt + δz_pt(θ,φ) where each CMB photon first enters the true vacuum. Two identities carry the argument: the comoving-distance matching condition χ_AO = χ_BO (Eq. 8), which cancels the linear term and leaves δz0 quadratic in δz_pt, and the projection of the resulting redshift power spectrum Pδz0(k)—built from the finite-bubble-statistics spectrum Pδt(k)—onto CMB multipoles via spherical Bessel functions (Eq. 13). The whole effect is a late-time analogue of the integrated Sachs-Wolfe effect: stochastic bubble nucleation sources metric perturbations near z̄_pt, and the second-order relation sets how strongly t

Load-bearing premise

The limit rests on assuming that a patchy phase transition changes only the expansion rate, not the paths photons take, so every photon that starts from the same redshift covers exactly the same spatial distance; if the transition bends photon trajectories at first order, the quadratic redshift relation and the r≲10⁻⁵(β/H⋆)² bound no longer follow.

What would settle it

One decisive check is to compute or simulate photon geodesics through a stochastic bubble-nucleation history at z≈0.1–0.3 and test the paper's matching condition χ_AO = χ_BO (Eq. 8) at first order in δz_pt: any linear-in-δz_pt contribution to δz0 invalidates the quadratic formula and the derived bound. Observationally, a low-multipole (ℓ<20) CMB temperature measurement with uncertainties smaller than the predicted D_l^TT,pt for a given (r, β/H⋆, z̄_pt) would confirm or exclude the signal; an absence of the predicted r²(β/H⋆)⁻⁴ scaling would likewise settle the claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • A completed slow phase transition (β/H⋆ ≲ 25) in the dark-energy-dominated era can release at most about 1% of the dark energy; anything larger would show up as an excess in the low-multipole CMB temperature spectrum.
  • Naive Hubble-expansion constraints would allow r as large as ~0.65 for a transition at z̄_pt = 0.3, while the CMB anisotropy bound is much stronger for β/H⋆ ≲ 200.
  • Negative-vacuum (r > 1) endpoints are not excluded, but only for fast transitions with β/H⋆ ≳ 300; if β/H⋆ ≲ 500, the universe must survive at least 14 Gyr before crunching.
  • Because the signal is quadratic in the transition-time fluctuation, the bound scales as r ∝ (β/H⋆)², so faster transitions evade the CMB constraint more easily.
  • The constraints are minimal: converting latent heat into free-streaming radiation homogenizes density perturbations, and the authors note that slow-moving final products would preserve perturbations and yield even stronger bounds.

Where Pith is reading between the lines

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

  • An immediate extension is to compute the same first-encounter-surface signal for transitions that reheat into massive or self-interacting particles; the paper notes such cases keep density perturbations alive, so the bounds presented here should be treated as lower limits on the observable signal.
  • Because the observable is δz_pt², the induced CMB map is not Gaussian: a skewness or bispectrum search at ℓ<20 could separate a phase-transition origin from ordinary adiabatic integrated Sachs-Wolfe anisotropy.
  • The same corrugated-entry-surface logic should apply at other redshifts—for example in the 1 eV to 10 keV window where other studies bound dark-sector phase transitions—so the quadratic redshift relation offers a template for a full timeline of first-order phase-transition constraints.
  • If a late-time transition is incomplete, the anisotropy calculation does not apply as written, but photons would still cross a patchwork of false and true vacuum; quantifying that regime is the natural next step and could close the remaining allowed window.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper considers a late-time first-order phase transition in a dark sector during dark-energy domination (z ≲ 0.3), with the latent heat converted into free-streaming dark radiation. It computes CMB temperature anisotropies generated by fluctuations in the redshift at which photons enter the true vacuum, described by a spatially varying δz_pt. Using Planck 2018 data through a simplified three-bin χ² analysis, it derives an approximate upper bound r ≲ 10^{-5}(β/H_⋆)^2 on the released fraction of dark energy, and it discusses transitions to negative vacuum energy (r > 1), concluding that β/H_⋆ ≳ 300 is required and that β/H_⋆ ≲ 500 still leaves at least 14 Gyr before a crunch. The central analytic results are Eq. (9), δz_0 ∝ δz_pt^2, and Eq. (11), with the derivation in Appendix A.

Significance. Late-time FOPTs are a plausible and understudied dark-sector signature, and translating the stochastic completion time into CMB anisotropies is a fresh and potentially useful angle. If the derivation were sound, the bound r ≲ 10^{-5}(β/H_⋆)^2 would be an important, broadly applicable constraint, and the negative-vacuum/crunch discussion is a nice addition. The paper is explicit about several approximations: latent heat as free-streaming radiation, fixed Planck fiducial parameters, a three-bin χ², and the use of a published P_δt(k) from Ref. [43]. It also provides an explicit scaling relation that can be checked numerically. However, the central quantitative claim currently rests on an unproven distance-matching condition and an unjustified Gaussian truncation, so the headline numbers are not yet established.

major comments (3)
  1. [Section III, Eq. (8)] The equality χ_AO = χ_BO is asserted with the sentence that 'the total comoving distance from the same initial redshift to us must remain the same,' but no derivation is given. In the two histories being compared, H(z) differs over a finite redshift interval; a comoving distance in a modified background is not an invariant quantity, and the photon redshift is determined by the null geodesic equation, not by an independent matching condition. This condition is precisely what removes the linear-in-δz_pt term in the expansion leading to Eq. (9). If a term ∝ r δz_pt survives in δz_0, then P_{δz0}(k) scales as r^2 P_{δt}(k) rather than the convolution in Eq. (12), changing the ℓ-shape and the β/H scaling of the bound in Eq. (15). The appendix uses Eq. (8) as input and therefore does not resolve the issue. The authors should either derive Eq. (8) from a consistent gauge/matching prescription o
  2. [Appendix A, Eq. (A10)] The four-point function is truncated to products of two-point functions with the statement that only cross-correlations between x and y contribute. This is a Gaussian approximation for δz_pt. However, the text in Section III states δz_pt ≥ 0, so δz_pt cannot be an exactly Gaussian field; in a bubble-nucleation process the connected part of the four-point function is generically nonzero and can be comparable to the disconnected part at relevant scales. No estimate or bound on the connected contribution is given. Because the normalization of D_ℓ^{TT,pt}, and hence the 2σ curve in Fig. 3, depends on this truncation, the numerical results are not fully supported.
  3. [Section IV and Fig. 3] The statement that 'for β/H_⋆ ≲ 500, the universe will not crunch for at least 14 Gyr' is obtained by combining the t_end formula with the upper bounds r ≤ 2 and r ≤ 3 from the CMB anisotropy estimate. Those bounds inherit the uncertainties in Eqs. (8) and (A10). The 14-Gyr claim should therefore be presented as conditional on the anisotropy analysis, not as a robust consequence of the phase-transition dynamics alone.
minor comments (4)
  1. [Section III, Eq. (4)] The text around Eq. (4) contains an inconsistency: the fiducial values are given as ΩΛ = 0.69, Ωm = 0.31 in one place, but the duplicated passage states ΩΛ = 0.31, Ωm = 0.69. The correct values should be used consistently.
  2. [Figure 2 caption] The caption includes stray editorial text ('SK : ¯zpt, also above in Figure with r=0.1') and duplicated figure text. This should be cleaned before submission.
  3. [Section III, Eq. (14)] The three-bin χ² uses Planck error bars that are correlated and derived from a baseline cosmology. The authors acknowledge the limitation, but the resulting upper bound should be described explicitly as an order-of-magnitude estimate rather than a rigorous confidence limit.
  4. [General] The statement δz_pt ≥ 0 and the subsequent treatment of δz_pt as a Gaussian random field in Appendix A should be reconciled; this is related to the major comment on Eq. (A10).

Circularity Check

0 steps flagged

No significant circularity: the central bound is derived from an external P_δt spectrum and a data χ², not from a fit or self-referential definition.

full rationale

The derivation chain is: (i) P_δt(k) from ref. [43] (external finite-bubble-statistics calculation), (ii) Eq. (8) matching condition defining δz0, (iii) Eq. (9) expansion giving δz0 ∝ r δz_pt², (iv) Eq. (11) power spectrum P_δz0 built from two P_δt factors, (v) Eq. (13) projection onto CMB and Eq. (14) χ² against Planck error bars, yielding Eq. (15). None of these steps fits a parameter to the target bound: r is bounded, not fitted, and P_δt is imported from a published paper, not adjusted. The one author overlap with ref. [43] (Tsai) is real, but ref. [43] is an independent parameter-free first-principles calculation with stated assumptions that do not include the present result, so it counts as real evidence and does not raise the circularity score. The distance-matching condition Eq. (8) is asserted rather than derived from perturbation theory, and the Appendix truncates the four-point function to Gaussian products (Eq. A10); these are physical/technical assumptions whose failure would change the numerical bound, but they are not circular in the sense of reducing to the output by construction. The paper also flags its own regime of validity (completed PT, gray region, free-streaming radiation), which is consistent with a non-circular derivation.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central claim rests on ref. [43]'s bubble-statistics power spectrum, the distance-matching condition of Eq. (8), the Gaussian truncation of the four-point function, and the free-streaming-radiation assumption. The one manually chosen parameter that maximizes the signal is v_w=1.

free parameters (2)
  • bubble wall velocity v_w = 1 (assumed)
    Set to 1 (relativistic) for all main results; P_δt ∝ v_w^3 in Eq. (3), so this choice maximizes the CMB signal and the strength of the bounds. No scan over v_w is shown.
  • inverse transition duration β/H⋆ = N/A (scanned over 10-500)
    The bounds are quoted as a function of this parameter; it is not fitted to data but is a model parameter of the FOPT. The headline bounds are conditional on the chosen range.
axioms (5)
  • domain assumption The comoving distance from a fixed initial redshift to the observer is invariant under the PT (Eq. 8).
    Used in Section III to derive δz0 ∝ δz_pt^2 (Eq. 9). If the PT alters the photon trajectory beyond the background expansion shift, the signal amplitude and shape change.
  • domain assumption The PT-time fluctuation field δt_c(x) is effectively Gaussian, so the four-point function in Eq. (A10) reduces to products of two-point functions.
    Appendix A: "Only the two-point correlations between perturbations at x- and y contribute..." The connected (non-Gaussian) part is dropped without an estimate of its size.
  • domain assumption The latent heat is converted to free-streaming dark radiation that homogenizes on the bubble scale immediately after the PT.
    Section III and footnote 1: this is the minimal-signal assumption; the authors note slow-moving fluids would produce larger anisotropies. The headline bounds are conditional on this choice.
  • domain assumption The power spectrum P_δt(k) from ref. [43] (Eq. 3 and Fig. 4) is valid for late-time PTs with β/H⋆≈10-500, including the sub-Hubble behavior.
    Adopted without rederivation in Section II and Appendix A. If the finite-bubble-statistics spectrum has a different normalization or shape at these redshifts, the r bounds rescale.
  • domain assumption Planck 2018 ΛCDM best-fit parameters are used and held fixed (H0=68, ΩΛ=0.69, Ωm=0.31).
    Section III: the PT signal is added incoherently to ΛCDM, justified by the claim that the allowed homogeneous redshift shift is <1%. This avoids a full cosmological fit.

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Cite this review

Pith. "Pith review of Boiling After the Dust Settles: Constraining First-Order Phase Transitions During Dark Energy Domination." pith.science (2026). https://pith.science/paper/EZXXKBN5

@misc{pith2026250907076,
  author       = {Pith},
  title        = {Pith review of: Boiling After the Dust Settles: Constraining First-Order Phase Transitions During Dark Energy Domination},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EZXXKBN5}},
  note         = {Machine review of arXiv:2509.07076}
}
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read the original abstract

A first-order phase transition could occur in the late universe when vacuum energy begins dominating the energy density ($z \lesssim 0.3$) and convert some latent heat into other forms such as invisible radiation. This generic possibility also has concrete motivation in particle physics models which invoke a multitude of vacua to address theoretical puzzles. The na\"{i}ve constraint on such an event comes from measurements of the Hubble expansion rate, but this can only probe transitions involving $\mathcal{O}(10)\%$ of the dark energy. In this work, we show that significantly tighter constraints appear when accounting for phase transition fluctuations affecting CMB photon propagation anisotropically, akin to the integrated Sachs-Wolfe effect. For instance, if a completed phase transition has $\beta/H_\star\lesssim 25$, current CMB data limits the associated vacuum energy released to less than $1\%$ of the dark energy. A transition to negative vacuum energy (quasi-anti-de Sitter) is allowed only for $\beta/H_\star \gtrsim 300$. For $\beta/H_\star \lesssim 500$, the universe will not crunch for at least $14$ Gyr.

Figures

Figures reproduced from arXiv: 2509.07076 by Runqing Wang, Seth Koren, Yuhsin Tsai.

Figure 1
Figure 1. Figure 1: FIG. 1: Sketch of the origin of CMB fluctuations from a [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Upper bounds on a FOPT converting a fraction [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Dimensionless power spectrum of PT time [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

discussion (0)

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Forward citations

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