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REVIEW 3 major objections 5 minor 206 references

After a supercooled phase transition the cosmos is not immediately matter-dominated: gradients left by bubble collisions keep the fluid near radiation until expansion by roughly the wall Lorentz factor γ⋆, or until the field thermalises.

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2026-08-01 12:38 UTC pith:TNPIBQJI

load-bearing objection The qualitative claim is right and important—no immediate matter era after a supercooled FOPT—but the headline delay amatter/a* ≃ γ* is an extrapolation that wants a sharper test. the 3 major comments →

arxiv 2607.19469 v1 pith:TNPIBQJI submitted 2026-07-21 hep-ph astro-ph.CO

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

Henda Mansour , Yann Gouttenoire , Felix Kahlhoefer This is my paper
classification hep-ph astro-ph.CO PACS 98.80.Cq
keywords first-order phase transitionssupercoolingequation of statebubble collisionsreheatinglattice simulationsgravitational wavesprimordial black holes
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 asks whether the universe enters an early matter-dominated era immediately after a supercooled first-order phase transition, as many reheating, dark-matter, primordial-black-hole and gravitational-wave studies assume. Using lattice simulations in one, two and three spatial dimensions, it shows the answer is generally no: bubble collisions leave behind a highly inhomogeneous scalar field whose persistent relativistic gradients give an equation of state between matter and radiation, controlled by the wall Lorentz factor at collision γ⋆. For γ⋆ ≳ 100 the post-collision fluid is within 10% of the radiation value ω = 1/3. Matter domination sets in only after an expansion a/a⋆ ≃ γ⋆ if the scalar modes free-stream, after a factor of a few if the field thermalises promptly without number-changing processes, or after ~10^3–4 if cannibal 3→2 reactions keep the gas warm. The delay reshapes relic dilution, gravitational-wave spectra, and the efficiency of primordial black hole formation.

Core claim

The central claim is that the equation of state after a supercooled first-order phase transition is neither matter-like (ω = 0) nor radiation-like (ω = 1/3) but an intermediate value set by the wall Lorentz factor at collision γ⋆. More relativistic walls deposit energy into higher momentum modes up to a cutoff k⋆ ≃ γ⋆mφ, and those hard modes keep the fluid radiation-like. The fitting function ω(γ⋆,a) (eq. 4.20), built from a broken power-law spectrum with a k³ infrared branch, a 1/k relativistic branch and a Gaussian UV cutoff, predicts that matter domination is delayed to a/a⋆ ≃ γ⋆ for free-streaming modes, to a factor of a few for prompt number-conserving thermalisation, and to ~10^3–4 whe

What carries the argument

The carrying object is the post-collision scalar power spectrum Δφ(k), modeled as a smooth broken power law (eq. 4.14): a causal k³ branch, a 1/k relativistic branch to the UV cutoff k⋆ ≃ γ⋆mφ set by the Lorentz-contracted wall width, and a Gaussian cutoff, with break momentum kp ≃ 0.5–0.7 mφ. The virial theorem ⟨ρkin⟩ ≃ ⟨ρgrad⟩ + ½⟨φV,φ⟩ and independent WKB mode redshift (relativistic modes dilute as a⁻², non-relativistic as a⁻³) feed the master formula ω = (1/d)∫d ln k (k/a)²Δφ / ∫d ln k [mφ² + (k/a)²]Δφ, yielding the time-dependent equation of state ω(γ⋆,a). Thermalisation is governed by the equilibrium EoS ωth(x) = K₂(x)/(3K₂(x) + xK₁(x)) for a scalar gas at x = mφ/T, with cannibal 3→2 r

Load-bearing premise

The results hinge on the assumed post-collision power spectrum — a k³ infrared branch, a 1/k relativistic branch, and a Gaussian cutoff at k⋆ ≃ γ⋆mφ — calibrated at γ⋆ ≲ 100 and extrapolated to γ⋆ up to 10^10, together with the free-streaming assumption that modes redshift independently until an uncomputed thermalisation epoch; if interactions or fragmentation reshuffle the spectrum at large γ⋆, the predicted delay amatter/a⋆ ≃ γ⋆ changes.

What would settle it

Run the single-bubble lattice experiment at γ⋆ ≳ 10^3 with resolution high enough to resolve the Lorentz-contracted wall, and measure the post-collision spectrum: if the break momentum kp departs from ~0.5–0.7 mφ, or the branch between kp and γ⋆mφ is not a 1/k power law with a Gaussian cutoff, the fitting function ω(γ⋆,a) and the delay amatter/a⋆ ≃ γ⋆ fail. Also run the expanding-background simulation at larger γ⋆ with negligible self-couplings: if ω(a) falls to zero before a/a⋆ = γ⋆, the free-streaming independent-mode-redshift picture is wrong.

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

If this is right

  • The common assumption of an immediate early matter-dominated era after a supercooled transition fails: for γ⋆ ≳ 100 the post-collision fluid has an equation of state within 10% of radiation, interpolating between ω = 0 and ω = 1/3.
  • If the scalar modes free-stream, the matter era begins only at a/a⋆ ≃ γ⋆; for strongly supercooled transitions γ⋆ can reach ~10^10, and a matter era requires the scalar to survive that long (Γφ ≪ H⋆/γ⋆²).
  • Prompt number-conserving thermalisation gives the earliest matter onset, amatter/a⋆ of only a few, fixed by the self-thermalisation temperature rather than γ⋆; cannibal number-changing interactions push the onset to ~10^3–4.
  • The delay shrinks the entropy injected at decay relative to the instantaneous-eMD approximation (eq. 5.22), weakening the dilution of pre-existing relics and lowering the maximal dark-matter mass reachable in dilution scenarios.
  • Gravitational-wave predictions change twice over: late entropy release redshifts the peak by D_late^(−1/3) and suppresses it by D_late^(−4/3), while the causal f³ infrared tail gains an f¹ stretch whose extent is set by when matter domination actually begins; primordial black hole formation from late-blooming patches becomes less efficient because pressure persists.

Where Pith is reading between the lines

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

  • If the 1/k branch extrapolates to γ⋆ ~ 10^10 as the paper assumes, the free-streaming delay will exceed the decay lifetime of most realistic scalars, so the default post-supercooling history may be an extended radiation-like phase rather than a matter era — entropy-dilution and PBH-enhancement scenarios would then apply only in a narrow parameter window.
  • The same broken-power-law mechanism should operate wherever relativistic expanding walls dump energy into a field — domain-wall annihilation, oscillon formation, or the end of inflation — making the γ⋆-to-EoS relation a semi-universal diagnostic that those simulations could test.
  • A concrete extension would be to compute the self-thermalisation rate of the post-collision spectrum: if the thermalisation epoch atherm grows with γ⋆, the cannibal branch of eq. (4.33) rather than free streaming would govern large-γ⋆ transitions, capping the delay near 10^4 instead of γ⋆.
  • The PBH consequence could be quantified by evolving curvature perturbations through the gradual EoS ω(a) of eq. (4.20) instead of an instantaneous matter step; the smoother pressure history should lower the collapse fraction from late-blooming overdensities.

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 / 5 minor

Summary. The paper asks whether the universe is matter-dominated immediately after a strongly supercooled first-order phase transition with delayed reheating. Using lattice simulations of colliding vacuum bubbles in 1+1, 2+1 and 3+1 dimensions, the authors show that the post-collision scalar field is highly inhomogeneous and has an equation-of-state parameter between matter and radiation that increases with the wall Lorentz factor at collision γ_*; for γ_* ≳ 100 it is within 10% of radiation. They model the post-collision power spectrum as a broken power law with a UV cutoff k_* ≃ γ_* m_φ, combine it with WKB redshifting of individual modes, and derive an analytic expression ω(γ_*, a) (Eq. 4.20). This predicts that, in the free-streaming case, matter domination begins only at a/a_* ≃ γ_*. They contrast this with prompt thermalisation (delay by a factor of a few, or ~10^3–10^4 with cannibalism) and discuss implications for entropy dilution, gravitational waves and primordial black holes.

Significance. If the central result holds, it corrects a common assumption in the supercooled-FOPT literature: a long-lived scalar after percolation does not automatically produce an early matter-dominated era, because persistent relativistic gradients keep ω close to 1/3. This directly affects PBH formation thresholds, GW spectral slopes and dark-matter dilution estimates. The paper has real strengths: it provides a public code (CoolBubble), performs simulations in three spatial dimensionalities, uses a virial-theorem framework to connect the power spectrum to the EoS, and is unusually honest about its limitations (atherm is a free parameter, the thermalisation rate is not computed, the expanding-universe test is limited). However, the headline quantitative prediction amatter/a_* ≃ γ_* rests on an extrapolation of a spectral template fitted at γ_* ≲ 100 to γ_* up to 10^10, and the direct expanding-universe check reaches only a/a_* ≃ 3.5 for γ_* ≃ 11. The qualitative conclusion is likely robust; the specific delay factor is not yet established to the same standard.

major comments (3)
  1. [§4.4, Eq. (4.22), Figs. 10–11] The central free-streaming prediction amatter/a_* ≃ γ_* is not directly tested in the regime where the transition occurs. The expanding-lattice run in Fig. 10 has ln(a) ∈ [0, 1.25], i.e. a/a_* ≲ 3.5, for γ_* ≃ 11, whereas the predicted transition is at a/a_* ≃ γ_*. The agreement shown in the upper-left panel of Fig. 11 therefore validates only the radiation-like plateau before the transition, not the onset of matter domination. Because Eq. (4.22) is the quantitative core of the abstract, this extrapolation should either be supported by an expanding simulation reaching a/a_* ≳ γ_* (or a controlled test of mode-independence in Eq. (4.6) over Δ ln a ∼ ln γ_*), or the claim should be explicitly labelled as an analytic extrapolation.
  2. [§4.2–4.3, Eqs. (4.13)–(4.21), Figs. 6, 9] The analytic EoS is calibrated to the same lattice data used for validation. The three parameters k_p, b and c are fitted from the simulations, and ⟨ω⟩ is shown in Figs. 6 and 9 without error bars. At γ_* ≃ 10 the interval between k_p ≃ 0.5–0.7 m_φ and k_* ≃ γ_* m_φ spans only about one decade, so the assumed 1/k branch and Gaussian cutoff are not uniquely determined by the data; Appendix C shows that tanh and arctan wall profiles give exponential rather than Gaussian form factors. Extrapolating to γ_* ∼ 10^10 therefore introduces an unquantified systematic uncertainty. The qualitative trend (larger γ_* → more radiation-like EoS) is robust, but the specific threshold γ_* ≳ 100 and the exact a/γ_* transition need either a first-principles derivation of the cutoff or a systematic exploration of alternative spectral templates.
  3. [§4.4, Eqs. (4.26), (4.33)] The thermalisation branch of the summary result depends on the free parameter atherm/a_*, and the thermalisation rate Γ_therm is not computed. Consequently Eq. (4.33) ranges from amatter/a_* ≃ γ_* to a few to 10^3–10^4 depending on an unspecified dynamical rate. The authors are transparent about this, but it means the paper gives a taxonomy of possibilities rather than a prediction for a generic supercooled model. I would ask for at least a schematic estimate of Γ_therm for the benchmark potential of Eq. (2.14) (for instance from the cubic self-coupling), or an explicit statement that the free-streaming branch is the only quantitative prediction and the thermalisation branches are illustrative.
minor comments (5)
  1. [§3.2, Fig. 6] Please add error bars or a discussion of run-to-run variance. The statement that results become nearly independent of d for γ_* > 10 is difficult to assess without uncertainty estimates.
  2. [§4.3, Eq. (4.15)] The text sets k_p ≃ m_φ before deriving Eq. (4.20), while the fits give k_p/m_φ ≃ 0.5–0.7. The difference should be discussed, since it affects the normalization of the non-relativistic branch.
  3. [§4.4, Eq. (4.26)] The optical-depth interpolation e^{-τ} is introduced without derivation. A sentence explaining that e^{-τ} represents the fraction of modes that have not yet thermalised would make the expression easier to interpret.
  4. [§5.3, Eq. (5.28)] The local slope formula uses ω_tot measured at horizon reentry. If ω_tot varies significantly during reentry, a time-integrated transfer function is needed; this standard caveat should be stated explicitly.
  5. [§4.4, Fig. 11] The text refers to the 'upper-left panel' of Fig. 11; since the figure has four panels with different γ_* values, please clarify which panel corresponds to γ_* = 11 and what the black points represent.

Circularity Check

1 steps flagged

Partially circular: the static EoS relation is a fit to the same lattice data used to validate it, and the free-streaming delay extrapolates that calibrated fit; but WKB kinematics, the expanding-lattice check, and the thermal branches give the central claim independent content.

specific steps
  1. fitted input called prediction [Sec. 2.2 (last paragraph), Sec. 4.3 before Eq. (4.18) and after Eq. (4.21), Fig. 9; feeding Eq. (4.33)]
    "The functional form of ω(γ⋆) can be motivated analytically, but contains free parameters that must be calibrated by simulations. ... We therefore parametrise the sum of the two regimes by introducing two positive dimensionless parameters b and c ... The best-fit curves to the simulation results of section 3.2 are shown in figure 9. The agreement confirms that eq. (4.21) provides a good description of ω(γ⋆) and predicts that for γ⋆ ≳ 100 the post-collision EoS is essentially radiation-like."

    The coefficients b,c entering the 'main analytical result' Eq. (4.20) are fitted to the same static-lattice EoS points that are then quoted as confirming Eq. (4.21); the agreement is therefore tautological for those points. The underlying spectrum template Eq. (4.14) also has kp fitted to the same simulations, and the cutoff k⋆ ≃ γ⋆mφ/√π is fixed by fitting the same spectra (App. C), so the γ⋆-dependence of the free-streaming delay amatter/a⋆ ≃ γ⋆ in Eq. (4.33) is largely encoded in these fitted inputs rather than emerging independently. The expanding-lattice run (Fig. 10) is a genuinely independent check but reaches only a/a⋆ ≈ 3.5 at γ⋆ ≃ 11, far short of the predicted transition; the γ⋆ ≳ 100 statements are extrapolations of the calibrated fit.

full rationale

The paper is largely self-contained: CoolBubble is the authors' own lattice code and is primary numerical evidence, not a self-citation. The self-citations present (e.g., refs. [12,63,78]) concern friction, PBHs, and curvature perturbations and are not load-bearing for the post-collision EoS derivation. No uniqueness theorem or ansatz is imported solely from the authors' prior work. The main EoS result is, however, a calibrated fit: b,c and kp are fitted to the same static lattice data used to demonstrate agreement, and the free-streaming delay amatter/a⋆ ≃ γ⋆ is an extrapolation of that fit to γ⋆ values up to 10^10, supported by only one expanding run at γ⋆ ≃ 11 reaching a/a⋆ ≃ 3.5. This is a fitting/extrapolation concern rather than a definitional identity or a self-citation chain. The WKB redshift mechanism and the thermalisation/cannibalism branches are independent physical ingredients, so the central qualitative conclusion does not reduce to its inputs. Score 4 reflects the one fitted-input step; it is not 6+ because the central claim has independent support from the expanding-lattice comparison and from thermal physics.

Axiom & Free-Parameter Ledger

7 free parameters · 7 axioms · 0 invented entities

The central claim rests on standard virial/WKB results plus a simulation-calibrated power-spectrum template. The main load-bearing inputs not derived in the paper are the broken-power-law spectrum with fitted kp, the fitted coefficients b and c, and the uncomputed thermalisation epoch. No new particles or forces are introduced.

free parameters (7)
  • b = 1.5 (d=3), 0.5 (d=2), 0.1 (d=1)
    Dimensionless coefficient for the nonrelativistic gradient-energy contribution in eqs. (4.18)–(4.19); obtained by fitting eq. (4.21) to the lattice EoS data in fig. 9.
  • c = 3.2 (d=3), 2.2 (d=2), 2.0 (d=1)
    Dimensionless coefficient for the nonrelativistic kinetic-energy contribution in eqs. (4.18)–(4.19); fitted together with b to the lattice data.
  • kp/mφ = 0.5–0.7
    Break momentum in the power-spectrum template eq. (4.14); determined by fitting the late-time lattice power spectra in figs. 7 and 16.
  • cφ = 4
    Potential-shape parameter in eq. (2.14); chosen as the benchmark, with robustness checked in appendix B. It changes the wall thickness and the scalar mass at the true minimum.
  • atherm/a⋆ = free parameter
    The epoch of thermalisation after the phase transition is not computed; it is treated as a free parameter in eq. (4.26) and in the summary eq. (4.33).
  • Mpl (expanding simulation) = 200 in lattice units
    The reduced Planck mass is lowered in the expanding-universe simulation (sec. 4.4) to make Hubble expansion appreciable within the box; this is a numerical compromise, not a physical input.
  • xfo = 30 (benchmark)
    Freeze-out value of x = mφ/T for the cannibal regime, chosen as a benchmark for fig. 11 and eq. (5.22).
axioms (7)
  • standard math Virial theorem for the average kinetic energy of a scalar field, eq. (2.8): ⟨ρkin⟩ ≃ ⟨ρgrad⟩ + (1/2)⟨φV,φ⟩.
    Used throughout to relate kinetic, gradient and potential energies when computing the equation of state.
  • standard math WKB mode evolution in an expanding universe, eqs. (4.4)–(4.6): ⟨|φk|²⟩ ∼ a^{-2}/k for k/a ≫ mφ and ∼ a^{-3}/mφ for k/a ≪ mφ.
    Basis for the redshift of the power spectrum and the derived delay amatter/a⋆ ≃ γ⋆.
  • domain assumption Runaway vacuum regime: plasma friction is negligible, so the wall Lorentz factor grows linearly with bubble radius, γ⋆ ≃ R⋆/Rn (eq. 2.24).
    The simulations and all main results are restricted to this regime; the paper explicitly does not treat the friction-dominated case.
  • domain assumption The scalar field dominates the energy density after the transition (α ≥ 1 and slow decay Γφ ≪ H).
    Needed for the scalar EoS to control the expansion history; the paper states this as the supercooled context.
  • ad hoc to paper Post-collision power spectrum has the smoothed broken-power-law form of eq. (4.14) with a k³ IR branch, 1/k mid branch, Gaussian UV cutoff, and kp ~ mφ.
    This template is fitted to the simulations and is the input to the analytic equation of state; it is not derived from first principles.
  • domain assumption In the free-streaming case, the scalar modes redshift independently and self-interactions are negligible until a separate thermalisation epoch.
    The central delay amatter/a⋆ ≃ γ⋆ follows from this assumption; thermalised cases are treated separately but the thermalisation rate itself is not computed.
  • ad hoc to paper Thermalisation can be described by an optical-depth transition e^{-τ} between the free-streaming and thermal EoS, eq. (4.26).
    Empirical interpolation used to model the crossover; the rate Γtherm and slope δ are introduced without a calculation.

pith-pipeline@v1.3.0-alltime-deepseek · 42806 in / 10528 out tokens · 108606 ms · 2026-08-01T12:38:12.730898+00:00 · methodology

0 comments
read the original abstract

We show that the answer is generally no, at least not immediately. Bubble collisions leave behind a highly inhomogeneous scalar field with persistent relativistic gradients, producing an equation of state between matter and radiation. Using lattice simulations in one, two and three spatial dimensions, we find that the equation of state is controlled by the wall Lorentz factor at collision $\gamma_\star$: walls with larger $\gamma_\star$ populate higher-momentum modes and drive the fluid closer to radiation. Matter domination begins only after these modes redshift, at $a/a_\star \simeq \gamma_\star$, or after the field thermalises through self-scattering and number-changing processes. This delay has direct implications for gravitational waves, primordial black holes and dark matter production.

discussion (0)

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Reference graph

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