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The terminal speed of a bubble wall in local equilibrium is the speed at which the pseudopotential's extrema become degenerate.

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2026-08-03 19:42 UTC pith:S3PBNEKN

load-bearing objection Elegant new pseudopotential method for LTE bubble velocities, cleanly derived and sub-percent validated in one model, but the abstract oversells the hybrid claim and the key T(φ,φ')≈T(φ,0) approximation is tested only in a weakly supercooled regime. the 4 major comments →

arxiv 2511.22711 v2 pith:S3PBNEKN submitted 2025-11-27 hep-ph

Bubble velocities in local equilibrium from a pseudopotential

Martin M\"unzenberg , Carlos Tamarit This is my paper
classification hep-ph
keywords bubble wall velocityfirst-order phase transitionlocal thermal equilibriumpseudopotentialhydrodynamic backreactionscalar field equation of motionelectroweak phase transitioncosmological phase transitions
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 proposes a new method to compute the terminal speed of expanding vacuum bubbles in a first-order cosmological phase transition when the surrounding plasma is in local thermal equilibrium. The central idea is a 'pseudopotential' for the scalar field whose shape changes with the wall velocity; the difference between its two extrema equals the net outward pressure on the wall. Stationary bubbles correspond exactly to degenerate pseudopotential extrema, giving a simple algebraic condition for the terminal velocity. This bypasses solving the scalar equation of motion, assuming a tanh wall profile, or using a simplified bag equation of state. In tests on a Standard Model plus singlet model, the method matches full numerical solutions to about 0.5%.

Core claim

The paper establishes that in local thermal equilibrium, the terminal velocity v_w of a planar bubble wall is the value for which the pseudopotential V(φ), defined as an integral of the finite-temperature force along the field, has degenerate extrema: V(φ₊) − V(φ₋) = 0, where φ₊ and φ₋ are the field values at the extrema, coinciding with the minima of the ordinary thermal potential. This condition directly expresses the balance between the vacuum driving pressure and the hydrodynamic backreaction from temperature gradients across the wall. The identification follows from conservation of an energy function for the scalar field once the plasma temperature is treated as a function of the field

What carries the argument

The pseudopotential is defined by V(φ) = ∫₀^φ dφ' (∂V(φ',T(φ'))/∂φ'), where T(φ) is obtained from the hydrodynamic conservation equations by setting field gradients to zero. Its derivative equals the ordinary finite-temperature force, so its extrema coincide with the minima of the thermal potential. The crucial property is that, when the temperature's dependence on field gradients is neglected, the energy E = ½(φ')² − V(φ) is conserved along a stationary wall profile, implying that stationary walls have degenerate pseudopotential extrema. The method replaces the second-order scalar differential equation with a single algebraic condition on the wall velocity and upstream temperature.

Load-bearing premise

The method assumes that the plasma temperature obtained from the hydrodynamic equations depends only on the scalar field and not on its spatial gradient; if gradient dependence is non-negligible, the pseudopotential energy is not conserved and the degenerate-extrema condition no longer identifies stationary walls.

What would settle it

Compute, for a local-equilibrium model with a first-order transition and strong coupling, the dimensionless quantity ΔT = (½ d²T/d(φ')²|φ'=0 φ'²)/T(φ,0) along the wall profile; if this quantity is not small (say, above a few percent), the terminal velocity from ΔV=0 should deviate from the full solution of the scalar equation of motion by more than 0.5%, contradicting the paper's central claim.

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

If this is right

  • Bubble velocities in local equilibrium can be computed by scanning the wall velocity and upstream temperature until the pseudopotential extrema become degenerate, a much faster procedure than solving the scalar field equation at every step.
  • The method works directly from the finite-temperature effective potential, removing the need for the bag equation of state or other simplified parametrisations of the plasma, and without imposing a tanh ansatz for the wall profile.
  • The computed net outward pressure as a function of wall velocity exhibits a peak near the sound speed for deflagrations; the slope of the pressure around the stationary solutions indicates that deflagrations are stable while detonations are unstable, in line with earlier simulation results.
  • In the benchmark model, predicted wall velocities agree with full equation-of-motion solutions to about 0.5%, and the temperature correction from field gradients stays below the per-mille level along the wall profile.
  • The method provides a practical tool for scanning particle-physics model parameters to predict gravitational-wave signals and baryogenesis efficiencies without expensive dynamical simulations.

Where Pith is reading between the lines

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

  • The pseudopotential degeneracy condition might be reinterpreted as a variational principle or an extremal principle for steady-state interfaces, which could lead to existence proofs for stationary solutions in more general hydrodynamic settings.
  • The method's accuracy depends on the smallness of the second-order gradient correction to the temperature; a direct calculation of that correction for models with strong mass variation could reveal how far the 0.5% accuracy extends across parameter space.
  • The paper notes the difficulty in probing stationary hybrid solutions due to numerical sensitivity near the sound-speed condition; a more robust root-finding approach for that constraint might uncover stationary hybrids in parameter regions not explored here.
  • If out-of-equilibrium corrections to the scalar equation can be expressed as field-dependent mass terms (as the paper suggests for certain condensate contributions), the pseudopotential approach could be extended to non-LTE regimes, giving a cheap estimate of how non-equilibrium physics shifts wall velocities.

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

4 major / 5 minor

Summary. The paper proposes a new method for computing terminal bubble-wall velocities in first-order phase transitions under local thermal equilibrium (LTE). The authors define a 'pseudopotential' V̂(φ) by integrating ∂V/∂φ along the field profile using the hydrodynamic temperature T(φ,0), and argue that the difference ΔV̂ = V̂(φ₊) − V̂(φ₋) equals the net outward pressure on the wall. They derive that for stationary bubbles the two relevant extrema must be degenerate, ΔV̂ = 0, so that the wall velocity can be obtained without solving the scalar equation of motion, without a tanh profile ansatz, and without a bag-like equation of state. The method is illustrated in a Standard Model plus N complex singlet model. For N = 4 and λ_HS ∈ [0.70, 0.95], the pseudopotential predictions agree with full solutions of the scalar equation of motion at the ~0.5% level (Fig. 4). The authors also compute the net pressure as a function of v_w, finding stable deflagrations, unstable detonations, and no stationary hybrids in the tested parameter space.

Significance. If the method holds beyond the single tested benchmark, it offers a computationally efficient and profile-independent route to wall velocities in LTE, avoiding both the scalar-field shooting and the bag-model approximation. The analytic derivation leading to Eq. (24) is clean, and the numerical validation against independent exact solutions is a genuine strength: no parameters are fitted to produce ΔV̂ = 0, and the comparison with the full EOM is a nontrivial check. The stability interpretation of the pressure curve is physically appealing and connects to known results in the literature. However, the advertised generality is conditioned on the approximation T(φ,φ') ≈ T(φ,0), whose domain of validity is tested only in weakly supercooled transitions with very small temperature corrections. The paper is honest about this caveat, but the central claim of a general method would be materially strengthened by a second, stronger benchmark or a quantitative error estimate.

major comments (4)
  1. [Sec. 3, Eqs. (18)–(24) and Sec. 5, Fig. 4] The central identity ΔV̂ = 0 relies on replacing T(φ,φ') by T(φ,0). The paper's own caveat after Eq. (18) and in Sec. 7 acknowledges this. The numerical validation, however, covers only one weakly supercooled model (N = 4, λ_HS ∈ [0.70, 0.95]) with ΔT/T < 10⁻⁵. The argument that T has no linear term in φ' (Eq. 35) controls only the local temperature correction; the force error entering the EOM (33) is δF = (∂V/∂T)[∂T/∂h(h,h') − ∂T/∂h(h,0)], a derivative of the quadratic correction that can become amplified in thin-wall or strongly supercooled transitions. Thus the claim 'this allows to compute bubble velocities without solving the EOM' is currently established only in a narrow regime. I recommend adding a benchmark with a stronger transition (e.g., larger λ_HS, smaller N, or a model with stronger supercooling) or providing a quantitative bound on the gradient correction in terms of the w
  2. [Abstract and Sec. 6, Fig. 5] The abstract states that the paper confirms 'the dip in outward pressure found in the literature for hybrid bubbles', but Sec. 6 finds no stationary hybrid configurations and the maximum backreaction pressure occurs near the speed of sound in the deflagration branch, not in the hybrid/Jouguet regime. The abstract therefore overstates the hybrid result. Either soften the abstract to say that the pressure curve shows a peak near c_s and that no stationary hybrids were found, or clarify what exactly is meant by 'dip'. This is a presentation issue, but it affects the paper's advertised conclusions.
  3. [Sec. 6, 'Hybrids' paragraph] The absence of stationary hybrids is not established as a robust result. The authors state that 'we could only probe negative values of |v₋|−cₛ⁻, never truly reaching zero', so the hybrid constraint |v₋| = cₛ⁻ could not be satisfied within numerical accuracy. The conclusion 'there is no configuration ... with ΔV̂ = 0' is therefore a numerical limitation rather than a definitive no-go. The paper should present this as an inconclusive finding, possibly with a discussion of the numerical uncertainty, rather than as a confirmed property of the model.
  4. [Sec. 3, Eqs. (21)–(24)] The derivation shows that if a static solution of the approximate EOM (21) exists, then the energy E of Eq. (22) is conserved and ΔV̂ = 0 follows. The converse — that ΔV̂ = 0 with a barrier between the two extrema guarantees a finite-energy kink solution — is assumed but not proved. In practice the numerical checks validate the sufficiency in the tested cases, but a brief statement of the conditions under which degeneracy implies existence (e.g., monotone profile, barrier height) would make the method more rigorous.
minor comments (5)
  1. [Abstract] The term 'dip in outward pressure' is inconsistent with the body text, which describes a peak in the backreaction force. Please align the wording.
  2. [Sec. 2, after Eq. (11)] The constants c₁ and c₂ are not explicitly related to the asymptotic values at z → ±∞; a reader may initially confuse them with the pressure parameters in the bag model. A one-sentence clarification would help.
  3. [Sec. 5, Fig. 4] The left panel shows Δv_w/v_exact; please state explicitly the reference value (deflagration vs. detonation) and whether the deviation is signed or absolute. The caption says 'relative deviation' but the text uses Δv_w = |v_pseudo − v_exact|.
  4. [Sec. 5, Eq. (33)] The boundary condition h''(z) → 0 at z → −∞ is mentioned but not derived; it follows from the requirement that h− minimizes V(h,T(h,h')). This could be stated explicitly.
  5. [Sec. 7] The phrase 'corrections ... are thus expected to be small' is an expectation, not a proof; given that the approximation is the main assumption, this sentence could be strengthened by referencing the quantitative bound requested above.

Circularity Check

0 steps flagged

No circularity found: the pseudopotential degeneracy criterion is derived from the scalar equation of motion and independently benchmarked against full EOM solutions; self-citations are contextual only.

full rationale

The central criterion, terminal LTE bubble-wall velocities satisfy ΔVhat = Vhat(φ+) − Vhat(φ−) = 0, is not imposed or fitted. It is derived in Sec. 3: the pseudopotential is defined in Eq. (18) as an integral of ∂V/∂φ at T(φ), Eq. (20) makes dVhat/dφ equal to the force in the scalar EOM, energy conservation for Eq. (21) then gives Eq. (24). No parameter is adjusted to force ΔVhat = 0; v+ and T+ are scanned with the nucleation and degeneracy conditions, and the resulting wall velocity is compared in Sec. 5 with an independent numerical solution of the full EOM, Eq. (33), giving sub-percent agreement. The self-citations to [20,21] (including a coauthor) provide background on LTE hydrodynamic obstruction and entropy conservation, but the derivation here is self-contained and does not reduce to those references. The main caveat, T(φ,φ') ≈ T(φ,0), is explicitly stated in Sec. 3 and Sec. 7 and tested numerically for N = 4, λ_HS ∈ [0.70,0.95]; this limits the demonstrated generality of the method but is an honest scope restriction, not circularity. No equation or prediction reduces by construction to its input, so no circular step can be exhibited.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

The method rests on LTE/perfect-fluid, planar-static wall, negligible gradient dependence of T, and standard hydrodynamical matching. Model inputs (N, lambda_HS, m_S, mu, nucleation criterion) are illustrative, not fitted to the target result. No invented physical entities; the pseudopotential is a derived auxiliary function, not a new physical object.

free parameters (5)
  • Number of complex singlets N = N=4
    Illustrative model choice for the numerical validation; not fitted to the pseudopotential result.
  • Singlet-Higgs quartic coupling lambda_HS = 0.70-0.95
    Scanned input controlling transition strength; not fitted to target.
  • Singlet mass ratio m_S^2/m_W^2 = 0.0625
    Chosen by hand as a benchmark (Sec. 4).
  • Renormalization scale mu = m_W
    Standard choice in the one-loop effective potential; not fitted.
  • Nucleation threshold S3/Tnuc = approx 140
    Standard criterion for one bubble per Hubble volume; not fitted to the velocity prediction.
axioms (7)
  • domain assumption Local thermal equilibrium; plasma is a perfect fluid with stress tensor omega u^mu u^nu - eta^mu nu p.
    Used throughout from Eq. (3); out-of-equilibrium friction is neglected.
  • domain assumption Planar, static wall approximation and negligible Hubble expansion.
    Standard for constant-velocity bubble walls; invoked in Sec. 2 before Eq. (10).
  • domain assumption Field profile is monotonic in z, making the change of variables z to phi invertible in Eq. (18).
    Needed to define the pseudopotential as an integral over field values.
  • ad hoc to paper Negligible gradient dependence of the temperature: T(phi,phi') approx T(phi,0).
    Central approximation; justified by quadratic onset and numerical check, but not proven generally.
  • domain assumption One-loop finite-temperature effective potential with high-T expansion (Eqs. 28-32) is adequate.
    Standard perturbative treatment; used to build the example model.
  • domain assumption Entropy current conservation and hydrodynamic matching with self-similar profiles recover Tnuc far from the wall.
    Standard hydrodynamic matching framework from Refs. [25,26]; used in Sec. 5 and Appendix A.
  • domain assumption For hybrids, the condition |v_-| = c_s^- from Landau-Lifshitz instability analysis.
    Imported from Refs. [41,42] and used to close the hybrid system in Appendix A.3.

pith-pipeline@v1.3.0-alltime-deepseek · 21130 in / 13335 out tokens · 115020 ms · 2026-08-03T19:42:00.417601+00:00 · methodology

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read the original abstract

We present a new method to estimate terminal bubble velocities during first-order phase transitions in a plasma in local equilibrium. The method relies on calculating the extrema of a modified potential function for the scalar field undergoing the transition. The shape of this function, which we refer to as the ``pseudopotential'', changes with the wall velocity, and if the dependence of the fluid temperature on scalar gradients is weak -- which is confirmed to hold with high accuracy in concrete examples -- the difference in pseudopotential between two appropriate extrema gives the net outward pressure acting on the bubble wall. It then follows that the correct terminal bubble velocities are those that lead to degenerate minima in the pseudopotential. This allows to compute bubble velocities without having to solve the equation of motion of the scalar field, and in contrast to other methods this can be done without relying on simplified equations of state for the plasma or without choosing a specific ansatz for the scalar field profile. We illustrate the method in a singlet extension of the Standard Model, computing the net outward pressure as a function of the wall velocity. We confirm the dip in outward pressure found in the literature for hybrid bubbles, which implies that stationary deflagrations are stable, while their detonation counterparts are unstable.

Figures

Figures reproduced from arXiv: 2511.22711 by Carlos Tamarit, Martin M\"unzenberg.

Figure 1
Figure 1. Figure 1: Different types of expanding bubbles with corresponding qualitative fluid velocity profile. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the physical meaning of the pseudopotential. Its extrema coincide with [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pseudopotential in the deflagration (left) and detonation (right) regime for [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Validity of taking h ′ (z) = 0 in T(h, h′ ), demonstrated by explicit numerical calculation for N = 4. The left figure shows the relative deviation between the physical wall velocity obtained from the pseudopotential and by solving Eq. (33) demanding h ′′(zmin) = 0. The small deviation confirms that the temperature dependence on h ′ has a negligible effect. This is illustrated in the right plot, showing th… view at source ↗
Figure 5
Figure 5. Figure 5: Upper plot: Effective pressure as a function of the wall velocity for three different values of the coupling constant λHS. Solid lines indicate solutions where the extrema of Vˆ (h) are separated by a barrier, while dashed lines correspond to cases without a barrier, i.e. when one of the extrema is a maximum of Vˆ (h). The grey contour area depicts the segment in which hybrid solutions are expected to exis… view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the geometry of a detonation. Red arrows indicate velocities measured in [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the geometry of a deflagration with a shock front preceding the bubble wall. [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of the geometry of a hybrid profile that includes both detonation and defla [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗

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