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arxiv: 2606.30740 · v1 · pith:AXC66JYVnew · submitted 2026-06-29 · ✦ hep-ph · astro-ph.CO

Dynamical evolution of the pressure on the bubble wall

Benoit Laurent , Miguel Vanvlasselaer This is my paper

Pith reviewed 2026-07-01 02:01 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.CO
keywords first-order phase transitionsbubble wallshydrodynamic obstructionheating waveJouguet velocitylocal thermal equilibriumgravitational waves
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The pith

Heating waves form too slowly to stop bubble walls near Jouguet speed

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

The paper investigates how pressure on an expanding bubble wall changes with time during a first-order phase transition. It shows that the heating wave in the plasma takes longer to form than the wall needs to accelerate, so the usual assumption of an already-steady fluid profile fails near the Jouguet velocity. This produces a higher threshold for the driving pressure that still allows deflagrations rather than detonations or runaway walls. The result matters for the gravitational-wave spectrum and for the efficiency of baryogenesis, both of which depend on the final wall velocity.

Core claim

In local thermal equilibrium the pressure on the bubble wall is found by solving the hydrodynamic equations without assuming a stationary profile. The formation time of the heating wave often exceeds the wall acceleration timescale, invalidating steady-state predictions near the Jouguet velocity. A revised criterion for the maximal driving pressure is derived analytically and confirmed by simulations; it shows that hydrodynamic obstruction is less restrictive than steady-state LTE predictions suggest.

What carries the argument

Time-dependent hydrodynamic evolution of the fluid pressure on the accelerating bubble wall under local thermal equilibrium; it computes the opposing pressure by following the delayed development of the heating wave instead of assuming an instantaneous steady state.

If this is right

  • Hydrodynamic obstruction is weaker than steady-state LTE calculations had indicated.
  • More walls reach detonation or runaway regimes than previously expected.
  • The boundary separating deflagration/hybrid from detonation/runaway shifts to higher driving pressures.
  • Gravitational-wave and baryogenesis predictions that rely on wall velocity must be recomputed.

Where Pith is reading between the lines

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

  • The dynamical treatment could be extended to models with significant departures from local thermal equilibrium to check whether the conclusion holds.
  • This shift in allowed velocities may enlarge the viable parameter space for dark-matter production linked to the same phase transition.
  • Full lattice simulations of the scalar field plus plasma could directly extract wall speeds to test the new criterion.

Load-bearing premise

Local thermal equilibrium holds throughout the wall's acceleration phase.

What would settle it

A hydrodynamic simulation that measures the heating-wave formation time against the wall-acceleration time near the Jouguet velocity; if formation is always faster than acceleration, the central claim is false.

Figures

Figures reproduced from arXiv: 2606.30740 by Benoit Laurent, Miguel Vanvlasselaer.

Figure 1
Figure 1. Figure 1: Left panel: Scatter plot of the value of α max p defined in Eq. (2) as a function of αp. The red markers are runaway solutions, while the blue markers are deflagration or hybrid solutions that have reached a terminal velocity. For the crosses (dots) α max p is computed using the first (second) term in the min of Eq. (3). The details of this scan are described in Section 4.2. α max p is computed from Eq. (4… view at source ↗
Figure 2
Figure 2. Figure 2: Left panel: Early evolution of a scalar field profile of a O(3)-symmetric bubble in vacuum. Right panel: Lorentz factor of the wall as a function of its position computed with the full dynamical simulation of the scalar field and with Eq. (14). which satisfies the boundary condition γ vac R (R0) = 1. An important consequence of this solution is that the bubble acceleration is quite fast. If the pressure is… view at source ↗
Figure 3
Figure 3. Figure 3: v−, v+, αθ,+ and the shock velocity ξsh as a function of the wall velocity. The plot is separated in three regimes: detonations, for wall velocity larger than the Jouguet velocity, deflagration, for wall velocity smaller than the speed of sound, and hybrid, for the regime in between. and α+ ≡ αθ¯(T+), αn ≡ αθ¯(Tn), αp ≡ ν∆Veff(Tn) 3w+(Tn) , (24) with ν = 1 + 1/c2 s,−. The two definitions αn and αp are rela… view at source ↗
Figure 4
Figure 4. Figure 4: Left panel: Wall velocity ξ LTE w versus phase transition strength Ψ for varying latent heat parameters αp. Solid lines represent full calculations with temperature-dependent sound speed param￾eters; dashed lines employ the simplifying assumption c 2 s,± = 1/3 while for solid lines we draw the speed velocity directly from the potential (62). Results span αp ∈ {0.001, 0.01, 0.05, 0.1} with each color denoti… view at source ↗
Figure 5
Figure 5. Figure 5: Study of the approach to scaling of the for [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left panel: Time to reach the steady state tSS/R0 as a function of the wall velocity ξw, comparing Criterion 1 (fluid velocity at the wall) and Criterion 2 (maximum fluid velocity). Right panel: Same quantity using Criterion 2 only, for four combinations of the parameters (αp, Ψ). Vertical dotted lines indicate the Jouguet velocity vJ for each curve. We typically observe a peak in the formation time close … view at source ↗
Figure 7
Figure 7. Figure 7: Pressure evolution P(t) as a function of rescaled time t/R0 for different constant wall velocities ξw ∈ [0.4, 0.85] with αp = 0.005, and Ψ = 0.95 (Left), Ψ = 0.9 (Right). The simulations show rapid transient dynamics at early times, followed by convergence to a quasi-steady state by t/R0 ≈ 10. Each color represents a different wall velocity. evolution of the pressure PLTE(t) for six different wall velociti… view at source ↗
Figure 8
Figure 8. Figure 8: Left panel: On the left panel we show the evolution of the pressure on the bubble using Eq. (29) for different accelerations. The vertical dashed line shows the Jouguet velocity. Right panel: We show the boundary in the plane (Ψ, αp) between the runaway (above the contour line) and the terminal velocity (below the contour line) regime for different acceleration histories. We compare this boundary with the … view at source ↗
Figure 9
Figure 9. Figure 9: Examples of solutions with [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Fraction of points classified incorrectly as a function of the fitting parameter [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Error in the terminal wall velocity as a function of the number of grid points [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: α SS p,max computed within the template model with c 2 s,+ = 1/3 and c 2 s,− = Ψ/(2+Ψ), together with the small 1 − Ψ expansion. We now have all the ingredients needed to determine what value of αn corresponds to this strongest-hybrid limit. First, one can eliminate α+ from Eqs. (70), which leads to a quartic equa￾tion in v+ with the solution v+ = 2 √ 3 sin  1 3 arcsin Ψ . (73) Using vsf,− = v+ and vsf,… view at source ↗
read the original abstract

First-order phase transitions in the early Universe are pivotal for gravitational wave production, baryogenesis, and dark matter generation. A central question is whether bubble walls reach a subjouguet or ultra-relativistic velocity - a distinction governed by hydrodynamic obstruction, where plasma heating counteracts the vacuum pressure driving the wall. Traditional analyses assume steady-state fluid profiles, but these may fail during the wall's acceleration phase. We study the dynamical evolution of the pressure on the bubble wall in local thermal equilibrium (LTE), combining analytical approximations with numerical hydrodynamic simulations. Our results reveal that the heating wave's formation time often exceeds the wall's acceleration timescale, invalidating steady-state predictions near the Jouguet velocity. We derive a revised criterion for the maximal driving pressure, which separates deflagration/hybrid regimes from detonations/runaway walls. This criterion, validated by simulations, shows that hydrodynamic obstruction is less restrictive than steady state LTE predictions suggest.

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 claims that steady-state fluid profiles fail to describe bubble wall dynamics during the acceleration phase of first-order phase transitions. Under the local thermal equilibrium (LTE) approximation, analytical approximations combined with numerical hydrodynamic simulations show that heating-wave formation time often exceeds the wall acceleration timescale near the Jouguet velocity. This yields a revised criterion for maximal driving pressure separating deflagration/hybrid from detonation/runaway regimes, with simulations indicating that hydrodynamic obstruction is less restrictive than steady-state LTE predictions.

Significance. If the dynamical result holds, the work would refine predictions for terminal wall velocities, with direct consequences for gravitational-wave spectra, baryogenesis efficiency, and dark-matter production mechanisms. The combination of analytics and simulations to test the steady-state assumption constitutes a concrete advance, and the revised criterion supplies a falsifiable threshold that can be checked against future lattice or hydrodynamic studies.

major comments (2)
  1. [Abstract] Abstract: the statement that simulations 'validate the revised criterion' is not accompanied by error bars on the extracted pressure evolution, convergence tests with respect to spatial resolution or time step, or explicit criteria for selecting post-acceleration data points; without these the claim that heating-wave formation exceeds acceleration timescale remains unquantified.
  2. [LTE closure section] LTE closure section: the central derivation solves the hydrodynamic equations under the assumption that local thermal equilibrium persists throughout the acceleration phase, yet no estimate or test is given for the ratio of thermalization time to acceleration time near the Jouguet velocity; if this ratio is O(1) the effective wall pressure and heating-wave speed would deviate from the reported LTE profiles.
minor comments (1)
  1. [Figures] Figure captions should explicitly state the numerical resolution and Courant number used for the hydrodynamic runs shown.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the changes planned for the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that simulations 'validate the revised criterion' is not accompanied by error bars on the extracted pressure evolution, convergence tests with respect to spatial resolution or time step, or explicit criteria for selecting post-acceleration data points; without these the claim that heating-wave formation exceeds acceleration timescale remains unquantified.

    Authors: We agree that the abstract's phrasing regarding validation would be strengthened by additional technical details. In the revised manuscript we will include error bars on the extracted pressure evolution, report convergence tests with respect to spatial resolution and time step, and state explicit criteria for selecting post-acceleration data points. These additions will allow a quantitative assessment of the heating-wave formation timescale relative to wall acceleration. revision: yes

  2. Referee: [LTE closure section] LTE closure section: the central derivation solves the hydrodynamic equations under the assumption that local thermal equilibrium persists throughout the acceleration phase, yet no estimate or test is given for the ratio of thermalization time to acceleration time near the Jouguet velocity; if this ratio is O(1) the effective wall pressure and heating-wave speed would deviate from the reported LTE profiles.

    Authors: The LTE closure is an explicit modeling choice stated in the manuscript. We will add to the revised text an order-of-magnitude estimate of the thermalization-to-acceleration time ratio, based on typical electroweak plasma scattering rates, showing that the ratio remains much less than unity near the Jouguet velocity for the parameter range considered. This discussion will clarify the regime of validity of the LTE profiles during the dynamical phase. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation from independent hydrodynamic evolution

full rationale

The paper obtains its revised maximal-driving-pressure criterion by numerically solving the hydrodynamic equations under the LTE assumption and comparing the resulting heating-wave formation timescale to the wall acceleration timescale. This comparison and the derived separation between deflagration/hybrid and detonation/runaway regimes emerge directly from the time-dependent fluid profiles; they are not obtained by fitting parameters to the same data used for steady-state benchmarks, nor do they rest on self-citation chains or uniqueness theorems imported from the authors' prior work. The central claim is therefore self-contained against external hydrodynamic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the validity of the LTE closure during the transient phase and on the numerical scheme capturing the correct heating-wave propagation speed.

axioms (1)
  • domain assumption Local thermal equilibrium (LTE) holds during the wall acceleration phase
    The entire analysis is performed under the LTE assumption stated in the abstract.

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discussion (0)

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Reference graph

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