⚛️ phenomenology

Benchmarking wall velocities in cosmological phase transitions: Fluid Ansatz and WallGo

This paper benchmarks two computational approaches for determining bubble wall velocities in cosmological phase transitions, finding that while the fluid Ansatz and WallGo agree for mild transitions, they diverge when scattering processes are included or transitions become strong, highlighting the need for higher-precision semi-classical computations and treatments beyond the WKB approximation for future gravitational wave observatories.

Original authors: Gláuber C. Dorsch, Marek Lewicki, Daniel A. Pinto

Published 2026-07-21

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CC BY 4.0

Original authors: Gláuber C. Dorsch, Marek Lewicki, Daniel A. Pinto

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, steaming pot of soup. In the very beginning, right after the Big Bang, this soup was hot and uniform, with all the ingredients mixed together perfectly. But as the universe expanded and cooled, something dramatic happened: it started to "freeze," much like water turning into ice. However, unlike water which freezes smoothly, the early universe underwent a "first-order phase transition." Think of this not as a smooth freeze, but as a sudden, violent snap where bubbles of a new, stable state (the "true vacuum") begin to form inside the old, unstable soup. These bubbles are like bubbles of air forming in boiling water, but instead of air, they are regions of space where the laws of physics have slightly changed.

As these bubbles expand, they push against the surrounding hot soup, creating a massive wall between the new and old states. The speed at which this wall moves is crucial. If it moves too slowly, the universe might not produce the matter we see today; if it moves too fast, it could create ripples in spacetime called gravitational waves that future telescopes might detect. To understand how fast this wall moves, scientists have to calculate how the particles in the hot soup push back against the wall, acting like a thick, sticky fluid slowing down a car. This is a complex problem involving the movement of trillions of particles, and for a long time, scientists have used two different "shortcuts" (or mathematical guesses) to solve it. One shortcut treats the soup like a flowing river (the "Fluid Ansatz"), while the other uses a grid of mathematical shapes to map the particles' behavior (the "WallGo" method).

This paper is a head-to-head race between these two shortcuts. The authors, Gláuber C. Dorsch, Marek Lewicki, and Daniel A. Pinto, wanted to see if these two different ways of calculating the wall's speed actually agree with each other. They set up a simulation using two different models of the early universe: one based on a standard version of particle physics with a "light" Higgs particle, and another based on a more complex theory called SMEFT that allows for stronger, more violent phase transitions.

The results are a mix of good news and a warning. When the phase transition is "mild"—meaning the energy released is small (specifically, when a parameter called $\alpha$ is less than 0.01)—both methods agree perfectly. It's as if two different navigation apps gave you the exact same route to the grocery store. This agreement is especially strong when they only look at one specific type of particle interaction (top quarks annihilating).

However, the story gets interesting when the phase transition gets "stronger" (when $\alpha$ gets larger, around 0.1 or more). In these high-energy scenarios, the two methods start to disagree significantly, with their predictions for the wall's speed differing by as much as 40% to 60%. The authors found that the "Fluid" method, which relies on assuming the particles' behavior is a simple, linear extension of their normal state, begins to break down. When they tried to fix this by adding more complex, "non-linear" corrections to the Fluid method, the predicted speed changed even more, moving further away from the WallGo result.

The paper suggests that this disagreement isn't just a calculation error; it might mean that the entire way we are modeling these strong transitions is hitting a wall. The "Fluid" method assumes the bubble wall is thick enough that particles move through it smoothly, but in strong transitions, the wall becomes very thin, making that assumption shaky. The authors conclude that while both methods are great for gentle transitions, we cannot trust them to give precise answers for the violent, strong transitions that future gravitational wave detectors are hunting for. To get the right answer for those extreme events, we might need to throw out the current shortcuts entirely and develop a completely new way to look at the physics.

Technical Summary: Benchmarking Wall Velocities in Cosmological Phase Transitions

Problem Statement
Accurately determining the terminal velocity ( $v_w$ ) of expanding vacuum bubbles during a cosmological first-order phase transition is critical for predicting observational signatures, such as gravitational waves (GWs) and baryon asymmetry. This velocity is determined by the balance between the vacuum driving pressure and the hydrodynamic friction exerted by the plasma. Calculating this friction requires solving the semi-classical Boltzmann equation for non-equilibrium particle distribution functions ( $\delta f$ ) in the vicinity of the bubble wall.

However, solving the full integro-differential Boltzmann equation is computationally prohibitive for model builders. Consequently, the community relies on approximations that reduce the problem to a tractable system of equations. Two dominant approaches have emerged:

The Fluid Ansatz: Assumes the non-equilibrium distribution retains the functional form of equilibrium distributions (Bose-Einstein/Fermi-Dirac) but with fluctuating parameters (chemical potential, temperature, velocity). This maps fluctuations to macroscopic hydrodynamic properties.
The Spectral Method (WallGo): Expands the non-equilibrium distribution function in a basis of orthogonal Chebyshev polynomials, implemented in the public code WallGo.

Despite their widespread use, no systematic comparison between these two methods had been performed to validate their consistency or identify regimes of divergence.

Methodology
The authors perform a rigorous benchmarking of the Fluid Ansatz against the WallGo spectral method across two theoretical frameworks:

SMEFT Model: A Standard Model Effective Field Theory extended with a dimension-six operator ( $\phi^6$ ), allowing for strong first-order phase transitions while maintaining the physical Higgs mass ( $m_h = 125$ GeV).
Light Higgs Model: A Standard Model-like potential with an artificially reduced Higgs mass ( $m_h \lesssim 80$ GeV), generating weaker first-order transitions.

The study proceeds in three stages:

Linear Comparison: Both methods are applied to calculate $v_w$ considering only top-quark annihilation ( $t\bar{t} \to gg$ ) and subsequently including scattering processes ( $tq \to tq$ , $tg \to tg$ ). The Fluid Ansatz is solved using a custom Python code implementing Green's method with momentum expansions up to order 3 (10 fluctuations), while WallGo uses a Chebyshev grid with $N=11$ and $M_{grid}=30$ .
Perturbativity Analysis: The authors introduce a perturbativity parameter $R$ , defined as the sum of dimensionless fluctuations. They monitor $R$ to identify the breakdown of the linearization approximation ( $\delta f \ll 1$ ) inherent in the Fluid Ansatz.
Non-Linear Extension: To investigate the source of discrepancies in strong transitions, the authors extend the Fluid Ansatz formalism to include second-order terms ( $O(\delta^2)$ ) in the Taylor expansion of the distribution function. This involves solving a non-linear Boundary Value Problem (BVP) to quantify the backreaction of quadratic fluctuations on the wall velocity.

Key Results

Agreement in Weak Regimes: For phase transitions with strength parameter $\alpha \lesssim 0.01$ , both methods yield essentially identical terminal velocities. The agreement is particularly robust when only the top-quark annihilation channel is considered.
Divergence in Stronger Regimes: As the transition strength increases ( $\alpha \gtrsim 0.1$ ), discrepancies emerge. When scattering processes are included, the Fluid Ansatz and WallGo results diverge by $O(40\% - 60\%)$ .
Breakdown of Linearization: The divergence correlates with the perturbativity parameter $R$ approaching unity ( $R \gtrsim 0.1$ ), indicating that the linear approximation of the distribution function in the Fluid Ansatz fails. Simultaneously, the wall thickness parameter $L_w T$ approaches unity, challenging the validity of the underlying WKB approximation required for the Boltzmann equation itself.
Impact of Non-Linear Terms: Including $O(\delta^2)$ terms in the Fluid Ansatz significantly shifts the predicted terminal velocity in strong transitions, generally reducing $v_w$ (increasing friction). Crucially, the authors find that while the direct second-order contribution to the pressure is quantitatively small ( $\sim 4\%$ of the total), its non-linear backreaction on the first-order fluctuations induces a large shift in the macroscopic wall dynamics. This shift worsens the agreement with WallGo rather than resolving it.
Convergence: The study confirms that WallGo solutions converge with sufficient grid resolution, while the Fluid Ansatz converges with a momentum expansion order of $\kappa=3$ (10 parameters) in weak regimes.

Significance and Claims
The paper claims that while both the Fluid Ansatz and WallGo are computationally efficient and reliable for weak to moderate phase transitions, the linearized Fluid Ansatz fundamentally breaks down in the strong transition regime ( $\alpha \gtrsim 0.01$ ).

The authors emphasize that the discrepancy is not merely a numerical artifact but stems from the failure of the linear expansion when macroscopic fluctuations become large. Furthermore, they highlight a deeper theoretical limitation: in the strong transition regime where these discrepancies are most pronounced, the wall thickness shrinks ( $L_w T \lesssim 3$ ), approaching the limit where the semi-classical WKB approximation (the foundation of the Boltzmann equation) is no longer valid.

Consequently, the paper concludes that achieving high-precision predictions for bubble wall velocities in strong phase transitions—precisely the regime targeted by future GW observatories—requires more than just refining the current semi-classical approaches. It suggests that a treatment beyond the WKB approximation may be necessary, as the current methods (both Fluid Ansatz and WallGo) may be operating outside their domain of validity in the most physically interesting scenarios.

Technical Summary: Benchmarking Wall Velocities in Cosmological Phase Transitions

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