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

A reliable computation of the bubble wall velocity during a cosmological phase transition requires an adequate modeling of the non-equilibrium dynamics in the vicinity of this expanding bubble. This task can be made computationally faster by imposing an \emph{Ansatz} on the shape of the non-equilibrium particle distribution function, thus simplifying the collision terms and making the Boltzmann equation solvable in terms of some out-of-equilibrium fluctuations. Two different \emph{Ansätze} have prevailed in the recent literature: the so-called fluid \emph{Ansatz} and an expansion in a basis of Chebyshev polynomials, consolidated in the public code \texttt{WallGo}. In this work we show that the two approaches yield essentially the same wall velocity in the regime of reasonably mild phase transitions, $α\lesssim 0.01$. Interestingly, the agreement is excellent when only top-quark annihilation is considered, but a noticeable discrepancy appears once scattering processes are included. We also investigate the limitations of linearizing the Boltzmann equation when the fluid \emph{Ansatz} is applied to stronger phase transitions, showing that non-linear contributions induce significant shifts in the predicted terminal velocity as $α\to 1$, even though the non-linear contribution to the wall pressure remain quantitatively small compared to the equilibrium and linearized non-equilibrium parts. We discuss possible consequences of this result for both \emph{Ansätze}, while also highlighting the possible limitations of the WKB approach itself when applied to the regime of strong transitions. Since strong phase transitions are precisely the primary targets for future gravitational waves observatories, our study emphasizes that not only higher precision computations of $v_w$ in the semi-classical approach are required, but a treatment beyond the WKB approximation may be needed.