The terminal wall velocity of a first-order phase transition bubble in the early Universe can be calculated from a set of fluid equations describing the scalar fields and the plasma’s state. We rederive these equations from the energy-momentum tensor conservation and the Boltzmann equation, without linearizing in the background temperature and fluid velocity. The resulting equations have a finite solution for any wall velocity. We propose a spectral method to integrate the Boltzmann equation, which is simple, efficient, and accurate. As an example, we apply this new methodology to the singlet scalar extension of the standard model. We find that all solutions are naturally categorized as deflagrations ($v_w\sim c_s$) or ultrarelativistic detonations (${\gamma }_w\gtrsim 10$). Furthermore, the contributions from out-of-equilibrium effects are, most of the time, subdominant. Finally, we use these results to propose several approximation schemes with increasing levels of complexity and accuracy. They can be used to considerably simplify the methodology, while correctly describing the qualitative behavior of the bubble wall.

• Received 11 May 2022
• Accepted 22 June 2022

DOI:https://doi.org/10.1103/PhysRevD.106.023501

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society