Fluid perturbations from expanding bubbles in first-order phase transitions
We study the power spectrum of the velocity field induced during a first-order phase transition occurring in the radiation-dominated era. We focus on the phase of bubble expansion, assuming that it ends with the onset of the sound-wave regime. The main result we present is a refined template for the velocity spectrum at the beginning of the sound-wave phase, which can be used for studying the resulting anisotropic stresses and gravitational wave production. In particular, we find that the breaks in the velocity spectrum are not associated to the bubble size and the sound shell thickness, as previously proposed, but to the position of the discontinuities. This distinction is particularly relevant for supersonic deflagrations, as it implies that the intermediate slope is more pronounced and the two breaks are more separated when the wall velocity approaches the Chapman-Jouget speed, instead of the sound speed. We also show that the asymptotic branches of the velocity power spectrum are determined by the integral over the single-bubble profiles at large scales, and by the discontinuities of the velocity profiles at small scales. Furthermore, we study the dependence of the two breaks and the intermediate slope on the distribution function of the times of bubble nucleation (exponential and simultaneous). All the results presented in this work have been included in the public Python package CosmoGW.