Hydrodynamic sound shell model

For a cosmological first-order phase transition in the early Universe, the associated stochastic gravitational wave background is usually dominated by sound waves from plasma fluid motions, which have been analytically modeled as a random superposition of freely propagating sound shells but with the force by the scalar field that produces the self-similar profile removed. In this Letter, we propose a new analytic sound shell model by focusing on the forced propagating contribution from the initial collision stage of sound shells when their self-similar profiles are still maintained by the moving bubble walls. We reproduce the causal $k^3$ scaling in the infrared consistent with numerical simulations, and also recover the broad dome in the power spectrum first observed in numerical simulations. The total sound waves should contain both contributions from forced collisions and free propagation of sound shells at early and late stages of the phase transition, respectively.