Primordial acoustic turbulence: three-dimensional simulations and gravitational wave predictions

Gravitational waves (GWs) generated by a first-order phase transition at the electroweak scale are detectable by future space-based detectors like LISA. The lifetime of the resulting shock waves plays an important role in determining the intensity of the generated GWs. We have simulated decaying primordial acoustic turbulence in three dimensions and make a prediction for the universal shape of the energy spectrum by using its self-similar decay properties and the shape of individual shock waves. The shape for the spectrum is used to determine the time dependence of the fluid kinetic energy and the energy containing length scale at late times. The inertial range power law is found to be close to the classically predicted $k^{-2}$ and approaches it with increasing Reynolds number. The resulting model for the velocity spectrum and its decay in time is combined with the sound shell model assumptions about the correlations of the velocity field to compute the GW power spectrum for flows that decay in less than the Hubble time. The decay is found to bring about a convergence in the spectral amplitude and the peak power law that leads to a power law shallower than the $k^9$ of the stationary case.