Decay of acoustic turbulence in two dimensions and implications for cosmological gravitational waves

We study decaying acoustic turbulence using numerical simulations of a relativistic fluid in two dimensions. Working in the limit of non-relativistic bulk velocities, with an ultra-relativistic equation of state, we find that the energy spectrum evolves towards a self-similar broken power law, with a high-wavenumber behaviour of $k^{-2.08 \pm 0.08}$, cut off at very high $k$ by the inverse width of the shock waves, and with a low-$k$ power law of $k^{2.50 \pm 0.31}$. The evolution of the energy and the integral length scale can be fitted with simple functions of time, derived from the self-similarity of the flow. We find them to develop proportional to $t^{-1.21 \pm 0.06}$ and $t^{0.32 \pm 0.03}$ respectively at late times. The model for the decay can be extended to three dimensions using the universality of the high-$k$ power law and the evolution laws for the kinetic energy and the integral length scale, and is used to build an estimate for the gravitational wave power spectrum resulting from a collection of shock waves, as might be found in the aftermath of a strong first order phase transition in the early universe. The power spectrum has a peak wavenumber set by the initial length scale of the acoustic waves, and a new secondary scale at lower wavenumber set by the integral scale after a Hubble time. The behaviour with wavenumber $k$ is $k^{2\beta + 1}$ at low $k$, where $\beta$ denotes the low-$k$ power law of the fluid energy spectrum, changing to a shallower $k^{(3 \beta - 1)/2}$ at intermediate $k$, and $k^{-3}$ at high $k$. The intermediate power law appears when the flow is short-lived in comparison to the Hubble time.