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Gravitational waves from a phase transition associated with the generation ofthe masses of elementary particles are within the reach of future space-baseddetectors such as LISA. A key determinant of the resulting power spectrum, notpreviously studied, is the lifetime of the acoustic turbulence which follows.We study decaying acoustic turbulence using numerical simulations of arelativistic fluid in two dimensions. Working in the limit of non-relativisticbulk velocities, with an ultra-relativistic equation of state, we find that theenergy spectrum evolves towards a self-similar broken power law, with ahigh-wavenumber behaviour of $k^{-2.08 \pm 0.08}$, cut off at very high $k$ bythe inverse width of the shock waves. Our model for the decay of acousticturbulence can be extended to three dimensions using the universality of thehigh-$k$ power law and the evolution laws for the kinetic energy and theintegral length scale. It is used to build an estimate for the gravitationalwave power spectrum resulting from a collection of shock waves, as might befound in the aftermath of a strong first order phase transition in the earlyuniverse. The power spectrum has a peak wavenumber set by the initial lengthscale of the acoustic waves, and a new secondary scale at a lower wavenumberset by the integral scale after a Hubble time. Between these scales adistinctive new power law appears. Our results allow more accurate predictionsof the gravitational wave power spectrum for a wide range of early universephase transition scenarios.

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