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We present results from large-scale numerical simulations of a first orderthermal phase transition in the early universe, in order to explore the shapeof the acoustic gravitational wave and the velocity power spectra. We comparethe results with the predictions of the recently proposed sound shell model.For the gravitational wave power spectrum, we find that the predicted $k^{-3}$behaviour, where $k$ is the wavenumber, emerges clearly for detonations. Thepower spectra from deflagrations show similar features, but exhibit a steeperhigh-$k$ decay and an extra feature not accounted for in the model. There aretwo independent length scales: the mean bubble separation and the thickness ofthe sound shell around the expanding bubble of the low temperature phase. It isthe sound shell thickness which sets the position of the peak of the powerspectrum. The low wavenumber behaviour of the velocity power spectrum isconsistent with a causal $k^{3}$, except for the thinnest sound shell, where itis steeper. We present parameters for a simple broken power law fit to thegravitational wave power spectrum for wall speeds well away from the speed ofsound where this form can be usefully applied. We examine the prospects for thedetection, showing that a LISA-like mission has the sensitivity to detect agravitational wave signal from sound waves with an RMS fluid velocity of about$0.05c$, produced from bubbles with a mean separation of about $10^{-2}$ of theHubble radius. The shape of the gravitational wave power spectrum depends onthe bubble wall speed, and it may be possible to estimate the wall speed, andconstrain other phase transition parameters, with an accurate measurement of astochastic gravitational wave background.

Shape of the acoustic gravitational wave power spectrum from a first order phase transition