Shape of the acoustic gravitational wave power spectrum from a first
  order phase transition
  We present results from large-scale numerical simulations of a first order
thermal phase transition in the early universe, in order to explore the shape
of the acoustic gravitational wave and the velocity power spectra. We compare
the 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. The
power spectra from deflagrations show similar features, but exhibit a steeper
high-$k$ decay and an extra feature not accounted for in the model. There are
two independent length scales: the mean bubble separation and the thickness of
the sound shell around the expanding bubble of the low temperature phase. It is
the sound shell thickness which sets the position of the peak of the power
spectrum. The low wavenumber behaviour of the velocity power spectrum is
consistent with a causal $k^{3}$, except for the thinnest sound shell, where it
is steeper. We present parameters for a simple broken power law fit to the
gravitational wave power spectrum for wall speeds well away from the speed of
sound where this form can be usefully applied. We examine the prospects for the
detection, showing that a LISA-like mission has the sensitivity to detect a
gravitational 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 the
Hubble radius. The shape of the gravitational wave power spectrum depends on
the bubble wall speed, and it may be possible to estimate the wall speed, and
constrain other phase transition parameters, with an accurate measurement of a
stochastic gravitational wave background.
