Gravitational waves from vacuum first-order phase transitions: from the
  envelope to the lattice
  We conduct large scale numerical simulations of gravitational wave production
at a first order vacuum phase transition. We find a power law for the
gravitational wave power spectrum at high wavenumber which falls off as
$k^{-1.5}$ rather than the $k^{-1}$ produced by the envelope approximation. The
peak of the power spectrum is shifted to slightly lower wave numbers from that
of the envelope approximation. The envelope approximation reproduces our
results for the peak power less well, agreeing only to within an order of
magnitude. After the bubbles finish colliding the scalar field oscillates
around the true vacuum. An additional feature is produced in the UV of the
gravitational wave power spectrum, and this continues to grow linearly until
the end of our simulation. The additional feature peaks at a length scale close
to the bubble wall thickness and is shown to have a negligible contribution to
the energy in gravitational waves, providing the scalar field mass is much
smaller than the Planck mass.
