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.