  We study the envelope approximation and its applicability to first-order
phase transitions in the early universe. We demonstrate that the power laws
seen in previous studies exist independently of the nucleation rate. We also
compare the envelope approximation prediction to results from large-scale phase
transition simulations. For phase transitions where the contribution to
gravitational waves from scalar fields dominates over that from the coupled
plasma of light particles, the envelope approximation is in agreement, giving a
power spectrum of the same form and order of magnitude. In all other cases the
form and amplitude of the gravitational wave power spectrum is markedly
different and new techniques are required.
