Gravitational waves from bubble collisions: analytic derivation
  We consider gravitational wave production by bubble collisions during a
cosmological first-order phase transition. In the literature, such spectra have
been estimated by simulating the bubble dynamics, under so-called thin-wall and
envelope approximations in a flat background metric. However, we show that,
within these assumptions, the gravitational wave spectrum can be estimated in
an analytic way. Our estimation is based on the observation that the two-point
correlator of the energy-momentum tensor $\langle T(x)T(y)\rangle$ can be
expressed analytically under these assumptions. Though the final expressions
for the spectrum contain a few integrations that cannot be calculated
explicitly, we can easily estimate it numerically. As a result, it is found
that the most of the contributions to the spectrum come from single-bubble
contribution to the correlator, and in addition the fall-off of the spectrum at
high frequencies is found to be proportional to $f^{-1}$. We also provide
fitting formulae for the spectrum.
