On the perturbative expansion at high temperature and implications for
  cosmological phase transitions
  We revisit the perturbative expansion at high temperature and investigate its
convergence by inspecting the renormalisation scale dependence of the effective
potential. Although at zero temperature the renormalisation group improved
effective potential is scale independent at one-loop, we show how this breaks
down at high temperature, due to the misalignment of loop and coupling
expansions. Following this, we show how one can recover renormalisation scale
independence at high temperature, and that it requires computations at two-loop
order. We demonstrate how this resolves some of the huge theoretical
uncertainties in the gravitational wave signal of first-order phase
transitions, though uncertainties remain stemming from the computation of the
bubble nucleation rate.
