The Exact Critical Bubble Free Energy and the Effectiveness of Effective Potential Approximations

To calculate the temperature at which a first-order cosmological phase transition occurs, one must calculate $F_c(T)$, the free energy of a critical bubble configuration. $F_c(T)$ is often approximated by the classical energy plus an integral over the bubble of the effective potential; one must choose a method for calculating the effective potential when $V^{\mathrm{\prime }\mathrm{\prime }}<0$. We test different effective potential approximations at one loop. The agreement is best if one pulls a factor of ${\mu }^4\mathrm{/}{T}^4$ into the decay rate prefactor [where $\mu^2 = V’‘(\phi_f)$],andtakestherealpartoftheeffectivepotentialintheregion$V’'<0$. We perform a similar analysis on the 1-dimensional kink.