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 ${V}^{\ensuremath{'}\ensuremath{'}}l0$. We test different effective potential approximations at one loop. The agreement is best if one pulls a factor $\frac{{\ensuremath{\mu}}^{4}}{{T}^{4}}$ into the decay rate prefactor [where ${\ensuremath{\mu}}^{2}={V}^{\ensuremath{'}\ensuremath{'}}({\ensuremath{\varphi}}_{f})$], and takes the real part of the effective potential in the region ${V}^{\ensuremath{'}\ensuremath{'}}l0$. We perform a similar analysis on the one-dimensional kink.