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 $F_c\mathrm{}\left(T\right)\mathrm{}$, the free energy of a critical bubble configuration. $F_c\mathrm{}\left(T\right)\mathrm{}$ 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 $V^{\prime \prime }<0\mathrm{}$. We test different effective potential approximations at one loop. The agreement is best if one pulls a factor $\frac{{\mu }^4}{T^4}$ into the decay rate prefactor [where ${\mu }^2=V^{\prime \prime }\left({\phi }_f\right)$], and takes the real part of the effective potential in the region $V^{\prime \prime }<0\mathrm{}$. We perform a similar analysis on the one-dimensional kink.