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To calculate the temperature at which a first-order cosmological phasetransition occurs, one must calculate $F_c(T)$, the free energy of a criticalbubble configuration. $F_c(T)$ is often approximated by the classical energyplus an integral over the bubble of the effective potential; one must choose amethod for calculating the effective potential when $V''<0$. We test differenteffective potential approximations at one loop. The agreement is best if onepulls a factor of $\mu^4/T^4$ into the decay rate prefactor [where $\mu^2 =V''(\phi_f)$], and takes the real part of the effective potential in the region$V''<0$. We perform a similar analysis on the 1-dimensional kink.Comment: 11 pages plus 3 figures in jyTeX; CALT-68-188

Exact critical bubble free energy and the effectiveness of effective potential approximations