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

Abstract

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 $V&#39;&#39;<0$. We test different effective potential approximations at one loop. The agreement is best if one pulls a factor of $μ^4/T^4$ into the decay rate prefactor [where $μ^2 = V&#39;&#39;(ϕ_f)$], and takes the real part of the effective potential in the region $V&#39;&#39;<0$. We perform a similar analysis on the 1-dimensional kink.