Subcritical bubble prehistory in weak first-order phase transition

Standard calculations of cosmological first-order phase transitions usually assume critical bubbles to nucleate on a homogeneous symmetric vacuum background. However, this assumption can fail in weak transitions, where thermal fluctuations trigger subcritical bubbles before the standard nucleation temperature $T_n$. Motivated by this possibility, we systematically examine whether the homogeneous nucleation background approximation is self-consistent. By evolving the Gelmini-Gleiser subcritical bubble kinetics and comparing it with the standard critical bubble nucleation picture, we identify the parameter regions in which the background becomes apparently mixed. A detailed scan of these regions shows that sizable subcritical volume fractions arise when the two phases are nearly degenerate at $T_n$, the potential barrier is low, the difference of free energy between the symmetric and broken phases is moderate and the transition strength is weak. Our analysis further yields a simple criterion, $\log_{10}\hat f_\xi(T_n)\simeq -1.95$, for a percent level subcritical bubble volume fraction. Parameter points above this boundary should be treated as mixed background candidates rather than as ordinary homogeneous bounce points.