Dynamical criterion for biased domain-wall formation

In the presence of a bias term, the conventional condition for forming a domain-wall network is $p_{\rm fv}>p_c\simeq 0.31$, with $p_{\rm fv}/(1-p_{\rm fv})=\exp(-\Delta V(0)/V_b(0))$, where $p_{\rm fv}$ is the false-vacuum fraction immediately after the phase transition, $\Delta V(0)$ is the zero-temperature energy splitting between the false and true vacua and $V_b(0)$ is the zero-temperature barrier height measured from the true vacuum. This criterion, however, cannot be generally valid, since it is insensitive to the dynamics of the phase transition. In this work, we derive a dynamical criterion for domain wall formation in the presence of a bias term. We evaluate $p_{\rm fv}$ at the freeze-out temperature of the false-vacuum correlation volumes $T_{\rm fo}$, obtaining a substantially stricter criterion. The same dynamical picture also yields a necessary consistency condition for applying scaling-regime gravitational-wave estimates, $T_{\rm fo}>T_{\rm ann}$, where $T_{\rm ann}$ is the annihilation temperature inferred from the scaling-regime dynamics.