condition (\[v/t\]) at $T_{0}$ was used in defining the phase transition. This means that the anomalous $B+L$ violating processes should be switched off at the end of the phase transition, i.e., the Universe is filled by the true vacuum by the expanded bubbles. The value $T_{0}$, usually called the lower metastability temperature or lower spinodial decomposition point [@MSSMcheck], can be determined using the Jacobian $det[\partial ^{2}V(h,\ T_{0})/\partial
h_{i}\partial h_{j}]_{h=0}=0$. However, in our work, it is safer to consider the condition (\[v/t\]) at $T_{c}$, that is defined in (\[TC\]).

Since the singlet field does not play an important role in the sphaleron processes [@amin], the criterion of a strong first-order phase transition in our case is given by$$\upsilon \left( T_{c}\right) /T_{c}\equiv \sqrt{\upsilon _{1}^{2}\left(
T_{c}\right) +\upsilon _{2}^{2}\left( T_{c}\right) }/T_{c}>1.  \label{cri}$$

In the general case where the relative phase $\theta \neq 0$, the field ground state at nonzero temperature should be written as $\left\{
v_{i}\right\} _{i=1,4}$= ($\upsilon _{1}$, $\upsilon _{2}$, $\upsilon
_{x}\cos \theta $, $\upsilon _{x}\sin \theta $) instead of ($\upsilon
_{1},\upsilon _{2},\upsilon _{x}$) and the relative phase. These 4 variables should be treated independently when looking for $\upsilon _{1,2,x}$ and $%
\theta $ at any temperature $T$. Then the phase transition could 