the singlet vev is much larger than this temperature \[$x(T<<x)\sim x(0)$\], and also, due to a possible two-stages phase transition realization, where the first stage $%
(0,0,0)\rightarrow (0,0,x(T))$, takes place around the temperature $T_{\ast
}\sim x$; therefore at lower temperatures $T<<T_{\ast }$, the singlet vev becomes almost temperature independent.

Another important remark, is that the Higgs vevs for this benchmark \[Fig. \[cs\]-a\] are increasing when the Universe gets cooled unlike the SM [andhall]{}, or MSSM [@MSSMcheck], and similar remark for (d). In cases (b) and (c), the Higgs vevs decrease, but with slower pace than in the SM or MSSM, which makes the critical temperature larger than in the SM. The modified cases (b) and (c) belong to the majority of the benchmarks mentioned before.

The effect of the increasing doublets’ vev values at high temperatures with respect to their zero temperature values has been mentioned in a similar work [@ham]. This behavior and the slow decrease of the doublets vevs with respect to the temperature, is a consequence of the interaction of the singlet with the doublet Higgs. These interactions have the effect of relaxing the shape of the potential in the direction of the doublets, and therefore enhancing 