NMSSM Inflation and Domain Walls from a Tri-critical Point of View

We study the conditions in which we could bring a universe filled with different $Z_3$ vacua created during the Next to Minimal Supersymmetric Standard Model (NMSSM) electroweak symmetry breaking at $\textit{O} (10)^2$ GeV and a three dimensional three states diluted Potts model together in the same universality class. Then we use Cardy-Jacobsen conjecture to prove that there might be a tri-critical point in the NMSSM electroweak epoch of early universe. We prove that due to the existence of this point there would be no cosmological domain wall problem. Moreover, at this point the heat capacity and correlation length diverge which lead to a huge amount of energy release at constant temperature and a new mechanism for cosmological structure formation. Also, the entropy decrease after the tri-critical phase transition could explain the problem of low initial entropy in early universe. Finally, we combine Cardy-Jacobsen and Yaffe-Svetitsky conjectures to show that there might be a tri-critical point for the $Z_3$ center symmetry breaking in a pure SU(3) theory at $\textit{O} (10)^2$ MeV. Near the pure SU(3) tri-critical point, unlike the electroweak epoch tri-critical point, entropy increases rather than decrease.