The electroweak phase transition in minimal supergravity models

We have explored the electroweak phase transition in minimal supergravity models by extending previous analysis of the one-loop Higgs potential to include finite temperature effects. Minimal supergravity is characterized by two higgs doublets at the electroweak scale, gauge coupling unification, and universal soft-SUSY breaking at the unification scale. We have searched for the allowed parameter space that avoids washout of baryon number via unsuppressed anomalous Electroweak sphaleron processes after the phase transition. This requirement imposes strong constraints on the Higgs sector. With respect to weak scale baryogenesis, we find that the generic MSSM is {\it not} phenomenologically acceptable, and show that the additional experimental and consistency constraints of minimal supergravity restricts the mass of the lightest CP-even Higgs even further to $m_h\lsim 32\GeV$ (at one loop), also in conflict with experiment. Thus, if supergravity is to allow for baryogenesis via any other mechanism above the weak scale, it {\it must} also provide for B-L production (or some other `accidentally' conserved quantity) above the electroweak scale. Finally, we suggest that the no-scale flipped $SU(5)$ supergravity model can naturally and economically provide a source of B-L violation and realistically account for the observed ratio $n_B/n_\gamma\sim 10^{-10}$.