  We study the possibility of relaxing the cosmological bound on the Higgs mass
coming from the requirement of non-erasure of the baryon asymmetry by
sphalerons. After reviewing the Standard Model case we obtain this bound in two
extensions of it: 1) The Standard Model with an additional gauge singlet, and
2) The Minimal Supersymmetric Standard Model. Taking fully into account all
experimental constraints and thermal screening effects we found that the
situation can be slightly improved with respect to the Standard Model but only
in case 1) a non negligible region in parameter space exists where the baryon
washout is avoided and the experimental bounds evaded. (Talk given at the NATO
Advanced Reseach Workshop: "Electroweak Physics and the early Universe", 23-25
March 1994, Sintra (Portugal).)
