Upper Bound on Hot Dark-Matter Density from So(10) Yukawa Unification

We study low-energy consequences of supersymmetric SO(10) models with Yukawa unification h(t) = h(N) and h(b) = h(tau). We find that it is difficult to reproduce the observed m(b)/m(tau) ratio when the third-generation right-handed neutrino is at an intermediate scale, especially for small tan beta. We obtain a conservative lower bound on the mass of the right-handed neutrino M(N) > 6 x 10(13) GeV for tan beta < 10. This bound translates into an upper bound on the tau-neutrino mass, and therefore on its contribution to the hot dark matter density of the present universe, OMEGA(nu)h2 < 0.004. Our analysis is based on the full two-loop renormalization group equations with one-loop threshold effects. However, we also point out that physics above the GUT-scale could modify the Yukawa unification condition h(b) = h(tau) for tan beta less-than-or-similar-to 10. This might affect the prediction of m(b)/m(tau) and the constraint on M(N).