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 $M_N > 6 \times 10^{13}$~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, $\Omega_\nu h^2 < 0.004$. 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 $\tan \beta \lsim 10$. This might affect the prediction of $m_b/m_\tau$ and the constraint on $M_N$.