Upper bound on Hot Dark Matter Density from $SO(10)$ Yukawa Unification

Abstract

We study low-energy consequences of supersymmetric $SO(10)$ models with Yukawa unification $h_t = h_N$ and $h_b = h_τ$. We find that it is difficult to reproduce the observed $m_b/m_τ$ ratio when the third-generation right-handed neutrino is at an intermediate scale, especially for small $\tan β$. We obtain a conservative lower bound on the mass of the right-handed neutrino $M_N > 6 \times 10^{13}$~GeV for $\tan β< 10$. This bound translates into an upper bound on the $τ$-neutrino mass, and therefore on its contribution to the hot dark matter density of the present universe, $Ω_νh^2 < 0.004$. 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 $h_b = h_τ$ for $\tan β\lsim 10$. This might affect the prediction of $m_b/m_τ$ and the constraint on $M_N$.