An Inverted Mass Hierarchy for Hot Dark Matter and the Solar Neutrino Problem.

The cosmological model in which 20% of the dark matter is shared by two nearly equal mass neutrinos fits the structure of the universe on all scales. This has been motivated a $\nu_\mu$-$\nu_{\tau}$ oscillation explanation of the deficit of atmospheric muon neutrinos. If the observed ratio of atmospheric $nu_\mu$ to $\nu_e$ has an alternative explanation, the cosmological model can be retained if the deficit of solar neutrinos is explained by $\nu_e$-$\nu_{\tau}$ oscillation. In this case an inverted mass hierarchy is required with $m_{\nu_{\mu}}\ll m_{\nu_e} \simeq m_{\nu_\tau}\approx 2.4$ eV. We show that if there exists an $L_e- L_{\tau}$ symmetry in nature, both the near mass degeneracy of \nue\ and \nut\ as well as the consistency of the above values for neutrino masses with the negative results for neutrinoless double beta decay search experiments are easily understood. We show that this symmetry implemented in the context of a high-scale left-right symmetric theory with the see-saw mechanism can lead to a simple theoretical understanding of the desired form of the mass matrix.