Neutrino Mass Hierarchy and Stepwise Spectral Swapping of Supernova Neutrino Flavors

We examine a phenomenon recently predicted by numerical simulations of supernova neutrino flavor evolution: the swapping of supernova ${\nu }_e$ and ${\nu }_{\mu ,\tau }$ energy spectra below (above) energy $\text{EC}$ for the normal (inverted) neutrino mass hierarchy. We present the results of large-scale numerical calculations which show that in the normal neutrino mass hierarchy case, $\text{EC}$ decreases as the assumed ${\nu }_e\rightleftharpoons {\nu }_{\mu ,\tau }$ effective $2\times 2$ vacuum mixing angle ($\simeq {\theta }_{13}$) is decreased. However, these calculations also indicate that $\text{EC}$ is essentially independent of the vacuum mixing angle in the inverted neutrino mass hierarchy case. With a good neutrino signal from a future Galactic supernova, the above results could be used to determine the neutrino mass hierarchy even if ${\theta }_{13}$ is too small to be detected in terrestrial neutrino oscillation experiments.