Probing neutrino mass hierarchies and ${\varphi }_{13}$ with supernova neutrinos

We investigate the feasibility of probing the neutrino mass hierarchy and the mixing angle ${\varphi }_{13}$ with the neutrino burst from a future supernova. An inverse power-law density $\rho \sim r^n$ with varying $n$ is adopted in the analysis as the density profile of a typical core-collapse supernova. The survival probabilities of ${\nu }_e$ and ${\bar{\nu }}_e$ are shown to reduce to two-dimensional functions of $n$ and ${\varphi }_{13}$. It is found that in the $n-{\sin }^2{\varphi }_{13}$ parameter space, the 3D plots of the probability functions exhibit highly non-trivial structures that are sensitive to the mass hierarchy, the mixing angle ${\varphi }_{13}$, and the value of $n$. The conditions that lead to observable differences in the 3D plots are established. With the uncertainty of $n$ considered, a qualitative analysis of the Earth matter effect is also included.