arXiv2015
Over cosmic distances, astrophysical neutrino oscillations average out to a classical flavor propagation matrix $\mathscr{P}$. Thus, flavor ratios injected at the cosmic source $W_e,W_μ,W_τ$ evolve to flavor ratios at Earthly detectors $w_e,w_μ,w_τ$ according to $\vec{w}=\mathscr{P} \vec{W}$. The unitary constraint reduces the Euclidean octant to a "flavor triangle". We prove a theorem that the area of the Earthly flavor triangle is proportional to Det$(\mathscr{P})$. One more constraint would further reduce the dimensionality of the flavor triangle at Earth (two) to a line (one). We discuss four motivated such constraints. The first is the possibility of a vanishing determinant for $\mathscr{P}$. We give a formula for a unique $δ(θ_{ij}$'s) that yields the vanishing determinant. Next we consider the thinness of the Earthly flavor triangle. We relate this thinness to the small deviations of the two angles $θ_{32}$ and $θ_{13}$ from maximal mixing and zero, respectively. Then we consider the confusion resulting from the tau neutrino decay topologies, which are showers at low energy, "double-bang" showers in the PeV range, and a mixture of showers and tracks at even higher energies. We examine the simple low-energy regime, where there are just two topologies, $w_{\rm shower}=w_e+w_τ$ and $w_{\rm track}=w_μ$. We apply the statistical uncertainty to be expected from IceCube to this model. Finally, we consider ramifications of the expected lack of $ν_τ$ injection at cosmic sources. In particular, this constraint reduces the Earthly triangle to a boundary line of the triangle. Some tests of this "no $ν_τ$ injection" hypothesis are given.