Aspects of the Flavor Triangle for Cosmic Neutrino Propagation

Over cosmic distances, astrophysical neutrino oscillations average out to a classical flavor propagation matrix $\mathscr{P}$. Thus, flavor ratios injected at the cosmic source $W_e,W_\mu,W_\tau$ evolve to flavor ratios at Earthly detectors $w_e,w_\mu,w_\tau$ 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 $\delta(\theta_{ij}$'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 $\theta_{32}$ and $\theta_{13}$ 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, $w_{\rm shower}=w_e+w_\tau$ and $w_{\rm track}=w_\mu$. We apply the statistical uncertainty to be expected from IceCube to this model. Finally, we consider ramifications of the expected lack of $\nu_\tau$ injection at cosmic sources. In particular, this constraint reduces the Earthly triangle to a boundary line of the triangle. Some tests of this "no $\nu_\tau$ injection" hypothesis are given.