Aspects of the flavor triangle for cosmic neutrino propagation

Over cosmic distances, astrophysical neutrino oscillations average out to a classical flavor propagation matrix P. Thus, flavor ratios injected at the cosmic source W₄, W_, W_ evolve to flavor ratios at Earthly detectors w₄, w_, w_ according to w=PW. 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 (P). One more constraint would further reduce the dimensionality of the flavor triangle at Earth (two) to a line (one). We discuss four such constraints. The first is the possibility of a vanishing determinant for P. We give a formula for a unique (₈₉) 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 ₃₂ and 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: wₒ₇₎ₖ₄ₑ=w₄+w_ and wₓₑ₀₂₊=w_. We apply the statistical uncertainty to be expected from IceCube to this model. Finally, we consider ramifications of the expected lack of _ injection at cosmic sources. In particular, this constraint reduces the Earthly triangle to a boundary line of the triangle. Some tests of this ``no _ injection'' hypothesis are given.