Constraining Astrophysical Neutrino Flavor Composition from Leptonic   Unitarity
  The recent IceCube observation of ultra-high-energy astrophysical neutrinos has begun the era of neutrino astronomy. In this work, using the unitarity of leptonic mixing matrix, we derive nontrivial unitarity constraints on the flavor composition of astrophysical neutrinos detected by IceCube. Applying leptonic unitarity triangles, we deduce these unitarity bounds from geometrical conditions, such as triangular inequalities. These new bounds generally hold for three flavor neutrinos, and are independent of any experimental input or the pattern of leptonic mixing. We apply our unitarity bounds to derive general constraints on the flavor compositions for three types of astrophysical neutrino sources (and their general mixture), and compare them with the IceCube measurements. Furthermore, we prove that for any sources without $\nu_\tau$ neutrinos, a detected $\nu_\mu$ flux ratio $< 1/4$ will require the initial flavor composition with more $\nu_e$ neutrinos than $\nu_\mu$ neutrinos. 
Xun-Jie Xu, Hong-Jian He, Werner Rodejohann
hep-ph astro-ph.HE hep-ex