  In this paper, we study the effect of active-neutrino-sterile-neutrino mixing
in the expected high-energy astrophysical neutrino flavor content.
Non-unitarity in the measurement of the three active neutrinos can be due to
the existence of sterile neutrino states. We introduce the concept of the
four-flavor tetrahedron in order to visualize the lack of unitarity in the
astrophysical neutrino three-flavor triangle. We demonstrate that
active-sterile neutrino mixings modify the allowed region of the astrophysical
flavor ratio from the standard case. However, a projection of the four-flavor
tetrahedron has restrictions of phase space similar to the three-flavor
triangle. On the other hand, the initial presence of astrophysical sterile
neutrinos drastically changes the scenario, and it allows an apparent unitarity
violation in the three-flavor triangle space. Using current global fit
constraints including the non-unitarity case, we also illustrate the allowed
astrophysical neutrino flavor ratios. Thus, the measurement of the high-energy
astrophyscal neutrino flavor content allows us to explore sterile neutrinos
independently of the sterile neutrino mass scale. These are topics of
investigation for current and future neutrino telescopes.
