  Astrophysical objects such as active-galactic nuclei (AGN) and gamma-ray
bursts (GRBs) can be sources of high energy, astrophysical neutrinos. The decay
of charged pions produces electron and muon-flavor neutrinos from the primary
decay of the pion, and from the secondary decay of the resulting charged
lepton. At low energies we expect the flavor ratio
$\Phi_{\nu_e}:\Phi_{\nu_\mu}:\Phi_{\nu_\tau}$ to be $1:2:0$ at the source. We
are interested in the flavor ratios as measured on Earth after the neutrinos
propagate over cosmic distance scales from the source. If we only consider
vacuum flavor transition probabilities between the three active flavors then we
expect a measured flavor ratio of $1:1:1$ up to small corrections from
$\theta_{13}$ and non-maximal $\theta_{23}$. When we include mixing with two
additional flavors of sterile neutrinos then we see corrections to this ratio
up to $\sim 30%$. Furthermore, if we consider energy-loss of the charged
leptons involved in the pion decay from cosmic sources then these flux ratios
depend on the neutrino energy. We examine these energy-dependent flavor ratios
using a specific model for sterile neutrino mixing, and compare to expected
ratios when only three neutrino flavors are considered. At energies $E_\nu >$ 1
TeV the flavor ratios observed in experiments such as IceCube can probe the
existence of sterile neutrinos.
