Observing EeV neutrinos through Earth: GZK and the anomalous ANITA events

Tau neutrinos are unique cosmic messengers, especially at extreme energies. When they undergo a charged-current interaction, the short lifetime of the produced tau gives rise to secondary tau neutrinos that carry a significant fraction of the primary neutrino energy. Here we apply this effect, known as tau neutrino regeneration, to extremely high energy neutrinos passing through Earth. We find that for most column depths, with the exception of propagation through the core, Earth-traversing tau neutrinos emerge at $O($PeV$)$ energies. We use these secondaries to estimate the expected signal from cosmogenic fluxes at IceCube and find a non-negligible contribution to the astrophysical component above 1 PeV . We also constrain the anomalous ANITA observations via the accompanying secondaries expected at IceCube. We calculate that ANITA should see fewer than ${10}^{-7}$ events in the reported direction, regardless of assumed source energy spectrum, ruling out the possibility that these events are astrophysical in origin under Standard Model assumptions.