IceCube PeV-EeV Neutrinos and Secret Interactions of Neutrinos

We show that the PeV neutrinos detected by IceCube put unique constraints on "secret" interactions of neutrinos with the cosmic neutrino background (C$\nu$B). The coupling must be $g <0.03$ for the mediating boson mass $m_{X} \lesssim 2$ MeV, $g/m_{X} < 5$ GeV$^{-1}$ for $m_{X} \gtrsim 20$ MeV, and $g/m_{X} < 0.07$ GeV$^{-1}$ in between. We also investigate the possibility that neutrino cascades degrade high-energy neutrinos to PeV energies by upgrading C$\nu$B where the energy flux of PeV neutrinos can coincide with the Waxman-Bahcall bound or the cosmogenic neutrino flux for protons, thanks to energy conservation. However a large coupling is required, which is disfavored by laboratory decay constraints. The suppression of PeV-EeV neutrinos is a testable prediction for the Askaryan Radio Array.