A search for extremely-high-energy neutrinos and first constraints on the ultra-high-energy cosmic-ray proton fraction with IceCube

We present a search for the diffuse extremely-high-energy neutrino flux using $12.6$ years of IceCube data. The non-observation of neutrinos with energies well above $10 \, \mathrm{PeV}$ constrains the all-flavor neutrino flux at $10^{18} \, \mathrm{eV}$ to a level of $E^2 \Phi_{\nu_e + \nu_\mu + \nu_\tau} \simeq 10^{-8} \, \mathrm{GeV} \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1} \, \mathrm{sr}^{-1}$, the most stringent limit to date. Using this data, we constrain the proton fraction of ultra-high-energy cosmic rays (UHECRs) above $\simeq 30 \, \mathrm{EeV}$ to be $\lesssim 70\,$% (at $90\,$% CL) if the cosmological evolution of the sources is comparable to or stronger than the star formation rate. This result complements direct air-shower measurements by being insensitive to uncertainties associated with hadronic interaction models. It is the first such result to disfavor the ``proton-only" hypothesis for UHECRs using neutrino data.