Relic Neutrino Background from Cosmic-Ray Reservoirs

We compute the flux of relic neutrino background (R$\nu$B) up-scattered by ultra-high-energy (UHE) cosmic rays (CRs) in clusters that act as CR-reservoirs. The long trapping times of UHECRs make this flux larger than that of R$\nu$B up-scattered by UHECRs on their way to Earth, which we also compute. We find that IceCube excludes R$\nu$B weighted overdensities larger than ${10}^{10}$ in clusters, and that PUEO, RNO-G, GRAND and IceCube-Gen2 will test values down to ${10}^8$. Our treatment incorporates the momentum transfer dependence of the neutrino-nucleus cross section, deep inelastic scattering, a mixed UHECR composition, and flavour information on the up-scattered R$\nu$B fluxes for both cases of neutrino mass spectrum with normal and inverted ordering, providing new handles to possibly disentangle the up-scattered R$\nu$B from cosmogenic neutrinos.