Impact of coherent scattering on relic neutrinos boosted by cosmic rays

Ultra-high-energy cosmic rays (UHECR) scattering off the cosmic relic neutrino background have recently gained renewed interest in the literature. Current data suggest that (UHECR) are predominantly made of heavy nuclei. Similar to the coherent elastic neutrino-nucleus scattering (CE$ν$NS) observed at low-energy neutrino experiments, the cross section of heavy nucleus scattering off relic neutrinos will be coherently enhanced since the energy of relic neutrinos can reach $\sim O(10)$ MeV in the rest frame of the UHECR. We calculate the diffuse flux of relic neutrinos boosted by UHECR after taking into account the contributions from both coherent and incoherent scatterings. Using current data from IceCube and Pierre Auger Observatory, we place constraints on the overdensity of relic neutrinos down to $\sim 10^8$. Since the flux of boosted relic neutrinos peaks at an energy of $\sim 200\, \text{PeV}$, we also entertain the possibility to explain the recently observed KM3NeT event with boosted relic neutrinos from UHECR.