The Cosmic Neutrino Background is within Reach of Future Neutrino Telescopes

The cosmic neutrino background (C$\nu$B) can be boosted to high energies due to scatterings with energetic cosmic rays (CRs) across cosmological scales. Previous calculations focused on neutral current incoherent and coherent elastic scatterings of cosmic-ray protons off relic neutrinos. However, charged current interactions and deep inelastic scatterings are also expected to occur, which enhances the boosted relic neutrino fluxes on Earth. Here, we compute the \textit{total} diffuse boosted cosmic neutrino background (DBC$\nu$B) arising from CRs at all redshifts in the Universe, accounting for neutral current and charged current elastic and deep inelastic scatterings. We find that IceCube already places an upper limit on the cosmic neutrino background overdensity in cosmological scales of ~$\mathcal{O}(100-1000)$ at $E_{\nu}=10^{10}$ GeV, for a lightest neutrino mass of $m_{\nu} \gtrsim 0.1$ eV. We further show that IceCube-Gen2 could test $\mathcal{O}(1-10)$ C$\nu$B overdensities, and the combination of $10$ future neutrino telescopes with similar sensitivity would allow us to test the $\Lambda$CDM expected C$\nu$B density for a lightest neutrino mass compatible with the KATRIN bound.