Suppression of boosted relic neutrinos by photon backgrounds during ultra-high-energy cosmic ray propagation

Constraining the cosmic neutrino background (C$\nu$B) represents a major experimental challenge in cosmology. Recent studies have suggested that relic neutrinos boosted by ultra-high-energy cosmic rays (UHECRs) may generate observable diffuse neutrino fluxes. Previous estimates have not effectively propagated the primary cosmic rays, often neglecting crucial energy losses and the unavoidable, competing interactions with diffuse photon backgrounds. Here we revisit these expectations using a realistic Monte Carlo propagation framework. This approach allows us to consistently incorporate cosmic ray energy losses, nuclear photodisintegration, and production of secondary neutrinos. We show that interactions with diffuse photon backgrounds strongly suppress the boosted relic neutrino flux predicted in simplified propagation scenarios. Furthermore, we demonstrate that to produce any observable suppression on the UHECR energy spectrum at Earth, or for the boosted C$\nu$B component to become comparable to the cosmogenic neutrino flux, the C$\nu$B density must be enhanced by a factor, the so-called overdensity, of extreme magnitude ($\eta \gtrsim {10}^8$).