On the energy of the protons producing the very high-energy astrophysical neutrinos

We study the distribution of the energy of the protons that can produce an astrophysical neutrino with a given observed energy, in the TeV–PeV range, both through pp or pγ interactions. Due to the increasing multiplicity of the pion production at high center of mass energies, the resulting average proton energies can be much larger than the often used approximate value E_p ≃ 20E_ν. Also the threshold of the pγ process can lead to a pronounced increase in the values of 〈E_p〉/E_ν for decreasing neutrino energies. The results depend sensitively on the assumed proton spectrum, since steeper spectra give less weight to the lower energy neutrino tail resulting from the decays of the abundant low-energy pions. In the pγ scenarios they also depend sensitively on the spectrum of the target photons. The results are in particular relevant to relate possible characteristics of the neutrino spectrum to those of the corresponding cosmic rays that generated them. We also discuss the associated production of gamma rays at the sources. (paper)