The cosmic radiation also contains neutrinos and antineutrinos in an extremely wide energy range. Neutrinos propagate on cosmological distances without attenuation due to their very small cross-sections with any target. However, at extremely high energies the resonant process ‘ν_x + ν̅_x → Z → anything’ happens with arather large cross-section on the cosmic antineutrino background (and vice versa)for any neutrino flavour x. The average kinetic energy of the ‘relic’ neutrino background is E_k,B = 0.25 m_eV and their density ρ = 5.6 x 10⁷ m^-3 each flavour. Assume all neutrinos to have the same mass mν = 100 m_eV.

(a) Evaluate the mean square speed of the relic neutrinos.

(b) Evaluate the incoming neutrino energy E_ν for the resonant process.

(c) Evaluate the mean free path recalling that σ(e⁺e^- → μ⁺μ^-) = 2.1 nb at the Z peak.