Various astrophysical processes produce antineutrinos, which then fly off into outer space. I assume there are pretty accurate estimates of the production rates. I can imagine three possible fates for such an antineutrino: (1) annihilating with a neutrino, (2) interacting with baryonic matter, (3) ending up as the only particle inside its own cosmological horizon. It seems like we ought to have pretty good estimates of the rate of the neutrino-antineutrino annihilation process. Each such annihilation produces two back-to-back photons. If the neutrino masses are on the order of 0.1 eV, then these are infrared photons with wavelengths on the order of 10^4 nm. Why can't we detect these photons and thereby determine the neutrino mass spectrum? Are the peaks too weak? Too spread out by Doppler broadening?