Upper bound on the dark matter total annihilation cross section.

We consider dark matter annihilation into standard model particles and show that the least detectable final states, namely, neutrinos, define an upper bound on the total cross section. Calculating the cosmic diffuse neutrino signal, and comparing it to the measured terrestrial atmospheric neutrino background, we derive a strong and general bound. This can be evaded if the annihilation products are dominantly new and truly invisible particles.