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. Our bound is much stronger than the unitarity bound at the most interesting masses, shows that dark matter halos cannot be significantly modified by annihilations, and can be improved by a factor of 10-100 with existing neutrino experiments.