On the detection of cosmological neutrinos by coherent scattering

There have been several proposals that the sea of cosmological relic neutrinos could be detected by the energy, momentum, or angular momentum transferred during their coherent interaction with matter. We show that all but one of these proposals is incorrect and that the one exception leads to an effect that is probably immeasurably small. We first review the existing limits on the cosmological neutrinos and describe the expectations from the standard hot big-bang model. We then prove a general theorem that if the time average of the neutrino flux is spatially homogeneous (which is expected for cosmological neutrinos), then to first order in the weak coupling the energy and momentum transfer to any microscopic or macroscopic target is zero. Similarly, the angular momentum transfer is zero unless the target has a nonzero polarization or current density (but in this case the effect is probably immeasurably small). No assumption is made concerning the isotropy of the neutrino flux. Finally, we reexamine the individual proposals using the language of geometrical optics and show where all but one of them was incorrect. In particular, we prove that the momentum and angular momentum transferred by refraction from a homogeneous incoming beam to a (microscopically) isotropic target vanish to order $n-1\mathrm{}$, with $n$ the index of refraction.