Superfluid response and the neutrino emissivity of neutron matter

We calculate the neutrino emissivity of superfluid neutron matter in the inner crust of neutron stars. We find that neutrino emission due to fluctuations resulting from the formation of Cooper pairs at finite temperature is highly suppressed in nonrelativistic systems. This suppression of the pair-breaking emissivity in a simplified model of neutron matter with interactions that conserve spin is of the order of v_F⁴ for density fluctuations and v_F² for spin fluctuations, where v_F is the Fermi velocity of neutrons. The larger suppression of density fluctuations arises because the dipole moment of the density distribution of a single component system does not vary in time. For this reason, we find that the axial current response (spin fluctuations) dominates. In more realistic models of neutron matter that include tensor interactions where the neutron spin is not conserved, neutrino radiation from bremsstrahlung reactions occurs at order v_F⁰. Consequently, even with the suppression factors due to superfluidity, this rate dominates near T_C. Present calculations of the pair-breaking emissivity are incomplete because they neglect the tensor component of the nucleon-nucleon interaction.