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 non-relativistic 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 $v_F^4$ for density fluctuations and $v_F^2$ 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 which include tensor interactions where the neutron spin is not conserved, neutrino radiation from bremsstrahlung reactions occurs at order $v_F^0$. 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.