Superfluid fraction in the crystalline crust of a neutron star: role of BCS pairing

The breaking of translational symmetry in the inner crust of a neutron star leads to the depletion of the neutron superfluid reservoir similarly to cold atomic condensates in optical lattices and in supersolids. This effect is studied in the general framework of the self-consistent time-dependent Hartree-Fock-Bogoliubov (HFB) theory, treating the crust as a perfect crystal. The superfluid fraction is derived in the Bardeen-Cooper-Schrieffer approximation for superfluid velocities much smaller than Landau's critical velocity within the linear-response theory. The different assumptions made in previous studies are clarified. Fully three-dimensional band-structure calculations of superfluid neutrons in a body-centered cubic lattice are carried out. Although the formation of Cooper pairs is essential for the occurrence of superfluidity, the superfluid fraction is found to be insensitive to the pairing gap, as in uniform neutron matter. In the intermediate region of the inner crust at the average baryon number density 0.03 fm$^{-3}$, only 8\% of the free neutrons are found to participate to the superflow. Such very low superfluid fraction challenges the classical interpretation of pulsar frequency glitches and calls for more systematic calculations within the full HFB approach.