Gravitationally bound BCS state as dark matter
  We explore the possibility that fermionic dark matter undergoes a BCS
transition to form a superfluid. This requires an attractive interaction
between fermions and we describe a possible source of this interaction induced
by torsion. We describe the gravitating fermion system with the Bogoliubov-de
Gennes formalism in the local density approximation. We solve the Poisson
equation along with the equations for the density and gap energy of the
fermions to find a self-gravitating, superfluid solution for dark matter halos.
In order to produce halos the size of dwarf galaxies, we require a particle
mass of $\sim 200\mathrm{eV}$. We find a maximum attractive coupling strength
before the halo becomes unstable. If dark matter halos do have a superfluid
component, this raises the possibility that they contain vortex lines.
