The role of the chemical potential in coupling superfluid dark matter to
  baryons
  Superfluid dark matter postulates that the centers of galaxies contain
superfluid condensates. An important quantity regarding these superfluids is
their chemical potential $ \mu $. Here, we discuss two issues related to this
chemical potential. First, there is no exactly conserved quantity associated
with this chemical potential due to the symmetry-breaking baryon-phonon
coupling. Second, $ \mu $ is sometimes introduced by shifting the phonon field
by $ \mu \cdot t $ which -- again due to the symmetry-breaking baryon-phonon
coupling -- introduces an explicit time dependence in the Lagrangian. We
investigate under which conditions introducing a chemical potential is
nevertheless justified and show how to correctly introduce it when these
conditions are met. We further propose a model that recovers superfluid dark
matter's zero-temperature equations of motion including a chemical potential
even if the aforementioned conditions for justifying a chemical potential are
not met.
