  Recently, a new form of dark matter has been suggested to naturally reproduce
the empirically successful aspects of Milgrom's law in galaxies. The dark
matter particle candidates are axion-like, with masses of order eV and strong
self-interactions. They Bose-Einstein condense into a superfluid phase in the
central regions of galaxy halos. The superfluid phonon excitations in turn
couple to baryons and mediate an additional long-range force. For a suitable
choice of the superfluid equation of state, this force can mimic Milgrom's law.
In this paper we develop in detail some of the main phenomenological
consequences of such a formalism, by revisiting the expected dark matter halo
profile in the presence of an extended baryon distribution. In particular, we
show how rotation curves of both high and low surface brightness galaxies can
be reproduced, with a slightly rising rotation curve at large radii in massive
high surface brightness galaxies, thus subtly different from Milgrom's law. We
finally point out other expected differences with Milgrom's law, in particular
in dwarf spheroidal satellite galaxies, tidal dwarf galaxies, and globular
clusters, whose Milgromian or Newtonian behavior depends on the position with
respect to the superfluid core of the host galaxy. We also expect ultra-diffuse
galaxies within galaxy clusters to have velocities slightly above the baryonic
Tully-Fisher relation. Finally, we note that, in this framework, photons and
gravitons follow the same geodesics, and that galaxy-galaxy lensing, probing
larger distances within galaxy halos than rotation curves, should follow
predictions closer to the standard cosmological model than those of Milgrom's
law.
