Exact Solution to Finite Temperature SFDM: Natural Cores without
  Feedback
  Recent high-quality observations of low surface brightness (LSB) galaxies
have shown that their dark matter (DM) halos prefer flat central density
profiles. On the other hand, the standard cold dark matter model simulations
predict a more cuspy behavior. One mechanism to reconcile the simulations with
the observed data is the feedback from star formation, this might be successful
in isolated dwarf galaxies but its success in LSB galaxies remains unclear.
Additionally, including too much feedback in the simulations is a double-edged
sword, in order to obtain a cored DM distribution from an initially cuspy one,
the feedback recipes usually require to remove a large quantity of baryons from
the center of galaxies, however, some feedback recipes produce twice more
satellite galaxies of a given luminosity and with much smaller mass to light
ratios from those that are observed. Therefore, one DM profile that produces
cores naturally and that does not require large amounts of feedback would be
preferable. We find both requirements to be satisfied in the scalar field dark
matter model. Here, we consider that the dark matter is an auto-interacting
real scalar field in a thermal bath at temperature T with an initial $Z_2$
symmetric potential, as the universe expands, the temperature drops so that the
$Z_2$ symmetry is spontaneously broken and the field rolls down to a new
minimum. We give an exact analytic solution to the Newtonian limit of this
system and show that it can satisfy the two desired requirements and that the
rotation curve profile is not longer universal.
