Non-relativistic approach for cosmological Scalar Field Dark Matter
  We derive non-relativistic equations of motion for the formation of
cosmological structure in a Scalar Field Dark Matter (SFDM) model corresponding
to a complex scalar field endowed with a quadratic scalar potential. Starting
with the full equations of motion written in the Newtonian gauge of scalar
perturbations, we separate out the fields involved into relativistic and
non-relativistic parts, and find the equations of motion for the latter that
can be used to build up the full solution. One important assumption will also
be that the SFDM field is in the regime of fast oscillations, under which its
behavior is exactly that of cold dark matter. The resultant equations are quite
similar to the Schr\"odinger-Poisson system of Newtonian boson stars plus
relativistic leftovers. We exploit that similarity to show how to simulate,
with minimum numerical effort, the formation of cosmological structure in SFDM
models and others alike, and ultimately prove their viability as complete dark
matter models.
