Beyond Schr\"{o}dinger-Poisson: Nonrelativistic Effective Field Theory
  for Scalar Dark Matter
  Massive scalar fields provide excellent dark matter candidates, whose
dynamics are often explored analytically and numerically using nonrelativistic
Schr\"{o}dinger-Poisson (SP) equations in a cosmological context. In this
paper, starting from the nonlinear and fully relativistic Klein-Gordon-Einstein
(KGE) equations in an expanding universe, we provide a systematic framework for
deriving the SP equations, as well as relativistic corrections to them, by
integrating out `fast modes' and including nonlinear metric and matter
contributions. We provide explicit equations for the leading-order relativistic
corrections, which provide insight into deviations from the SP equations as the
system approaches the relativistic regime. Upon including the leading-order
corrections, our equations are applicable beyond the domain of validity of the
SP system, and are simpler to use than the full KGE case in some contexts. As a
concrete application, we calculate the mass-radius relationship of solitons in
scalar dark matter and accurately capture the deviations of this relationship
from the SP system towards the KGE one.
