Distribution function approach to redshift space distortions. Part IV:
  perturbation theory applied to dark matter
  We develop a perturbative approach to redshift space distortions (RSD) using
the phase space distribution function approach and apply it to the dark matter
redshift space power spectrum and its moments. RSD can be written as a sum over
density weighted velocity moments correlators, with the lowest order being
density, momentum density and stress energy density. We use standard and
extended perturbation theory (PT) to determine their auto and cross
correlators, comparing them to N-body simulations. We show which of the terms
can be modeled well with the standard PT and which need additional terms that
include higher order corrections which cannot be modeled in PT. Most of these
additional terms are related to the small scale velocity dispersion effects,
the so called finger of god (FoG) effects, which affect some, but not all, of
the terms in this expansion, and which can be approximately modeled using a
simple physically motivated ansatz such as the halo model. We point out that
there are several velocity dispersions that enter into the detailed RSD
analysis with very different amplitudes, which can be approximately predicted
by the halo model. In contrast to previous models our approach systematically
includes all of the terms at a given order in PT and provides a physical
interpretation for the small scale dispersion values. We investigate RSD power
spectrum as a function of \mu, the cosine of the angle between the Fourier mode
and line of sight, focusing on the lowest order powers of \mu and multipole
moments which dominate the observable RSD power spectrum. Overall we find
considerable success in modeling many, but not all, of the terms in this
expansion.
