Distribution function approach to redshift space distortions. Part II:
  N-body simulations
  Measurement of redshift-space distortions (RSD) offers an attractive method
to directly probe the cosmic growth history of density perturbations. A
distribution function approach where RSD can be written as a sum over density
weighted velocity moment correlators has recently been developed. We use Nbody
simulations to investigate the individual contributions and convergence of this
expansion for dark matter. If the series is expanded as a function of powers of
mu, cosine of the angle between the Fourier mode and line of sight, there are a
finite number of terms contributing at each order. We present these terms and
investigate their contribution to the total as a function of wavevector k. For
mu^2 the correlation between density and momentum dominates on large scales.
Higher order corrections, which act as a Finger-of-God (FoG) term, contribute
1% at k~0.015h/Mpc, 10% at k~0.05h/Mpc at z=0, while for k>0.15h/Mpc they
dominate and make the total negative. These higher order terms are dominated by
density-energy density correlations which contribute negatively to the power,
while the contribution from vorticity part of momentum density auto-correlation
is an order of magnitude lower. For mu^4 term the dominant term on large scales
is the scalar part of momentum density auto-correlation, while higher order
terms dominate for k>0.15h/Mpc. For mu^6 and mu^8 we find it has very little
power for k<0.15h/Mpc. We also compare the expansion to the full 2D P^ss(k,mu)
as well as to their multipoles. For these statistics an infinite number of
terms contribute and we find that the expansion achieves percent level accuracy
for kmu<0.15h/Mpc at 6th order, but breaks down on smaller scales because the
series is no longer perturbative. We explore resummation of the terms into FoG
kernels, which extend the convergence up to a factor of 2 in scale. We find
that the FoG kernels are approximately Lorentzian.
