Distribution function approach to redshift space distortions
  We develop a phase space distribution function approach to redshift space
distortions (RSD), in which the redshift space density can be written as a sum
over velocity moments of the distribution function. These moments are density
weighted and their lowest orders are density, momentum density, and stress
energy density. The series expansion is convergent on large scales. We perform
an expansion of these velocity moments into helicity modes, which are
eigenmodes under rotation around the axis of Fourier mode direction,
generalizing the scalar, vector, tensor decomposition of perturbations to an
arbitrary order. We show that only equal helicity moments correlate and derive
the angular dependence of the individual contributions to the redshift space
power spectrum in terms of angle mu between wave vector and line of sight. We
show that the dominant term of mu^2 dependence on large scales is the
cross-correlation between the density and scalar part of momentum density,
which can be related to the time derivative of the matter power spectrum.
Additional terms contributing and dominating on small scales are the vector
part of momentum density-momentum density correlations, the energy
density-density correlations, and the scalar part of anisotropic stress
density-density correlations. Similarly, we identify 7 terms contributing to
mu^4 dependence. Some of the advantages of the distribution function approach
are that the series expansion converges on large scales and remains valid in
multi-stream situations. We finish with a brief discussion of implications for
RSD in galaxies relative to dark matter, highlighting the issue of scale
dependent bias of velocity moments correlators.
