## Abstract

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. In this paper we use results of N-body 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 μ, cosine of the angle between the Fourier mode and line of sight, then 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 μ2 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 ~ 0.015hMpc-1, 10% at k ~ 0.05hMpc-1 at z = 0, while for k > 0.15hMpc-1 they dominate and make the total negative. These higher order terms are dominated by density-energy density correlations which contributes negatively to the power, while the contribution from vorticity part of momentum density auto-correlation adds to the total power, but is an order of magnitude lower. For μ4 term the dominant term on large scales is the scalar part of momentum density auto-correlation, while higher order terms dominate for k > 0.15hMpc-1. For μ6 and μ8 we find it has very little power for k < 0.15hMpc-1, shooting up by 2-3 orders of magnitude between k < 0.15hMpc-1 and k < 0.4hMpc-1. We also compare the expansion to the full 2-d Pss(k,μ), as well as to the monopole, quadrupole, and hexadecapole integrals of Pss(k,μ). For these statistics an infinite number of terms contribute and we find that the expansion achieves percent level accuracy for kμ < 0.15hMpc-1 at 6-th 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 with velocity dispersions around 600 km/s at z = 0.