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. 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 μ 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 ∼ 0.015hMpc ^-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 μ ⁴ term the dominant term on large scales is the scalar part of momentum density auto-correlation, while higher order terms dominate for k > 0.15hMpc ^-1. For μ ⁶ and μ ⁸ 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 P ^ss(k,μ), as well as to the monopole, quadrupole, and hexadecapole integrals of P ^ss(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.