Peculiar Velocity Decomposition, Redshift Space Distortion and Velocity Reconstruction in Redshift Surveys. II. Dark Matter Velocity Statistics

Massive spectroscopic redshift surveys open a promising window to accurately measure peculiar velocity at cosmological distances through redshift space distortion (RSD). In paper I of this series of work we proposed to decompose peculiar velocity into three eigen-modes (v_\delta, v_S and v_B) in order to facilitate the RSD modeling and peculiar velocity reconstruction. In the current paper we measure the dark matter RSD related statistics of the velocity eigen-modes through a set of N-body simulations, including the velocity power spectra, correlation functions, one-point probability distribution functions, cumulants and the damping functions describing the Finger of God effect. (1) The power spectrum measurement shows that these velocity components have distinctly different spatial distribution and redshift evolution. In particular, we measure the window function \tilde{W}(k,z), which describes the impact of nonlinear evolution on the v_\delta-density relation. We confirm that it can induce a significant systematic error of O(10%) in RSD cosmology. We demonstrate that \tilde{W} can be accurately described by a simple fitting formula with one or two free parameters. (2) The correlation function measurement shows that the correlation length is O(100), O(10) and O(1) Mpc for v_\delta, v_S and v_B respectively. These correlation lengths determine where we can treat the velocity fields as spatially uncorrelated. (3) The velocity PDFs and cumulants quantify non-Gaussianities of the velocity fields. We confirm speculation in paper I that v_\delta is largely Gaussian, nevertheless with non-negligible non-Gaussianity, v_B is significantly non-Gaussian. We also measure the damping functions. Despite the observed non-Gaussianities, the damping functions and hence the FOG effect are all well approximated as Gaussian ones at scales of interest.