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 Zhang et al. [Phys. Rev. D 87, 063526 (2013)] of this series of work, we proposed decomposing peculiar velocity into three eigenmodes (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 eigenmodes through a set of N-body simulations. These statistics include the velocity power spectra, correlation functions, one-point probability distribution functions, cumulants, and the damping functions describing the Finger of God effect. We have carried out a number of tests to quantify possible numerical artifacts in these measurements and have confirmed that these numerical artifacts are under control. Our major findings are as follows: (1) The power spectrum measurement shows that these velocity components have distinctly different spatial distribution and redshift evolution, consistent with predictions in Paper I. In particular, we measure the window function (W) over tilde (k, z). (W) over tilde describes the impact of nonlinear evolution on the v(delta)-density relation. We confirm that the approximation (W) over tilde = 1 can induce a significant systematic error of O(10%) in RSD cosmology. We demonstrate that (W) over tilde 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. Hence, they are important properties in RSD modeling. (3) The velocity probability distribution functions and cumulants quantify non-Gaussianities of the velocity fields. We confirm speculation in Paper I that v(delta) is largely Gaussian, but with non-negligible non-Gaussianity. We confirm that v(B) is significantly non-Gaussian. We also measure the damping functions. Despite the observed non-Gaussianities, the damping functions and hence the Finger of God effect are all well approximated as Gaussian ones at scales of interest.