On the statistical theory of self-gravitating collisionless dark matter flow: Scale and redshift variation of velocity and density distributions

Abstract

The statistics of velocity and density fields are crucial for cosmic structure formation and evolution. This paper extends our previous work on the two-point second-order statistics for the velocity field [Xu, Phys. Fluids 35, 077105 (2023)] to one-point probability distributions for both density and velocity fields. The scale and redshift variation of density and velocity distributions are studied by a halo-based non-projection approach. First, all particles are divided into halo and out-of-halo particles so that the redshift variation can be studied via generalized kurtosis of distributions for halo and out-of-halo particles, respectively. Second, without projecting particle fields onto a structured grid, the scale variation is analyzed by identifying all particle pairs on different scales r. We demonstrate that: (i) the Delaunay tessellation can be used to reconstruct the density field. The density correlation, spectrum, and dispersion functions were obtained, modeled, and compared with the N-body simulation; (ii) the velocity distributions are symmetric on both small and large scales and are non-symmetric with a negative skewness on intermediate scales due to the inverse energy cascade on small scales with a constant rate εu; (iii) on small scales, the even-order moments of pairwise velocity ΔuL follow a two-thirds law ∝(−εur)2/3, while the odd-order moments follow a linear scaling ⟨(ΔuL)2n+1⟩=(2n+1)⟨(ΔuL)2n⟩⟨ΔuL⟩∝r; (iv) the scale variation of the velocity distributions was studied for longitudinal velocities uL or uL′, pairwise velocity (velocity difference) ΔuL = uL′ − uL, and velocity sum ΣuL = uL′ + uL. Fully developed velocity fields are never Gaussian on any scale, despite that they can initially be Gaussian; (v) on small scales, uL and ΣuL can be modeled by a X distribution to maximize the entropy of the system. The distribution of ΔuL can be different; (vi) on large scales, ΔuL and ΣuL can be modeled by a logistic or a X distribution, while uL has a different distribution; and (vii) the redshift variation of the velocity distributions follows the evolution of the X distribution involving a shape parameter α(z) decreasing with time.