PNNL

Abstract

This paper presents a statistical theory for the flow of dark matter that is validated by N-body simulations. More complex than incompressible flow, dark matter flow is self-gravitating, long-range, and collisionless with a scale dependent flow behavior. The peculiar velocity field is of constant divergence on small scale and irrotational on large scale. The statistical measures, i.e. correlation, structure, dispersion, and spectrum functions are modeled on both small and large scales, respectively. Kinematic relations between statistical measures are fully developed for incompressible, constant divergence, and irrotational flow. Incompressible and constant divergence flow share same kinematic relations for even order correlations. The limiting correlation of velocity $\rho_L=1/2$ on the smallest scale ($r=0$) is a unique feature of collisionless flow ($\rho_L=1$ for incompressible flow). On large scale, transverse velocity correlation has an exponential form $T_2\propto e^{-r/r_2}$ with a constant comoving scale $r_2$=21.3Mpc/h that maybe related to the size of sound horizon. All other correlation, structure, dispersion, and spectrum functions for velocity, density, and potential fields are derived analytically from kinematic relations for irrotational flow. On small scale, longitudinal structure function follows one-fourth law of $S^l_2\propto r^{1/4}$. All other statistical measures can be obtained from kinematic relations for constant divergence flow. Vorticity is negatively correlated for scale $r$ between 1 and 7Mpc/h. Divergence is negatively correlated for $r$>30Mpc/h that leads to a negative density correlation.