The two-thirds law for pairwise velocity and origin of critical MOND acceleration from distributions of density, velocity, and acceleration in dark matter flow

The statistics of density, velocity, and acceleration plays an essential role for probing large scale structure formation and direct/indirect detection experiments. In this paper, the scale and redshift dependence of density, velocity and acceleration distributions in dark matter flow are extensively studied. A halo-based non-projection approach is proposed to perform the statistical analysis: i) all halos are identified with particles divided into halo and out-of-halo particles such that distributions can be studied separately; ii) instead of projecting particle field onto structured grid that involves information loss, analysis is performed over all pairs on a given scale; iii) scale and redshift dependence of distributions can be studied by the variation of generalized kurtosis. For density distribution, Delaunay tessellation is used to reconstruct the density field. The particle overdensity  has an asymmetric distribution with a long tail 3    . The log-density  evolves from Gaussian to a bimodal distribution at z=0, with two peaks corresponding to high density from halo particles and low density from out-of-halo particles. Without projecting onto grid, density correlation is obtained directly by counting all pairs on a given scale and cannot be positive on all scales due to normalization. Density spectrum and fluctuation functions are obtained a) Electronic mail: zhijie.xu@pnnl.gov; zhijiexu@hotmail.com 2 from correlations and compared with theory. Simple analytical models for all second order density statistics are developed. The scale dependence of velocity field is studied for longitudinal velocity L u or ' L u , velocity difference ' L L L u u u    (or pairwise velocity) and sum ' L L L u u u    of two longitudinal velocities. Fully developed velocity fields are never Gaussian on any scale despite that they can be initially Gaussian. L u  and L u have the same distribution on small scale and the correlation L  between L u and ' L u approaches 1/2. While on large scale, L u  and L u  have the same distribution with correlation 0 L   . The first moment of L u  (mean pairwise velocity) is analytically modelled on both small and large scales. The second moment of L u  (structure function) approaches 2 2u on small scale and a two-thirds law can be identified for a reduced structure function that is   3 2 3 u r    , where u  is the constant rate of energy cascade. A constant length scale is introduced 3 0 s u r u   , below which the two-thirds law is valid. The twothirds law can be generalized to all even moments of pairwise velocity   L u  , while odd moments   1 n L u r    satisfy generalized stable clustering hypothesis (GSCH). The distributions of velocities are analytically modeled. On small scale, both L u and L u  can be modelled by a X distribution to maximize system entropy. While explicit form for distribution of L u  is still unknown, the moments and kurtosis of L u  can be analytically estimated based on a halo-size dependent correlation coefficient L  . On intermediate scale, distributions of L u and L u  becomes significantly non-symmetric with non-zero skewness, a necessary feature for inverse energy cascade. On large scale, both L u  and L u  approach the same distribution that can be