The scale and redshift variation of density and velocity distributions in dark matter flow and two-thirds law for pairwise velocity

The scale and redshift variation of density and velocity distributions in dark matter flow and two-thirds law for pairwise velocity A halo-based non-projection approach is proposed to study the scale and redshift dependence of density and velocity distributions (PDF) in dark matter flow. All particles are divided into halo and out-of-halo particles such that PDF can be studied separately. Without projecting particle fields onto grid, scale dependence is analyzed by counting all pairs on different scales $r$. Redshift dependence is studied via generalized kurtosis. From this analysis, we can demonstrate: i) Delaunay tessellation can be used to reconstruct density field. Density correlations/spectrum are obtained, modeled and compared with theory; ii) $m$th moment of pairwise velocity can be analytically modelled. On small scale, even order moments can be modelled by a two-thirds law \(\langle(\Delta u_L)^{2n}\rangle\propto{(-\epsilon_ur)}^{2/3}\), while odd order moments \(\langle(\Delta u_L)^{2n+1}\rangle=(2n+1)\langle(\Delta u_L)^{2n}\rangle\langle\Delta u_L\rangle\propto{r}\) and satisfy a generalized stable clustering hypothesis (GSCH); iii) Scale dependence is studied for longitudinal velocity \(u_L\) or \(u_L^{'}\), pairwise velocity (velocity difference) \(\Delta u_L=u_L^{'}-u_L\) and velocity sum \(\Sigma u_L=u^{'}_L+u_L\). Fully developed velocity fields are never Gaussian on any scale; iv) On small scale, both \(u_L\) and \(\Sigma u_L\) can be modelled by a \(X\) distribution to maximize system entropy. Distributions of \(\Delta u_L\) is different with its moments analytically derived; v) On large scale, both \(\Delta u_L\) and \(\Sigma u_L\) can be modelled by a logistic function; vi) Redshift evolution of velocity distributions follows prediction of X distribution with a decreasing shape parameter \(\alpha(z)\) to continuously maximize system entropy. Condensed slides for all applications "Cascade Theory for Turbulence, Dark Matter, and Supermassive Black Holes" Applications of cascade and statistical theory for dark matter and SMBH evolution: Dark matter particle mass ,size, and properties from energy cascade in dark matter flow: 1) arxiv 2) zenodo slides Origin of MOND acceleration & deep-MOND from acceleration fluctuation & energy cascade: 1) arxiv 2) zenodo slides The baryonic-to-halo mass relation from mass and energy cascade in dark matter flow: 1) arxiv 2) zenodo slides Universal scaling laws and density slope for dark matter halos from rotation curves: 1) arxiv 2) zenodo slides A unified theory for dark matter halo mass function and density profile: 1) arxiv 2) zenodo slides Energy cascade for distribution and evolution of supermassive black holes (SMBHs): 2) zenodo slides The two relevant datasets and accompanying presentation can be found at: Dark matter flow dataset Part I: Halo-based statistics from cosmological N-body simulation Dark matter flow dataset Part II: Correlation-based statistics from cosmological N-body simulation. A comparative study of Dark matter flow & hydrodynamic turbulence and its applications The same dataset also available on Github at: Github: dark_matter_flow_dataset and zenodo at: Dark matter flow dataset from cosmological N-body simulation. Cascade and statistical theory developed by these datasets: Inverse mass cascade in dark matter flow and effects on halo mass functions: 1) arxiv 2) zenodo slides Inverse mass cascade and effects on halo deformation, energy, size, and density profiles: 1) arxiv 2) zenodo slides Inverse energy cascade in dark matter flow and effects of halo shape: 1) arxiv 2) zenodo slides The mean flow, velocity dispersion, energy transfer and evolution of dark matter halos: 1) arxiv 2) zenodo slides Two-body collapse model and generalized stable clustering hypothesis for pairwise velocity 1) arxiv 2) zenodo slides Energy, momentum, spin parameter in dark matter flow and integral constants of motion: 1) arxiv 2) zenodo slides Maximum entropy distributions of velocity, speed, and energy in dark matter flow: 1) arxiv 2) zenodo slides Halo mass functions from maximum entropy distributions in dark matter flow: 1) arxiv 2) zenodo slides The statistical theory of dark matter flow for velocity, density, and potential fields: 1) arxiv 2) zenodo slides High order kinematic and dynamic relations for velocity correlations in dark matter flow: 1) arxiv 2) zenodo slides Evolution of density and velocity distributions and two-thirds law for pairwise velocity: 1) arxiv 2) zenodo slidesPaper and presentation slide