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.

Applications of cascade and statistical theory for dark matter and bulge-SMBH evolution:

1. Dark matter particle mass ,size, and properties from energy cascade in dark matter flow: 1) arxiv 2) zenodo slides
2. Origin of MOND acceleration & deep-MOND from acceleration fluctuation & energy cascade: 1) arxiv 2) zenodo slides
3. The baryonic-to-halo mass relation from mass and energy cascade in dark matter flow: 1) arxiv 2) zenodo slides
4. Universal scaling laws and density slope for dark matter haloes: 1) arxiv 2) zenodo slides 3) paper
5. Dark matter halo mass functions and density profiles from mass/energy cascade: 1) arxiv 2) zenodo slides 3) paper
6. Energy cascade for distribution and evolution of supermassive black holes (SMBHs): 2) zenodo slides

Condensed slides for all applications "Cascade Theory for Turbulence, Dark Matter, and bulge-SMBH evolution "

The two relevant datasets and accompanying presentation can be found at: 

1. Dark matter flow dataset Part I: Halo-based statistics from cosmological N-body simulation 
2. Dark matter flow dataset Part II: Correlation-based statistics from cosmological N-body simulation.
3. 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:

1. Inverse mass cascade in dark matter flow and effects on halo mass functions: 1) arxiv 2) zenodo slides 
2. Inverse mass cascade and effects on halo deformation, energy, size, and density profiles: 1) arxiv 2) zenodo slides
3. Inverse energy cascade in dark matter flow and effects of halo shape: 1) arxiv 2) zenodo slides
4. The mean flow, velocity dispersion, energy transfer and evolution of dark matter halos: 1) arxiv 2) zenodo slides
5. Two-body collapse model and generalized stable clustering hypothesis for pairwise velocity 1) arxiv 2) zenodo slides
6. Energy, momentum, spin parameter in dark matter flow and integral constants of motion: 1) arxiv 2) zenodo slides
7. Maximum entropy distributions of dark matter in ΛCDM cosmology: 1) arxiv 2) zenodo slides 3) paper
8. Halo mass functions from maximum entropy distributions in dark matter flow: 1) arxiv 2) zenodo slides
9. On the statistical theory of self-gravitating collisionless dark matter flow: 1) arxiv 2) zenodo slides 3) paper
10. High order kinematic and dynamic relations for velocity correlations in dark matter flow: 1) arxiv 2) zenodo slides
11. Evolution of density and velocity distributions and two-thirds law for pairwise velocity: 1) arxiv 2) zenodo slides