The statistical theory of self-gravitating collisionless dark matter flow for velocity, density, and potential fields

The statistical theory of self-gravitating collisionless dark matter flow for velocity, density, and potential fields N-body simulation is an invaluable tool to understand cosmic velocity field. However, simulation samples velocity only at particle locations that leads to information loss when projecting particle fields onto structured grid. Here we extract two-point statistics without field projection. These statistics, i.e. correlation, structure, dispersion functions in real space and spectrum functions in Fourier space, are modeled on both small and large scales. Kinematic relations between statistical measures are fully developed for incompressible, constant divergence, and irrotational flow. The nature of dark matter flow is identified by these relations. Much more complex than incompressible flow, peculiar velocity of dark matter flow is of constant divergence on small scale and irrotational on large scale. 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). Assuming gravity is the only interaction and no radiation produced, this feature leads to an increase in particle mass converted from kinetic energy upon "annihilation". On large scale, transverse velocity correlation has an exponential form \(T_2\propto e^{-r/r_2}\) with a 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 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 negative density correlation. 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