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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., Comment: Reformatted with data source provided

Inverse energy cascade in self-gravitating collisionless dark matter flow and effects of halo shape Halo-mediated mass and energy cascades are key to understand dark matter flow. Both cascades origin from the mass exchange between halo and out-of-halo sub-systems. Kinetic energy can be from the motion of halos and particle motion in halos. Similarly, potential energy can be due to the inter- and intra-halo interactions. Intra-halo virial equilibrium is established much faster than inter-halo. Change of energy of entire system comes from virilization in halos. At statistically steady state, continuous mass exchange is required to sustain growth of total halo mass \(M_h\propto a^{1/2}\)and energy \(E\propto a^{3/2}\), where a is scale factor. Inverse cascade is identified for kinetic energy that is transferred from the smallest scale to larger mass scales. This is sustained by the direct cascade of potential energy from large to small scale. Both energies have a scale- and time-independent flux in propagation range that is proportional to mass flux. Energy cascade is mostly facilitated by the mass cascade, which can be quantitatively described by the mass accretion of typical halos. Halo radial, angular momentum, and angular velocity are also modelled and an inverse cascade is identified for the coherent radial and rotational motion in halos. In hydrodynamic turbulence, vortex stretching (shape changing) along its axis of spin enables energy cascade from large to small length scales. However, change in halo shape is not the dominant mechanism for energy cascade as the halo moment of inertial gained from shape changing is less than 2 times. Large halos exhibit preference for prolateness over oblateness and most halos have spin axis perpendicular to major axis. Since mass cascade is local in mass space, halo shape evolves continuously in mass space with large halos formed by incrementally inheriting structure from their progenitor halos. A unique evolution path of halo structure is found that gradually approaches sphere with increasing size. Condensed slides for all applications "Cascade Theory for Turbulence, Dark Matter, and Supermassive Black Holes" Applicationsof 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 fromacceleration 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_datasetandzenodo 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)arxiv2)zenodo slides Inverse mass cascade and effects on halo deformation, energy, size, and density profiles: 1) arxiv 2) zenodo slides Inverse energy cascade indark matter flow and effects of halo shape: 1) arxiv 2) zenodo slides The mean flow, velocity dispersion, energy transfer and evolution ofdark matter halos: 1) arxiv 2) zenodo slides Two-body collapse model and generalized stable clustering hypothesis for pairwise velocity1) 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 indark 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 ofdensity andvelocity distributions and two-thirds law for pairwise velocity: 1) arxiv 2) zenodo slides

Inverse mass cascade in dark matter flow and effects on halo deformation, energy, size, and density profiles Inverse mass cascade is a key feature of the intermediate statistically steady state for self-gravitating collisionless dark matter flow (SG-CFD). This paper focus on effects of mass cascade on halo energy, momentum, dispersion, size, and density. Halo with fast mass accretion has an expanding core. Mass cascade forms a new layer of mass that deforms the original halo and induces nonzero radial flow (outwards in core and inwards in outer regions). The inward/outward flow leads to an extra length scale (scale radius) that is not present in isothermal profile. Halo concentration c=3.5 can be derived for fast growing halos. For cusp-core controversy, a double-power-law density is proposed as a result of nonzero radial flow. The inner/outer density are controlled by halo deformation rate and halo growth, respectively. The slower deformation at center, the steeper density. For fast growing halos, radial flow at center is simply Hubble flow that leads to the existence of central core. Mass cascade leads to nonzero halo surface energy/tension and radial flow that enhances dispersion in outer region. An effective exponent of gravity ne= -1.3 (not -1) is obtained due to halo surface energy. Evolution of halo size follows a geometric Brownian motion and lognormal distribution. The Brownian motion of particles in evolving halos leads to Fokker-Planck equations for particle distribution that is dependent on the radial and osmotic flow. Complete solutions of particle distribution are presented based on a simple model of osmotic flow. The proposed model agrees with simulation for various halo group sizes. With reference pressure/density defined at center, equation of state can be established for relative pressure/density. Pressure, density, and dispersion at halo center are presented. The core size xcis obtained where Hubble flow is dominant. Simple closures are proposed for self-consistent halo density. Condensed slides for all applications "Cascade Theory for Turbulence, Dark Matter, and Supermassive Black Holes" Applicationsof 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 fromacceleration 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_datasetandzenodo 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)arxiv2)zenodo slides Inverse mass cascade and effects on halo deformation, energy, size, and density profiles: 1) arxiv 2) zenodo slides Inverse energy cascade indark matter flow and effects of halo shape: 1) arxiv 2) zenodo slides The mean flow, velocity dispersion, energy transfer and evolution ofdark matter halos: 1) arxiv 2) zenodo slides Two-body collapse model and generalized stable clustering hypothesis for pairwise velocity1) 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 indark 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 ofdensity andvelocity distributions and two-thirds law for pairwise velocity: 1) arxiv 2) zenodo slides

Inverse mass cascade is a key feature of statistically steady state for self-gravitating collisionless dark matter flow (SG-CFD). Continuous mass transfer from small to large mass scales (inverse) is formulated. Direct effect of mass cascade on halo mass function is presented. Mass cascade is local, two-way, and asymmetric in mass space. Halos inherit/pass their mass from/to halos of similar size. Two regimes are identified: a propagation range with scale-independent rate of mass transfer and a deposition range with cascaded mass consumed to grow halos. Dimensional analysis leads to a power-law mass function in propagation range with a geometry exponent ${\lambda}$. A fundamental merging frequency $f_0{\sim}m_p^{\lambda-1}a^{-1}$ is identified, where $a$ is scale factor. Particle mass $m_p$ can be determined if that frequency is known. Rate of mass transfer ${\epsilon}_m{\sim}a^{-1}$ is independent of halo mass, a key feature of propagation range. Typical halos grow as $m_h{\sim}a^{3/2}$ and halo lifespan scales as ${\sim}m_h^{-\lambda}$. Chain reaction of mass cascade provides non-equilibrium dark matter flow a mechanism to continuously release energy and maximize entropy. Continuous injection of mass ("free radicals") at the smallest scale is required to sustain the everlasting inverse mass cascade such that total halo mass $M_h$ increases as $a^{1/2}$. These "radicals" might be directly generated at the smallest Planck scale or by a direct cascade from large to small scales. Entire mass cascade can be formulated by random walk in mass space, where halos migrate with an exponential distribution of waiting time. This results in a heterogeneous diffusion model, where Press-Schechter mass function can be fully derived without relying on any specific collapse models. A double-$\lambda$ mass function is proposed with different $\lambda$ for two ranges and agrees with N-body simulations., Comment: Reformatted with data source provided