Universal scaling laws and density slope for dark matter haloes

Universal scaling laws and density slope for dark matter haloes

Small scale challenges suggest some missing pieces in our current understanding of dark matter. A cascade theory for dark matter is proposed to provide extra insights, similar to the cascade phenomenon in hydrodynamic turbulence. The kinetic energy is cascaded in dark matter from small to large scales involving a constant rate \(\varepsilon_u\) \(\approx -4.6\times 10^{-7}m^2/s^3\). Confirmed by N-body simulations, the energy cascade leads to a two-thirds law for kinetic energy \(v_r^2\) on scale \(r\) such that \(v_r^2 \propto (\varepsilon_u r)^{2/3}\). Equivalently, a four-thirds law can be established for halo density $\rho_s$ enclosed in the scale radius \(r_s\) such that \(\rho_s \propto \varepsilon_u^{2/3}G^{-1}r_s^{-4/3}\), which can be confirmed by galaxy rotation curves. Critical properties of dark matter might be obtained by identifying key constants on relevant scales. First, the largest halo scale \(r_l\) can be determined by \(-u_0^3/\varepsilon_u\), where \(u_0\) is the velocity dispersion. Second, the smallest scale \(r_{\eta}\) is dependent on the nature of dark matter. For collisionless dark matter, length scale \(r_{\eta} \propto (-{G\hbar/\varepsilon_{u}}) ^{1/3}\approx 10^{-13}m\) was found along with the mass scale \(m_{X}\propto (-{\varepsilon _{u} \hbar ^{5}}{G^{-4}})^{1/9} \approx 10^{12}GeV\), where \(\hbar\) is the Planck constant. An uncertainty principle for momentum and acceleration fluctuations is also postulated. For self-interacting dark matter, \(r_{\eta} \propto \varepsilon_{u}^2 G^{-3}(\sigma/m)^3\), where \(\sigma/m\) is the cross-section of interaction. On halo scale, the energy cascade leads to an asymptotic density slope \(\gamma=-4/3\) for fully virialized haloes with a vanishing radial flow, which might explain the nearly universal halo density. Based on the continuity equation, halo density is analytically shown to be closely dependent on the radial flow and mass accretion, such that simulated haloes can have different limiting slopes. A modified Einasto density profile is proposed accordingly.

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