Two-body collapse model for gravitating flow of dark matter and generalized stable clustering hypothesis for pairwise velocity

Two-body collapse model for self-gravitating flow of dark matter and generalized stable clustering hypothesis for pairwise velocity

Analytical tools are valuable to study gravitational collapse. However, solutions are hard to find due to the highly non-linear nature. Only a few simple but powerful tools exist so far. Two examples are the spherical collapse model (SCM) and stable clustering hypothesis (SCH). We present a new analytical tool based on the elementary step of inverse mass cascade in dark matter flow, i.e. a two-body collapse model (TBCM). TBCM plays the same role as harmonic oscillator in dynamics and can be fundamental to understand structure evolution. For convenience, TBCM is formulated for gravity with any potential exponent \(n\) in a static background with a fixed damping (\(n\)=-1 for Newtonian gravity). The competition between gravity, expanding background (or damping), and angular momentum classifies two-body collapse into: 1) free fall collapse for weak angular momentum, where free fall time is greater if same system starts to collapse at earlier time; 2) equilibrium collapse for weak damping that persists longer in time, whose perturbative solutions lead to power-law evolution of system energy and momentum. Two critical values \(\beta_{s1}=1\) and \(\beta_{s2}=1/3\pi\) are identified that quantifies the competition between damping and gravity. Value \(\beta_{s2}\) only exists for discrete values of potential exponent \(n=(2-6m)/(1+3m)=\) -1,-10/7... for integer \(m\). Critical density ratio (\(\Delta_c=18\pi^2\)) is obtained for \(n\)=-1 that is consistent with SCM. TBCM predicts angular velocity \(\propto Hr^{-3/2}\) for two-body system of size r. The isothermal density is a result of infinitesimal halo lifetime or extremely fast mass accretion. TBCM is able to demonstrate SCH, i.e. mean pairwise velocity (first moment) \(\langle\Delta u\rangle=-Hr\). A generalized SCH is developed for higher order moments \(\langle\Delta u^{2m+1}\rangle=-(2m+1)\langle\Delta u^{2m}\rangle Hr\) that is validated by N-body simulation. Energy evolution in TBCM is independent of particle mass and energy equipartition does not apply. TBCM can be considered as a non-radial SCM. Both models predict the same critical density ratio, while TBCM contains much richer information.

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