Published February 8, 2022 | Version 1.0
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Evolution of energy, momentum, and spin parameter in dark matter flow and integral constants of motion

  • 1. Pacific Northwest National Lab

Description

Evolution of energy, momentum, and spin parameter in dark matter flow and integral constants of motion

N-body equations of motion in comoving system and expanding background are reformulated in a transformed system with static background and fixed damping. The energy and momentum evolution in dark matter flow are rigorously formulated for both systems. The energy evolution in transformed system has a simple form that is identical to the damped harmonic oscillator. The cosmic energy equation can be easily derived in both systems. For entire N-body system, 1) combined with the two-body collapse model (TBCM), kinetic and potential energy increase linearly with time $t$ such that \(K_p=\varepsilon_ut\) and \(P_y=-7\varepsilon_ut/5\), where \(\varepsilon_u\) is a constant rate of energy cascade; 2) an effective gravitational potential exponent \(n_e=-10/7\ne-1\) (\(n_e=-1.38\) from simulation) can be identified due to surface energy of fast growing halos; 3) the radial momentum \(G\propto a^{3/2}\) and angular momentum \(H\propto a^{5/2}\), where \(a\) is the scale factor. On halo scale, 1) halo kinetic and potential energy can be modelled by two dimensionless constants \(\alpha_s^*\) and \(\beta_s^*\). Both constants are independent of time and halo mass; 2) both halo radial and angular momentum \(\propto a^{3/2}\) and can be modeled by two mass-dependent coefficients \(\tau_s^*\) and \(\eta_s^*\); 3) halo spin parameter is determined by \(\alpha_s^*\) and \(\eta_s^*\) and decreases with halo mass with derived values of 0.09 and 0.031 for small and large halos. Finally, the radial and angular momentum are closely related to the integral constants of motion \(I_m\), i.e. the integral of velocity correlation or the \(m\)th derivative of energy spectrum at long wavelength limit. On large scale, angular momentum is negligible, \(I_2\)=0 reflects the conservation of linear momentum, while \(I_4\) reflects the fluctuation of radial momentum \(G\). On halo scale, \(I_4\) is determined by both momentum that are comparable with each other.

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

Notes

Paper and presentation slides

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00_Evolution of energy, momentum, and spin parameter in dark matter flow and integral constants of motion.pdf

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