Dark matter halo mass functions and density profiles from mass and energy cascade

Abstract

Without relying on a spherical or ellipsoidal collapse model, we analytically derive the halo mass function and cuspy halo density (inner slope of -4/3) based on the mass and energy cascade theory in dark matter flow. The hierarchical halo structure formation leads to halo or particle random walk with a position-dependent waiting time $\ddot{I}{„}_g$. The inverse mass cascade from small to large scales leads to the halo random walk in mass space with $\ddot{I}{„}_g\propto m_h^{-\hat{I}»}$, where $m_h$ is the halo mass and $\hat{I}»$ is a halo geometry parameter with predicted value of 2/3. The corresponding Fokker-Planck solution for halo random walk in mass space gives rise to the halo mass function with a power-law behavior on small scale and exponential decay on large scale. This can be further improved by considering two different $\hat{I}»$ for haloes below and above a critical mass scale $m_h^{\ast }$, i.e. a double-$\hat{I}»$ halo mass function. A double-${\hat{I}}^3$ density profile can be derived based on the particle random walk in 3D space with a position-dependent waiting time $\ddot{I}{„}_g\propto \hat{I}¦(r{)}^{-1}\propto r^{-{\hat{I}}^3}$, where $\hat{I}¦$ is the gravitational potential and $r$ is the particle distance to halo center. Theory predicts ${\hat{I}}^3=2/3$ that leads to a cuspy density profile with an inner slope of -4/3, consistent with the predicted scaling laws from energy cascade. The Press-Schechter mass function and Einasto density profile are special cases of proposed models. The small scale permanence can be identified due to the scale-independent rates of mass and energy cascade, where density profiles of different halo masses and redshifts converge to the $-4/3$ scaling law ($\ddot{I}{}_h\propto r^{-4/3}$) on small scales. Theory predicts halo number density scales with mass as $\propto m_h^{-1.9}$, while halo mass density scales as $\propto m_h^{4/9}$. Results were compared against the Illustris simulations.

14 pages 20 figures