Dark matter halo mass functions and density profiles from mass and energy cascade (2210.01200v3)

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 $\tau_g$. The inverse mass cascade from small to large scales leads to the halo random walk in mass space with $\tau_g\propto m_h^{-\lambda}$, where $m_h$ is the halo mass and $\lambda$ 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 $\lambda$ for haloes below and above a critical mass scale $m_h^*$, i.e. a double-$\lambda$ halo mass function. A double-$\gamma$ density profile can be derived based on the particle random walk in 3D space with a position-dependent waiting time $\tau_g \propto \Phi(r)^{-1} \propto r^{-\gamma}$, where $\Phi$ is the gravitational potential and $r$ is the particle distance to halo center. Theory predicts $\gamma=2/3$ 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 ($\rho_h \propto r^{-4/3}$) 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.