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

Halo abundance and structure play a central role for modeling structure formation and evolution. 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 [Formula: see text]. First, the inverse mass cascade from small to large scales leads to the halo random walk in mass space with [Formula: see text], where [Formula: see text] is the halo mass and [Formula: see text] 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 [Formula: see text] for haloes below and above a critical mass scale [Formula: see text], i.e. a double-[Formula: see text] halo mass function. Second, a double-[Formula: see text] density profile can be derived based on the particle random walk in 3D space with a position-dependent waiting time [Formula: see text], where [Formula: see text] is the gravitational potential and r is the particle distance to halo center. Theory predicts [Formula: see text] 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 just special cases of proposed models. The small scale permanence can be identified due to the scale-independent rate of mass and energy cascade, where density profiles of different halo masses and redshifts converge to the [Formula: see text] scaling law ([Formula: see text]) on small scales. Theory predicts the halo number density scales with halo mass as [Formula: see text], while the halo mass density scales as [Formula: see text]. Results were compared against the Illustris simulations. This new perspective provides a theory for nearly universal halo mass functions and density profiles.