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Without relying on a spherical or ellipsoidal collapse model, we analyticallyderive 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. Thehierarchical halo structure formation leads to halo or particle random walkwith a position-dependent waiting time $\tau_g$. The inverse mass cascade fromsmall 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 ahalo geometry parameter with predicted value of 2/3. The correspondingFokker-Planck solution for halo random walk in mass space gives rise to thehalo mass function with a power-law behavior on small scale and exponentialdecay 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. adouble-$\lambda$ halo mass function. A double-$\gamma$ density profile can bederived based on the particle random walk in 3D space with a position-dependentwaiting time $\tau_g \propto \Phi(r)^{-1} \propto r^{-\gamma}$, where $\Phi$ isthe gravitational potential and $r$ is the particle distance to halo center.Theory predicts $\gamma=2/3$ that leads to a cuspy density profile with aninner slope of -4/3, consistent with the predicted scaling laws from energycascade. The Press-Schechter mass function and Einasto density profile arespecial cases of proposed models. The small scale permanence can be identifieddue to the scale-independent rates of mass and energy cascade, where densityprofiles of different halo masses and redshifts converge to the $-4/3$ scalinglaw ($\rho_h \propto r^{-4/3}$) on small scales. Theory predicts halo numberdensity scales with mass as $\propto m_h^{-1.9}$, while halo mass densityscales as $\propto m_h^{4/9}$. Results were compared against the Illustrissimulations.

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