Universal scaling laws and density slopes for dark matter halos

Smalls scale challenges suggest some missing pieces in our current understandings of dark matter. A cascade theory for dark matter is proposed to provide extra insights, similar to the cascade in hydrodynamic turbulence. The kinetic energy is cascaded in dark matter from small to large scales with a constant rate ε_u~-4.6x10^-7m²/s³. Confirmed by N-body simulations, the energy cascade leads to a two-thirds law for kinetic energy v_r² on scale r such that v_r² ~(ε_ur)^2/3. Equivalently, a four-thirds law can be established for mean halo density ρ_s enclosed in the scale radius r_s such that ρ_s~ ε_u^2/3G^-1r_s^-4/3, which can be confirmed by galaxy rotation curves. Critical properties of dark matter might be obtained by identifying key constants on relevant scales. First, the largest halo scale r_l can be determined by -u₀³/ε_u, where u₀ is the velocity dispersion. Second, the smallest scale r_η is dependent on the nature of dark matter. For collisionless dark matter, r_η~(-Gh/ε_u)^1/3, where h is the Planck constant. For self-interacting dark matter, r_η~ε_u² G^-3(σ/m)³, where σ/m is the cross-section of interaction. On halo scale, the energy cascade leads to an asymptotic density slope γ=-4/3 for fully virialized haloes with a vanishing radial flow, which might explain the nearly universal halo density. Based on the continuity equation, halo density slope is analytically shown to be closely dependent on the radial flow and mass accretion, such that simulated haloes can have different limiting slopes. A modified Einasto density profile is proposed accordingly.