Universal scaling laws and density slopes for dark matter haloes.

Smalls scale challenges suggest some missing pieces in our current understanding of dark matter. A cascade theory for dark matter is proposed to provide extra insights, similar to the cascade phenomenon in hydrodynamic turbulence. The kinetic energy is cascaded in dark matter from small to large scales involves a constant rate [Formula: see text] ([Formula: see text]). Confirmed by N-body simulations, the energy cascade leads to a two-thirds law for kinetic energy [Formula: see text] on scale r such that [Formula: see text]. Equivalently, a four-thirds law can be established for mean halo density [Formula: see text] enclosed in the scale radius [Formula: see text] such that [Formula: see text], which was 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 [Formula: see text] can be determined by [Formula: see text], where [Formula: see text] is the velocity dispersion. Second, the smallest scale [Formula: see text] is dependent on the nature of dark matter. For collisionless dark matter, [Formula: see text], where [Formula: see text] is the Planck constant. An uncertainty principle for momentum and acceleration fluctuations is also postulated. For self-interacting dark matter, [Formula: see text], where [Formula: see text] is the cross-section of interaction. On halo scale, the energy cascade leads to an asymptotic density slope [Formula: see text] for fully virialized haloes with a vanishing radial flow, which might explain the nearly universal halo density. Based on the continuity equation, halo density 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.