Dark matter properties from universal scaling laws and energy cascade in dark matter flow

We present a new approach to estimate dark matter particle mass, size, density, and many other properties based on a cascade theory for the flow of dark matter, similar to the cascade in hydrodynamic turbulence. The energy cascade from small to large scales with a constant rate ε_u≈-4.6x10^-7m²/s³ is a fundamental feature of dark matter flow. Energy cascade leads to a two-thirds law for kinetic energy v_r² ≈(ε_ur)^2/3 on scale r, as confirmed by N-body simulations. This is equivalent to a four-thirds law for mean halo density ρ_s enclosed in the halo scale radius r_s such that ρ_s ≈ε_u^2/3G^-1r_s^-4/3, as confirmed by galaxy rotation curves. The smallest length scale in dark matter flow is dependent on the nature of dark matter. For collisionless dark matter, particle mass is estimated to be m_X≈(-ε_uh⁵G⁴)^1/9≈10¹²GeV with a size of r_η ≈(-ε_uhG)^1/3≈10^-13m, where h is the Planck constant. An uncertainty principle for momentum and acceleration fluctuations is also postulated to explain the physical origin of the rate of energy cascade ε_u. For self-interacting dark matter, the smallest structure has a sale r_η ≈ε_u²G^-3(σ/m)³, where σ/m is the cross-section. On halo scale, the energy cascade leads to an asymptotic density slope γ=-4/3 for fully virialized halos with a vanishing radial flow, which might explain the nearly universal halo density. A modified Einasto density profile is proposed accordingly. Accompanying slides and datasets for this work can be found at https://doi.org/10.5281/zenodo.6569901.