Universal scaling laws and density slopes for dark matter haloes

Abstract

AbstractSmalls 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 $$\varepsilon _u$$ ε u ($$\approx -\,4.6\times 10^{-7}{\text { m}}^2/{\text { s}}^3$$ ≈ - 4.6 × 10 - 7 m 2 / s 3 ). Confirmed by N-body simulations, the energy cascade leads to a two-thirds law for kinetic energy $$v_r^2$$ v r 2 on scale r such that $$v_r^2 \propto (\varepsilon _u r)^{2/3}$$ v r 2 ∝ ( ε u r ) 2 / 3 . Equivalently, a four-thirds law can be established for mean halo density $$\rho _s$$ ρ s enclosed in the scale radius $$r_s$$ r s such that $$\rho _s \propto \varepsilon _u^{2/3}G^{-1}r_s^{-4/3}$$ ρ s ∝ ε u 2 / 3 G - 1 r s - 4 / 3 , 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 $$r_l$$ r l can be determined by $$-u_0^3/\varepsilon _u$$ - u 0 3 / ε u , where $$u_0$$ u 0 is the velocity dispersion. Second, the smallest scale $$r_{\eta }$$ r η is dependent on the nature of dark matter. For collisionless dark matter, $$r_{\eta } \propto (-{G\hbar /\varepsilon _{u}}) ^{1/3}\approx 10^{-13}\,{\text {m}}$$ r η ∝ ( - G ħ / ε u ) 1 / 3 ≈ 10 - 13 m , where $$\hbar$$ ħ is the Planck constant. An uncertainty principle for momentum and acceleration fluctuations is also postulated. For self-interacting dark matter, $$r_{\eta } \propto \varepsilon _{u}^2 G^{-3}(\sigma /{{m}})^3$$ r η ∝ ε u 2 G - 3 ( σ / m ) 3 , where $$\sigma /m$$ σ / m is the cross-section of interaction. On halo scale, the energy cascade leads to an asymptotic density slope $$\gamma =-\,4/3$$ 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 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.