On the statistical theory of self-gravitating collisionless dark matter flow: high order kinematic and dynamic relations

Abstract: To better understand the collisionless dark matter flow on different scales, statistical theory involving kinematic and dynamic relations must be developed for different types of flow, e.g. incompressible, constant divergence, and irrotational flow. This paper extends our previous work on the second-order statistics (Phys. Fluids 35, 077105) to high order statistics. Kinematic and dynamic relations were developed for dark matter flow on different scales. The results were validated by N-body simulations. On large scales, we found i) third-order velocity correlations can be related to density correlation or pairwise velocity; ii) the $p$th-order velocity correlations follow $\propto a^{(p+2)/2}$ for odd $p$ and $\propto a^{p/2}$ for even $p$, where $a$ is the scale factor; iii) the overdensity $\delta$ is proportional to density correlation on the same scale; iv) velocity dispersion on a given scale $r$ is proportional to the overdensity on the same scale. On small scales, i) a self-closed velocity evolution is developed by decomposing the velocity into motion in haloes and motion of haloes; ii) the evolution of vorticity and enstrophy are derived from the evolution of velocity; iii) dynamic relations are derived to relate second- and third-order correlations; iv) while the first moment of pairwise velocity follows $\langle\Delta u_L\rangle=-Har$ ($H$ is the Hubble parameter), the third moment follows $\langle(\Delta u_L)^3\rangle\propto\varepsilon_uar$ that can be directly compared with simulations and observations, where $\varepsilon_u\approx10^{-7}$m$^2$/s$^3$ is the constant rate for energy cascade; v) the $p$th order velocity correlations follow $\propto a^{(3p-5)/4}$ for odd $p$ and $\propto a^{3p/4}$ for even $p$. Finally, the combined kinematic and dynamic relations lead to exponential and one-fourth power-law velocity correlations on large and small scales, respectively.