Maximum entropy distributions of dark matter in $\Lambda$CDM cosmology

Small-scale challenges to $\Lambda$CDM cosmology require a deeper understanding of dark matter physics.This paper aims to develop maximum entropy distributions for dark matter particle velocity (denoted by $X$), speed (denoted by $Z$), and energy (denoted by $E$) that are especially relevant on small scales where system approaches full virialization. For systems involving long-range interactions, a spectrum of halos of different sizes is required to form to maximize system entropy. While velocity in halos can be Gaussian, the velocity distribution throughout entire system, involving all halos of different sizes, is non-Gaussian. With the virial theorem for mechanical equilibrium, we applied maximum entropy principle to the statistical equilibrium of entire system, such that maximum entropy distribution of velocity (the $X$ distribution) could be analytically derived. The halo mass function was not required in this formulation, but it did indeed result from the maximum entropy. The predicted $X$ distribution involves a shape parameter $\alpha$ and a velocity scale, $v_0$. The shape parameter $\alpha$ reflects the nature of force ($\alpha\rightarrow0$ for long-range force or $\alpha\rightarrow\infty$ for short-range force). Therefore, the distribution approaches Laplacian with $\alpha\rightarrow0$ and Gaussian with $\alpha\rightarrow\infty$. For an intermediate value of $\alpha$, the distribution naturally exhibits a Gaussian core for $v\ll v_0$ and exponential wings for $v\gg v_0$, as confirmed by N-body simulations. From this distribution, the mean particle energy of all dark matter particles with a given speed, $v$, follows a parabolic scaling for low speeds ($\propto v^2$ for $v\ll v_0$ in halo core region, i.e., "Newtonian") and a linear scaling for high speeds ($\propto v$ for $v\gg v_0$ in halo outskirt, i.e., exhibiting "non-Newtonian" behavior in MOND due to long-range gravity).