Maximum entropy distributions of dark matter in ΛCDM cosmology

Physical and Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, WA, 99354 USA ¹

Received: 16 March 2023
Accepted: 4 May 2023

Abstract

Context. Small-scale challenges to ΛCDM cosmology require a deeper understanding of dark matter physics.

Aims. This paper aims to develop the 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.

Methods. For systems involving long-range interactions, a spectrum of halos of different sizes is required to form to maximize system entropy. While the velocity in halos can be Gaussian, the velocity distribution throughout the entire system, involving all halos of different sizes, is non-Gaussian. With the virial theorem for mechanical equilibrium, we applied the maximum entropy principle to the statistical equilibrium of entire system, such that the 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.

Results. The predicted X distribution involves a shape parameter α and a velocity scale, v₀. The shape parameter α reflects the nature of force (α → 0 for long-range force or α → ∞ for short-range force). Therefore, the distribution approaches Laplacian with α → 0 and Gaussian with α → ∞. For an intermediate value of α, the distribution naturally exhibits a Gaussian core for v ≪ v₀ and exponential wings for v ≫ v₀, 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 (∝v² for v ≪ v₀ in halo core region, i.e., “Newtonian”) and a linear scaling for high speeds (∝v for v ≫ v₀ in halo outskirt, i.e., exhibiting “non-Newtonian” behavior due to long-range gravity). We compared our results against N-body simulations and found a good agreement.