Statistical Mechanics of Collisionless Orbits. I. Origin of Central Cusps in Dark-Matter Halos

We present an equilibrium statistical mechanical theory of collisionless self-gravitational systems with isotropic velocity distributions. Compared with existing standard theories, we introduce two changes: (1) the number of possible microstates is computed in energy (orbit) space rather than phase space and (2) low occupation numbers are treated more appropriately than they are using Stirling's approximation. Combined, the two modifications predict that the relaxed parts of collisionless self-gravitating systems, such as dark-matter halos, have a differential energy distribution N (ε) ∝ [exp(ϕ 0 − ε)-1], dubbed "DARKexp." Such systems have central power-law density cusps ρ( r ) ∝ r −1 , which suggests a statistical mechanical origin of cusps in simulated dark-matter halos.