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For comoving position $\mathbf{x}$, canonical momentum $\mathbf{p}=am{\mathbf{x}}^{\prime }$, and Newtonian potential $\varphi$, the dark-matter phase-space density obeys

This is the gravitational, expanding-universe specialization of the Vlasov equation.

The Vlasov equation describes collisionless evolution of a phase-space distribution function in a self-consistent force field: ${\partial }_{t}f+\mathbf{v}\cdot {\nabla }_{x}f+\mathbf{F}\cdot {\nabla }_{p}f=0$. The force can be electromagnetic or gravitational; the Collisionless dark-matter Vlasov equation is the cosmological gravitational specialization.