Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 3 / ii / Solution

Integrating the Collisionless dark-matter Vlasov equation over momentum makes the force term a vanishing momentum-space boundary term. Since $\rho =m{a}^{-3}\int f{d}^{3}p$, the zeroth moment isMultiplying by $p^i/(am)$ before integrating gives the first moment. Decompose the second velocity moment asusing the velocity-dispersion tensor of collisionless matter. Combining the result with the continuity equation givesThe single-stream pressureless-fluid equations follow when ${\sigma }^{ij}=0$.