= Solution
Integrating the <Collisionless dark-matter Vlasov equation> over momentum makes the force term a vanishing momentum-space boundary term. Since $\rho=m a^{-3}\int f\,d^3p$, the zeroth moment is
$$
\boxed{\rho'+3\mathcal H\rho+\partial_i(\rho v^i)=0}.
$$
Multiplying by $p^i/(am)$ before integrating gives the first moment. Decompose the second velocity moment as
$$
\langle u^iu^j\rangle=v^iv^j+\sigma^{ij}
$$
using the <velocity-dispersion tensor of collisionless matter>. Combining the result with the continuity equation gives
$$
\boxed{v^{i\prime}+\mathcal Hv^i+v^j\partial_jv^i
=-\partial^i\phi-\frac1\rho\partial_j(\rho\sigma^{ij})}.
$$
The single-stream pressureless-fluid equations follow when $\sigma^{ij}=0$.
Back to article page
