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 $\sigma^{ij}=0$.

Solution

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

Liouville transport of the phase-space density along the particle trajectories gives
$$
\frac{df}{d\eta}
=\frac{\partial f}{\partial\eta}
+\mathbf x'\mathbin\cdot\nabla_xf
+\mathbf p'\mathbin\cdot\nabla_pf=0.
$$
Using the equations of motion produces the <Collisionless dark-matter Vlasov equation>
$$
\boxed{\frac{\partial f}{\partial\eta}
+\frac{\mathbf p}{am}\mathbin\cdot\nabla_xf
-am\nabla\phi\mathbin\cdot\nabla_pf=0}.
$$

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

{c}
{wiki=Vlasov_equation}

For comoving position $\mathbf x$, canonical momentum $\mathbf p=am\mathbf x'$, and Newtonian potential $\phi$, the dark-matter phase-space density obeys
$$
\partial_\eta f+\frac{\mathbf p}{am}\mathbin\cdot\nabla_xf-am\nabla\phi\mathbin\cdot\nabla_pf=0.
$$


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