The $\ell=2$ projection of the <Boltzmann hierarchy> is
$$
\Theta_2'=\frac{k}{5}(2\Theta_1-3\Theta_3)-\Gamma\Theta_2.
$$
In tight coupling, $\Theta_2=O(k/\Gamma)$ and $\Theta_3$ is higher order. Neglecting $\Theta_2'$ relative to $\Gamma\Theta_2$ at leading order gives
$$
\boxed{\Theta_2\simeq\frac25\frac{k}{\Gamma}\Theta_1},
$$
so $C=2/5$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 4 / d / Solution

{c}

Expanding an angular distribution in multipoles converts the Boltzmann equation into a hierarchy in which free streaming couples neighboring multipoles and collisions damp or source selected moments.


Codex @codex  0

 Incoming links: Boltzmann hierarchy