Write $\mu=\widehat{\mathbf k}\mathbin\cdot\widehat{\mathbf p}$ and use the <Legendre polynomial recurrence relation>. The definition in the question is inverted by
$$
\Theta(\mu)=\sum_{\ell\geq0}(2\ell+1)(-i)^\ell
\Theta_\ell P_\ell(\mu).
$$
For Newtonian-gauge potentials the scanned geodesic equation is the standard identity
$$
\frac{d\log\epsilon}{d\eta}
=-\frac{d\Psi}{d\eta}+\Phi'+\Psi'
=\Phi'-\widehat{\mathbf p}\mathbin\cdot\nabla\Psi.
$$
Its monopole and dipole are $S_0=\Phi'$ and $S_1=k\Psi/3$. Projecting the <photon Boltzmann hierarchy> onto $P_0$ and $P_1$ therefore gives
$$
\boxed{\Theta_0'+k\Theta_1=\Phi'},
$$
$$
\boxed{\Theta_1'=\frac{k}{3}
(\Theta_0+\Psi-2\Theta_2)}.
$$
The first is the photon <continuity equation>; the second is the <photon Euler equation>, with the <photon quadrupole> providing the anisotropic-stress term. Indeed, with photon density contrast $\delta_\gamma=4\Theta_0$ and velocity divergence $\theta_\gamma=3k\Theta_1$,
$$
\delta_\gamma'=-\frac43\theta_\gamma+4\Phi',
\qquad
\theta_\gamma'=k^2(\delta_\gamma/4+\Psi-2\Theta_2).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 312 / 3 / ii / Solution

The dipole member of the photon Boltzmann hierarchy is the photon Euler equation. In Newtonian gauge and without collisions,
$$
\Theta_1'=\frac{k}{3}(\Theta_0+\Psi-2\Theta_2).
$$


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 Incoming links: Photon Euler equation