= Solution
The collision rate $\kappa=a n_e\sigma_T$ is the scattering probability per unit <conformal time>. <Thomson scattering> approximately conserves photon number and energy in the electron rest frame, but redistributes propagation directions. Consequently the collision term drives the photon angular distribution toward an isotropic monopole plus the dipole associated with the baryon/electron bulk velocity. It damps the photon-baryon relative velocity and the quadrupole and higher directional anisotropies. Photon and baryon momentum-transfer terms have equal and opposite forces when weighted by their background enthalpies.
The metric terms describe the change in photon energy along a geodesic in a time-dependent spatial metric. The trace perturbation supplies an isotropic redshift, while the scalar trace-free perturbation supplies a direction-dependent redshift proportional to $3\mu^2-1$. Brightness is proportional to the fourth power of temperature, accounting for the factor four relative to the temperature shift. These are gravitational sources, not scattering terms.
When $\kappa\gg k,\mathcal H$, angular anisotropy is scattered away faster than free streaming or background expansion can regenerate it. In the <photon Boltzmann hierarchy>, a moment of order $\ell\geq2$ has damping of order $\kappa\Delta_\ell$ and is sourced by neighboring moments with coefficients of order $k$. Successive moments are therefore suppressed by powers of $k/\kappa=k\tau_c$. This is the <tight-coupling approximation>. At leading perfect-fluid order the quadrupole can be neglected; its first nonzero correction must be retained to calculate the diffusion damping in part (iii).
The printed equations do not all share a consistent velocity convention. To fix the intended one, take the Fourier velocity to be $v^i=ik^i\theta$. Ignoring the metric source, the <synchronous perfect-fluid density equation> then gives
$$
\delta_\gamma'=\frac43k^2\theta_\gamma,\qquad\delta_b'=k^2\theta_b.
$$
The PDF's baryon continuity sign is opposite to this convention. Keeping it literally would make the adiabatic ratio and common velocity in part (ii) incompatible for any nonstationary perturbation. Likewise, the momentum-transfer terms for these potentials are $-\kappa(\theta_\gamma-\theta_b)$ and $-R\kappa(\theta_b-\theta_\gamma)$, without the printed $k^{-2}$.
For isotropic scattering in this convention the brightness source is $\kappa[\delta_\gamma+4ik\mu\theta_b-\Delta]$. Its dipole relaxes toward the baryon velocity. The printed source involving $\theta_\gamma-\theta_b$ is not this source, and cannot yield the stated Euler drag by taking its dipole. The following solutions use the consistent continuity and momentum equations. These repairs do not change the requested tight-coupling reduction because the opposite drag forces cancel from its total-momentum equation. The velocity-divergence convention in https://arxiv.org/html/astro-ph/9506072v1#S5.SS5[the standard photon and baryon evolution equations] is converted to this potential convention by $\vartheta=-k^2\theta$.
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