= Solution
In the decoupled <perfect fluid in general relativity> limit, photons have $w=1/3$ and no <anisotropic stress>, while cold baryons have $w\simeq0$ with a small pressure correction $c_b^2\delta_b$. The density equations express particle and energy conservation; <Thomson scattering> transfers momentum rather than leading-order rest-frame energy. The photon Euler equation consequently gains a drag toward the baryon <velocity potential>, and the baryon Euler equation gains the equal-and-opposite drag weighted by their inertias.
Let $\Gamma_T=an_e\sigma_T>0$. With the <velocity potential> used in part (i), these forces are $-\Gamma_T(\theta_\gamma-\theta_b)$ and $-R\Gamma_T(\theta_b-\theta_\gamma)$, with $R=4\bar\rho_\gamma/(3\bar\rho_b)$. Multiplication by the photon and baryon enthalpies, $4\bar\rho_\gamma/3$ and $\bar\rho_b$, makes their sum zero. Note that this paper's $R$ is the reciprocal of the usual <baryon loading parameter>. <Thomson scattering> also relaxes the photon quadrupole, giving the $-\Gamma_T\sigma_\gamma$ term in the stipulated simplified hierarchy. This quadrupole is an <anisotropic stress>, omitted by a perfect-fluid model; finite relaxation creates <Silk damping>.
\b[The printed baryon density sign and collision normalization are inconsistent with part (i).] The density equation must be $\delta_b'-k^2\theta_b=-h'/2$. As a direct check, with $h'=0$, nonzero common velocity, and initially $\delta_\gamma=4\delta_b/3$, the printed plus sign gives $(\delta_\gamma-4\delta_b/3)'=8k^2\theta/3$, violating the tight-coupling adiabatic condition immediately. Likewise division of the collision forces by $k^2$ belongs to neither this potential convention nor the usual common velocity-divergence convention; the physical relaxation rate is $\Gamma_T$, independent of $k$. Both printed drag terms still cancel in the weighted sum used below, so that particular algebraic combination is unaffected by this normalization error.
For a nonrelativistic distribution, $T^{ij}=\int d^3p\,p^ip^j f/E$ is down by $v^2$ relative to the rest-mass energy density, and the corresponding <anisotropic stress> is also suppressed. The streaming terms coupling successive angular moments are proportional to $kv$, and stresses from the velocity distribution disappear in the cold limit. Thus, on scales where velocity dispersion is negligible, density and bulk velocity suffice; the photon <Boltzmann hierarchy> has no analogous small-speed suppression. This argument needs a cold distribution or a controlled small-dispersion approximation. Collisionless nonrelativistic matter with appreciable dispersion or multistreaming need not be an exact perfect fluid, and near photon decoupling the higher photon moments cannot all remain negligible.
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