= Photon opacity normalization for a velocity potential
{title2=$\mathbf v=\nabla\theta,\quad\theta'_\gamma\vert_{\rm drag}=-a n_e\sigma_T(\theta_\gamma-\theta_b)$}
Let $\mathbf v=i\mathbf k\theta$ for a nonzero <Fourier mode>, so its <velocity divergence> is $-k^2\theta$. In units $c=1$, the conformal <Thomson scattering> rate is $q=a n_e\sigma_T$, with proper <electron number density> $n_e$ and <scale factor> $a$. The photon drag term is $-q(\mathbf v_\gamma-\mathbf v_b)$. Taking a divergence and dividing the entire equation by $-k^2$ therefore gives
$$
\theta'_\gamma\big|_{\rm drag}=-q(\theta_\gamma-\theta_b).
$$
No additional $k^{-2}$ survives. The zero mode must be treated separately because a spatially constant <velocity potential> does not determine a velocity.
To display <momentum conservation>, define $R_\gamma=4\bar\rho_\gamma/(3\bar\rho_b)$, the inverse of the usual <baryon loading parameter>. Neglect baryon thermal <pressure>. With Newtonian potential $\Psi$ and photon shear convention $\sigma_\gamma$, the two Euler equations can be written
$$
\theta'_\gamma=-\frac{\delta_\gamma}{4}+\sigma_\gamma-\Psi-q(\theta_\gamma-\theta_b),\qquad
\theta'_b+\mathcal H\theta_b=-\Psi+R_\gamma q(\theta_\gamma-\theta_b).
$$
Weighting the photon equation by $R_\gamma$ and adding the baryon equation cancels the drag. In the <tight-coupling approximation>, $\theta_\gamma\simeq\theta_b=\theta$, hence
$$
(1+R_\gamma)\theta'+\mathcal H\theta
=-(1+R_\gamma)\Psi-R_\gamma(\delta_\gamma/4-\sigma_\gamma).
$$
For adiabatic acoustic fluctuations the \b[squared] <sound speed> is $c_s^2=R_\gamma/[3(1+R_\gamma)]$, so $c_s=\sqrt{R_\gamma/[3(1+R_\gamma)]}$. This tends to $1/\sqrt3$ in the negligible-baryon limit. Calling the squared expression the <sound speed> loses the square root.
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