= Photon opacity normalization for a velocity potential
{title2=$\mathbf v=\nabla\theta,\quad\theta'_\gamma\vert_{\rm drag}=-a n_e\sigma_T(\theta_\gamma-\theta_b)$}
With Fourier convention $\mathbf v=i\mathbf k\theta$, the velocity divergence is $-k^2\theta$. Thomson drag in the velocity equation is the conformal scattering rate $q=a n_e\sigma_T$ times the velocity difference. Taking its divergence and dividing the whole divergence equation by $-k^2$ leaves the same rate multiplying the difference of velocity potentials; an additional $k^{-2}$ is not retained. Weighting the photon Euler equation by $R=4\bar\rho_\gamma/(3\bar\rho_b)$ and adding the baryon equation cancels the drag. In tight coupling the remaining equation is $(1+R)\theta'+\mathcal H\theta+R(\delta_\gamma/4-\sigma_\gamma)=0$, with sound speed $R/[3(1+R)]$.
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