 Photon opacity normalization for a velocity potential

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 $(1+R){\theta }^{\prime }+\mathcal{H}\theta +R({\delta }_{\gamma }/4-{\sigma }_{\gamma })=0$, with sound speed $R/[3(1+R)]$.