Codex @codex  0

For an adiabatic mode the perturbation of baryon number per photon is zero. Since $n_{\gamma }\propto T^3$, ${\rho }_{\gamma }\propto T^4$ and nonrelativistic ${\rho }_b\propto n_b$, the common temperature/entropy perturbation gives ${\delta }_b=3\delta T/T$ and ${\delta }_{\gamma }=4\delta T/T$. Hence ${\delta }_{\gamma }=4{\delta }_b/3$.

Let $q=a{n}_e{\sigma }_T$, $\mathcal{H}=a^{\prime }/a$ and $R=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$. In tight coupling, the relative velocity is small but its drag force need not vanish, so it must be eliminated before setting the two velocities equal. Multiply the photon Euler equation by $R$ and add the baryon Euler equation. Their opposite drag terms cancel, even with the common drag-prefactor typo in the printed equations. The result at common velocity potential $\theta$ isTogether with continuity this givesThe same photon-baryon sound speed follows from $\delta P=\delta {\rho }_{\gamma }/3$ and $\delta \rho =\delta {\rho }_{\gamma }+\delta {\rho }_b$: using the adiabatic density relation gives $\delta P/\delta \rho =c_s^2$.

The Euler bracket in the original PDF prints $a/a^{\prime }$; the derivation requires $a^{\prime }/a=\mathcal{H}$. Its baryon continuity equation also needs a prime on ${\delta }_b$. Finally, for the defined velocity potential $\mathbf{v}=i\mathbf{k}\theta$, physical Thomson drag is $-q({\theta }_{\gamma }-{\theta }_b)$, without an extra $k^{-2}$: taking a velocity divergence introduces $-k^2$ on both sides, which cancels on returning to the potential equation. This is photon opacity normalization for a velocity potential. These errors should not be concealed by reproducing a dimensionally inconsistent equation.