= Solution
An initially <adiabatic cosmological perturbation> leaves the local photon-to-baryon number ratio unchanged. Since radiation density scales as the fourth power of temperature and nonrelativistic particle number as the third, its <density contrasts> obey $\delta_b=3\delta_\gamma/4$. With the corrected common-velocity continuity equations,
$$
\left(\delta_b-\tfrac34\delta_\gamma\right)'=k^2(\theta_b-\theta_\gamma)\simeq0,
$$
so the adiabatic relation persists in <tight coupling>.
Use the paper's ratio $R=4\bar\rho_\gamma/(3\bar\rho_b)$, which is the reciprocal of the commonly used <baryon loading parameter>. Multiply the photon Euler equation by $R$ and add the baryon Euler equation. The collision forces cancel exactly:
$$
R\theta_\gamma'+\theta_b'
=-R(\tfrac14\delta_\gamma-\sigma_\gamma)-\mathcal H\theta_b-c_{sb}^2\delta_b.
$$
This cancellation works even with the common erroneous $k^{-2}$ in the printed forces, but a physical collision rate uses the repaired forces in part (i). At leading tight-coupling order put $\theta_\gamma=\theta_b=\theta$ and substitute $\delta_b=3\delta_\gamma/4$. Then
$$
\boxed{\begin{aligned}
\delta_\gamma'&=\frac43k^2\theta,\\
\theta'&=-\frac{R}{1+R}(\tfrac14\delta_\gamma-\sigma_\gamma)
-\frac1{1+R}(\mathcal H\theta+\tfrac34c_{sb}^2\delta_\gamma).
\end{aligned}}
$$
Define the pressure-to-inertia sound speed, neglecting the very small baryon thermal pressure, by
$$
\boxed{c_s^2=\frac{R}{3(1+R)}=\frac1{3(1+3\bar\rho_b/(4\bar\rho_\gamma))}.}
$$
The combined velocity equation consequently becomes
$$
\boxed{\theta'=-3c_s^2(\tfrac14\delta_\gamma-\sigma_\gamma)
-\frac{3c_s^2}{R}(\mathcal H\theta+\tfrac34c_{sb}^2\delta_\gamma).}
$$
This is the requested <photon-baryon acoustic oscillator> system. Photon pressure supplies the restoring force, while both photon enthalpy and baryon mass supply inertia. The retained $c_{sb}^2$ term adds the small thermal-baryon correction rather than being part of the quoted leading $c_s^2$.
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