 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,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: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 ${\delta }_b=3{\delta }_{\gamma }/4$. ThenDefine the pressure-to-inertia sound speed, neglecting the very small baryon thermal pressure, byThe combined velocity equation consequently becomesThis 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 $c_s^2$.