Photon-baryon diffusion damping equation

When gravity and expansion can be neglected and $R$ is constant, the photon monopole satisfiesThe first damping contribution comes from photon-baryon velocity slip; the second comes from the photon quadrupole. This is a local form of Silk damping.

For the simplified quadrupole equation ${\sigma }_{\gamma }^{\prime }+4k^2{\theta }_{\gamma }/15=-{\sigma }_{\gamma }/{\tau }_c$, neglecting its time derivative and gravitational driving gives ${\sigma }_{\gamma }=-{\tau }_c{\delta }_{\gamma }^{\prime }/5$. A radiation-dominated inertia then obeys ${\delta }_{\gamma }^{\prime \prime }+4{\tau }_c{k}^2{\delta }_{\gamma }^{\prime }/15+k^2{\delta }_{\gamma }/3=0$. On slowly varying subhorizon scales the amplitude is damped by $\exp [-(2/15)k^2\int {\tau }_{c}d\tau ]$. This coefficient belongs to that truncated collision model, rather than the polarization-inclusive Silk damping equation.

Removing the first derivative from the truncated photon diffusion equation gives the displayed transformed oscillator. Constant mean free time changes its oscillation frequency only at second order. For a slowly varying mean free time and subhorizon wavelengths, derivative corrections to the phase and prefactor are small; the leading solution is an acoustic sine/cosine multiplied by $\exp [-k^2/k_D^2]$, where $k_D^{-2}=2\int {\tau }_{c}d\tau /15$. Dropping the derivative correction requires slow variation as well as a small mean free time; it is not an exact solution for arbitrary time dependence.