Shear-only photon diffusion damping ($k_D^{-2}=\frac{2}{15}\int {\tau }_{c}d\tau$)

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.