= Slowly varying shear-damped photon oscillator
{title2=$\delta_\gamma=e^{-\frac2{15}k^2\int\tau_c d\tau}y,\quad y''+[k^2/3-\frac2{15}k^2\tau_c'+O(\tau_c^2)]y=0$}
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.
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