= Finite-width last-scattering damping
{title2=$\int g(\tau)e^{ik\mu\tau}d\tau\propto e^{-(k\mu\sigma_\tau)^2/2}$}
A finite visibility width averages photon sources over emission time and radial position. For a Gaussian visibility profile, a plane-wave radial phase has the displayed damping factor; its power has the square of this amplitude factor. An acoustic source also varies with frequency $kc_s$, so its rapid oscillations are averaged. This averaging is distinct from <Silk damping>, which smooths the perturbations by photon diffusion before decoupling. Angular projection and source evolution prevent a universal isotropic damping coefficient based only on radial width.
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