Finite-width last-scattering damping ($\int g(\tau )e^{ik\mu \tau }d\tau \propto e^{-(k\mu {\sigma }_{\tau }{)}^2/2}$)

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 $k{c}_s$, 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.