Diffusion damping calibration. In the absence of both diffusion damping and gravitational sources, the rms temperature fluctuation at recombination (short-dashed line) exhibits simple acoustic oscillations. These are mapped onto anisotropies on the sky in a near one to one fashion (solid line). The inclusion of diffusion terms in the Boltzmann equation allows for a simple numerical calibration of its effects.

Reionization damping calibration. By removing the relative Doppler effect from a reionized Boltzmann calculation and comparing the result to the same model (here standard CDM $\Omega_0=1$, $h=0.5$, $\Omega_b h^2=0.0125$) with no reionization, the effects of rescattering damping are isolated. The reionization damping envelope is fit by two parameters, the optical depth during reionization and the horizon scale at last scattering (see Eq.~24).

Diffusion scale calibration. Analytic estimates of $k_D(\Omega_0 h^2, \Omega_b h^2)$ based on the tight-coupling approximation trace the results to reasonable accuracy and explains their general behavior (HSc, Eq.~E4). The fitting function of Eq.~(17) tracks the numerical calibration to better than the $1\%$ level.