Uncovering the Potential Envelope. The potential envelope is obscured by diffusion damping. By numerically removing the damping, one sees that the intrinsic fluctuations follow the analytic estimates of ${\cal P}_\ell$ reasonably well. By multiplying by the numerically-calibrated damping function ${\cal D}_\ell^2$, one recovers the form of the full calculation even at very small angles. The model here is standard CDM.

Constraining $\Omega_0$ with the damping tail. By measuring the anisotropy power in at some scale $\ell$ in the damping tail (here averaged over 10\% in $\ell$) and comparing it to a reference scale (here $\ell=2$), one determines the ratio of intrinsic powers ${\cal P}_\ell/{\cal P}_2$ before damping necessary to reproduce the observation (here $\Omega_0=1$ in standard CDM). Since this is a strong function of the assumed $\Omega_0$, only order of magnitude knowledge of the model-dependent intrinsic power is needed (e.g. square, estimated from Eq.~(32)) to reject values of $\Omega_0$. Multiple measurements in the damping tail largely removes this ambiguity (curve intersection). For simplicity, we have fixed $h=0.5$, $\Omega_bh^2=0.0125$ and $\Omega_\Lambda=0$.

Reionization damping in standard CDM. Damping described by the envelope ${\cal R}_\ell$ is the main effect of late reionization in CDM type models. Hence employing either the numerical calibration of ${\cal R}_\ell$ and the fit to it from Eq.~(24) to filter the results of a standard recombination (SR, no reionization) calculation approximate the full calculation to better than 1\% in power. The scatter at low $\ell$ is a numerical artifact from finite sampling of the $C_\ell$ integral in $k$-space (see Eq.~(4)).

Reionization and the Doppler Effect. For early ionization, the Doppler effect due to the relative electron-photon velocity can regenerate fluctuations around the horizon scale at the last scattering epoch. By comparing the standard recombination (SR) result filtered by reionization damping ${\cal R}_\ell^2$ to the full calculation, we can uncover such effects.