A Small-Scale Modification to the Lensing Kernel.

Calculations of the Cosmic Microwave Background lensing power implemented into the standard cosmological codes such as CAMB and CLASS usually treat the surface of last scatter as an infinitely thin screen. However, since the CMB anisotropies are smoothed out on scales smaller than the diffusion length due to the effect of Silk damping, the photons which carry information about the small-scale density distribution come from slightly earlier times than the standard recombination time. The dominant effect is the scale dependence of the mean redshift associated with the fluctuations during recombination. We find that fluctuations at $k = 0.01 {\rm \ Mpc^{-1}}$ come from a characteristic redshift of $z \approx 1090$, while fluctuations at $k = 1 {\rm \ Mpc^{-1}}$ come from a characteristic redshift of $z \approx 1200$. We then estimate the corrections to the lensing kernel due to the finite size of the thickness of the surface of last scatter.

We conclude that neglecting it would result in a deviation from the true value of the lensing kernel at the half percent level at small scales. For future high signal-to-noise CMB experiments (e.g., CMB-S4), this corresponds to a $\sim 0.3 \sigma$ shift on scales $k \sim 1 {\rm \ Mpc^{-1}}$.