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 $k = 0.3 {\rm \ Mpc^{-1}}$ come from a characteristic redshift of $z \approx 1130$. We then estimate the corrections to the lensing kernel and the related power spectra due to this effect. 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 CMB scales. For an all-sky, noise-free experiment, this corresponds to a $\sim 0.1 \sigma$ shift in the observed temperature power spectrum on small scales ($2500 \lesssim l \lesssim 4000$).