Small-scale modification to the lensing kernel

Calculations of the cosmic microwave background (CMB) 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 $k=0.01{\mathrm{Mpc}}^{-1}$ come from a characteristic redshift of $z\approx 1090$, while fluctuations at $k=0.3{\mathrm{Mpc}}^{-1}$ 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 ($2500\lesssim l\lesssim 4000$).

Figure 1

The CMB anisotropies can be broken into small scale, intermediate scale and large scale. Due to diffusion damping, the small-scale information is provided by photons which last scattered at redshifts larger than the redshift of recombination. In contrast, at smaller redshifts, the information on small scales is lost, and large-scale information is provided by late-time photons.

Figure 2

Top two panels: Damping scale and visibility function as a function of conformal time. Bottom two panels: Damping factor and the normalized product between the damping factor and the visibility function for different wave numbers $k$. Smaller-scale modes are most likely to have scattered at earlier times.

Figure 3

Time of the last scattering of photons as a function of CMB scale. The blue curve is derived numerically by computing the peak of the product $g(k,\eta )=D(k,\eta )v(\eta )$ for each wave number $k$. The orange curve uses our approximate result from Eq. (4).

Figure 4

Fractional difference in the lensing kernel at redshifts $z=1$, $z=5$ and $z=10$ assuming that the distance to last scattering depends on the CMB scale $k$. The deviations from the standard value of the kernel increase with redshift and decrease with scale.

Figure 5

Percentage difference in the lensed temperature power spectrum assuming a scale-dependent distance to last scattering. This result is derived by using the full-sky nonperturbative approximation in Ref. [25] to second order.