Internal delensing of cosmic microwave background acoustic peaks

We present a method to delens the acoustic peaks of the cosmic microwave background (CMB) temperature and polarization power spectra internally, using lensing maps reconstructed from the CMB itself. We find that when delensing CMB acoustic peaks with a lensing potential map derived from the same CMB sky, a large bias arises in the delensed power spectrum. The cause of this bias is that the noise in the reconstructed potential map is derived from, and hence correlated with, the CMB map when delensing. This bias is more significant relative to the signal than an analogous bias found when delensing CMB $B$ modes. We calculate the leading term of this bias, which is present even in the absence of lensing. We also demonstrate one method to remove this bias, using reconstructions from CMB angular scales within given ranges to delens CMB scales outside of those ranges. Some details relevant for a realistic analysis are also discussed, such as the importance of removing mask-induced effects for successful delensing, and a useful null test, obtained from randomizing the phases of the reconstructed potential. Our findings should help current and next-generation CMB experiments obtain tighter parameter constraints via the internal removal of lensing-induced smoothing from temperature and $E$-mode acoustic peaks.

Figure 1

Top panel: The mean temperature power spectra, $D_l=l(l+1)C_l/(2\pi )$, of lensed and delensed CMB maps from 2000 simulations are shown as red and green points, respectively. For this case, there is no noise added to simulations and the delensing is done with the input potential maps. Bottom panel: The mean lensed minus delensed power spectra from 2000 simulations are shown as blue points, and the errors represent the error on the mean. The expected theory prediction of lensed minus unlensed power spectra is shown by the green curve. At high $\ell$, the blue points fall slightly below the green theory line due to the difference between delensing and inverse-lensing (see text for details).

Figure 2

Top panel: Blue points show lensed minus delensed power spectra when delensing with the input potential map with Gaussian reconstruction noise added corresponding to the expected noise when reconstructing from a CMB temperature map with $1\mu \mathrm{K}$-arcmin white noise. In this case, only some of the lensing signal is removed, as seen by the difference between the points and green curve, the latter of which shows the theory prediction of lensed minus unlensed power spectra. Middle panel: Same as top, but delensing with the reconstructed potential map. The correlation between the reconstruction noise and the CMB map being delensed results in a large bias. Bottom panel: Same as middle panel, except the bias has been removed by using the reconstruction from CMB $\ell$-modes in a given range to delens $\ell$-modes outside of that range. This procedure avoids correlated noise bias at the expense of some signal-to-noise, however more delensing can be recovered with modifications to this procedure (see text).

Figure 3

Lensed minus delensed power spectra using a reconstructed potential plus bias fix as in the bottom panel of Fig. 2, now with error bars representative of a CMB-S4 type experiment with $f_{\mathrm{sky}}=0.4$ and $1\mu \mathrm{K}$-arcmin white noise.

Figure 4

The points show lensed minus delensed temperature maps when delensing with a reconstructed potential map that has not been mean-field subtracted. The red curve shows the result from the bottom panel of Fig. 2, where the mean-field has been subtracted. Even though modes with $L\le 100$ have been removed in both cases, the significant difference between pink and red curves in this plot demonstrates the importance of mean-field subtraction for delensing acoustic peaks.

Figure 5

The blue points show the power spectrum of lensed minus delensed temperature maps, when delensing with a reconstructed potential map whose phases have been randomized. The peaks are now out of phase with the theory expectation for delensing with the correct potential map shown by the green curve in the case of no noise. Thus delensing with a reconstructed potential map where the phases have been randomized can serve as a null test to verify delensing is being done properly with the original reconstructed map.