We present a method to delens the acoustic peaks of the 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.
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