Lensing power spectrum of the Cosmic Microwave Background with deep polarization experiments

Precise reconstruction of the cosmic microwave background lensing potential can be achieved with deep polarization surveys by iteratively removing lensing-induced $B$ modes. We introduce a lensing spectrum estimator and its likelihood for such optimal iterative reconstruction. Our modelling share similarities to the state-of-the-art likelihoods for quadratic estimator-based (QE) lensing reconstruction. In particular, we generalize the $N_L^{(0)}$ and $N_L^{(1)}$ lensing biases, and design a realization-dependent spectrum debiaser, making this estimator robust to uncertainties in the data modelling. We demonstrate unbiased recovery of the cosmology using map-based reconstructions, focussing on lensing-only cosmological constraints and neutrino mass measurement in combination with CMB spectra and acoustic oscillation data. We find this spectrum estimator is essentially optimal and with a diagonal covariance matrix. For a CMB-S4 survey, this likelihood can double the constraints on the lensing amplitude compared to the QE on a wide range of scales, while at the same time keeping numerical cost under control and being robust to errors.