A hybrid approach to cosmic microwave background lensing reconstruction from all-sky intensity maps

On the basis of realistic simulations, we propose a hybrid method to reconstruct the lensing potential power spectrum, directly on Planck-like cosmic microwave background frequency maps. This involves the use of a large Galactic mask and the treatment of strong inhomogeneous noise. For l <= 100, we show that a full-sky inpainting method, which was previously described, still allows a minimal variance reconstruction, with a bias that must be accounted for by a Monte Carlo method but that does not couple to the deflection field. For l >= 100, we develop a method based on tiling the cut-sky with local 10 degrees x 10 degrees overlapping tangent planes (referred to in the following as patches). We tackle various issues related to their size/position, non-periodic boundaries, and irregularly sampled data of the planes after the sphere-to-plane projection. We show that the predominant noise term of the quadratic lensing estimator determined from an apodized patch can still be recovered directly from the data. To prevent any loss of spatial accuracy, we developed a tool that allows the efficient determination of the complex Fourier series coefficients from a bi-dimensional irregularly sampled dataset, without performing any interpolation. We show that our multi-patch approach enables the lensing power spectrum to be reconstructed with a very small bias, thanks to the omission of a Galactic mask and smaller noise inhomogeneities, as well as an almost minimal variance. At each stage, the data quality can be assessed and simple bi-dimensional spectra compiled, which allows the control of local systematic errors.