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Gravitational lensing distorts the cosmic microwave background (CMB) anisotropies and imprints a characteristic pattern onto it. The distortions depend on the projected matter density between today and the redshift z1100. In this paper we develop a method for a direct reconstruction of the projected matter density from the CMB anisotropies. This reconstruction is obtained by averaging over quadratic combinations of the derivatives of the CMB field. We test the method using simulations and show that it can successfully recover a projected density profile of a cluster of galaxies if there are measurable anisotropies on scales smaller than the characteristic cluster size. In the absence of a sufficient small scale power the reconstructed maps have a low signal to noise ratio on individual structures, but can give a positive detection of the power spectrum or when cross correlated with other maps of large scale structure. We develop an analytic method to reconstruct the power spectrum including the effects of noise and beam smoothing. Tests with Monte Carlo simulations show that we can recover the input power spectrum both on large and small scales, provided that we use maps with sufficiently low noise and high angular resolution.
Zaldarriaga et al. (Mon,) studied this question.