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We demonstrate a method for reconstructing the weak lensing potential fromthe Lyman-$\alpha$ forest data. We derive an optimal estimator for the lensingpotential on the sky based on the correlation between pixels in real space.This method effectively deals with irregularly spaced data, holes in thesurvey, missing data and inhomogeneous noise. We demonstrate an implementationof the method with simulated spectra and weak lensing. It is shown that with asource density of $>\sim 0.5$ per square arcminutes and $\sim 200$ pixels ineach spectrum ($\lambda / \Delta\lambda = 1300$) the lensing potential can bereconstructed with high fidelity if the relative absorption in the spectralpixels is signal dominated. When noise dominates the measurement of theabsorption in each pixel the noise in the lensing potential is higher, but forreasonable numbers of sources and noise levels and a high fidelity map thelensing potential is obtainable. The lensing estimator could also be applied tolensing of the Cosmic Microwave Background (CMB), 21 cm intensity mapping (IM)or any case in which the correlation function of the source can be accuratelyestimated.

Reconstructing the Gravitational Lensing Potential from the Lyman-$\alpha$ Forest