Reconstructing the Gravitational Lensing Potential from the
  Lyman-$\alpha$ Forest
  We demonstrate a method for reconstructing the weak lensing potential from
the Lyman-$\alpha$ forest data. We derive an optimal estimator for the lensing
potential on the sky based on the correlation between pixels in real space.
This method effectively deals with irregularly spaced data, holes in the
survey, missing data and inhomogeneous noise. We demonstrate an implementation
of the method with simulated spectra and weak lensing. It is shown that with a
source density of $>\sim 0.5$ per square arcminutes and $\sim 200$ pixels in
each spectrum ($\lambda / \Delta\lambda = 1300$) the lensing potential can be
reconstructed with high fidelity if the relative absorption in the spectral
pixels is signal dominated. When noise dominates the measurement of the
absorption in each pixel the noise in the lensing potential is higher, but for
reasonable numbers of sources and noise levels and a high fidelity map the
lensing potential is obtainable. The lensing estimator could also be applied to
lensing of the Cosmic Microwave Background (CMB), 21 cm intensity mapping (IM)
or any case in which the correlation function of the source can be accurately
estimated.
