Unbiased Cluster Lens Reconstruction

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Abstract
We consider the problem of determining a galaxy cluster mass distribution using the weak gravitational distortion of background galaxies. From the measurements of the shapes of the weakly lensed background galaxies, one can measure the shear field, γα, and hence the gradient of the dimensionless surface density, κ, in the foreground cluster lens. We present several new algorithms to recover κ from shear estimates on a finite region and compare how they perform with realistically noisy data. The reconstruction methods studied here are divided into two classes: direct reconstruction and regularized inversion techniques. The direct reconstruction techniques express the surface density as a two-dimensional integral of the shear field. This allows one to construct an estimator for κ as a discrete sum over background galaxy ellipticities, which is straightforward to implement and allows a rigorous yet simple estimate of the noise arising from random intrinsic background galaxy ellipticities. We study three types of direct reconstruction methods: (1) κ estimators that measure the surface density at any given target point relative to the mean value in some reference region; (2) a method that explicitly attempts to minimize the rotational part of ∇κ that is due to noise; and (3) a novel, exact Fourier-space inverse gradient operator. We also develop two "regularized maximum-likelihood" methods, one of which employs the conventional discrete Laplacian operator as a regularizer and the other of which uses regularization of all components in Fourier space. We compare the performance of all the estimators by means of simulations and noise power analysis. A general feature of these unbiased methods is an enhancement of the low-frequency noise power that, for some of the methods, can be quite severe. We find the best performance is provided by the maximum-likelihood method with Fourier space regularization, although some of the other methods perform almost as well.
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