Unbiased Cluster Lens Reconstruction
Gordon Squires, Nick Kaiser · The Astrophysical Journal · 1996
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, γ α , 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.