Shapelets Multiple Multipole Shear Measurement

The measurement of weak gravitational lensing is currently limited to a precision of ∼10% by instabilities in galaxy shape measurement techniques and uncertainties in their calibration. The potential of large, on-going and future cosmic shear surveys will only be realised with the development of more accurate image analysis methods. We present a description of several possible shear measurement methods using the linear “shapelets” decomposition. Shapelets provides a complete reconstruction of any galaxy image, including higher-order shape moments that can be used to generalise the KSB method to arbitrary order. Many independent shear estimators can then be formed for each object, using linear combinations of shapelet coefficients. These estimators can be treated separately, to improve their overall calibration; or combined in more sophisticated ways, to eliminate various instabilities and a calibration bias. We apply several methods to simulated astronomical images containing a known input shear, and demonstrate the dramatic improvement in shear recovery using shapelets. A complete IDL software package to perform image analysis and manipulation in shapelet space can be downloaded from www.astro.caltech.edu/∼rjm/shapelets/. 1. Requirements for a shear estimator Mass fluctuations along the line of sight to a distant galaxy distort its apparent shape via weak gravitational lensing. If we can measure the “shear” field γ from the observed shapes of galaxies, we can map out the intervening mass distribution. But how should the galaxies’ shapes be measured? A monochromatic image of the sky is simply a two-dimensional function of surface brightness, in which the galaxies are isolated peaks. We would like to form local shear estimators γ̂ from some combination of the pixel values around each peak. The estimators are merely required to trace the true shear signal when averaged over a galaxy population: 〈γ̂〉 = γ. Individual estimators will inevitably be noisy, because of galaxies’ wide range of intrinsic ellipticities and morphologies. Furthermore, we are primarily interested in distant (and therefore faint) galaxies. Additional biases from observational noise can therefore be limited by forcing γ̂ to be a linear (or only mildly non-linear) combination of the pixel values. The standard shear measurement method applied to most current weak lensing data was invented by Kaiser, Squires & Broadhurst (1995; KSB). KSB provides a formalism to correct for smearing by a Point-Spread Function (PSF), and to form a shear estimator γ̂ ≡ e/P γ . It uses a galaxy’s Gaussian-weighted quadrupole ellipticity e, because the unweighted ellipticity does not converge in the presence of observational noise. Unfortunately, the weight function complicates PSF correction, and there is no obvious choice for its scale size. It is important to note that such an ellipticity by itself would not be a valid shear estimator. It does not respond linearly with shear; nor is it expected to, and this is a separate issue from the 0.85 calibration factor of Bacon et al. (2001). The necessary