Shapes and Shears, Stars and Smears: Optimal Measurements for Weak Lensing

TL;DR

The paper tackles the challenge of extracting weak gravitational lensing signals from noisy galaxy images by developing a geometrically grounded, end-to-end framework for optimal shape measurement and shear estimation. It introduces a conformal-shear geometry and an adaptive, Gaussian-weighted Laguerre expansion that enables precise PSF correction, deconvolution, and noise-aware aggregation of measurements across exposures and bands. Key contributions include a rigorous derivation of shape estimators with calculable uncertainties, an optimal weighting scheme tied to intrinsic shape distributions, and a comprehensive treatment of PSF bias, dilution, and selection/centroid biases using Laguerre-based methods. The approach promises improved control of systematic errors and calibrated uncertainties, with practical methods for nanoscopic shear measurements applicable to cosmic shear and precision cosmology.

Abstract

We present the theoretical and analytical bases of optimal techniques to measure weak gravitational shear from images of galaxies. We first characterize the geometric space of shears and ellipticity, then use this geometric interpretation to analyse images. The steps of this analysis include: measurement of object shapes on images, combining measurements of a given galaxy on different images, estimating the underlying shear from an ensemble of galaxy shapes, and compensating for the systematic effects of image distortion, bias from PSF asymmetries, and `"dilution" of the signal by the seeing. These methods minimize the ellipticity measurement noise, provide calculable shear uncertainty estimates, and allow removal of systematic contamination by PSF effects to arbitrary precision. Galaxy images and PSFs are decomposed into a family of orthogonal 2d Gaussian-based functions, making the PSF correction and shape measurement relatively straightforward and computationally efficient. We also discuss sources of noise-induced bias in weak lensing measurements and provide a solution for these and previously identified biases.

Figures (9)

• Figure 1: The shaded surface is an embedding of the shear manifold into Euclidean space. The radius vector along this surface is the conformal shear $\eta$; the radius upon projection onto the $xy$ plane is the distortion $\delta$ (or ellipticity $e$). At small $\eta$ the manifold is tangent to the $\delta$ plane, and at large $\eta$ approaches the unit cylinder. Two shear vectors ${\bf s}_1$ and ${\bf s}_2$ of length $\eta=1$ are plotted from the origin, both on the true shear manifold and in the $\delta$ plane. The result of adding the two vectors is also plotted; displacements do not commute in this non-Euclidean space.
• Figure 2: The scheme for ellipticity measurement and its errors are illustrated schematically: in (a), the triangle marks the true shape of a target galaxy, located in the e plane. The shape is determined by shearing the image until the galaxy appears round. In the ellipticity plane, we are moving the object along the vector to the origin. Panel (b) shows the location of the target (and the original coordinate grid) in the e plane after application of the shear that makes it round. The shaded region represents the uncertainty region for the shape in the sheared view---the error region must be circular because the image is round. Finally in (c) we undo the applied shear, shifting the target and the error region back to the original shape. This mapping, however, shrinks the error region by a factor $1-e^2$ ($\sqrt{1-e^2}$) in the radial (azimuthal) axes.
• Figure 3: The left panel shows a model for intrinsic distribution $P(e)$ of galaxy shapes over the ${\bf e}$ plane---it must have circular symmetry. When each galaxy is sheared by $\delta_+=0.05$, the galaxy distribution shifts to the right as in the middle panel. The right-hand panel shows the change in population under the applied distortion; this is the signal which we wish to detect. Shape noise arises from the Poisson fluctuations in the population, which is proportional to the left-hand panel's $P(e)$. The optimal weight for $\delta_+$ determination is the ratio of the right-hand to the left-hand panel.
• Figure 4: The distribution of intrinsic ellipticities for modestly bright galaxies ($m_R\approx 20$) is plotted; we plot $2\pi e P(e)$ rather than $P(e)$ as the nature of the population is more apparent. The distribution is highly dependent upon surface brightness $\mu_R$, presumably reflecting the difference between spheroid- and disk-dominated galaxies. The dashed line is the distribution for an isotropic population of circular disks. The high-$\mu$ galaxies are more useful for distortion measurements. The heavy histogram combines all surface brightnesses in the magnitude range $20<m_R<21$. Though it is difficult to tell from this plot, $P(e)$ is finite and increasing as $e\rightarrow 0$. Optimal weighting takes advantage of this structure to reduce the noise in the distortion measurement.
• Figure 5: The upper panel plots the approximate, simplified closed-sum estimate of the calibration factor ${\cal R}$ (Equation [\ref{['nresp1']}]) relative to the exact form (Equation [\ref{['resp3']}]), as a function of the ellipticity measurement noise at $e=0$. The weight function is the "easy" Equation (\ref{['goodwt']}) and the factors $k_0$ and $k_1$ adopt the simple heuristic approximation (\ref{['k0k1gauss']}). The heavy (black) curve is for the population of $20<m_R<21$ galaxies, the upper (green) curve is for a low-SB sample ($20<\mu_R<21$), and the lower (red) curve for a high-SB sample ($17<\mu_R<19$), for which the intrinsic $P(e)$ distributions are plotted in Figure \ref{['pe']}. The simple formulation yields a calibration accurate to 5% or better in all cases, but 1% accuracy is difficult to achieve. The lower panel shows the uncertainty in the distortion determination when the galaxy shapes are combined with optimum weights (solid lines), the "easy" weights (\ref{['goodwt']}, dotted), and equal weighting (dashed). The line weights/colors code the galaxy sample, as above. Note that with optimal or easy weighting, the distortion errors continue to shrink even when measurement error is well below the canonical shape noise of $0.3$.