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

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 expressed as "Laguerre expansions," a 2d generalization of the Edgeworth expansion, making the PSF correction and shape measurement relatively straightforward and computationally efficient. We also discuss sources of noise-induced bias in weak lensing measurements-selection biases, and "centroid" biases arising from noise rectification-and provide a solution for these and previously identified biases.

Observational probes of cosmic acceleration

“…Many improvements of these methods have been made -e.g., in computing better conversion factors from shear to quadrupole moments 50 (Semboloni et al, 2006b). Elliptical-weighted moments and the concept of shear-covariance were introduced by Bernstein and Jarvis (2002) and have been used extensively in SDSS (Hirata and Seljak, 2003b). Further progress was made by moving to moments in Fourier space, where the PSF "correction" becomes trivial (one divides by the Fourier transform of the PSF, at least in the regions where it is nonzero).…”

“…, and so "non-Gaussianity corrections" were introduced (Bernstein and Jarvis, 2002;Hirata and Seljak, 2003b) that yielded shear calibration errors of a few percent. But these methods were heuristic, and moreover they suffer from a fundamental limitation: Q elfit ij [f ] depends on very high-wavenumber Fourier modes u of the image, which are not preserved by the PSF, i.e.G(u) = 0.…”

“…Rules of thumb can be obtained by considering nearly circular Gaussians. Propagating instrument noise through the elliptical Gaussian fitting method, Bernstein and Jarvis (2002) find, in the absence of a PSF,…”

“…It does however have a minimum value: the ellipticity of an individual galaxy has an RMS variation of e rms ∼ 0.4 per component, so there is a limiting "shape noise" contribution to the shear measurement uncertainty of σ γ ≈ 0.2. There are some ideas for how to circumvent this limit using the color-or scale-dependence of ellipticity (Lombardi and Bertin, 1998;Jarvis and Jain, 2008) or taking advantage of the non-Gaussianity of the ellipticity distribution (Kaiser, 2000;Bernstein and Jarvis, 2002), but there are no clear routes to large improvement for optical galaxies. For galaxies imaged in the HI 21cm line, one might be able to use kinematic signatures to distinguish random orientation from lensing shear (e.g.…”

“…The noise rectification bias was first recognized in the context of WL by Kaiser (2000), who showed that because the centroiding of a galaxy is more accurate on the "short" than the "long" axis of the PSF there is a preference for the measured second moment of the galaxy to be elongated along the PSF, even if the PSF correction method is perfect in the deterministic case. This was generalized by Bernstein and Jarvis (2002) to incorporate other noise-related biases and by to include the effect on shear calibration errors. Equation (126) provides a unified framework for computing all of these biases to order (S/N ) −2 .…”