Gravitational lensing by foreground halos with masses as small as 1010 M☉ can induce observable second-order shape distortions in background galaxy images in addition to the well-known elliptical shape distortion produced by first-order shear. Furthermore, the orientation of the second-order distortions is correlated with the orientation of the shear-induced elliptical (quadrupolar) shape. Because of its angular dependence, we refer to the most prominent second-order distortion as a sextupole distortion. The correlation between first-order and second-order shapes provides a sensitive signal by which to identify galaxies that may have been lensed. Galaxy images appear to be slightly curved by these lensing events. In this paper we develop a general theoretical lensing framework based on a lensing distortion map. Tools to infer map coefficients from the galaxy images are described and applied to the Hubble Deep Field-North. Instrumental PSFs, camera charge diffusion, and image composition methods are modeled in the coefficient determination process. Estimates of Poisson counting noise for each galaxy are used to cut galaxies with signals too small to reliably establish curvature. These noise estimates are confirmed using a Fisher matrix analysis. Background shape distortions may also be represented by maps. Curved galaxies are found to be spatially clumped, as might arise from sextupole lensing by overdensities in a foreground dark matter halo distribution. We calculate the cross section for sextupole lensing by Moore and NFW halos and estimate the total cross section of the field based on these halo cross sections.
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