We investigate the weak lensing shear due to dark matter galaxy halos whose mass distributions , as projected on the sky , are nearly elliptical .
The shear pattern due to these halos is anisotropic about the lens centers and we quantify the level of anisotropy by comparing the mean shear experienced by sources located closest to the major axes of the lenses , \left < \gamma \right > _ { major } , to that experienced by sources located closest to the minor axes , \left < \gamma \right > _ { minor } .
We demonstrate that the degree of anisotropy is independent of angular scale and show that in the case of substantially flattened halos ( \epsilon = 0.7 ) , the value of \left < \gamma \right > _ { minor } is of order 40 % of the value of \left < \gamma \right > _ { major } when all sources within \pm 45 ^ { \circ } of the axial direction vectors of the lenses are included in the calculation .
In the case of halos that are flattened at more realistic level ( \epsilon = 0.3 ) , the value of \left < \gamma \right > _ { minor } is of order 75 % of the value of \left < \gamma \right > _ { major } .
We compute the degree to which the anisotropy in the lensing signal is degraded due to a noisy determination of the position angles of the lens galaxies and find that provided the typical 1- \sigma error on the orientation of the lenses is less than 15 ^ { \circ } , more than 90 % of the true lensing signal will be recovered in the mean .
We discuss our results in the context of detecting anisotropic galaxy–galaxy lensing in large , ground–based data sets and conclude that a modest net flattening of dark matter halos should be detectable at a statistically significant level .
The forthcoming Sloan Digital Sky Survey ( SDSS ) data will necessarily provide a very useful data set for this analysis , but a detection of anisotropic galaxy–galaxy lensing is not dependent upon the very large sky coverage of the SDSS .
Rather , we argue that a significant detection of this effect can also be obtained from an imaging survey that is of order two magnitudes fainter than SDSS and which covers only a relatively small area of sky , of order one square degree .