We use a series of Monte Carlo simulations to investigate the theory of galaxy-galaxy lensing by non-spherical dark matter haloes .
The simulations include a careful accounting of the effects of multiple deflections on the galaxy-galaxy lensing signal .
In a typical observational data set where the mean tangential shear of sources with redshifts z _ { s } \simeq 0.6 is measured with respect to the observed symmetry axes of foreground galaxies with redshifts z _ { l } \simeq 0.3 , we find that the signature of anisotropic galaxy-galaxy lensing differs substantially from the simple expectation that one would have in the absence of multiple deflections .
In general , the observed ratio of the mean tangential shears , \gamma ^ { + } ( \theta ) / \gamma ^ { - } ( \theta ) , is strongly suppressed compared to the function that one would measure if the intrinsic symmetry axes of the foreground galaxies were known .
Depending upon the characteristic masses of the lenses , the observed ratio of the mean tangential shears may be consistent with an isotropic signal ( despite the fact that the lenses are non-spherical ) , or it may even be reversed from the expected signal ( i.e. , the mean tangential shear for sources close to the observed minor axes of the lenses may exceed the mean tangential shear for sources close to the observed major axes of the lenses ) .
These effects are caused primarily by the fact that the images of the lens galaxies have , themselves , been lensed and therefore the observed symmetry axes of the lens galaxies differ from their intrinsic symmetry axes .
We show that the effects of lensing of the foreground galaxies on the observed function \gamma ^ { + } ( \theta ) / \gamma ^ { - } ( \theta ) can not be eliminated simply by the rejection of foreground galaxies with very small image ellipticities , nor by simply focusing the analysis on sources that are located very close to the observed symmetry axes of the foreground galaxies .
We conclude that any attempt to use a measurement of \gamma ^ { + } ( \theta ) / \gamma ^ { - } ( \theta ) to constrain the shapes of dark matter galaxy haloes must include Monte Carlo simulations that take multiple deflections properly into account .