Analytic Cross Sections for Substructure Lensing

I study the gravitational lensing properties of a small mass clump perturbing a strong lens system. A clump near one of the lensed images changes the magnification of that image, an effect that has been detected in 4-image lenses. I show that the problem of a clump anywhere along the line of sight to a lens can be mapped onto an equivalent problem of a clump in a simple convergence and shear field. For clumps modeled as singular isothermal spheres (SIS), I derive simple analytic estimates of the cross section for magnification perturbations of a given strength. The results yield two interesting conceptual points. First, lensed images with positive parity are always made brighter by SIS clumps; images with negative parity can be brightened but are much more likely to be dimmed. Second, the clumps need not lie within the lens galaxy; they can be moved in redshift by several tenths and still have a significant lensing effect. Isolated small halos are expected to be common in hierarchical structure formation models, but it is not yet known whether they are abundant enough compared with clumps inside lens galaxies to affect the interpretation of substructure lensing.