Finite source effects in strong lensing: Implications for the substructure mass scale

Flux ratio 'anomalies' in quadruply imaged gravitational lenses can be explained with galactic substructure of the sort predicted by ΛCDM, but the strength and uniqueness of that hypothesis need to be further assessed. A good way to do that is to use the physical scale associated with the size of the source quasar, and its dependence on wavelength. We develop a toy model to study finite source effects in substructure lensing. Treating substructure as a singular isothermal sphere allows us to compute the images of a finite source analytically, and then to explore how the image configurations and magnifications depend on source position and size. Although simplified, our model yields instructive general principles: image positions and magnifications are basically independent of source size until the source is large enough to intersect a substructure caustic; even sources that are much larger than the substructure Einstein radius can be perturbed at a detectable level; and, most importantly, there is a tremendous amount to be learned from comparing image positions and magnifications at wavelengths that correspond to different source sizes. In a separate analysis, we carefully study four observed radio lenses to determine which of the images are anomalous. In B0712+472, the evidence for a radio flux ratio anomaly is marginal, but if the anomaly is real then image C is probably the culprit. In B1422+231, the anomaly is in image A. Interestingly, B2045+265 and B1555+375 both appear to have two anomalous images. Coincidentally, in each system one of the anomalies is in image C, and the other is in either image A or image B (both possibilities lead to acceptable models). It remains to be seen whether ΛCDM predicts enough substructure to explain multiple anomalies in multiple lenses. When we finally join our modelling results and substructure theory, we obtain lower bounds on the masses of the substructures responsible for the observed anomalies. The mass bounds are broadly consistent with expectations for ΛCDM. Perhaps more importantly, we outline various systematic effects in the mass bounds; poor knowledge of whether the substructure lies within the main lens galaxy or elsewhere along the line of sight appears to be the dominant systematic.