Multipole Models of Four-Image Gravitational Lenses with Anomalous Flux   Ratios
  Anomalous radio flux ratios in four-image gravitational lens systems can be explained by adding a clumpy cold dark matter (CDM) component to the potential of the lens galaxy. Evans & Witt (2003) recently suggested that smooth multipole perturbations can provide a reasonable alternative to CDM substructure in some but not all cases. We generalize their method in two ways so as to determine whether multipole models can explain highly anomalous systems. We carry the multipole expansion to higher order, and also include external tidal shear as a free parameter. Fitting for the shear proves crucial to finding a physical (positive-definite density) model. For B1422+231, working to order k = 5 (and including shear) yields a model that is physical but implausible. Going to higher order (k >~ 9) reduces global departures from ellipticity at the cost of introducing small scale wiggles near the bright images. These localized undulations are more pronounced in B2045+265, where k ~ 17 multipoles are required to smooth out large scale deviations from elliptical symmetry. Such modes surely cannot be taken at face value; they must indicate that the models are trying to reproduce some other sort of structure. Our formalism naturally finds models that fit the data exactly, but we use B0712+472 to show that measurement uncertainties have little effect on our results. Finally, we consider the system B1933+503, where two sources are lensed by the same foreground galaxy. The additional constraints provided by the images of the second source render the multipole model unphysical. We conclude that external shear must be taken into account to obtain plausible models, and that a purely smooth angular structure for the lens galaxy does not provide a viable alternative to the prevailing CDM clump hypothesis. 
Arthur B. Congdon, Charles R. Keeton (Rutgers)