Galaxy-lens determination of $H_0$: the effect of the ellipse+shear modeling assumption.

Galaxy lenses are frequently modeled as an elliptical mass distribution with external shear and isothermal spheres to account for secondary and line-of-sight galaxies. There is statistical evidence that some fraction of observed quads are inconsistent with these assumptions, and require a dipole-like contribution to the mass with respect to the light. Simplifying assumptions about the shape of mass distributions can lead to the incorrect recovery of parameters such as $H_0$. We create several tests of synthetic quad populations with different deviations from an elliptical shape, then fit them with an ellipse+shear model, and measure the recovered values of $H_0$. Kinematic constraints are not included. We perform two types of fittings -- one with a single point source and one with an array of sources emulating an extended source. We carry out two model-free comparisons between our mock quads and the observed population. One result of these comparisons is a statistical inconsistency not yet mentioned in the literature: the image distance ratios with respect to the lens center of observed quads appear to span a much wider range than those of synthetic or simulated quads. Bearing this discrepancy in mind, our mock populations can result in biases on $H_0$ $\sim10\%$.