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 $H_0$ $\sim10\%$.
