Over-constrained Models of Time Delay Lenses Redux: How the Angular Tail Wags the Radial Dog

The two properties of the radial mass distribution of a gravitational lens that are well-constrained by Einstein rings are the Einstein radius R_E and xi2 = R_E alpha(R_E)/(1-kappa_E), where alpha(R_E) and kappa_E are the second derivative of the deflection profile and the convergence at R_E. However, if there is a tight mathematical relationship between the radial mass profile and the angular structure, as is true of ellipsoids, an Einstein ring can appear to strongly distinguish radial mass distributions with the same xi2. This problem is beautifully illustrated by the ellipsoidal models in Millon et al. (2019). When using Einstein rings to constrain the radial mass distribution, the angular structure of the models must contain all the degrees of freedom expected in nature (e.g., external shear, different ellipticities for the stars and the dark matter, modest deviations from elliptical structure, modest twists of the axes, modest ellipticity gradients, etc.) that work to decouple the radial and angular structure of the gravity. Models of Einstein rings with too few angular degrees of freedom will lead to strongly biased likelihood distinctions between radial mass distributions and very precise but inaccurate estimates of H0 based on gravitational lens time delays.