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The two properties of the radial mass distribution of a gravitational lensthat 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 secondderivative of the deflection profile and the convergence at R_E. However, ifthere is a tight mathematical relationship between the radial mass profile andthe angular structure, as is true of ellipsoids, an Einstein ring can appear tostrongly distinguish radial mass distributions with the same xi2. This problemis beautifully illustrated by the ellipsoidal models in Millon et al. (2019).When using Einstein rings to constrain the radial mass distribution, theangular structure of the models must contain all the degrees of freedomexpected in nature (e.g., external shear, different ellipticities for the starsand the dark matter, modest deviations from elliptical structure, modest twistsof the axes, modest ellipticity gradients, etc.) that work to decouple theradial and angular structure of the gravity. Models of Einstein rings with toofew angular degrees of freedom will lead to strongly biased likelihooddistinctions between radial mass distributions and very precise but inaccurateestimates of H0 based on gravitational lens time delays.

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