Bridging the Gap Between Simply Parametrized and Free-Form Pixelated Models of Galaxy Lenses: The Case of WFI 2033-4723 Quad

We study the radial and azimuthal mass distribution of the lensing galaxy in WFI2033-4723. Mindful of the fact that modeling results depend on modeling assumptions, we examine two very different recent models: simply parametrized (SP) models from the H0LiCOW collaboration, and pixelated free-form (FF) GLASS models. In addition, we fit our own models which are a compromise between the astrophysical grounding of SP, and the flexibility of FF approaches. Our models consist of two offset parametric mass components, and generate many solutions, all fitting the quasar point image data. Among other results, we show that to reproduce point image properties the lensing mass must be lopsided, but the origin of this asymmetry can reside in the main lens plane or along the line of sight. We also show that there is a degeneracy between the slope of the density profile and the magnitude of external shear, and that the models from various modeling approaches are connected not by the mass sheet degeneracy, but by a more generalized transformation. Finally, we discuss interpretation degeneracy which afflicts all mass modeling: inability to correctly assign mass to the main lensing galaxy vs. nearby galaxies or line of sight structures. While this may not be a problem for the determination of $H_0$, interpretation degeneracy may become a major issue for the detailed study of galaxy structure.