# Publications

### Liliya L.R. Williams

**Model-free analysis of quadruply imaged gravitationally lensed systems and substructured galaxies**Woldesenbet, Addishiwot G.; Williams, Liliya L. R.,

*MNRAS*
**Abstract**

Multiple image gravitational lens systems, and especially quads, are invaluable in determining the amount and distribution of mass in galaxies. This is usually done by mass modelling using parametric or free-form methods. An alternative way of extracting information about lens mass distribution is to use lensing degeneracies and invariants. Where applicable, they allow one to make conclusions about whole classes of lenses without model fitting. Here, we use approximate, but observationally useful invariants formed by the three relative polar angles of quad images around the lens centre to show that many smooth elliptical+shear lenses can reproduce the same set of quad image angles within observational error. This result allows us to show in a model-free way what the general class of smooth elliptical+shear lenses looks like in the three-dimensional (3D) space of image relative angles, and that this distribution does not match that of the observed quads. We conclude that, even though smooth elliptical+shear lenses can reproduce individual quads, they cannot reproduce the quad population. What is likely needed is substructure, with clump masses larger than those responsible for flux ratio anomalies in quads, or luminous or dark nearby perturber galaxies.