Host group degeneracy in gravitational lensing time delay determination of H₀

Massive elliptical galaxies, that serve as lenses in gravitational lensing time delay measurements of the Hubble parameter H₀, often reside in a host group. We consider degeneracies in the modeling of the group halo. When the group effect on imaging can be summarized by its flexion (the next order term beyond shear in the tidal expansion), the posterior likelihood map can develop disjoint local minima, associated with an approximate discrete symmetry of a dominant flexion term. Monte-Carlo Markov Chain (MCMC) algorithms that are not designed to explore a rich posterior landscape can miss some of the minima, introducing systematic bias. We study mock data and demonstrate that the bias in H₀ can exceed 10%, and pulls the inference value of H₀ above its truth value, for a reason that can be traced to the structure of a mismodeled flexion term. MCMC algorithms that are designed to cope with a rich posterior landscape can uncover the structure. If the group is X-ray bright enough, X-ray data may also help to resolve the degeneracy, by pinpointing the group's center of mass. Finally, we show that some implementations in the literature used an inaccurate kinematical prior, mis-modeling the group velocity dispersion by as much as 20%.