On the Choice of Lens Density Profile in Time Delay Cosmography

Time delay lensing is a mature and competitive cosmological probe. However, it is limited in accuracy by the well-known problem of the mass-sheet degeneracy: too rigid assumptions on the density profile of the lens can potentially bias the inference on cosmological parameters. I investigate the degeneracy between the choice of the lens density profile and the inference on the Hubble constant, focusing on double image systems. By expanding lensing observables in terms of the local derivatives of the lens potential around the Einstein radius, and assuming circular symmetry, I show that three degrees of freedom in the radial direction are necessary to achieve a few percent accuracy in the time-delay distance. Additionally, while the time delay is strongly dependent on the second derivative of the potential, observables typically used to constrain lens models in time-delay studies, such as image position and radial magnification information, are mostly sensitive to the first and third derivatives, making it very challenging to accurately determine time-delay distances with lensing data alone. Tests on mock observations show that the assumption of a power-law density profile results in a 5% average bias on $H_0$, with a 6% scatter. Using a more flexible model and adding unbiased velocity dispersion constraints allows to obtain an inference with 1% accuracy. A power-law model can still provide 3% accuracy if velocity dispersion measurements are used to constrain its slope. Although this work is based on the assumption of axisymmetry, its main findings can be generalized to cases with moderate ellipticity.