  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.
