Mass-sheet degeneracy, power-law models and external convergence: Impact
  on the determination of the Hubble constant from gravitational lensing
  The light travel time differences in strong gravitational lensing systems
allows an independent determination of the Hubble constant. This method has
been successfully applied to several lens systems. The formally most precise
measurements are, however, in tension with the recent determination of $H_0$
from the Planck satellite for a spatially flat six-parameters $\Lambda CDM$
cosmology. We reconsider the uncertainties of the method, concerning the mass
profile of the lens galaxies, and show that the formal precision relies on the
assumption that the mass profile is a perfect power law. Simple analytical
arguments and numerical experiments reveal that mass-sheet like transformations
yield significant freedom in choosing the mass profile, even when exquisite
Einstein rings are observed. Furthermore, the characterization of the
environment of the lens does not break that degeneracy which is not physically
linked to extrinsic convergence. We present an illustrative example where the
multiple imaging properties of a composite (baryons + dark matter) lens can be
extremely well reproduced by a power-law model having the same velocity
dispersion, but with predictions for the Hubble constant that deviate by $\sim
20%$. Hence we conclude that the impact of degeneracies between parametrized
models have been underestimated in current $H_0$ measurements from lensing, and
need to be carefully reconsidered.
