TDCOSMO VI: Distance Measurements in Time-delay Cosmography under the
  Mass-sheet transformation
  Time-delay cosmography with gravitationally lensed quasars plays an important
role in anchoring the absolute distance scale and hence measuring the Hubble
constant, $H_{0}$, independent of traditional distance ladder methodology. A
current potential limitation of time delay distance measurements is the
"mass-sheet transformation" (MST) which leaves the lensed imaging unchanged but
changes the distance measurements and the derived value of $H_0$. In this work
we show that the standard method of addressing the MST in time delay
cosmography, through a combination of high-resolution imaging and the
measurement of the stellar velocity dispersion of the lensing galaxy, depends
on the assumption that the ratio, $D_{\rm s}/D_{\rm ds}$, of angular diameter
distances to the background quasar and between the lensing galaxy and the
quasar can be constrained. This is typically achieved through the assumption of
a particular cosmological model. Previous work (TDCOSMO IV) addressed the
mass-sheet degeneracy and derived $H_{0}$ under the assumption of $\Lambda$CDM
model. In this paper we show that the mass sheet degeneracy can be broken
without relying on a specific cosmological model by combining lensing with
relative distance indicators such as supernovae type Ia and baryon acoustic
oscillations, which constrain the shape of the expansion history and hence
$D_{\rm s}/D_{\rm ds}$. With this approach, we demonstrate that the mass-sheet
degeneracy can be constrained in a cosmological-model-independent way, and
hence model-independent distance measurements in time-delay cosmography under
mass-sheet transformations can be obtained.
