TDCOSMO V: strategies for precise and accurate measurements of the
  Hubble constant with strong lensing
  Strong lensing time delays can measure the Hubble constant H$_0$ independent
of any other probe. Assuming commonly used forms for the radial mass density
profile of the lenses, a 2\% precision has been achieved with 7 Time-Delay
Cosmography (TDCOSMO) lenses, in tension with the H$_0$ from the cosmic
microwave background. However, without assumptions on the radial mass density
profile -- and relying exclusively on stellar kinematics to break the
mass-sheet degeneracy -- the precision drops to 8\% with the current data of
the 7 TDCOSMO lenses, insufficient to resolve the H$_0$ tension. With the
addition of external information from 33 Sloan Lens ACS (SLACS) lenses, the
precision improves to 5\%, {\it if} the deflectors of TDCOSMO and SLACS lenses
are drawn from the same population. We investigate the prospects to improve the
precision of time-delay cosmography without relying on mass profile assumptions
to break the mass sheet degeneracy. Our forecasts are based on the hierarchical
framework introduced by Birrer et al. (2020). With existing samples and
technology, 3.3\% precision on H$_0$ can be reached by adding spatially
resolved kinematics of the 7 TDCOSMO lenses. The precision improves to 2.5\%
with the further addition of kinematics for 50 non-time-delay lenses from SLACS
and the Strong Lensing Legacy Survey (SL2S). Expanding the samples to 40 time
delay and 200 non-time delay lenses will improve the precision to 1.5\% and
1.2\%, respectively. Time-delay cosmography can reach sufficient precision to
resolve the Hubble tension at 3-5$\sigma$, without assumptions on the radial
mass profile of lens galaxies. By obtaining this precision with and without
external datasets, we will test the consistency of the samples and enable
further improvements based on even larger future samples of time delay and
non-time-delay lenses (e.g. from the Rubin, Euclid, and Roman Observatories).
