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Strong lensing time delays can measure the Hubble constant H$_0$ independentof any other probe. Assuming commonly used forms for the radial mass densityprofile of the lenses, a 2\% precision has been achieved with 7 Time-DelayCosmography (TDCOSMO) lenses, in tension with the H$_0$ from the cosmicmicrowave background. However, without assumptions on the radial mass densityprofile -- and relying exclusively on stellar kinematics to break themass-sheet degeneracy -- the precision drops to 8\% with the current data ofthe 7 TDCOSMO lenses, insufficient to resolve the H$_0$ tension. With theaddition of external information from 33 Sloan Lens ACS (SLACS) lenses, theprecision improves to 5\%, {\it if} the deflectors of TDCOSMO and SLACS lensesare drawn from the same population. We investigate the prospects to improve theprecision of time-delay cosmography without relying on mass profile assumptionsto break the mass sheet degeneracy. Our forecasts are based on the hierarchicalframework introduced by Birrer et al. (2020). With existing samples andtechnology, 3.3\% precision on H$_0$ can be reached by adding spatiallyresolved 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 SLACSand the Strong Lensing Legacy Survey (SL2S). Expanding the samples to 40 timedelay and 200 non-time delay lenses will improve the precision to 1.5\% and1.2\%, respectively. Time-delay cosmography can reach sufficient precision toresolve the Hubble tension at 3-5$\sigma$, without assumptions on the radialmass profile of lens galaxies. By obtaining this precision with and withoutexternal datasets, we will test the consistency of the samples and enablefurther improvements based on even larger future samples of time delay andnon-time-delay lenses (e.g. from the Rubin, Euclid, and Roman Observatories).

TDCOSMO V: strategies for precise and accurate measurements of the Hubble constant with strong lensing