Correcting for Selection Biases in the Determination of the Hubble Constant from Time-Delay Cosmography

The time delay between multiple images of strongly lensed quasars has been used to infer the Hubble constant. The primary systematic uncertainty for time-delay cosmography is the mass-sheet transform (MST), which preserves the lensing observables while altering the inferred $H_0$. The TDCOSMO collaboration used velocity dispersion measurements of lensed quasars and lensed galaxies to infer that mass sheets are present, which decrease the inferred $H_0$ by 8$\%$. Here, we test the assumption that the density profiles of galaxy-galaxy and galaxy-quasar lenses are the same. We use a composite star-plus-dark-matter mass profile for the parent deflector population and model the selection function for galaxy-galaxy and galaxy-quasar lenses. We find that a power-law density profile with an MST is a good approximation to a two-component mass profile around the Einstein radius, but we find that galaxy-galaxy lenses have systematically higher mass-sheet components than galaxy-quasar lenses. For individual systems, $\lambda_\mathrm{int}$ correlates with the ratio of the half-light radius and Einstein radius of the lens. By propagating these results through the TDCOSMO methodology, we find that $H_0$ is lowered by a further $\sim$3\%. Using the velocity dispersions from \citet{slacs9} and our fiducial model for selection biases, we infer $H_0 = 66\pm4 \ \mathrm{(stat)} \pm 1 \ \mathrm{(model \ sys)} \pm 2 \ \mathrm{(measurement \ sys)} \ \mathrm{km} \ \mathrm{s}^{-1} \ \mathrm{Mpc}^{-1}$ for the TDCOSMO plus SLACS dataset. The first residual systematic error is due to plausible alternative choices in modeling the selection function, and the second is an estimate of the remaining systematic error in the measurement of velocity dispersions for SLACS lenses. Accurate time-delay cosmography requires precise velocity dispersion measurements and accurate calibration of selection biases.