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Gravitational lensing offers a competitive method to measure $H_0$ with thegoal of 1% precision. A major obstacle comes in the form of lensingdegeneracies, such as the mass sheet degeneracy (MSD), which make it possiblefor a family of density profiles to reproduce the same lensing observables butreturn different values of $H_0$. The modeling process artificially selects onechoice from this family, potentially biasing $H_0$. The effect is morepronounced when the profile of a given lens is not perfectly described by thelens model, which will always be the case to some extent. To explore this, wequantify the bias and spread in $H_0$ by creating quads from two-component massmodels and fitting them with a power-law ellipse+shear model. We find that thebias does not correspond to the estimate one would calculate by transformingthe profile into a power law near the image radius. We also emulate the effectof including stellar kinematics by performing fits where the slope isconstrained to the true value. Informing the fit using the true value near theimage radius can introduce substantial bias (0-23% depending on the model). Weconfirm using Jeans arguments that kinematic constraints can result in a biasedvalue of $H_0$, though the degree of bias depends on the region kinematicmodeling probes in specific lenses. We conclude that lensing degeneraciesmanifest through commonplace modeling approaches in a more complicated way thanis assumed in the literature. If stellar kinematics incorrectly break the MSD,their inclusion may introduce more bias than their omission.

Galaxy-lens determination of $H_0$: constraining density slope in the context of the mass sheet degeneracy