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

## TL;DR

This paper investigates how lensing degeneracies, especially the mass sheet degeneracy, bias the measurement of the Hubble constant ($H_0$) in gravitational lensing, highlighting the complexities and potential pitfalls of current modeling approaches.

## Contribution

It demonstrates that common modeling assumptions and kinematic constraints can introduce significant biases in $H_0$ estimates due to lensing degeneracies.

## Key findings

- Bias in $H_0$ can reach up to 23% depending on the model.
- Including true slope constraints can introduce substantial bias.
- Lensing degeneracies are more complex than previously assumed.

## Abstract

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