Comments on the mass sheet degeneracy in cosmography analyses

TL;DR

This paper discusses modeling degeneracies in strong lensing measurements of the Hubble constant, emphasizing the importance of accounting for line of sight effects and residual degeneracies even with multiple sources, to avoid bias in cosmographic analyses.

Contribution

It highlights the necessity of including all line of sight convergence terms and shows that multiple sources do not fully resolve the mass sheet degeneracy, especially with differential external convergence.

Findings

External convergence terms are crucial for accurate $H_0$ measurement.

Multiple sources do not completely eliminate the mass sheet degeneracy.

Differential external convergence complicates internal-external degeneracy analysis.

Abstract

We make a number of comments regarding modeling degeneracies in strong lensing measurements of the Hubble parameter $H_0$. The first point concerns the impact of weak lensing associated with different segments of the line of sight. We show that external convergence terms associated with the lens-source and observer-lens segments need to be included in cosmographic modeling, in addition to the usual observer-source term, to avoid systematic bias in the inferred value of $H_0$. Specifically, we show how an incomplete account of some line of sight terms biases stellar kinematics as well as ray tracing simulation methods to alleviate the mass sheet degeneracy. The second point concerns the use of imaging data for multiple strongly-lensed sources in a given system. We show that the mass sheet degeneracy is not fully resolved by the availability of multiple sources: some degeneracy remains because of differential external convergence between the different sources. Similarly, differential external convergence also complicates the use of multiple sources in addressing the approximate mass sheet degeneracy associated with a local ("internal") core component in lens galaxies. This internal-external degeneracy is amplified by the non-monotonicity of the angular diameter distance as a function of redshift. For a rough assessment of the weak lensing effects, we provide estimates of external convergence using the nonlinear matter power spectrum, paying attention to non-equal time correlators.