Comments on the mass sheet degeneracy in cosmography analyses

Article
Report number	arXiv:2201.05111
Title	Comments on the mass sheet degeneracy in cosmography analyses
Author(s)	Teodori, Luca (Weizmann Inst.) ; Blum, Kfir (Weizmann Inst.) ; Castorina, Emanuele (Milan U.) ; Simonović, Marko (CERN) ; Soreq, Yotam (Technion)
Publication	2022-07-15
Imprint	2022-01-13
Number of pages	16
Note	16 pages, 6 figures. v3: corrected mistake in Sec.III, changing details in weak lensing bias due to kinematics and imaging combination
In:	JCAP 2207 (2022) 027
DOI	10.1088/1475-7516/2022/07/027
Subject category	hep-ph ; Particle Physics - Phenomenology ; astro-ph.GA ; Astrophysics and Astronomy ; astro-ph.CO ; Astrophysics and Astronomy
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.
Copyright/License	preprint: (License: arXiv nonexclusive-distrib 1.0) publication: © 2022-2024 The Author(s) (License: CC-BY-4.0)

$RMS external convergence terms, for lens redshift $z_{\rm l}=0.59$, presented as functions of the source redshift $z_{\rm s}$. The orange band around $\kappa^{\rm s}$ shows a rough estimate of the theoretical uncertainty, obtained by varying the cutoff of the matter power spectrum calculation from $k_{\rm cutoff}=5~{\rm Mpc}^{-1}$ to $20~{\rm Mpc}^{-1}$; the default in the calculation is $10~{\rm Mpc}^{-1}$. Modifying $k_{\rm cutoff}$ has a similar effect on the other weak convergence terms in the plot. Details of the calculation are given in App.~\ref{sec:kappa_variance}. Note that $\kappa^{\rm s}, \kappa^{\rm ls}, \kappa^{\rm l}$ are statistically independent (albeit correlated) cosmological random variables; thus, for example, the RMS value of $\kappa^{\rm s}-\kappa^{\rm l}$ is not simply shifted by a constant from the RMS value of $\kappa^{\rm s}$, even though the RMS of $\kappa^{\rm l}$ is a constant (given that the plot is done at constant fixed $z_{\rm l}$). Note that this plot is not expected to be accurate beyond the $\mathcal{O}(1)$ level. More accurate results would require ray tracing techniques to capture bias from excess of structure along the LOS~\cite{Keeton:1996tq,Holder:2002hq,Dalal:2004as,Momcheva:2005ex,Suyu:2009by,2017MNRAS.467.4220R,2018ApJ...867..107W,H0LiCOWX2019,2020MNRAS.498.1406T}. Code: \href{https://github.com/lucateo/Comments_MSD/blob/main/Notebooks/delta_kappa_nonlinear.ipynb}{\faGithub}.$ ${\rm Top:} 50\% and 90\% joint probability quantiles for $\kappa^{\rm s}$ and $\kappa^{\rm ls}$, using the RMS values from Fig.~\ref{fig:kappa_ls_s_l} at $z_{\rm s}=2$. {\rm Bottom:} Conditional probability $P\left(\kappa^{\rm ls}|\kappa^{\rm s}=0.034\right)$. The reference value of $0.034$ is approximately equal to the RMS of $\kappa^{\rm s}$ at $z_{\rm s}=2$ in Fig.~\ref{fig:kappa_ls_s_l}. Note that this plot (like Fig.~\ref{fig:kappa_ls_s_l}) relies on a simplified model of the non-equal time matter power spectrum, and is not expected to be accurate beyond the $\mathcal{O}(1)$ level.$ ${\rm Top:} 50\% and 90\% joint probability quantiles for $\kappa^{\rm s}$ and $\kappa^{\rm ls}$, using the RMS values from Fig.~\ref{fig:kappa_ls_s_l} at $z_{\rm s}=2$. {\rm Bottom:} Conditional probability $P\left(\kappa^{\rm ls}|\kappa^{\rm s}=0.034\right)$. The reference value of $0.034$ is approximately equal to the RMS of $\kappa^{\rm s}$ at $z_{\rm s}=2$ in Fig.~\ref{fig:kappa_ls_s_l}. Note that this plot (like Fig.~\ref{fig:kappa_ls_s_l}) relies on a simplified model of the non-equal time matter power spectrum, and is not expected to be accurate beyond the $\mathcal{O}(1)$ level.$ Show more plots

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