The ellipticity parameterization for an NFW profile: An overlooked angular structure in strong lens modeling

¹ STAR Institute, University of Liège, Quartier Agora – Allée du Six Août, 19c, 4000 Liège, Belgium
e-mail: mgomer@uliege.be
² Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA
³ Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA
⁴ SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
⁵ Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA
⁶ Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
⁷ Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland

Received: 2 March 2023
Accepted: 3 October 2023

Abstract

Galaxy-scale gravitational lenses are often modeled with two-component mass profiles where one component represents the stellar mass and the second is a Navarro Frenk White (NFW) profile representing the dark matter. Outside of the spherical case, the NFW profile is costly to implement, and so it is approximated via two different methods; ellipticity can be introduced via the lensing potential (NFWp) or via the mass by approximating the NFW profile as a sum of analytical profiles (NFWm). While the NFWp method has been the default for lensing applications, it gives a different prescription of the azimuthal structure, which we show introduces ubiquitous gradients in ellipticity and boxiness in the mass distribution rather than having a constant elliptical shape. Because an unmodeled azimuthal structure has been shown to be able to bias lens model results, we explored the degree to which this azimuthal structure that was introduced can affect the model accuracy. We constructed input profiles using composite models using both the NFWp and NFWm methods and fit these mocks with a power-law elliptical mass distribution (PEMD) model with external shear. As a measure of the accuracy of the recovered lensing potential, we calculated the value of the Hubble parameter H₀ one would determine from the lensing fit. We found that the fits to the NFWp input return H₀ values that are systematically biased by about 3% lower than the NFWm counterparts. We explored whether such an effect is attributable to the mass sheet transformation (MST) by using an MST-independent quantity, ξ₂. We show that, as expected, the NFWm mocks are degenerate with PEMD through an MST. For the NFWp, an additional bias was found beyond the MST due to the azimuthal structure exterior to the Einstein radius. We recommend modelers use an NFWm prescription in the future, such that the azimuthal structure can be introduced explicitly rather than implicitly.