Gravitational lensing by an ellipsoidal Navarro–Frenk–White dark-matter halo: An analytic solution and its properties

Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Praha 8, Czech Republic

Received: 29 March 2024
Accepted: 4 June 2024

Abstract

Context. The analysis of gravitational lensing by galaxies and galaxy clusters typically relies on ellipsoidal lens models to describe the deflection of light by the involved dark-matter halos. These models are most often based on the isothermal density profile – not an optimal description of the halo, but easy to use because it leads to an analytic deflection-angle formula.

Aims. Dark-matter halos are better described by the Navarro–Frenk–White (hereafter NFW) density profile. We set out to study lensing by a general triaxial ellipsoidal NFW halo, with the aim of providing an analytic model that would be more consistent with the current understanding of dark-matter halos.

Methods. We computed the conversion between the properties of a triaxial ellipsoidal lens model and its elliptical surface-density profile. In the case of the NFW lens model, its angular scale is defined by the projected scale semi-major axis of the halo, while its lensing regime depends on two parameters: the projected eccentricity e and the convergence parameter κ_s. We employed the Bourassa & Kantowski formalism to compute the complex scattering function of the model, which yields the deflection-angle components when separated into its real and imaginary parts.

Results. We present the obtained closed-form expressions for the deflection-angle components, valid for an arbitrary eccentricity of the surface-density profile. We use them to compute and describe the lensing properties of the model, including: the shear, its components, and the phase; the critical curves, caustics, and the parameter-space mapping of their different geometries; the deformations and orientations of images.

Conclusions. The analytically solved ellipsoidal NFW lens model is available for implementation in gravitational lensing software. The techniques introduced here such as the image-plane analysis can prove to be useful for understanding the properties of other lens models as well.