Gravitational lensing galaxies are commonly modeled with elliptical density profiles, to which angular complexity is sometimes added through a multipole expansion—encoding deformations of the elliptical isodensity contours. The formalism that is widely used in current studies and software packages, however, employs perturbations that are defined with respect to a circle. In this work, we show that this popular formulation (the “circular multipoles”) leads to perturbation patterns that depend on the axis ratio and do not agree with physical expectations (from studies of galaxy isophotal shapes) when applied to profiles that are not near circular. We propose a more appropriate formulation, the “elliptical multipoles,” representing deviations from ellipticity suited for any axis ratio. We solve for the lensing potentials associated with the elliptical multipoles of any order $m$, assuming that the reference profile is near isothermal. We implement these solutions into the lens modeling package lenstronomy, and assess the importance of the multipole formulation by comparing flux-ratio perturbations in mock lensed systems with quadruply imaged quasars: we show that elliptical multipoles typically produce smaller flux-ratio perturbations than their circular counterparts.

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• Cosmology
• Dark matter
• Gravitational lenses
• Galaxies
• Differential equations
• Perturbative methods