Generalised model-independent characterisation of strong gravitational lenses

Abstract

In galaxy-galaxy strong gravitational lensing, Einstein rings are generated when the lensing galaxy has an axisymmetric lensing potential and the source galaxy is aligned with its symmetry centre along the line of sight. Using a Taylor expansion around the Einstein radius and eliminating the unknown source, I derive a set of analytic equations that determine differences of the deflection angle of the perturber weighted by the convergence of the axisymmetric lens and ratios of the convergences at the positions of the arcs from the measurable thickness of the arcs. In the same manner, asymmetries in the brightness distributions along an arc determine differences in the deflection angle of the perturber if the source has a symmetric brightness profile and is oriented parallel to or orthogonal to the caustic. These equations are the only model-independent information retrievable from observations to leading order in the Taylor expansion. General constraints on the derivatives of the perturbing lens are derived such that the perturbation does not change the number of critical curves. To infer physical properties such as the mass of the perturber or its position, models need to be inserted. The same conclusions about the scale of detectable masses (of the order of 10 8 M ) and model-dependent degeneracies as in other approaches are then found and supported by analysing B1938+666 as an example. Yet, the model-independent equations show that there is a fundamental degeneracy between the main lens and the perturber that can only be broken if their relative position is known. This explains the degeneracies between lens models already found in simulations from a more general viewpoint. Hence, apart from the radii and brightness distributions of the arcs, independent information on the axisymmetric lens or the perturber has to be employed to disentangle the axisymmetric lens and the perturber. Depending on the properties of the pertuber, this degeneracy can be broken by characterising the surrounding of the lens or by measuring the time delay between quasar images embedded in the perturbed Einstein ring of the host galaxy.

Extended lens reconstructions with grale: exploiting time-domain, substructural, and weak lensing information

“…That the source plane offset has no direct meaning can also be seen by noticing that both a given lensing potential ψ0(θ) and ψ1(θ) = ψ0(θ) + a · θ (A1) correspond to the same images and mass density, but with a shift in source plane (Seitz et al 1998). This can be seen to be a special case of the more general equation (21) in Wagner (2018) which shows the change in potential that corresponds to a shift in source plane position.…”

“…It was soon found to be a special case of several classes of invariance transformations that leave the observables in multiple-image configurations invariant (Gorenstein et al 1988a;Schneider & Sluse 2014). Making the lens reconstruction independent of specific (parametric) lens models, it was found that these degeneracies that had been treated as global transformations of the entire lens and source plane properties, can be further generalised, such that they locally apply to each system of multiple images individually (Liesenborgs et al 2008a;Liesenborgs & De Rijcke 2012;Wagner 2018). It is clear that these degeneracies cause difficulties in constraining the mass density of a specific lensing object at a certain redshift from such lensing observations, with the integration of mass along the line of sight further confounding the issue (Wagner 2019).…”

A Model-Independent Characterisation of Strong Gravitational Lensing by Observables

“…Equation (31) implies a degeneracy between the deflection angle of the perturber and the convergence of the axisymmetric lens. As further detailed in [24], this degeneracy can be interpreted as an exact source-position transformation, introduced in [75]. As explained in [27], this degeneracy originates from an a priori arbitrary split of the total deflection potential into a main lens and a perturber.…”

“…[62] already described the possibility to extract the local lens properties contained in A(x) by transforming multiple images onto each other. In [23,47], we realised this idea for unresolved and resolved multiple images that contain at least two non-parallel vectors, as indicated by the black arrows in the figures of Table 1. We assume to have m reference points within each multiple image i which are located at positions x iα , α = 1, ..., m. Using the quadrupole of an unresolved image as observables, the three reference points are the centre of light and the end points of the semi-major and semi-minor axis of the quadrupole.…”

“…In [24], we investigated which properties of a perturber to an axisymmetric mass-density distribution could be determined from the observables shown in Table 2 (right) without a lens model. Due to the underlying axisymmetry, we used polar coordinates (r, ϕ) and perturbed the symmetric deflection potential, ψ a (r), around the Einstein radius r E with a small deflection potential ψ p (r, ϕ), ψ p (r, ϕ) |ψ a (r)|, such that the total deflection potential ψ(r, ϕ) reads…”