Generalised model-independent characterisation of strong gravitational lenses – VI. The origin of the formalism intrinsic degeneracies and their influence on H0

Abstract

We give a physical interpretation of the formalism intrinsic degeneracies of the gravitational lensing formalism that we derived on a mathematical basis in part IV of this series. We find that all degeneracies occur due to the partition of the mass density along the line of sight. Usually, it is partitioned into a background (cosmic) density and a foreground deflecting object. The latter can be further partitioned into a main deflecting object and perturbers. Weak deflecting objects along the line of sight are also added, either to the deflecting object or as a correction of the angular diameter distances, perturbing the cosmological background density. A priori, this is an arbitrary choice of reference frame and partition. They can be redefined without changing the lensing observables which are sensitive to the integrated deflecting mass density along the entire line of sight. Reformulating the time delay equation such that this interpretation of the degeneracies becomes easily visible, we note that the source can be eliminated from this formulation, which simplifies reconstructions of the deflecting mass distribution or the inference of the Hubble constant, H 0 . Subsequently, we list necessary conditions to break the formalism intrinsic degeneracies and discuss ways to break them by model choices or including non-lensing observables like velocity dispersions along the line of sight with their advantages and disadvantages. We conclude with a systematic summary of all formalism intrinsic degeneracies and possibilities to break them.

A Model-Independent Characterisation of Strong Gravitational Lensing by Observables

“…In principle, there are other ways to describe the deflection of light rays in our universe which are detailed in [27]. For instance, the approach in which the background metric accounts for the entire deflecting mass-density distribution is an equivalent ansatz.…”

“…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. Without additional assumptions or information on the mass-density profile of the axisymmetric lens at r E or on the deflection angle of the perturber, Equations (31) and (32) are under-constrained.…”

“…Based on the physical prerequisites of Section 2 and the detailed derivation in [27], we set up the standard gravitational lensing formalism for one lens plane as follows. First, the light propagation along the line of sight is divided into the light propagation through the flat FLRW metric and the light deflection at the lens plane.…”

“…The shear thus represents the non-local part of the deflection potential that is not included in the Poisson equation. It enters ψ(x) via the boundary conditions, as detailed in [27]. g(x) = (g 1 (x), g 2 (x)) is the reduced shear, which is the quantity constrained by observables, as detailed in Section 2.3.…”

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

“…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).…”