## 1. Introduction

## 2. Materials and Methods

#### 2.1. Observables in Multiple Images

#### 2.2. The Standard Gravitational Lensing Formalism

#### 2.3. Local Lens Properties from a Taylor Expansion around the Centre of Light of the Multiple Images

#### The Fold Case

#### 2.4. Local Lens Properties from a Taylor Expansion around a Critical Point

#### 2.4.1. The Fold Case

#### 2.4.2. The Cusp Case

#### 2.4.3. The Giant-Arc Case

#### 2.5. Lensing Distance Ratios in a General Friedmann Universe

## 3. Results

#### 3.1. Data Preprocessing

#### 3.2. Comparison to Lens Modelling Approaches for the Cluster-Scale Lens CL0024

- whether neighbouring MISs influence the reconstruction of local lens properties at the image positions of MIS 1 in Lenstool and Grale and thus introduce correlations between MISs that are not accounted for in our approach,
- whether the local lens properties at MIS 1 as obtained by our local approach and by the two global reconstructions coincide, or if image distortions beyond leading order play a significant role (apart from potential correlations between MISs mentioned under 1.),
- whether the light-traces-mass (LTM) assumption used in Lenstool is corroborated and, if so, which mass-to-light ratio (MLR) is favoured for the cluster member galaxies, and
- whether the small-scale fine-tuning of the mass density implemented in Grale overfits the free-form lens reconstruction to the multiple-image constraints.

- non-local influences of neighbouring MISs are negligible at the current precision of the quantities in Equation (12). Thus, the local lens properties are mainly derived from the local constraints of the respective multiple images. Hence, we can reconstruct the morphology of the source up to an overall scale as detailed in Section 2.3. The back-projections of all images to the source plane by their model-independently reconstructed $A\left({\mathit{x}}_{i}\right)$ and their pixel-wise averaged source are shown in Figure 4 (for comparison with model-based source reconstructions, see [90,92]; a physical analysis of the Lenstool-reconstructed source can be found in [93,94]).
- the leading-order local lens properties of our model-independent approach mostly agree to local lens properties as obtained by Lenstool and Grale within their 1$-\sigma $ confidence bounds.
- the LTM assumption is corroborated by the high degree of agreement of the local lens properties as obtained by Lenstool and the other two approaches that do not make any assumptions about the relation between dark and luminous matter. A comparison of the local lens properties of all approaches favours a constant MLR for the brightest cluster member galaxies over a non-constant MLR. This result was also found in [89,95]. A non-constant MLR that reproduces the fundamental plane (see [96] and references therein) yields a higher degree of agreement between the redshifts as estimated by Lenstool and the photometric redshifts of the additional MISs. This indicates that spectroscopic redshift observations are necessary to further constrain the redshifts and the MLR.
- the small-scale fine-tuning of the mass density in Grale does not significantly change the local lens properties, nor does it tighten the confidence bounds. As it does not significantly alter the pathways of the critical curves, either, it can be omitted for run-time optimisation.

#### 3.3. Transfer of the Method to Unresolved Multiple Images for the Galaxy-Scale Lens B0128

#### 3.4. Degeneracies Arising in the Detection of Perturbers for the Galaxy-Scale Lens B1938+666

#### 3.5. Lensing Distance Ratios from the Pantheon Sample of Supernovae

#### 3.6. Usage of the Time-Delay Equation to Infer ${H}_{0}$

## 4. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$A\left(\mathit{x}\right)$ | distortion matrix around point $\mathit{x}$ in the lens plane |

$\mathit{\alpha}\left(\mathit{x}\right)$ | deflection angle in the lens plane |

c | speed of light |

$D({z}_{\mathrm{l}},{z}_{\mathrm{s}})$ | lensing distance ratio, s. Equation (37) |

${D}_{\mathrm{A}}({z}_{1},{z}_{2})$ | angular diameter distance between two redshifts, s. Equation (1) |

${D}_{\mathrm{L}}({z}_{1},{z}_{2})$ | luminosity distance between two redshifts |

$E\left(z\right)$ | cosmic expansion function dependent on redshift z |

${f}_{ij}^{\left(\kappa \right)}$ | ratio of $1-\kappa \left(\mathit{x}\right)$ between multiple image i and j, s. Equation (12) |

${f}_{ij}^{\left(\varphi \right)}$ | ratio of ${\varphi}_{11}\left(\mathit{x}\right)$ between multiple image i and j, s. Equation (13) |

FLRW metric | Friedmann–Lemaître–Robertson–Walker metric |

FRB | fast radio burst |

$\mathit{g}\left(\mathit{x}\right)$ | reduced shear at position $\mathit{x}$, s. Equation (12) |

$\mathit{\gamma}\left(\mathit{x}\right)$ | shear at position $\mathit{x}$ |

$\Gamma $ | ratio describing the lensing geometry along the line of sight, s. Equation (6) |

${H}_{0}$ | Hubble–Lemaître constant |

HST | Hubble Space Telescope |

$\kappa \left(\mathit{x}\right)$ | convergence at position $\mathit{x}$ |

$\Lambda $CDM | cold-dark-matter model with cosmological constant $\Lambda $ |

LTM | light traces mass (assumption) |

MERLIN | Multi-Element Radio Linked Interferometer Network |

MIS | multiple-image set |

MLR | mass-to-light ratio |

MSD | mass-sheet degeneracy |

NFW | Navarro–Frenk–White (lens model), s. Appendix A.1 |

${\Omega}_{K}$ | curvature density parameter of the $\Lambda $CDM at $z=0$ |

${\Omega}_{\Lambda}$ | cosmological constant parameter of the $\Lambda $CDM at $z=0$ |

${\Omega}_{m}$ | matter density parameter of the $\Lambda $CDM at $z=0$ |

${\Omega}_{r}$ | radiation density parameter of the $\Lambda $CDM at $z=0$ |

$\varphi (\mathit{x},\psi )$ | Fermat potential, s. Equation (6) |

PSF | point spread function |

$\psi \left(\mathit{x}\right)$ | projected Newtonian deflection potential in the lens plane |

QSO | quasi-stellar object, quasar |

SIE | singular isothermal ellipse (lens model), s. Appendix A.2 |

SIS | singular isothermal sphere (lens model) |

SN | supernova |

SNR | signal-to-noise ratio |

VLA | Very Large Array |

VLBI | very-long-baseline interferometry |

$\mathit{x}$ | positions in the lens plane |

${\mathit{x}}_{0}$ | critical point (either a cusp or a fold singularity) |

${\mathit{x}}_{\mathrm{c}}$ | cusp critical point |

${\mathit{x}}_{\mathrm{f}}$ | fold critical point |

${\mathit{x}}_{i\alpha}$ | reference point $\alpha $ in image i |

${\mathit{x}}_{\mathrm{t}}$ | expansion point for Taylor series of the Fermat potential |

$\mathit{y}$ | positions in the source plane |

${\mathit{y}}_{0}$ | caustic point (either a cusp or a fold singularity) |

${z}_{\mathrm{l}}$ | redshift of the lens |

${z}_{\mathrm{s}}$ | redshift of the source object |

## Appendix A. The Fermat Potential at Critical Curves of Axisymmetric and Elliptical Lenses

#### Appendix A.1. Axisymmetric Lens

#### Appendix A.2. Elliptical Lens

## Appendix B. Transformation of a Cusp Configuration into Its Special Coordinate System

## Appendix C. Brief Characterisation of Lenstool and Grale

#### Appendix C.1. Lenstool

#### Appendix C.2. Grale

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1. | The expansion point is a different one in the two cases, yet, the underlying principle to determine ${\tilde{\varphi}}_{122}\left({\mathit{x}}_{0}\right)$ from the off-diagonal entries of the quadrupole moment of images A and B is the same. |

2. | Based on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Centre (CADC/NRC/CSA). |

3. | As we consider the late-time cosmology with $z\in \left[0,2.3\right]$, ${\Omega}_{r}=0$ in Equation (2) to good approximation. |

4. | The reconstruction of ${\varphi}_{ij}$ might include angular diameter distances between us, the source, and the lens, which correlates the imprecisions between ${\varphi}_{ij}$ and $D({z}_{\mathrm{l}},{z}_{\mathrm{s}})$. For the following estimates, these correlations are ignored. |

5. | Similarly, extended unresolved multiple images can be used to arrive at the same result. |

**Figure 1.**Strong gravitational lenses can cause a deflection of light rays from a source S (grey circle) in the source plane ${P}_{\mathrm{s}}$ into two images (yellow circles) in the lens plane ${P}_{\mathrm{l}}$. The upper image is observed at the angular position ${x}_{1}$ on the sky. The deflecting mass density (grey cloud) is assumed to consist of all masses along the line of sight (red circles) projected into ${P}_{\mathrm{l}}$, such that the light deflection is assumed to happen instantaneously at ${P}_{\mathrm{l}}$ and not at the individual masses (as indicated by the grey dashed paths). Differences in the arrival times of the light rays from the images at the observer O (indicated by the yellow arrows) occur due to the different light paths.

**Figure 2.**A linear transformation ${T}_{ij}$ between resolved or unresolved multiple images located around ${\mathit{x}}_{i}$ and ${\mathit{x}}_{j}$ (black circles) in the image plane of the same source located around $\mathit{y}$ (black circle) in the source plane is equal to a product of distortion matrices, $A\left(\mathit{x}\right)$. Thus, without employing any source properties, the local lens properties represented by the matrix entries of $A\left(\mathit{x}\right)$ are inferred from the ${T}_{ij}$.

**Figure 3.**Galaxy cluster CL0024 and its multiple-image sets (MISs): MIS 1 (yellow ellipses) with five images with six reference points each (red circles in image details) to apply our approach as described in Section 2.3 and with MISs 2 to 11 (blue points) as predicted by [91], but not spectroscopically confirmed. MIS 3, 4, 5, 8, and 10 were used to set up lens models that were compared to the model-independent approach. Image credits: NASA/ESA/HST.

**Figure 4.**Back-projected multiple images of MIS 1 to the source plane using the distortion matrix of local lens properties of Equation (8) for each image, normalised such that the transformation between 1.1. and 1.2 has unit determinant. Comparing the back-projections for images 1.1 and 1.5, we see that highly-resolved, large images with high signal-to-noise ratio (SNR) yield higher resolutions in the source reconstruction than smaller images with a low SNR.

**Figure 5.**Observational data of B0128 (

**left**): MERLIN 5 GHz band showing four images of a quasar at ${z}_{\mathrm{s}}=3.124$ (

**centre**) and VLBA 8.4 GHz details of all multiple images, revealing that images A, C, and D are resolved into three sub-components on milli-arcsecond scale, while B is most likely scatter-broadened. North is up and East is to the left. Schematic of applications of our model-independent approach (

**right**): on image scale, the positions of the sub-components are used as reference points to apply our approach detailed in Section 2.3, on sub-component scale, the quadrupole moments of images A, C, and D are employed to determine local lens properties for the sub-components 1 and 3 using our approach detailed in Section 2.3. Image credits: [63,98].

**Figure 6.**Observational data of B1938+666: F160W band filter of HST NIC1 (

**left**); H-band of the Keck NIRC2 camera with Laser Guide Star Adaptive Optics System (

**centre**); K’-band of the Keck NIRC2 camera with Laser Guide Star Adaptive Optics System (

**right**). North is up and East is to the left. Image credits: part of Figure 1 from [86].

**Figure 7.**Comparison between the relative imprecision of the data-based lensing distance ratio (

**left**) and the relative imprecision of the Planck-model-based distance ratio (

**right**) for lens redshifts ${z}_{\mathrm{l}}=0.25$ (

**top row**); ${z}_{\mathrm{l}}=0.5$ (

**centre row**); ${z}_{\mathrm{l}}=1.0$ (

**bottom row**). $D\left({z}_{\mathrm{s}}\right)$ is the lensing distance ratio defined in Equation (37) for fixed ${z}_{\mathrm{l}}$. Note the difference in the orders of magnitudes on the ordinate.

**Figure 8.**Summary of our approach as outlined in Section 2, its prerequisites (yellow boxes), working principles and resulting lens properties (grey boxes). Each result also has its remaining degeneracies and there are cases to which the approach cannot be applied.

**Table 1.**Types of multiple images and quantitative observables that can be extracted from each image. Each type can occur in different multiple-image configurations shown in Table 2. Image credits: NASA/ESA/HST.

Point Image | Unresolved Image | Resolved Image |
---|---|---|

angular position on the sky | centre of light, quadrupole | vectors of distances between features |

**Table 2.**Special multiple-image configurations of a common source. The figure shows an observed example, the schematic underneath the observables for unresolved images. Point-like and resolved images can also form these configurations. The bottom rows list the quantitative observables for each case. For arcs, we assume the centre of brightness of the lens is known. Image credits: NASA/ESA/HST.

Fold | Cusp | Giant Arcs |
---|---|---|

rel. distances ${\delta}_{ij}$ rel. orientations ${\phi}_{i}$ (time delays) | rel. distances ${\delta}_{ij}$ rel. orientations ${\tilde{\phi}}_{i}$ (time delays) | radial distances ${r}_{i}$, arc lengths ${r}_{i}{\phi}_{i}$ (time delays) |

**Table 3.**Synopsis of approximations to Equation (15) and embedding of the observables for fold and cusp critical points, ${\mathit{x}}_{\mathrm{f}}$ and ${\mathit{x}}_{\mathrm{c}}$, in the coordinate system of Equation (18). The critical point is marked by a black dot in the figures. Degeneracies arise in the sign of the slope of the critical curve at ${\mathit{x}}_{\mathrm{f}}$. Depending on the parity of image A, there are two possible configurations for a cusp.

Fold | Cusp | |
---|---|---|

(Positive) | (Negative) | |

coordinate system specifications ${\varphi}_{222}\left({\mathit{x}}_{0}\right)>0$, | coordinate system specifications ${\varphi}_{122}\left({\mathit{x}}_{0}\right)<0$, ${\varphi}_{2222}\left({\mathit{x}}_{0}\right)>0$, ${\varphi}_{122}{\left({\mathit{x}}_{0}\right)}^{2}-{\textstyle \frac{1}{3}}{\varphi}_{11}\left({\mathit{x}}_{0}\right){\varphi}_{2222}\left({\mathit{x}}_{0}\right)\ne 0$ | |

leading order ${\varphi}_{\mathrm{T}}(\mathit{x},\psi )=$ $-\mathit{x}\mathit{y}+{\textstyle \frac{1}{2}}{\varphi}_{11}\left({\mathit{x}}_{0}\right){x}_{1}^{2}+{\textstyle \frac{1}{2}}{\varphi}_{112}\left({\mathit{x}}_{0}\right){x}_{1}^{2}{x}_{2}$ $+{\textstyle \frac{1}{2}}{\varphi}_{122}\left({\mathit{x}}_{0}\right){x}_{1}{x}_{2}^{2}+{\textstyle \frac{1}{6}}{\varphi}_{222}\left({\mathit{x}}_{0}\right){x}_{2}^{3}$ | leading order ${\varphi}_{\mathrm{T}}(\mathit{x},\psi )=$ $-\mathit{x}\mathit{y}+{\textstyle \frac{1}{2}}{\varphi}_{11}\left({\mathit{x}}_{0}\right){x}_{1}^{2}$ $+{\textstyle \frac{1}{2}}{\varphi}_{122}\left({\mathit{x}}_{0}\right){x}_{1}{x}_{2}^{2}+{\textstyle \frac{1}{24}}{\varphi}_{2222}\left({\mathit{x}}_{0}\right){x}_{2}^{4}$ |

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