# Generalised model-independent characterisation of strong gravitational lenses VI: the origin of the formalism intrinsic degeneracies and their influence on $H_0$

## TL;DR

This paper provides a physical interpretation of the formalism intrinsic degeneracies in gravitational lensing, linking them to the partitioning of mass density along the line of sight, and discusses methods to break these degeneracies for better mass and Hubble constant inference.

## Contribution

It offers a physical understanding of lensing degeneracies based on mass partitioning and proposes strategies to break these degeneracies using additional observables.

## Key findings

- Degeneracies arise from mass density partitioning along the line of sight.
- Reformulating the time delay equation simplifies mass reconstruction and $H_0$ inference.
- Including non-lensing observables can help break the degeneracies.

## 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.