# Generalised model-independent characterisation of strong gravitational lenses V: reconstructing the lensing distance ratio by supernovae for a general Friedmann universe

## TL;DR

This paper presents a model-independent method to determine the lensing distance ratio using supernovae data, enabling precise lens property measurements without assuming a specific cosmological model.

## Contribution

It introduces a novel, data-driven approach to reconstruct the lensing distance ratio and lens properties in a general Friedmann universe without relying on predefined cosmological parameters.

## Key findings

- Lensing distance ratio $D$ can be determined with 1.7% relative imprecision.
- The method allows for model-independent lens characterization.
- Distance measures are valid for $z\le2.3$ in any homogeneous, isotropic universe.

## Abstract

We determine the cosmic expansion rate from supernovae of type Ia to set up a data-based distance measure that does not make assumptions about the constituents of the universe, i.e. about a specific parametrisation of a Friedmann cosmological model. The scale, determined by the Hubble constant $H_0$, is the only free cosmological parameter left in the gravitational lensing formalism. We investigate to which accuracy and precision the lensing distance ratio $D$ is determined from the Pantheon sample. Inserting $D$ and its uncertainty into the lensing equations for given $H_0$, esp. the time-delay equation between a pair of multiple images, allows to determine lens properties, esp. differences in the lensing potential ($\Delta \phi$), without specifying a cosmological model. We expand the luminosity distances into an analytic orthonormal basis, determine the maximum-likelihood weights for the basis functions by a globally optimal $\chi^2$-parameter estimation, and derive confidence bounds by Monte-Carlo simulations. For typical strong lensing configurations between $z=0.5$ and $z=1.0$, $\Delta \phi$ can be determined with a relative imprecision of 1.7%, assuming imprecisions of the time delay and the redshift of the lens on the order of 1%. With only a small, tolerable loss in precision, the model-independent lens characterisation developed in this paper series can be generalised by dropping the specific Friedmann model to determine $D$ in favour of a data-based distance ratio. Moreover, for any astrophysical application, the approach presented here, provides distance measures for $z\le2.3$ that are valid in any homogeneous, isotropic universe with general relativity as theory of gravity.