Generalised model-independent characterisation of strong gravitational lenses V: lensing distance ratio in a general Friedmann universe

Determining the cosmic expansion history from a sample of supernovae of type Ia, data-based cosmic distance measures can be set up that make no assumptions about the constituents of the universe, i.e. about a specific cosmological model. The overall scale, usually determined by the Hubble constant $H_0$, is the only free parameter left. We investigate to which accuracy and precision the lensing distance ratio D of our distance to the lens, to the source, and their relative distance can be determined from the most recent Pantheon sample. Subsequently inserting D and its uncertainty into the gravitational 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. Alternatively, given \Delta \Phi$\,$ between multiple images, e.g. by a lens model, $H_0$ can be determined. For typical strong gravitational lensing configurations between z=0.5 and z=1.0, we find that \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%. Using a $\Lambda$CDM model, the relative imprecision of \Delta \Phi$\,$ is 1.4%. Minimum relative imprecisions for $H_0$ amount to 20% and 10% for galaxy- and galaxy-cluster-scale lenses when including measurements of velocity dispersions in a single-lens-plane model. 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. For any astrophysical application, the approach presented here, provides distance measures up to z=2.3 that are valid in any homogeneous, isotropic universe with general relativity as theory of gravity.