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

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 parametrization of a Friedmann cosmological model. The scale, determined by the Hubble constant H₀, 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 H₀, especially the time-delay equation between a pair of multiple images, allows to determine lens properties, especially differences in the lensing potential (Δϕ), 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 χ²-parameter estimation, and derive confidence bounds by Monte Carlo simulations. For typical strong lensing configurations between z = 0.5 and 1.0, Δϕ can be determined with a relative imprecision of 1.7 per cent, assuming imprecisions of the time delay and the redshift of the lens on the order of 1 per cent. 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 ≤ 2.3 that are valid in any homogeneous, isotropic universe with general relativity as theory of gravity.