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

Abstract

We determine the cosmic expansion rate from supernovae of type Ia to set up a databased 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 (∆φ), 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 χ 2 -parameter estimation, and derive confidence bounds by Monte-Carlo simulations. For typical strong lensing configurations between z = 0.5 and z = 1.0, ∆φ 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 ≤ 2.3 that are valid in any homogeneous, isotropic universe with general relativity as theory of gravity.nation with a cosmological model that assigns the redshift to a cosmic distance. So far, the cosmological standard model, as most precisely measured by Planck Collaboration et al. (2018), is inserted into these distance measures.As one example, in the gravitational lensing formalism, angular diameter distances between the observer and the lens D l , the observer and the source Ds, and the distance between the lens and the source D ls appear in the lensing equations to scale the (multiple) images, the source, and the deflection potential with respect to each other. The so-called lensing distance ratio D(z l , zs) = D l Ds D ls ,appears, for instance, in the time delay τij between two multiple images i and j of the same background galaxy located

A Model-Independent Characterisation of Strong Gravitational Lensing by Observables

“…We review the precision up to which the lensing distance ratio in Equation (6) can be determined from the Pantheon sample of SNe type Ia, [41], in Section 3.5. The details to set up the angular diameter distance measure from the sample are found in [26]. Section 3.6 sets up theoretical limits on the precision to which H 0 can be determined by Equation (16).…”

“…The term for the spatial curvature with Ω K is discarded, as [64] found that the curvature is compatible with zero. Alternatively, any data-based expansion function can be inserted, as, for instance set up for our approach in [26]. Section 2.5 details these data-based distance measures.…”

“…Replacing the model-based angular diameter distances in Equation (3) by the data-based ones, we obtain the data-based lensing distance ratio, as detailed in [26]. Leaving H 0 as a free parameter, we can use time delay differences to constrain its value, if the difference of the Fermat potentials between the image positions is known.…”

“…As outlined in Section 2.5 and detailed in [26], we used the Pantheon sample of SNe type Ia of [41] and H 0 from [107] to set up data-based angular diameter distances for the lensing distance ratio in Equation (6). We abbreviate the lensing distance ratio by…”

“…In our research, as being developed in [21][22][23][24][25][26][27], we pursue the question which properties of a strong gravitational lens can be directly inferred from the observables that describe a multiple-image configuration.…”