The cosmological lens equation and the equivalent single-plane gravitational lens

The gravitational lens equation resulting from a single (non-linear) massconcentration (the main lens) plus inhomogeneities of the large-scale structureis shown to be strictly equivalent to the single-plane gravitational lensequation without the cosmological perturbations. The deflection potential (and,by applying the Poisson equation, also the mass distribution) of the equivalentsingle-plane lens is derived. If the main lens is described by ellipticalisopotential curves plus a shear term, the equivalent single-plane lens will beof the same form. Due to the equivalence shown, the determination of the Hubbleconstant from time delay measurements is affected by the same mass-sheetinvariance transformation as for the single-plane lens. If the lens strength isfixed (e.g., by measuring the velocity dispersion of stars in the main lens),the determination of $H_0$ is affected by inhomogeneous matter between us andthe lens. The orientation of the mass distribution relative to the imagepositions is the same for the cosmological lens situation and the single-planecase. In particular this implies that cosmic shear cannot account for amisalignment of the observed galaxy orientation relative to the best-fittinglens model.Comment: TeX, 11 pages, submitted to MNRA