Degeneracies in parameter estimates for models of gravitational lens systems

Observations of gravitational lens systems may yield useful estimates for characteristics of the lenses and for cosmological parameters. However, the nature of these observations restricts the types of characteristics that can be determined. We demonstrate some of these inherent restrictions, i.e., degeneracies in parameter estimation, through analysis of the effects of "prismatic," "similarity," "magnification," and "combined" transformations. These so-called invariance transformations show that changes over continuous ranges in the values of parameters describing the source and the lens models can leave the predicted values of observable properties of gravitational images unchanged, These properties include the relative positions, luminosities, magnifications, and time delays. The prismatic transformation adds a constant deflection field, leaving predictions of all these image properties invariant. The similarity transformation scales distances but preserves angles, leaving predictions of relative image positions and magnifications invariant. The magnification transformation scales angles and adds a uniform magnification, analogous to a magnifying lens, also leaving predictions of relative image positions and magnifications unchanged. The combination of the latter two transformations, with appropriate choice of scaling factors, leaves the predictions of all observable image properties invariant. We show, in two cases, how the combined transformation leads to degeneracies in the estimation of parameters characterizing models of gravitational lenses. In both cases, the redshift of the source, and the redshift and apparent position of the "center" of the lens, are assumed known. In the first case, we consider circularly symmetric lenses for which the radial dependence of the projected surface mass density is not known; in the second, we consider a lens of arbitrary mass distribution superposed on a uniform, but unknown, surface mass density. In both cases, knowledge of certain additional properties of the lens is needed to obtain (point) estimates of the Hubble constant rather than only bounds on its value.