Redshift Determinations from a Self-consistent Grid-based Lens Model for the Hubble Frontiers Field Cluster RXC J2248.7-4431 (AS1063)

We present an iterative method to construct a freeform lens model that self-consistently reproduces the sky positions, geometrically inferred redshifts, and relative brightnesses of all multiply lensed images toward a galaxy cluster. This method is applied to the cluster RXC J2248.7-4431 (z = 0.348) from the Hubble Frontier Fields program, toward which 10 multiply lensed sources with accurate spectroscopic redshifts and 6 others with inexact photometric redshifts have been identified. Using the spectroscopically secure systems to define an initial lens model, we compute the geometric redshifts of the photometric systems. We then iterate the lens model by incorporating the photometric systems at redshifts shifted by incremental amounts toward their geometric redshifts inferred from the previous step; on convergence, we find geometric redshifts in good agreement with the spectroscopically determined redshifts, but they can depart significantly from the photometrically determined redshifts. In the final lens model, all 16 lensed sources tightly follow the cosmological form of the angular diameter distance relation. Furthermore, although they are not used as model constraints, our lens model predicts relative brightnesses between image pairs for a given set of multiply lensed images in reasonable agreement with observations, thus providing independent validation of this model. Our method for inferring the redshifts and intrinsic brightnesses of multiply lensed sources will become especially important in the era of the James Webb Space Telescope, when deep infrared detections will typically be unmatched optically such that photometric redshifts will be very uncertain.