Consistent use of type Ia supernovae highly magnified by galaxy clusters to constrain the cosmological parameters

We discuss how Type Ia supernovae (SNe) strongly magnified by foreground galaxy clusters should be self-consistently treated when used in samples fitted for the cosmological parameters. While the cluster lens magnification of a SN can be well constrained from sets of multiple images of various background galaxies with measured redshifts, its value is typically dependent on the fiducial set of cosmological parameters used to construct the mass model. In such cases, one should not naively demagnify the observed SN luminosity by the model magnification into the expected Hubble diagram, which would create a bias, but instead take into account the cosmological parameters a priori chosen to construct the mass model. We quantify the effect and find that a systematic error of typically a few percent, up to a few dozen percent per magnified SN may be propagated onto a cosmological parameter fit unless the cosmology assumed for the mass model is taken into account (the bias can be even larger if the SN is lying very near the critical curves). We also simulate how such a bias propagates onto the cosmological parameter fit using the Union2.1 sample supplemented with strongly magnified SNe. The resulting bias on the deduced cosmological parameters is generally at the few percent level, if only few biased SNe are included, and increases with the number of lensed SNe and their redshift. Samples containing magnified Type Ia SNe, e.g., from ongoing cluster surveys, should readily account for this possible bias.