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 to
begin with. In such cases, one should not naively demagnify the observed SN
luminosity by the model magnification into the expected Hubble diagram, which
would then create a bias, but 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 increasing 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.
