Cosmology Requirements on Supernova Photometric Redshift Systematics for
  Rubin LSST and Roman Space Telescope
  Some million Type Ia supernovae (SN) will be discovered and monitored during
upcoming wide area time domain surveys such as the Vera C. Rubin Observatory
Legacy Survey of Space and Time (LSST). For cosmological use, accurate
redshifts are needed among other characteristics; however the vast majority of
the SN will not have spectroscopic redshifts, even for their host galaxies,
only photometric redshifts. We assess the redshift systematic control necessary
for robust cosmology. Based on the photometric vs true redshift relation
generated by machine learning applied to a simulation of 500,000 galaxies as
observed with LSST quality, we quantify requirements on systematics in the mean
relation and in the outlier fraction and deviance so as not to bias dark energy
cosmological inference. Certain redshift ranges are particularly sensitive,
motivating spectroscopic followup of SN at $z\lesssim0.2$ and around
$z\approx0.5$-0.6. Including Nancy Grace Roman Space Telescope near infrared
bands in the simulation, we reanalyze the constraints, finding improvements at
high redshift but little at the low redshifts where systematics lead to strong
cosmology bias. We identify a complete spectroscopic survey of SN host galaxies
for $z\lesssim0.2$ as a highly favored element for robust SN cosmology.
