Euclid Preparation IV. Impact of undetected galaxies on weak-lensing
  shear measurements
  In modern weak-lensing surveys, the common approach to correct for residual
systematic biases in the shear is to calibrate shape measurement algorithms
using simulations. These simulations must fully capture the complexity of the
observations to avoid introducing any additional bias. In this paper we study
the importance of faint galaxies below the observational detection limit of a
survey. We simulate simplified Euclid VIS images including and excluding this
faint population, and measure the shift in the multiplicative shear bias
between the two sets of simulations. We measure the shear with three different
algorithms: a moment-based approach, model fitting, and machine learning. We
find that for all methods, a spatially uniform random distribution of faint
galaxies introduces a shear multiplicative bias of the order of a few times
$10^{-3}$. This value increases to the order of $10^{-2}$ when including the
clustering of the faint galaxies, as measured in the Hubble Space Telescope
Ultra-Deep Field. The magnification of the faint background galaxies due to the
brighter galaxies along the line of sight is found to have a negligible impact
on the multiplicative bias. We conclude that the undetected galaxies must be
included in the calibration simulations with proper clustering properties down
to magnitude 28 in order to reach a residual uncertainty on the multiplicative
shear bias calibration of a few times $10^{-4}$, in line with the
$2\times10^{-3}$ total accuracy budget required by the scientific objectives of
the Euclid survey. We propose two complementary methods for including faint
galaxy clustering in the calibration simulations.
