Noise bias in weak lensing shape measurements
  Weak lensing experiments are a powerful probe of cosmology through their
measurement of the mass distribution of the universe. A challenge for this
technique is to control systematic errors that occur when measuring the shapes
of distant galaxies. In this paper we investigate noise bias, a systematic
error that arises from second order noise terms in the shape measurement
process. We first derive analytical expressions for the bias of general Maximum
Likelihood Estimators (MLEs) in the presence of additive noise. We then find
analytical expressions for a simplified toy model in which galaxies are modeled
and fitted with a Gaussian with its size as a single free parameter. Even for
this very simple case we find a significant effect. We also extend our analysis
to a more realistic 6-parameter elliptical Gaussian model. We find that the
noise bias is generically of the order of the inverse-squared signal-to-noise
ratio (SNR) of the galaxies and is thus of the order of a percent for galaxies
of SNR of 10, i.e. comparable to the weak lensing shear signal. This is nearly
two orders of magnitude greater than the systematics requirements for future
all-sky weak lensing surveys. We discuss possible ways to circumvent this
effect, including a calibration method using simulations discussed in an
associated paper.
