Optimal PSF modeling for weak lensing: complexity and sparsity
  We investigate the impact of point spread function (PSF) fitting errors on
cosmic shear measurements using the concepts of complexity and sparsity.
Complexity, introduced in a previous paper, characterizes the number of degrees
of freedom of the PSF. For instance, fitting an underlying PSF with a model
with low complexity will lead to small statistical errors on the model
parameters, however these parameters could suffer from large biases.
Alternatively, fitting with a large number of parameters will tend to reduce
biases at the expense of statistical errors. We perform an optimisation of
scatters and biases by studying the mean squared error of a PSF model. We also
characterize a model sparsity, which describes how efficiently the model is
able to represent the underlying PSF using a limited number of free parameters.
We present the general case and illustrate it for a realistic example of PSF
fitted with shapelet basis sets. We derive the relation between complexity and
sparsity of the PSF model, signal-to-noise ratio of stars and systematic errors
on cosmological parameters. With the constraint of maintaining the systematics
below the statistical uncertainties, this lead to a relation between the
required number of stars to calibrate the PSF and the sparsity. We discuss the
impact of our results for current and future cosmic shear surveys. In the
typical case where the biases can be represented as a power law of the
complexity, we show that current weak lensing surveys can calibrate the PSF
with few stars, while future surveys will require hard constraints on the
sparsity in order to calibrate the PSF with 50 stars.
