Optimal point spread function modeling for weak lensing: complexity and sparsity

Context. We address the issue of controling the systematic errors in shape measurements for weak gravitational lensing. Aims. We make a step to quantify the impact of systematic errors in modeling the point spread function (PSF) of observations, on the determination of cosmological parameters from cosmic shear. Methods. We explore the impact of 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 of low complexity produces small statistical errors on the model parameters, although these parameters could be affected by large biases. Alternatively, fitting a large number of parameters (i.e. a high complexity) tends to reduce biases at the expense of increasing the statistical errors. We attempt to find a trade-off between scatters and biases by studying the mean squared error of a PSF model. We also characterize the 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 give an illustration for a realistic example of a PSF modeled by a shapelet basis set. Results. We derive a relation between the complexity and the sparsity of the PSF model, the signal-to-noise ratio of stars and the systematic errors in the cosmological parameters. By insisting that the systematic errors are below the statistical uncertainties, we derive a relation between the number of stars required to calibrate the PSF and the sparsity of the PSF model. We discuss the impact of our results for current and future cosmic shear surveys. In the typical case where the sparsity can be represented by a power-law function of the complexity, we demonstrate that current ground-based 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.