Optimal PSF modelling for weak lensing : complexity and sparsity

Context. Controlling shape measurement systematics in weak gravitational lensing. Aims. Quantifying the effect of systematic errors in modelling the Point Spread Function (PSF) on cosmological parameter measurements 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, characterises 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 (MSE) of a PSF model. We also characterise 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 a PSF fitted with the shapelet basis set. Results. 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 ground 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.