PSF calibration requirements for dark energy from cosmic shear

Context. The control of systematic effects when measuring background galaxy shapes is one of the main challenges for cosmic shear analyses. Aims. Study the fundamental limitations on shear accuracy due to the measurement of the point spread function (PSF) from the finite number of stars that are available. We translate the accuracy required for cosmological parameter estimation to the minimum number of stars over which the PSF must be calibrated. Methods. We characterise the error made in the shear arising from errors on the PSF. We consider different PSF models, from a simple elliptical gaussian to various shapelet parametrisations. First we derive our results analytically in the case of infinitely small pixels (i.e. infinitely high resolution), then image simulations are used to validate these results and investigate the effect of finite pixel size in the case of the elliptical gaussian PSF. Results. Our results are expressed in terms of the minimum number of stars required to calibrate the PSF in order to ensure that systematic errors are smaller than statistical errors when estimating the cosmological parameters. On scales smaller than the area containing this minimum number of stars, there is not enough information to model the PSF. This means that these small scales should not be used to constrain cosmology unless the instrument and the observing strategy are optimised to make this variability extremely small. The minimum number of stars varies with the square of the star Signal-to-Noise Ratio, with the complexity of the PSF and with the pixel size. In the case of an elliptical gaussian PSF and in the absence of dithering, 2 pixels per PSF full width at half maximum (FWHM) implies a 20% increase of the minimum number of stars compared to the ideal case of infinitely small pixels; 0.9 pixels per PSF FWHM implies a factor 100 increase. Conclusions. In the case of a good resolution and a typical Signal-to-Noise Ratio distribution of stars, we find that current surveys need the PSF to be calibrated over a few stars, which may explain residual systematics on scales smaller than a few arcmins. Future all-sky cosmic shear surveys require the PSF to be calibrated over a region containing about 50 stars. Due to the simplicity of our models these results should be interpreted as optimistic and therefore provide a measure of a systematic 'floor' intrinsic to shape measurements.