  The control of systematic effects when measuring galaxy shapes is one of the
main challenges for cosmic shear analyses. In this context, we study the
fundamental limitations on shear accuracy due to the measurement of the Point
Spread Function (PSF) from the finite number of stars. In order to do that, we
translate the accuracy required for cosmological parameter estimation to the
minimum number of stars over which the PSF must be calibrated. We first 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 an
elliptical gaussian PSF. 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. 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. 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.
