We studied the effect of primordial non-Gaussianity with varied bispectrum shapes on the number counts of signal-to-noise peaks in wide field cosmic shear maps. The two cosmological contributions to this particular weak lensing statistic, namely the chance projection of Large Scale Structure and the occurrence of real, cluster-sized dark matter halos, have been modeled semi-analytically, thus allowing to easily introduce the effect of non-Gaussian initial conditions. We performed a Fisher matrix analysis by taking into account the full covariance of the peak counts in order to forecast the joint constraints on the level of primordial non-Gaussianity and the amplitude of the matter power spectrum that are expected by future wide field imaging surveys. We find that positive-skewed non-Gaussianity increases the number counts of cosmic shear peaks, more so at high signal-to-noise values, where the signal is mostly dominated by massive clusters as expected. The increment is at the level of ~1 for f_NL=10 and ~10 for f_NL=100 for a local shape of the primordial bispectrum, while different bispectrum shapes give generically a smaller effect. For a future survey on the model of the proposed ESA space mission Euclid and by avoiding the strong assumption of being capable to distinguish the weak lensing signal of galaxy clusters from chance projection of Large Scale Structures we forecasted a 1-sigma error on the level of non-Gaussianity of ~30-40 for the local and equilateral models, and of ~100-200 for the less explored enfolded and orthogonal bispectrum shapes.
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