  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.
