Statistical connection of peak counts to power spectrum and moments in weak lensing field

The number density of local maxima of weak lensing field, referred to as weak-lensing peak counts, can be used as a cosmological probe. However, its relevant cosmological information is still unclear. We study the relationship between the peak counts and other statistics in weak lensing field by using 1000 ray-tracing simulations. We construct a local transformation of lensing field $\cal K$ to a new Gaussian field $y$, named local-Gaussianized transformation. We calibrate the transformation with numerical simulations so that the one-point distribution and the power spectrum of $\cal K$ can be reproduced from a single Gaussian field $y$ and monotonic relation between $y$ and $\cal K$. Therefore, the correct information of two-point clustering and any order of moments in weak lensing field should be preserved under local-Gaussianized transformation. We then examine if local-Gaussianized transformation can predict weak-lensing peak counts in simulations. The local-Gaussianized transformation is insufficient to explain weak-lensing peak counts in the absence of shape noise. The prediction by local-Gaussianized transformation underestimates the simulated peak counts with a level of $\sim20-30\%$ over a wide range of peak heights. Local-Gaussianized transformation can predict the weak-lensing peak counts with a $\sim10\%$ accuracy in the presence of shape noise. Our analyses suggest that the cosmological information beyond power spectrum and its moments would be necessary to predict the weak-lensing peak counts with a percent-level accuracy, which is an expected statistical uncertainty in upcoming wide-field galaxy surveys.