The self similarity of weak lensing peaks

We study the statistics of weak lensing convergence peaks, such as their abundance and two-point correlation function (2PCF), for a wide range of cosmological parameters ${\mathrm{\Omega }}_m$ and ${\sigma }_8$ within the standard $\mathrm{\Lambda }$CDM paradigm, focusing on intermediate-height peaks with signal-to-noise ratio (SNR) of $1.5$ to $3.5$. We find that the cosmology dependence of the peak abundance can be described by a one-parameter fitting formula that is accurate to within $\sim 3\mathrm{\%}$. The peak 2PCFs are shown to feature a self-similar behaviour: if the peak separation is rescaled by the mean inter-peak distance, catalogues with different minimum peak SNR values have identical clustering, which suggests that the peak abundance and clustering are closely interconnected. A simple fitting model for the rescaled 2PCF is given, which together with the peak abundance model above can predict peak 2PCFs with an accuracy better than $\sim 5\mathrm{\%}$. The abundance and 2PCFs for intermediate peaks have very different dependencies on ${\mathrm{\Omega }}_m$ and ${\sigma }_8$, implying that their combination can be used to break the degeneracy between these two parameters.