  In weak-lensing cosmological studies, peak statistics is sensitive to
nonlinear structures and thus complementary to cosmic shear two-point
correlations. In this paper, we explore a new approach, namely, the peak
steepness statistics, with the overall goal to understand the cosmological
information embedded there in comparison with the commonly used peak height
statistics. We perform the analyses with ray-tracing simulations considering
different sets of cosmological parameters $\Omega_{\rm m}$ and $\sigma_8$. A
theoretical model to calculate the abundance of high peaks based on steepness
is also presented, which can well describe the main trend of the peak
distribution from simulations. We employ $\Delta\chi^2$ and Fisher analyses to
study the cosmological dependence of the two peak statistics using our limited
sets of simulations as well as our theoretical model. Within our considerations
without including potential systematic effects, the results show that the
steepness statistics tends to have higher sensitivities to the cosmological
parameters than the peak height statistics and this advantage is diluted with
the increase of the shape noise. Using the theoretical model, we investigate
the physical reasons accounting for the different cosmological information
embedded in the two statistics. Our analyses indicate that the projection
effect from large-scale structures plays an important role to enhance the gain
from the steepness statistics. The redshift and cosmology dependence of dark
matter halo density profiles also contributes to the differences between the
two statistics.

Cosmology and Nongalactic Astrophysics