KiDS-450: Cosmological Constraints from Weak Lensing Peak Statistics -
  II: Inference from Shear Peaks using N-body Simulations
  We study the statistics of peaks in a weak lensing reconstructed mass map of
the first 450 square degrees of the Kilo Degree Survey. The map is computed
with aperture masses directly applied to the shear field with an NFW-like
compensated filter. We compare the peak statistics in the observations with
that of simulations for various cosmologies to constrain the cosmological
parameter $S_8 = \sigma_8 \sqrt{\Omega_{\rm m}/0.3}$, which probes the
($\Omega_{\rm m}, \sigma_8$) plane perpendicularly to its main degeneracy. We
estimate $S_8=0.750\pm0.059$, using peaks in the signal-to-noise range $0 \leq
{\rm S/N} \leq 4$, and accounting for various systematics, such as
multiplicative shear bias, mean redshift bias, baryon feedback, intrinsic
alignment, and shear-position coupling. These constraints are $\sim25\%$
tighter than the constraints from the high significance peaks alone ($3 \leq
{\rm S/N} \leq 4$) which typically trace single-massive halos. This
demonstrates the gain of information from low-S/N peaks. However we find that
including ${\rm S/N} < 0$ peaks does not add further information. Our results
are in good agreement with the tomographic shear two-point correlation function
measurement in KiDS-450. Combining shear peaks with non-tomographic
measurements of the shear two-point correlation functions yields a $\sim20\%$
improvement in the uncertainty on $S_8$ compared to the shear two-point
correlation functions alone, highlighting the great potential of peaks as a
cosmological probe.
