We study the cosmological information of weak lensing ( WL ) peaks , focusing on two other statistics besides their abundance : the stacked tangential-shear profiles and the peak-peak correlation function .
We use a large ensemble of simulated WL maps with survey specifications relevant to future missions like \it { Euclid } and lsst , to measure and examine the three peak probes .
We find that the auto-correlation function of peaks with high signal-to-noise ( { \mathcal { S } } / { \mathcal { N } } ) ratio measured from fields of size 144 deg ^ { 2 } has a maximum of \sim 0.3 at an angular scale \vartheta \sim 10 arcmin .
For peaks with smaller { \mathcal { S } } / { \mathcal { N } } , the amplitude of the correlation function decreases , and its maximum occurs on smaller angular scales .
The stacked tangential-shear profiles of the peaks also increase with their { \mathcal { S } } / { \mathcal { N } } .
We compare the peak observables measured with and without shape noise and find that for { \mathcal { S } } / { \mathcal { N } } \sim 3 only \sim 5 \% of the peaks are due to large-scale structures , the rest being generated by shape noise .
The correlation function of these small peaks is therefore very weak compared to that of small peaks measured from noise-free maps , and also their mean tangential-shear profile is a factor of a few smaller than the noise-free one .
The covariance matrix of the probes is examined : the correlation function is only weakly covariant on scales \vartheta < 30 arcmin , and slightly more on larger scales ; the shear profiles are very correlated for \vartheta > 2 arcmin , with a correlation coefficient as high as 0.7 .
The cross-covariance of the three probes is relatively weak : the peak abundance and profiles have the largest correlation coefficient \sim 0.3 .
Using the Fisher-matrix formalism , we compute the cosmological constraints for \ { { \Omega _ { m } } , { \sigma _ { 8 } } , w, { n _ { s } } \ } considering each probe separately , as well as in combination .
We find that the peak-peak correlation and shear profiles yield marginalized errors which are larger by a factor of 2 - 4 for \ { { \Omega _ { m } } , { \sigma _ { 8 } } \ } than the errors yielded by the peak abundance alone , while the errors for \ { w, { n _ { s } } \ } are similar .
By combining the three probes , the marginalized constraints are tightened by a factor of \sim 2 compared to the peak abundance alone , the least contributor to the error reduction being the correlation function .
This work therefore recommends that future WL surveys use shear peaks beyond their abundance in order to constrain the cosmological model .