Constraining $\sigma_8$ with Lensing Statistics in Low and High Density Regions

Lensing studies are typically carried out around high density regions, such as groups and clusters, where the lensing signals are significant and indicative of rich density structures. However, a more comprehensive test of the cosmological model should also include the lensing effect in low density regions. In this work, we incorporate the stacked weak lensing signals around the low density positions, alongside galaxy-galaxy lensing and galaxy-galaxy two point correlation function to perform a joint cosmological analysis on $\sigma_8$. The low density positions are constructed from the DR9 data release of the DESI legacy imaging survey, using galaxies with r-band absolute magnitude cut M$<$-21.5 and in the redshift range of 0.18$<$z$<$0.28. In doing so, we simultaneously parameterize photometric redshift errors and halo mass uncertainties while building mock catalogs from simulations using the method of SubHalo Abundance Matching (SHAM). For the weak lensing measurements, we use the shear estimators derived from the DECaLS DR8 imaging data, processed by the Fourier_Quad pipeline. The survey boundaries and masks are fully taken into account. Our analysis achieves a total significance of $31.1\sigma$ detection for lensing in the low density positions, which significantly improve the $\sigma_8$ constraint compared to galaxy-galaxy lensing and galaxy-galaxy two point correlation function by 14$\%$. For flat $\Lambda$CDM model, we constrain $\sigma_8$ =$0.824^{+0.015}_{-0.015}$, which shows a good agreement with the PLANCK result. Additionally, the halo mass uncertainty $\sigma_{\lg M}$ and photometric redshift error $\sigma_z$ are constrained to be $0.565^{+0.086}_{-0.070}$ and $0.004^{+0.004}_{-0.003}$ respectively, which are somewhat different from our expectations due to the significant degeneracy of the two parameters.