New constraints on ${\sigma }_8$ from a joint analysis of stacked gravitational lensing and clustering of galaxy clusters

The joint analysis of clustering and stacked gravitational lensing of galaxy clusters in large surveys can constrain the formation and evolution of structures and the cosmological parameters. On scales outside a few virial radii, the halo bias, $b$, is linear and the lensing signal is dominated by the correlated distribution of matter around galaxy clusters. We discuss a method to measure the power spectrum amplitude ${\sigma }_8$ and $b$ based on a minimal modelling. We considered a sample of $\sim 120000$ clusters photometrically selected from the Sloan Digital Sky Survey in the redshift range $0.1<z<0.6$. The auto-correlation was studied through the two-point function of a subsample of $\sim 70000$ clusters; the matter-halo correlation was derived from the weak lensing signal of the subsample of $\sim 1200$ clusters with Canada-France-Hawaii Lensing Survey data. We obtained a direct measurement of $b$, which increases with mass in agreement with predictions of the $\mathrm{\Lambda }$CDM paradigm. Assuming ${\mathrm{\Omega }}_{\mathrm{M}}=0.3$, we found ${\sigma }_8=0.78\pm 0.16$. We used the same clusters for measuring both lensing and clustering and the estimate of ${\sigma }_8$ did require neither the mass-richness relation, nor the knowledge of the selection function, nor the modelling of $b$. With an additional theoretical prior on the bias, we obtained ${\sigma }_8=0.75\pm 0.08$.