The cluster mass function from weak gravitational lensing

We present the first measurement of the mass function of galaxy clusters based directly on cluster masses derived from observations of weak gravitational lensing. To investigate the degree of sample incompleteness resulting from the X-ray-based selection of the target clusters, we use a sample of 50 clusters with weak-lensing mass measurements to empirically determine the relation between lensing mass and X-ray luminosity and the scatter about this relation. We use a complete, volume-limited subsample of 35 X-ray-luminous clusters of galaxies at 0: 15 < z < 0.3 to constrain the abundance of very massive (M greater than or similar to 10(15) h(-1) M circle dot) clusters. From this, we constrain sigma(8)(Omega(m)/0.3)(0.37) = 0.67(-0.05)(+0.04) (68% confidence limits), agreeing well with constraints from the 3 year WMAP CMB measurements and estimates of cluster abundances based on X-ray observations, but somewhat lower than constraints from cosmic shear'' weak-lensing measurements in random fields.