The mass function of galaxy clusters is a powerful tool to constrain cosmological parameters , e.g. , the mass fluctuation on the scale of 8 h ^ { -1 } Mpc , \sigma _ { 8 } , and the abundance of total matter , \Omega _ { m } .
We first determine the scaling relations between cluster mass and cluster richness , summed r -band luminosity and the global galaxy number within a cluster radius .
These relations are then used to two complete volume-limited rich cluster samples which we obtained from the Sloan Digital Sky Survey ( SDSS ) .
We estimate the masses of these clusters and determine the cluster mass function .
Fitting the data with a theoretical expression , we get the cosmological parameter constraints in the form of \sigma _ { 8 } ( \Omega _ { m } / 0.3 ) ^ { \alpha } = \beta and find out the parameters of \alpha = 0.40–0.50 and \beta = 0.8–0.9 , so that \sigma _ { 8 } = 0.8–0.9 if \Omega _ { m } = 0.3 .
Our \sigma _ { 8 } value is slightly higher than recent estimates from the mass function of X-ray clusters and the Wilkinson Microwave Anisotropy Probe ( WMAP ) data , but consistent with the weak lensing statistics .