We present precise constraints on the normalization of the power spectrum of mass fluctuations in the nearby universe , \sigma _ { 8 } , as a function of the mean local matter density , \Omega _ { m } .
Using the observed local X-ray luminosity function of galaxy clusters from the extended BCS and REFLEX studies , a mass-luminosity relation determined from Chandra and ROSAT X-ray data and weak gravitational lensing observations , and the mass function predicted by the Hubble Volume simulations of Evrard et al . , we obtain \sigma _ { 8 } = ( 0.508 \pm 0.019 ) \Omega _ { m } ^ { - ( 0.253 \pm 0.024 ) } , with \Omega _ { m } < 0.34 at 68 per cent confidence .
The degeneracy between \sigma _ { 8 } and \Omega _ { m } can be broken using Chandra measurements of the X-ray gas mass fractions in dynamically relaxed clusters .
Using this information and including Gaussian priors on the mean baryon density of the universe and the Hubble constant , we obtain \sigma _ { 8 } = 0.695 \pm 0.042 and \Omega _ { m } = 0.287 \pm 0.036 , for an assumed flat \Lambda CDM cosmology ( marginalized 68 per cent confidence limits ) .
Our results are in good agreement with some recent studies based on the local X-ray temperature function of clusters , the redshift evolution of the X-ray luminosity and temperature functions of clusters , early results from the Sloan Digitized Sky Survey , the most recent results from studies of cosmic shear , and combined analyses of the 2dF galaxy redshift survey and cosmic microwave background anisotropies .