  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\pm0.019)
Omega_m^-(0.253\pm0.024), 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\pm0.042 and
Omega_m=0.287\pm0.036, for an assumed flat LambdaCDM 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.
