The growth of structure in the Universe is tightly correlated with the cosmological parameters. Galaxy clusters as tracers of the large scale structure are the ideal objects to witness this evolution. The X-ray bright, hot gas in the potential well of a galaxy cluster enables systematic X-ray studies of samples of galaxy clusters to constrain cosmological parameters. HIFLUGCS consists of the 64 X-ray brightest clusters in the Universe, building up a local sample of galaxy clusters. Here we utilize this sample to determine, for the first time, individual hydrostatic mass estimates for all the clusters of the sample and, by making use of the completeness of the sample, we quantify constraints on the two interesting cosmological parameters, OmegaM and sigma8. In paper I we describe the data analysis procedure and compared the individual mass estimates with other references. Now we apply the total hydrostatic and gas mass estimates from the X-ray analysis to a Bayesian cosmological likelihood analysis and leave several parameters free to be constrained. We find OmegaM = 0.30+-0.01 and sigma8 = 0.79+-0.03 (statistical uncertainties, 68% credibility level) using our default analysis strategy combining both, a mass function analysis and the gas mass fraction results. The main sources of biases that we also correct here are (1) the influence of galaxy groups, (2) the hydrostatic mass bias, (3) the extrapolation of the total mass, (4) the theoretical halo mass function and (5) other physical effects. We find that galaxy groups introduce a strong bias, since their number density seems to be over predicted by the halo mass function. On the other hand, baryonic effects as incorporated by recent hydrodynamical simulations do not result in a significant change in the constraints. The total systematic uncertainties (20%) clearly dominate the statistical uncertainties on cosmological parameters.
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