The main uncertainty in current determinations of the power spectrum normalization, sigma_8, from abundances of X-ray luminous galaxy clusters arises from the calibration of the mass-temperature relation. We use our weak lensing mass determinations of 30 clusters from the hitherto largest sample of clusters with lensing masses, combined with X-ray temperature data from the literature, to calibrate the normalization of this relation at a temperature of 8 keV, M_{500c,8 keV}=(8.7 +/- 1.6) h^{-1} 10^{14} M_sun. This normalization is consistent with previous lensing-based results based on smaller cluster samples, and with some predictions from numerical simulations, but higher than most normalizations based on X-ray derived cluster masses. Assuming the theoretically expected slope alpha=3/2 of the mass-temperature relation, we derive sigma_8 = 0.88 +/-0.09 for a spatially-flat LambdaCDM universe with Omega_m = 0.3. The main systematic errors on the lensing masses result from extrapolating the cluster masses beyond the field-of-view used for the gravitational lensing measurements, and from the separation of cluster/background galaxies, contributing each with a scatter of 20%. Taking this into account, there is still significant intrinsic scatter in the mass-temperature relation indicating that this relation may not be very tight, at least at the high mass end. Furthermore, we find that dynamically relaxed clusters are 75 +/-40% hotter than non-relaxed clusters.
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