  The thermal properties of hydrodynamical simulations of galaxy clusters are
usually compared to observations by relying on the emission-weighted
temperature T_ew, instead of on the spectroscopic X-ray temperature T_spec,
which is obtained by actual observational data. Here we show that, if the
intra-cluster medium is thermally complex, T_ew fails at reproducing T_spec. We
propose a new formula, the spectroscopic-like temperature, T_sl, which
approximates T_spec better than a few per cent. By analyzing a set of
hydrodynamical simulations of galaxy clusters, we also find that T_sl is lower
than T_ew by 20-30 per cent. As a consequence, the normalization of the M-T
relation from the simulations is larger than the observed one by about 50 per
cent. If masses in simulated clusters are estimated by following the same
assumptions of hydrostatic equilibrium and beta-model gas density profile, as
often done for observed clusters, then the M-T relation decreases by about 40
per cent, and significantly reduces its scatter. Based on this result, we
conclude that using the observed M-T relation to infer the amplitude of the
power spectrum from the X--ray temperature function could bias low sigma_8 by
10-20 per cent. This may alleviate the tension between the value of sigma_8
inferred from the cluster number density and those from cosmic microwave
background and large scale structure.
