= Cluster temperature function normalization
{title2=$k_BT\propto M^{2/3}[\Delta_v\rho_c(z)]^{1/3}$}
The <virial theorem> and <hydrostatic equilibrium> relate a cluster's gas <temperature> to $GM/R$, while $M=(4\pi/3)\Delta_v\rho_cR^3$. A calibrated mass-temperature relation converts the <Press-Schechter halo mass function> into $dn/dT=(dn/dM)|dM/dT|$. Its exponential rare-cluster tail is sensitive to the <density contrast> <variance> on cluster scales, hence to the power-spectrum amplitude, often parameterized by $\sigma_8$. <Mass> calibration, <temperature> scatter, nonthermal <pressure>, sample selection and the matter density must be included rather than treating <temperature> counts as an independent measurement of amplitude alone.
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