We use giga-particle N-body simulations to study galaxy cluster populations in Hubble volumes of \Lambda CDM ( \Omega _ { m } = 0.3 , \Omega _ { \Lambda } = 0.7 ) and \tau CDM ( \Omega _ { m } = 1 ) world models .
Mapping past light-cones of locations in the computational space , we create mock sky surveys of dark matter structure to z \simeq 1.4 over 10 , 000 sq deg and to z \simeq 0.5 over two full spheres .
Calibrating the Jenkins mass function at z = 0 with samples of \sim 1.5 million clusters , we show that the fit describes the sky survey counts to \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \mathchar 536 $ } \hss } \raise 2.0 pt% \hbox { $ \mathchar 316 $ } } 20 \% acccuracy over all redshifts for systems more massive than poor galaxy groups ( 5 \times 10 ^ { 13 } \hbox { $ h ^ { -1 } $ } { M } _ { \odot } ) .

Fitting the observed local temperature function determines the ratio \beta of specific thermal energies in dark matter and intracluster gas .
We derive a scaling with power spectrum normalization \beta \propto \sigma _ { 8 } ^ { 5 / 3 } , and find that the \Lambda CDM model requires \sigma _ { 8 } = 1.04 to match \beta = 1.17 derived from gas dynamic cluster simulations .
We estimate a 10 \% overall systematic uncertainty in \sigma _ { 8 } , 4 \% arising from cosmic variance in the local sample and the bulk from uncertainty in the absolute mass scale of clusters .

Considering distant clusters , the \Lambda CDM model matches EMSS and RDCS X–ray-selected survey observations under economical assumptions for intracluster gas evolution .
Using transformations of mass-limited cluster samples that mimic \sigma _ { 8 } variation , we explore Sunyaev–Zel ’ dovich ( SZ ) search expectations for a 10 sq deg survey complete above 10 ^ { 14 } \hbox { $ h ^ { -1 } $ } { M } _ { \odot } .
Cluster counts are shown to be extremely sensitive to \sigma _ { 8 } uncertainty while redshift statistics , such as the sample median , are much more stable .
Redshift information is crucial to extract the full cosmological diagnostic power of SZ cluster surveys .

For \Lambda CDM , the characteristic temperature at fixed sky surface density is a weak function of redshift , implying an abundance of hot clusters at z > 1 .
Assuming constant \beta , four kT > 8 { keV } clusters lie at z > 2 and 40 kT > 5 { keV } clusters lie at z > 3 on the whole sky .
Too many such clusters can falsify the model ; detection of clusters more massive than Coma at z > 1 violates \Lambda CDM at 95 \% confidence if their surface density exceeds 0.003 per sq deg , or 120 on the whole sky .