We study Sunyaev-Zel ’ dovich Effect ( SZE ) cluster counts in different cosmologies .
It is found that even without the full knowledge of the redshift distribution of SZE clusters , one can still readily distinguish a flat universe with a cosmological constant from an open universe .
We divide clusters into a low redshift group ( with redshift z \leq 0.5 ) and a high redshift group ( with z \geq 1 ) , and compute the ratio of r = N ( z \leq 0.5 ) / N ( z \geq 1 ) , where N ( z \leq 0.5 ) is the number of flux-limited ( S _ { \nu } ^ { lim } ) SZE clusters with z \leq 0.5 and N ( z \geq 1 ) is the number of flux-limited SZE clusters with z \geq 1 .
With about the same total number of SZE clusters N ( z \geq 0 ) , the r value for a flat universe with a non-zero cosmological constant and that for an open universe occupy different regions in the S _ { \nu } ^ { lim } - r plot for the most likely cosmological parameters 0.25 \leq \Omega _ { 0 } \leq 0.35 and 0.2 \leq \Gamma \leq 0.3 , where \Omega _ { 0 } is the matter density parameter of the universe , and \Gamma is the shape parameter of the power spectrum of linear density fluctuations .
Thus with a deep SZE cluster survey , the ratio r can reveal , independent of the normalization of the power spectrum , whether we are living in a low-density flat universe or in an open universe .
Within the flat universe scenario , the SZE cluster-normalized \sigma _ { 8 } is studied , where \sigma _ { 8 } is the r.m.s .
density fluctuation within the top-hat scale 8 \hbox { Mpc } h ^ { -1 } where h is the Hubble constant in units of 100 \hbox { kms } ^ { -1 } \hbox { Mpc } ^ { -1 } .
A functional relation \sigma _ { 8 } \propto \Omega _ { 0 } ^ { -0.13 } is found .
Combined with the X-ray cluster-normalized \sigma _ { 8 } \propto \Omega _ { 0 } ^ { -0.52 + 0.13 \Omega _ { 0 } } , one can put constraints on both \Omega _ { 0 } and \sigma _ { 8 } simultaneously .