Evolution of the cluster population has been recognized as a powerful cosmological tool .
While the present–day abundance of X-ray clusters is degenerate in \sigma _ { 8 } , n and \Omega _ { 0 } , Oukbir and Blanchard ( 1992 , 1997 ) have pointed out that the number density evolution of X-ray clusters with redshift can be used to determine \Omega _ { 0 } .
Here , we clarify the origin of this statement by identifying those parameters to which the evolution of cluster number density is most sensitive .
We find that the evolution is controlled by only two parameters : the amplitude of fluctuations , \sigma _ { M } , on the scale associated with the mass under consideration , R = 9.5 h ^ { 1 / 3 } \Omega _ { 0 } ^ { -1 / 3 } M _ { 15 } ^ { 1 / 3 } h ^ { -1 } Mpc , and the cosmological background density , \Omega _ { 0 } .
In contrast , evolution is remarkably insensitive to the slope of the power spectrum .
We verify that the number density evolution of clusters is a powerful probe of the mean density of the universe , under the condition that \sigma _ { M } is chosen to reproduce current-day abundances .
Comparison of the cluster abundance at z \sim 0.5 - 0.6 , from the EMSS , to the present-day abundance , from the ROSAT BCS sample , unambiguously reveals the existence of significant negative evolution .
This number evolution , in conjunction with the absence of any negative evolution in the luminosity-temperature relation , provides robust evidence in favor of a critical density universe ( \Omega _ { 0 } = 1 ) , in agreement with the analysis by Sadat et al .
( 1998 ) .