Scale Invariance of Rich Cluster Abundance: A Possible Test for Models   of Structure Formation
  We investigate the dependence of cluster abundance $n(>M,r_{cl})$, i.e., the number density of clusters with mass larger than $M$ within radius $r_{cl}$, on scale parameter $r_{cl}$. Using numerical simulations of clusters in the CDM cosmogonic theories, we notice that the abundance of rich clusters shows a simple scale invariance such that $n[>(r_{cl}/r_0)^{\alpha}M, r_{cl}]= n(>M,r_0)$, in which the scaling index $\alpha$ remains constant in a scale range where halo clustering is fully developed. The abundances of scale $r_{cl}$ clusters identified from IRAS are found basically to follow this scaling, and yield $\alpha \sim 0.5$ in the range $1.5 < r_{cl} < 4 h^{-1}$Mpc. The scaling gains further supports from independent measurements of the index $\alpha$ using samples of X-ray and gravitational lensing mass estimates. We find that all the results agree within error limit as: $\alpha \sim 0.5 - 0.7$ in the range of $1.5 < r_{cl} < 4 h^{-1}$Mpc. These numbers are in good consistency with the predictions of OCDM ($\Omega_M=0.3$) and LCDM ($\Omega_M+\Omega_{\Lambda} =1$), while the standard CDM model has different behavior. The current result seems to favor models with a low mass density. 
Wen Xu, Li-Zhi Fang, Xiang-Ping Wu