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., thenumber density of clusters with mass larger than $M$ within radius $r_{cl}$, onscale parameter $r_{cl}$. Using numerical simulations of clusters in the CDMcosmogonic theories, we notice that the abundance of rich clusters shows asimple 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 scalerange where halo clustering is fully developed. The abundances of scale$r_{cl}$ clusters identified from IRAS are found basically to follow thisscaling, 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. Wefind 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 goodconsistency with the predictions of OCDM ($\Omega_M=0.3$) and LCDM($\Omega_M+\Omega_{\Lambda} =1$), while the standard CDM model has differentbehavior. The current result seems to favor models with a low mass density.Comment: 20 pages, 7 figures, accepted for publication in Ap