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 .