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 cold dark matter (CDM) cosmogonic theories, we notice that the abundance of rich clusters shows a simple scale invariance such that n[>(r(cl/)r(0))M-alpha,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 similar to 0.5 in the range 1.5 < r(cl) < 4 h(-1) Mpc. The scaling gains further support 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 limits as alpha similar to 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 open CDM (OCDM) (Omega(M) = 0.3) and low-density, flat CDM model with a nonzero lambda (LCDM) (Omega(M) + Omega(A) = 1), whereas the standard CDM model has different behavior. The current result seems to favor models with a low mass density.
