THE EFFICIENCY OF GALAXY FORMATION IN CLUSTERS

We examine whether the observed decrease in the efficiency of galaxy formation between poor and rich X-ray clusters is consistent with galaxy formation scenarios involving hierarchical clustering, such as cold dark matter (CDM). To calculate the efficiency of galaxy formation we adopt the common biased galaxy formation scenario in which galaxies form only at peaks in the primordial distribution of density fluctuations that exceed some critical density threshold. The efficiency of galaxy formation in a cluster can then be defined as the fractional mass within a cluster contained within galaxy peaks exceeding this threshold. The distribution of cluster and galaxy scale overdensities is described as a bivariate Gaussian random field. For relatively flat primordial spectra, such as the CDM power spectrum between galaxy and cluster mass scales, there is a significant amount of statistical correlation between the two length scales. The probability that a random galaxy scale peak exceeds the biasing threshold depends strongly on the mass and amplitude of any background protocluster fluctuation. A greater statistical correlation is present between galaxy and poor cluster masses than between galaxy and rich cluster masses due to the closer proximity in filter length. This is the fundamental reason why galaxy formation is more efficient in poor clusters than in rich clusters. We find that the observed trend in the efficiency of galaxy formation and the observed values can be reproduced within the context of CDM biased galaxy formation scenarios or other scenarios involving flat spectra if X-ray luminous clusters are typically 2-3 sigma objects. This is consistent with the observed number density of X-ray clusters and their virial properties only if the biasing parameter b is close to unity.