Galaxy clusters and the amplitude of primordial fluctuations

The distributions of velocity dispersion, gas temperature, and mass-to- light ratio for Abell clusters can be used to constrain the amplitude of primordial density fluctuations within hierarchical clustering theories for the formation of cosmic structure. We have calculated these distributions in the "standard" cold dark matter (CDM) cosmogony with the dual aim of testing the validity of this model and of fixing the value of the biasing parameter, b, which quantifies the segregation between galaxies and mass. To compare the model predictions with optical data, we construct catalogs of "galaxies" from N-body simulations, we subject them to projection effects similar to those in Abell's cluster catalog, and we estimate velocity dispersions using the techniques employed in analyses of real clusters. We find that a significant fraction of rich clusters identified in projection do not correspond to rich three-dimensional clusters, but result instead from superpositions of foreground groups on poorer clusters. A similar fraction of true rich clusters are missed in the projected catalogs. Projection effects have a dramatic effect on the predicted distribution of cluster velocity dispersions. The observed distribution, including a significant tail above ~1500 km s^-1^ is reproduced by our strongly biased CDM models (b ~ 2-2. 5) even though these models predict only a negligible abundance of clusters with true dispersions larger than 1000 km s^-1^. The temperature of the X-ray-emitting gas in clusters is not affected by projection effects, but a comparison with theoretical predictions is hampered by selection effects in existing data and by uncertainties in the thermodynamic state of the gas. Combining our simulations with recent hydrodynamical models, we find that CDM models with b ~ 2-2. 5 provide an acceptable match to present data. The most reliable constraint on b may come from the mean mass-to-light ratio of clusters which appears less sensitive to systematic observational errors than the distributions of velocity dispersions and X-ray temperatures. This comparison also suggests b ~ 2-3. Finally, we discuss theoretical expectations and possible observational problems associated with the study of clusters at intermediate and high redshifts. Optical studies of high- redshift clusters are unlikely to constrain b significantly, but X-ray studies seem more promising. Our results differ from those of other authors largely as a result of our detailed modeling of observational procedures.