The correlation function of flux-limited X-ray clusters

We show that the spatial correlation function of a flux-limited sample of X-ray-selected clusters of galaxies will exhibit a correlation scale that is smaller than the correlation scale of a volume-limited, richness-limited sample of comparable apparent spatial density. The flux-limited sample contains clusters of different richnesses at different distances: poor groups are found nearby and rich clusters at greater distances. Since the cluster correlation strength is known to increase with richness, the flux-limited sample averages over the correlations of poor and rich clusters. On the other hand, a volume-limited, richness-limited sample has a minimum richness threshold, and a constant mixture of richnesses with redshift. Using the observed correlation scale for rich (R ≥ 1) clusters, r₀(R ≥ 1) = 21 ± 2 h^-1 Mpc, that was determined from previous volume-limited studies, we derive for the ROSAT flux-limited X-ray cluster sample r₀ (flux-limited) ≈ 14 h^-1 Mpc, in agreement with the recently observed value of 13.7 ± 2.3 h^-1 Mpc.