On the accuracy of the high‐redshift cluster luminosity function

We study the reliability of the statistical background subtraction method for computing the Ks-band luminosity function of cluster galaxies at z ∼ 1 using mock Red-sequence Cluster Survey cluster catalogues constructed from GALFORM semi-analytic galaxies. The underlying cluster luminosity function in the mocks is compatible with recent estimates at z ∼ 1 by several authors. We simulate different samples where the number of clusters with Ks-band photometry goes from five to a maximum of 50, in order to find the most suitable observational sample to carry out this study; the current observational status in the near-infrared wavelength range has been reached using five real clusters at z ∼ 1. We compute the composite luminosity function for several samples of galaxy clusters with masses ≃1.5 x 10 14 M ⊙ assuming a flux-limited, complete sample of galaxies down to Ks = 21.0 mag. We find that the Schechter fit parameters Ks* and a for a sample of galaxies with no redshift information are rather poorly constrained if both parameters are allowed to vary freely; if a is fixed at a fiducial value, then Ks* shows significantly improved stochastic uncertainties but can be influenced by systematic deviations. We find a significantly improved accuracy in the luminosity function parameters when adding photometric redshift information for bright cluster galaxies. The impact of a 10-fold increase in the number of clusters with available Ks-band photometry is that of decreasing stochastic errors in Ks* and a by factors of ≃2 and ≃4, respectively, for accuracies of up to (ΔKS* /Ks*) stochastic = 0.02 and (Δα/α) stochastic = 0.09. The dwarf-to-giant ratios inferred from the luminosity functions of red-sequence galaxies in the mock catalogue agree very well with the underlying values; however, there is an indication that the semi-analytic model overpredicts the abundance of dwarf galaxies by up to a factor of 3 with respect to recent measurements. Finally, we find that in order to use estimates of Ks* to study the formation redshift of cluster galaxies at z = 1, the sample would need to contain 520 z ∼ 1 clusters, for an accuracy of ∼2 Gyr at the 68 per cent confidence level. However, combining this method with other estimates may significantly reduce the sample size, and allow important new constraints on galaxy formation models.