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We analyse the evolution of the red sequence in a sample of galaxy clustersat redshifts $0.8<z<1.5$ taken from the HAWK-I Cluster Survey (HCS). Thecomparison with the low-redshift ($0.04<z<0.08$) sample of the WIde-fieldNearby Galaxy-cluster Survey (WINGS) and other literature results shows thatthe slope and intrinsic scatter of the cluster red sequence have undergonelittle evolution since $z=1.5$. We find that the luminous-to-faint ratio andthe slope of the faint end of the luminosity distribution of the HCS redsequence are consistent with those measured in WINGS, implying that there is nodeficit of red galaxies at magnitudes fainter than $M_V^*$ at high redshifts.We find that the most massive HCS clusters host a population of bright redsequence galaxies at $M_V < -22.0$ mag, which are not observed in low-massclusters. Interestingly, we also note the presence of a population of verybright ($M_V < -23.0$ mag) and massive (${\log(M_*/M_\odot)} > 11.5$) redsequence galaxies in the WINGS clusters, which do not include only thebrightest cluster galaxies and which are not present in the HCS clusters,suggesting that they formed at epochs later than $z=0.8$. The comparison withthe luminosity distribution of a sample of passive red sequence galaxies drawnfrom the COSMOS/UltraVISTA field in the photometric redshift range$0.8<z_{phot}<1.5$ shows that the red sequence in clusters is more developed atthe faint end, suggesting that halo mass plays an important role in setting thetime-scales for the build-up of the red sequence.

The Accelerated Build-up of the Red Sequence in High Redshift Galaxy Clusters