The colour-magnitude relation as a constraint on the formation of rich cluster galaxies

The colours and magnitudes of early-type galaxies in galaxy clusters are strongly correlated. The existence of such a correlation has been used to infer that early-type galaxies must be old passively evolving systems. Given the dominance of early-type galaxies in the cores of rich clusters, this view sits uncomfortably with the increasing fraction of blue galaxies found in clusters at intermediate redshifts, and with the late formation of galaxies favoured by cold dark matter type cosmologies. In this paper, we make a detailed investigation of these issues and examine the role that the colour–magnitude relation can play in constraining the formation history of galaxies currently found in the cores of rich clusters. We start by considering the colour evolution of galaxies after star formation ceases. We show that the scatter of the colour–magnitude relation places a strong constraint on the spread in age that is allowed for the bulk of the stellar population. In the extreme case that the stars are formed in a single event, the spread in age cannot be more than 4 Gyr. Although the bulk of stars must be formed in a short period, continuing formation of stars in a fraction of the galaxies is not so strongly constrained. We examine a model in which star formation occurs over an extended period of time in most galaxies with star formation being truncated randomly. This model is consistent with the formation of stars in a few systems until look-back times of ∼ 5 Gyr. An extension of this type of star formation history allows us to reconcile the small present-day scatter of the colour–magnitude relation with the observed blue galaxy fractions of intermediate redshift galaxy clusters. In addition to setting a limit on the variations in luminosity-weighted age between the stellar populations of cluster galaxies, the colour–magnitude relation can also be used to constrain the degree of merging between pre-existing stellar systems. This test relies on the slope of the colour–magnitude relation: mergers between galaxies of unequal mass tend to reduce the slope of the relation and to increase its scatter. We show that random mergers between galaxies very rapidly remove any well-defined colour–magnitude correlation. This model is not physically motivated, however, and we prefer to examine the merger process using a self-consistent merger tree. In such a model there are two effects. First, massive galaxies preferentially merge with systems of similar mass. Secondly, the rate of mass growth is considerably smaller than for the random merger case. As a result of both of these effects, the colour–magnitude correlation persists through a larger number of merger steps. The passive evolution of galaxy colours and their averaging in dissipationless mergers provide opposing constraints on the formation of cluster galaxies in a hierarchical model. At the level of current constraints, a compromise solution appears possible. The bulk of the stellar population must have formed before z = 1, but cannot have formed in mass units much less than about half the mass of a present-day L* galaxy. In this case, the galaxies are on average old enough that stellar population evolution is weak, yet formed recently enough that mass growth resulting from mergers is small.