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We use a sample of 37 of the densest clusters and protoclusters across $1.3\le z \le 3.2$ from the Clusters Around Radio-Loud AGN (CARLA) survey to studythe formation of massive cluster galaxies. We use optical $i'$-band andinfrared 3.6$\mu$m and 4.5$\mu$m images to statistically select sources withinthese protoclusters and measure their median observed colours; $\langlei'-[3.6] \rangle$. We find the abundance of massive galaxies within theprotoclusters increases with decreasing redshift, suggesting these objects mayform an evolutionary sequence, with the lower redshift clusters in the samplehaving similar properties to the descendants of the high redshiftprotoclusters. We find that the protocluster galaxies have an approximatelyunevolving observed-frame $i'-[3.6]$ colour across the examined redshift range.We compare the evolution of the $\langle i'-[3.6] \rangle$ colour of massivecluster galaxies with simplistic galaxy formation models. Taking the fullcluster population into account, we show that the formation of stars within themajority of massive cluster galaxies occurs over at least 2Gyr, and peaks at $z\sim 2$-3. From the median $i'-[3.6]$ colours we cannot determine the starformation histories of individual galaxies, but their star formation must havebeen rapidly terminated to produce the observed red colours. Finally, we showthat massive galaxies at $z>2$ must have assembled within 0.5Gyr of themforming a significant fraction of their stars. This means that few massivegalaxies in $z>2$ protoclusters could have formed via dry mergers.

The formation history of massive cluster galaxies as revealed by CARLA