Evolution of the Physical Properties of the Most Massive Galaxies in Clusters and their Protohalos

We investigated the evolution of the physical properties of the brightest galaxies in clusters and their protohalos from $z = 4$ to $z = 0$. Galaxy clusters and groups are identified using a halo-based group finder applied to the COSMOS2020 galaxy catalog. We construct evolution chains from low redshift clusters to higher redshift groups via the abundance matching method. The region of protohalos corresponding to clusters is defined on the basis of a characteristic radius. Our analysis encompasses a wide range of physical properties, including stellar mass, luminosity, star formation rate (SFR), specific star formation rate (sSFR), color ($g - r$), and stellar age. The evolution trends of the most massive galaxies (MMGs) in higher redshift groups and their corresponding protohalos are generally consistent. The stellar mass of MMGs shows an increasing trend across the entire redshift range. By considering the stellar mass growth as in-situ and ex-situ components, we find that in-situ star formation is efficient at $z \sim 2$, while ex-situ accretion becomes the primary growth channel at later times. At $z \gtrsim 2$, MMGs undergo an intense star formation phase of approximately $10^{2}\ \rm M_{\odot}yr^{-1}$, but are generally quenched at lower redshifts. Stellar age analysis suggests that most stars in MMGs formed at $z > 2$. Our results present a coherent picture of MMG evolution across cosmic epochs, which is broadly consistent with the current theoretical framework of galaxy formation and evolution. Moreover, our work provides an intriguing way to trace galaxy evolution through the construction of cluster evolutionary chains in observations.