The radio galaxy K-z relation: The $\mathsf{10^{12}}~$M$\mathsf{_\odot}$ mass limit - Masses of galaxies from the L$\mathsf{_{K}}$ luminosity, up to z $\mathsf{> 4}$

The narrow K - z relation of powerful radio galaxies in the Hubble K -band diagram is often attributed to the stellar populations of massive elliptical galaxies. Because it extends over a large range of redshifts ($0 z , used to estimate valid photometric redshifts. The baryonic mass of the initial gas cloud $M_{{\rm bar}}$ is then derived. The K - z relation is remarkably reproduced by our evolutionary scenario for elliptical galaxies of baryonic mass $M_{{\rm bar,max}} \simeq 10^{12}~M_\odot$, at all z up to $z = 4$. $M_{{\rm bar,max}}$ is also the maximum mass limit of all types of galaxies. Using another initial mass function (IMF), even a top-heavy one, does not alter our conclusions. The high value of $M_{{\rm bar,max}}$ observed at $z > 4$ implies that massive clouds were already formed at early epochs. We also find that the $M_{{\rm bar,max}}$ limit is similar to the critical mass $M_{{\rm crit}}$ of a self-gravitating cloud regulated by cooling (Rees & Ostriker [CITE]; Silk [CITE]). Moreover, the critical size $r_{{\rm crit}}\simeq 75~{\rm kpc}$ is remarkably close to the typical diameter of Ly α haloes surrounding distant radio galaxies. This confirms the validity of the method of baryonic mass determination based on the K -band luminosity. A puzzling question that remains to be answered is the short time-scale of mass-accumulation required to form such massive galaxies at $z = 4$. We discuss the dispersion of the K - z relation in terms of uncertainties on the mass limit. The link between the presence of the active nucleus and a large stellar mass is also discussed.