Here we investigate the evolution of a Milky Way (MW) -like galaxy with the aim of predicting the properties of its progenitors all the way from $z sim 20$ to $z = 0$. We apply GAMESH (Graziani et al. 2015) to a high resolution N-Body simulation following the formation of a MW-type halo and we investigate its properties at $z sim 0$ and its progenitors in $0 < z < 4$. Our model predicts the observed galaxy main sequence, the mass-metallicity and the fundamental plane of metallicity relations in $0 < z < 4$. It also reproduces the stellar mass evolution of candidate MW progenitors in $0 lesssim z lesssim 2.5$, although the star formation rate and gas fraction of the simulated galaxies follow a shallower redshift dependence. We find that while the MW star formation and chemical enrichment are dominated by the contribution of galaxies hosted in Lyman $alpha$-cooling halos, at z > 6 the contribution of star forming mini-halos is comparable to the star formation rate along the MW merger tree. These systems might then provide an important contribution in the early phases of reionization. A large number of mini-halos with old stellar populations, possibly Population~III stars, are dragged into the MW or survive in the Local Group. At low redshift dynamical effects, such as halo mergers, tidal stripping and halo disruption redistribute the baryonic properties among halo families. These results are critically discussed in light of future improvements including a more sophisticated treatment of radiative feedback and inhomogeneous metal enrichment.
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