The Co-Evolution Between Galaxies and Dark Matter Halos

The current cosmological paradigm asserts that dark matter halos provide the gravitational scaffolding for galaxy formation through a combination of hierarchical structure formation and non-linear local (g)astrophysical processes. This close relationship, known as the galaxy-halo connection, suggests that the growth and assembly of dark matter halos impact the properties of galaxies. While the stellar mass of galaxies correlates strongly with the mass of their halos, it is important to note that the galaxy-halo connection encompasses a broader distribution of galaxy and halo properties. This distribution can be constrained using data from astronomical observations and cosmological $N$-body simulations, a technique known as semi-empirical modeling. By operating at the intersection of observational data and the cosmological structure formation model, the semi-empirical modeling provides valuable insights into galaxy formation and evolution from a cosmological perspective. In this proceeding, we utilize a new sEM-emPIRical modEl, EMPIRE, to explore the star formation, SF, history of central galaxies across cosmic epochs, spanning from dwarfs to massive ellipticals. EMPIRE aims to constrain the multivariate distribution that links galaxy and halo properties. Our findings reveal distinct growth stages for progenitors of central massive galaxies. Evidence suggests that cold streams played a significant role in sustaining SF at higher $z$, while virial shock heating became more prominent at lower $z$. The maximum star formation efficiency, SFE, occurs at a factor of $\sim1.5-2$ below $M_{\rm vir \; shocks}$ for $z\lesssim1$. Furthermore, at higher redshifts, $z>1$, this peak tends towards higher masses, $M_{\rm vir}\sim 2\times 10^{12} M_{\odot}$. Notably, at redshifts higher than $z\sim2$, the peak of SFE aligns comfortably within the region characterized by cold streams.