  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.
