H$_2$-based star formation laws in hierarchical models of galaxy
  formation
  We update our recently published model for GAlaxy Evolution and Assembly
(GAEA), to include a self-consistent treatment of the partition of cold gas in
atomic and molecular hydrogen. Our model provides significant improvements with
respect to previous ones used for similar studies. In particular, GAEA (i)
includes a sophisticated chemical enrichment scheme accounting for
non-instantaneous recycling of gas, metals, and energy; (ii) reproduces the
measured evolution of the galaxy stellar mass function; (iii) reasonably
reproduces the observed correlation between galaxy stellar mass and gas
metallicity at different redshifts. These are important prerequisites for
models considering a metallicity dependent efficiency of molecular gas
formation. We also update our model for disk sizes and show that model
predictions are in nice agreement with observational estimates for the gas,
stellar and star forming disks at different cosmic epochs. We analyse the
influence of different star formation laws including empirical relations based
on the hydro-static pressure of the disk, analytic models, and prescriptions
derived from detailed hydro-dynamical simulations. We find that modifying the
star formation law does not affect significantly the global properties of model
galaxies, neither their distributions. The only quantity showing significant
deviations in different models is the cosmic molecular-to-atomic hydrogen
ratio, particularly at high redshift. Unfortunately, however, this quantity
also depends strongly on the modelling adopted for additional physical
processes. Useful constraints on the physical processes regulating star
formation can be obtained focusing on low mass galaxies and/or at higher
redshift. In this case, self-regulation has not yet washed out differences
imprinted at early time.
