  We present first results of the evolution of cold cosmic gas obtained through
a set of state-of-the-art numerical simulations (ColdSIM). We model
time-dependent atomic and molecular non-equilibrium chemistry coupled to HI and
H$_2$ self-shielding, various UV backgrounds as suggested by the recent
literature, H$_2$ dust grain catalysis, photoelectric heating, cosmic-ray
heating, as well as hydrodynamics, star formation and feedback effects. By
means of such non-equilibrium calculations we are finally able to reproduce the
latest HI and H$_2$ observational data. The neutral-gas mass density parameter
results around $\Omega_{\rm neutral} \!\sim\! 10^{-3}$ and increases from lower
to higher redshift ($z$). The molecular-gas mass density parameter shows peak
values of $ \Omega_{\rm H_2} \! \sim \! 10^{-4}$, while expected H$_2$
fractions can be as high as 50% of the cold gas mass at $ z\!\sim$4-8, in line
with the latest high-$z$ measurements. Both observed HI and H$_2$ trends are
well reproduced by our non-equilibrium H$_2$-based star formation modelling.
H$_2$ depletion times remain below the Hubble time and comparable to the
dynamical time at all epochs. These findings suggest that, besides HI,
non-equilibrium H$_2$ analyses are key probes for assessing the cold gas and
the role of UV background radiation. Abridged.
