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We present first results of the evolution of cold cosmic gas obtained througha set of state-of-the-art numerical simulations (ColdSIM). We modeltime-dependent atomic and molecular non-equilibrium chemistry coupled to HI andH$_2$ self-shielding, various UV backgrounds as suggested by the recentliterature, H$_2$ dust grain catalysis, photoelectric heating, cosmic-rayheating, as well as hydrodynamics, star formation and feedback effects. Bymeans of such non-equilibrium calculations we are finally able to reproduce thelatest HI and H$_2$ observational data. The neutral-gas mass density parameterresults around $\Omega_{\rm neutral} \!\sim\! 10^{-3}$ and increases from lowerto higher redshift ($z$). The molecular-gas mass density parameter shows peakvalues 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$4-8, in linewith the latest high-$z$ measurements. Both observed HI and H$_2$ trends arewell reproduced by our non-equilibrium H$_2$-based star formation modelling.H$_2$ depletion times remain below the Hubble time and comparable to thedynamical time at all epochs. These findings suggest that, besides HI,non-equilibrium H$_2$ analyses are key probes for assessing the cold gas andthe role of UV background radiation. Abridged.

HI and H$_2$ gas evolution over cosmic times: ColdSIM