  We use a pressure-based model for splitting cold hydrogen into its atomic
(HI) and molecular (H2)components to tackle the co-evolution of HI, H2, and
star formation rates (SFR) in ~3e7 simulated galaxies in the Millennium
simulation. The main prediction is that galaxies contained similar amounts of
HI at redshift z=1-5 than today, but substantially more H2, in quantitative
agreement with the strong molecular line emission already detected in a few
high redshift galaxies and approximately consistent with inferences from
studies of the damped Lyman-alpha absorbers seen in the spectra of quasars. The
cosmic H2/HI-ratio is predicted to evolve monotonically as Omega(H2)/Omega(HI)
(1+z)^1.6. This decline of the H2/HI-ratio as a function of cosmic time is
driven by the growth of galactic disks and the progressive reduction of the
mean cold gas pressure. Finally, a comparison between the evolutions of HI, H2,
and SFRs reveals two distinct cosmic epochs of star formation: an early epoch
(z>3), driven by the evolution of Omega(HI+H2), and a late epoch (z<3), driven
by the evolution of Omega(H2)/Omega(HI).
