Although observationally fare, damped Ly alpha absorption systems dominate the mass density of neutral gas in the Universe. 11 high-redshift damped Ly alpha systems covering 2.8 less than or equal to z less than or equal to 4.4 were discovered in 26 QSOs from the APMz>4 SO survey, extending these absorption system surveys to the highest redshifts currently possible. Combining our new data set with previous surveys, we find that the cosmological mass density in neutral gas, Omega(g) does not rise as steeply prior to z similar to 2 as indicated by previous studies. There is evidence in the observed Omega(g), for a flattening at z similar to 2 and a possible turnover at z similar to 3. When combined with the decline at z>3.5 in number density per unit redshift of damped systems with column densities log N(Ht)greater than or equal to 21 atom cm(-2), these results point to an epoch at 2 greater than or similar to 3 prior to which the highest column density damped systems are still forming. We find that, over the redshift range 2<z<4, the total mass in neutral gas is marginally comparable to the total visible mass in stars in present-day galaxies. However, if one considers the total mass visible in stellar discs alone, i.e. excluding galactic bulges, the two values are comparable. We are observing a mass of neutral gas that is comparable to the mass of visible disc stars. Lanzetta, Wolfe & Turnshek found that Omega(2 approximate to 3.5) was twice Omega(z approximate to 2), implying that a much larger amount of star formation must have taken place between z=3.5 and 2 than is indicated by metallicity studies. This created a 'cosmic G-dwarf problem'. The more gradual evolution of Omega(g), that we find alleviates this. These results have profound implications for theories of galaxy formation.