We study the H _ { 2 } molecular content in high redshift damped Lyman- \alpha systems ( DLAs ) as a function of the H i column density .
We find a significant increase of the H _ { 2 } molecular content around \log N ( { H \textsc { i } } ) ( cm ^ { -2 } ) \sim 21.5 - 22 , a regime unprobed until now in intervening DLAs , beyond which the majority of systems have \log N ( H _ { 2 } ) > 17 .
This is in contrast with lines of sight towards nearby stars , where such H _ { 2 } column densities are always detected as soon as \log N ( { H \textsc { i } } ) ¿ 20.7 .
This can qualitatively be explained by the lower average metallicity and possibly higher surrounding UV radiation in DLAs .
However , unlike in the Milky Way , the overall molecular fractions remain modest , showing that even at a large N ( H i ) only a small fraction of overall H i is actually associated with the self-shielded H _ { 2 } gas .
Damped Lyman- \alpha systems with very high- N ( { H \textsc { i } } ) probably arise along quasar lines of sight passing closer to the centre of the host galaxy where the gas pressure is higher .
We show that the colour changes induced on the background quasar by continuum ( dust ) and line absorption ( H i Lyman and H _ { 2 } Lyman & Werner bands ) in DLAs with \log N ( { H \textsc { i } } ) \sim 22 and metallicity \sim 1/10 solar is significant , but not responsible for the long-discussed lack of such systems in optically selected samples .
Instead , these systems are likely to be found towards intrinsically fainter quasars that dominate the quasar luminosity function .
Colour biasing should in turn be severe at higher metallicities .