X-shooter observations of strong H$_2$-bearing DLAs at high redshift
  We present results from spectroscopic observations with X-shooter at the Very
Large Telescope of seven H2-bearing DLAs at high redshifts (z$_{\rm abs}\sim
2.5-3$). These DLAs were originally selected from the presence of strong H$_2$
lines directly seen at the DLA redshift in low-resolution, low S/N SDSS
spectra. We confirm the detection of molecular hydrogen in all of them. We
measure the column densities of HI, H$_2$ in various rotational levels, and
metal species, and associated dust extinction. The metallicities, obtained from
undepleted species, are in the range logZ=-0.8 to -0.2. We discuss the chemical
enrichment in these clouds and compare their properties with that of other
molecular-rich systems selected by other means. In particular, we show that
three different methods of pre-selection of H$_2$-bearing DLAs in the SDSS have
their own biases but complement each other mostly in terms of chemical
enrichment. We use the rotational excitation of H$_2$ molecules together with
the fine-structure energy levels of neutral carbon to constrain the physical
conditions in the gas with the help of numerical modeling as well as analytical
expressions for the surface density at which atomic to molecular conversion
happens. We find that the H$_2$-bearing medium revealed by the studied DLAs has
typical values for the kinetic temperature, hydrogen density, and UV radiation
field of, respectively, T$\sim$100K, $n_{\rm H}\sim 100$ cm$^{-3}$, and I$_{\rm
UV}$ about twice the intensity of the Draine field. Detailed studies combining
different selections should, therefore, bring important clues to understand the
HI-H$_2$ transition at high redshift.
