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We present results of analysis of physical conditions (number density, intensity of UV field, kinetic temperature) in the cold H$_2$-bearing interstellar medium of local and high redshift galaxies. Our measurements based on the fit to the observed population of H$_2$ rotational levels and CI fine-structure levels with the help of grids of numerical models calculated with the PDR Meudon code. A joint analysis of low H$_2$ rotational levels and CI fine-structure levels allows to break the degeneracy in the $I_{UV}-n_{rm{H}}$ plane and provides significantly tighter constraints on the number density and intensity of UV field. Using archive data from the VLT/UVES, KECK/HIRES, HST/STIS and FUSE telescopes we selected 12 high redshift damped Ly$alpha$ systems (DLAs) in quasar spectra and 14 H$_2$ absorption systems along the lines of sight towards stars in the Milky-Way and the Magellanic Clouds galaxies. These systems have strong H$_2$ components with the column density $log N({rm{H_2}})/{rm{cm}}^{-2}>18$ and associated CI absorptions. We find that H$_2$-bearing medium in high redshift DLAs and in local galaxies has similar values of the kinetic temperatures $T_{rm{kin}}sim100$K and number density $10-500$ cm$^{-3}$. However, the intensity of incident UV radiation in DLAs is varied in the wide range ($0.1-100$ units of Mathis field), while it is $sim0.1-3$ units of Mathis field for H$_2$ systems in Milky-Way and LMC and SMC galaxies. The large dispersion of measured UV flux in DLAs is probably a consequence that DLA sample probes the galaxies selected from the overall galaxy population at high-z and therefore corresponds to the wide range of the physical conditions.
Following the first pioneering efforts in the 1990s that have focused on the detection of the molecular interstellar medium in high redshift galaxies, recent years have brought great advances in our understanding of the actual physical properties of
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