CO-dark molecular gas at high redshift: very large H$_2$ content and high pressure in a low metallicity damped Lyman-alpha system

We present a detailed analysis of a H$_2$-rich, extremely strong intervening Damped Ly-$\alpha$ Absorption system (DLA) at $z_{\rm abs}=2.786$ towards the quasar J$\,$0843+0221, observed with the Ultraviolet and Visual Echelle Spectrograph on the Very Large Telescope. The total column density of molecular (resp. atomic) hydrogen is $\log N$(H$_2$)=$21.21\pm0.02$ (resp. $\log N$(H$\,$I)=$21.82\pm0.11$), making it to be the first case in quasar absorption lines studies with H$_2$ column density as high as what is seen in $^{13}$CO-selected clouds in the Milky-Way. We find that this system has one of the lowest metallicity detected among H$_2$-bearing DLAs, with $\rm [Zn/H]=-1.52^{+0.08}_{-0.10}$. This can be the reason for the marked differences compared to systems with similar H$_2$ column densities in the local Universe: $(i)$ the kinetic temperature, $T\sim$120~K, derived from the $J=0,1$ H$_2$ rotational levels is at least twice higher than expected; $(ii)$ there is little dust extinction with A$_V 2\times 10^{23} $ cm$^{-2}$/(km/s\,K), in the very low metallicity gas. Low CO and high H$_2$ contents indicate that this system represents "CO-dark/faint" gas. We investigate the physical conditions in the H$_2$-bearing gas using the fine-structure levels of C$\,$I, C$\,$II, Si$\,$II and the rotational levels of HD and H$_2$. We find the number density to be about $n \sim 260-380\,$cm$^{-3}$, implying a high thermal pressure of $(3-5) \times 10^4\,$cm$^{-3}\,$K. We further identify a trend of increasing pressure with increasing total hydrogen column density. This independently supports the suggestion that extremely strong DLAs (with $\log\,$N(H) $\sim 22$) probe high-z galaxies at low impact parameters.