Probing the physical environment of the most high-redshift H$_2$-DLAs through numerical models

Damped Lyman-$\alpha$ absorbers (DLAs) with molecular hydrogen have been probed in detail through both spectroscopic observations and numerical modelling. However, such H$_2$ absorbers are quite sparse at very high redshifts. We identify six of the most distant known H$_2$-DLAs (redshift between 3 and 4.5), with medium/high-resolution spectroscopic observations reported in the literature, and perform detailed numerical modelling followed by Bayesian analysis to constrain their physical properties mainly using the H$_2$ rotational level population and CI fine structure levels. Our modelling approach involves setting up a constant-pressure multiphase cloud irradiated from both sides, in comparison to most models which employ constant density. This enables us to use all observed atomic and molecular species as constraints to build a more realistic model of the DLA. Our results indicate high interstellar radiation field strength $\sim$ 10$^2$ to 10$^3$ G$_0$ for some sightlines, which is suggestive of in situ star formation. The cosmic ray ionization rate for all DLAs is constrained between 10$^{-17}$ and 10$^{-14}$ s$^{-1}$, consistent with recent estimates for high-redshift sightlines. Total hydrogen density and temperature lie in the ranges 50 to 4 $\times$ 10$^4$ cm$^{-3}$ and 35-200 K in the innermost part of the absorbers. The corresponding gas pressure in our DLA models lies between 10$^{3.5}$ and 10$^{6.4}$ cm$^{-3}$ K, with three sightlines having a higher pressure than the range typical of high-redshift H$_2$-DLAs.