Physical conditions in high$-$z optically thin C III absorbers: Origin of cloud sizes and associated correlations

We present detailed photoionization models of well aligned optically thin C III absorption components at $2.1 < z < 3.4$. Using our models we estimate density ($n_{\rm \tiny H}$), metallicity ($[C/H]$), total hydrogen column density and line-of-sight thickness ($L$) in each C III components. We estimate the systematic errors in these quantities contributed by the allowed range of the quasar spectral index used in the ultraviolet background radiation calculations. Our inferred $n_{\rm \tiny H}$ and overdensity ($\Delta$) are much higher than the measurements available in the literature and favor the absorption originating from gas associated with circumgalactic medium and probably not in hydrostatic equilibrium. We also notice $n_{\rm \tiny H}$, $L$ and $[C/H]$ associated with C III components show statistically significant redshift evolution. To some extent, these redshift evolutions are driven by the appearance of compact, high $n_{\rm \tiny H}$ and high $[C/H]$ components only in the low$-z$ end. We find more than 5$\sigma$ level correlation between $[C/H]$ and $L$, $L$ and neutral hydrogen column density (N (HI)), N (HI) and $[C/H]$. We show $L$ versus $[C/H]$ correlation can be well reproduced if $L$ is governed by the product of gas cooling time and sound speed as expected in the case of cloud fragmentation under thermal instabilities. This allows us to explain other observed correlations by simple photoionization considerations. Studying the optically thin C III absorbers over a large $z$ range and probably correlating their $z$ evolution with global star formation rate density evolution can shed light into the physics of cold clump formation and their evolution in the circumgalactic medium.