Cosmic evolution of the [CII]-to-molecular gas relation

The [CII] 158 $\mu$m line is widely used to trace star formation and the gas contents of high-redshift galaxies. However, it remains unclear under which physical conditions it reliably traces the molecular reservoir, and whether a unique conversion factor ${\alpha }_{[\mathrm{C}\mathrm{I}\mathrm{I}]}$ can be applied across cosmic time. We investigate the evolution of the relation between the [CII] luminosity and molecular gas mass from $z\simeq 10$ to $z\simeq 0.2$ using the Vintergatan simulation, a high-resolution cosmological zoom-in of a Milky Way-like galaxy. We post-process the snapshots with the Skirt radiative transfer code to generate synthetic [CII] data cubes. We measure global and spatially resolved (100 pc) relations between [CII] luminosity ($L_{[\mathrm{C}\mathrm{I}\mathrm{I}]}$), star formation rate (SFR), and molecular gas mass ($M_{\mathrm{m}\mathrm{o}\mathrm{l}}$). We follow the redshift evolution of the [CII]-to-molecular gas conversion factor ${\alpha }_{[\mathrm{C}\mathrm{I}\mathrm{I}]}$, and link these trends to the evolution of the interstellar medium (ISM) phases. The global $L_{[\mathrm{C}\mathrm{I}\mathrm{I}]}$-$M_{\mathrm{m}\mathrm{o}\mathrm{l}}$ and $L_{[\mathrm{C}\mathrm{I}\mathrm{I}]}$-SFR relations evolve from a steep, [CII]-deficient regime at very low metallicity to an almost linear behaviour, similar to calibrations at $z\approx 2$, once the ISM reaches $Z\gtrsim 0.05$-$0.1Z_{\odot }$ at $z\lesssim 5$. Over this evolution, ${\alpha }_{[\mathrm{C}\mathrm{I}\mathrm{I}]}$ spans nearly three orders of magnitude, from $\gtrsim {10}^4$ down to $\approx 10{\mathrm{M}}_{\odot }{\mathrm{L}}_{\odot }^{-1}$, even though the [CII] emission remains spatially correlated with the molecular gas. A unique, redshift-independent ${\alpha }_{[\mathrm{C}\mathrm{I}\mathrm{I}]}$ therefore cannot recover molecular gas masses across the regimes we explore. [CII] remains a viable tracer of molecular gas at very high redshifts, but only when used with conversion factors that explicitly account for metallicity, ISM phase mix, and merger events.