ALMACAL XIII. Evolution of the CO luminosity function and the molecular gas mass density out to $z$ ~ 6
Cold molecular gas, largely traced by CO emission, is the primary fuel for star formation, making it essential for understanding galaxy evolution. ALMA has made significant progress in the study of the cosmic evolution of cold molecular gas. Here, we exploit the ALMACAL survey to address issues relating to small sample sizes and cosmic variance, utilising calibration data from ALMA to compile a statistically significant and essentially unbiased sample of CO-selected galaxies. By employing a novel statistical approach to emission-line classification using semi-analytical models, we place strong constraints on the CO luminosity function and the cosmic evolution of molecular gas mass density ($ρ_{H_2}$) back to $z \sim 6$. The cosmic molecular gas mass density increases with redshift, peaking around $z \sim 1.5$, then slowly declines towards higher redshifts by $\sim 1$ dex. Our findings confirm the key role of molecular gas in fuelling star formation. The new $ρ_{H_2}$ estimates allow us to revisit the cosmic baryon cycle, showing that the ratio of molecular gas-to-stellar mass density is consistent with the so-called 'bathtub model' of baryons, which implies a continuous replenishment of gas. The cosmic gas depletion timescale, estimated on a global scale, is shown to be fairly constant at all redshifts. We emphasise the importance of surveys using multiple small fields rather than a single contiguous area to mitigate the effects of cosmic variance.