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Context. Cold molecular gas tracers, such as C I and CO lines, have been widely used to infer specific characteristics of the interstellar medium (ISM) and to derive star formation relations among galaxies.
Aims. However, there is still a lack of systematic studies of the star formation scaling relation of CO and [C I] lines across cosmic time on multiple physical scales.
Methods. We used observations of the ground state transitions of [C I], CO, and [C II], for 885 sources collected from the literature, to infer possible correlations between line luminosities of $ \rm L{\prime}_{[CI](1-0)} $, L'_CO(1 - 0)_, and $ \rm L{\prime}_{[CII]} $ with star formation rates (SFRs). With linear regression, we fit the relations between SFR and molecular mass derived from CO, C I, and C II lines.
Results. The relation between [C I]- and CO-based total molecular masses is weakly superlinear. Nevertheless, they can be calibrated against each other. For α_CO = 0.8 and 4.0 M_⊙(K km s^–1 pc²)^–1 we derived α_[CI] = 3.9 and ∼ 17 M_⊙(K km s^–1 pc²)^–1, respectively. Using the lmfit package, we derived relation slopes of SFR-L'_[CI](1 - 0)_, SFR-L'_CO(1 - 0)_, and SFR-L'_[CII](1 - 0)_ to be β = 1.06 ± 0.02, 1.24 ± 0.02, and 0.74 ± 0.02, respectively. With a Bayesian inference linmix method, we find consistent results.
Conclusions. Our relations for [C I](1-0) and CO(1-0) indicate that they trace similar molecular gas contents, across different redshifts and different types of galaxies. This suggests that these correlations do not have strong evolution with cosmic time.