Constraining the Molecular Kennicutt-Schmidt Relation with Multi-Transition CO Observations of Nearby Galaxies

The relationship between the star formation rate surface density and the molecular gas surface density in galaxies is key to understanding galaxy evolution. To investigate the molecular Kennicutt-Schmidt (K-S) relation and its dependence on gas density, we analyze a uniform sample of 36 nearby galaxies from the AMISS survey, focusing on the CO(1-0), CO(2-1), and CO(3-2) transitions, which trace progressively denser and warmer molecular gas. Using statistical methods that combine binning with Markov Chain Monte Carlo (MCMC) fitting, we derive the slope, scatter, and intercept of the $\Sigma_{\mathrm{SFR}}$-$\Sigma_{\mathrm{CO}}$ relation for each transition. We find power-law slopes of 1.26, 1.14, and 1.07 for CO(1-0), CO(2-1), and CO(3-2), respectively, consistent with a trend toward increasingly linear star formation relations at higher-J transitions. This behavior supports the idea that denser gas is more directly linked to ongoing star formation and is consistent with previous findings of near-linear correlations between HCN or high-J CO luminosities and global SFR. The observed trend suggests an underlying relation between gas and SFR volume densities with a power-law index of $\sim$1.5, indicating enhanced star formation efficiency in denser environments. These findings underscore the critical role of dense gas in regulating star formation and highlight the importance of tracer selection and excitation conditions when interpreting the K-S relation across different environments.