ATOMS: ALMA Three-millimeter Observations of massive Star-forming regions -XX. Probability distribution function of integrated intensity for dense molecular gas tracers

We report the observations of J=1-0 of HCN, HCO+, H13CO+, and H13CN, HC3N (J=11-10) emission towards 135 massive star-forming clumps, as part of the ATOMS (ALMA Three-millimeter Observations of Massive Star-forming regions) Survey. We present the integrated intensity probability distribution function for these molecular tracers, modeled as a combination of a log-normal distribution and a power-law tail. The molecular line luminosities for the power-law tail segment, Lmol(p), have been calculated. We have investigated the correlation between the bolometric luminosity, Lbol, and the power-law part of the molecular line luminosity, Lmol(p). Our findings suggest that the scaling relationships between Lbol and Lmol(p) for HCN and HCO+ are sublinear, indicating that these molecules might not be the most effective tracers for the dense gas. In contrast, H13CN and HC3N exhibit a nearly linear relationship between Lbol and Lmol(p), indicating that they can well trace gravitationally bound dense gas. The ratios of Lbol-to-Lmol(p), serving as indicators of star formation efficiency within massive star-forming clumps, exhibit a weak anti-correlation with the power-law index in the I-PDF. In addition, the star formation efficiency is also weakly anti-correlated with the exponent U of the corresponding equivalent density distribution. Our results implie that clumps with substantial gas accumulation may still display low star formation efficiencies.