a) (Left panel) Star-formation rate per free-fall time (SFR_ff; shown in the left Y axis) derived from the observed SFE_dense in different populations of galaxies assuming a characteristic gas density for the HCN cloud n_HCN(H₂) = 3 × 10⁴ cm^-3. We show on the right Y axis the value of n_HCN(H₂) derived from SFE_dense assuming a constant SFR_ff = 0.02. This value lies within the most likely range for SFR_ff ~ 0.011–0.028 as determined by the star-formation model of Krumholz & McKee (2005). We also highlight the average value of SFR_ff  ~  0.0058 as determined by Krumholz & Tan (2007) from a compilation of galactic and extragalactic HCN observations. Symbols are as in Fig. 3. Errorbars on SFR_ff (±42%) and L_FIR (±30%) are shown. b) (Right panel) Same as a) but obtained based on the revised values of Σ_dense (in LIRGs/ULIRGs) and Σ_SFR (in normal galaxies) discussed in Sect. 6.