The observational study of star formation relations in galaxies is central to unraveling the physical processes at work on local and global scales. We wish to expand the sample of extreme starbursts, represented by local LIRGs and ULIRGs, with high quality observations in the 1-0 line of HCN. We study if a universal law can account for the star formation relations observed for the dense molecular gas in normal star forming galaxies and extreme starbursts. We have used the IRAM 30m telescope to observe a sample of 19 LIRGs in the 1-0 lines of CO, HCN and HCO+. The analysis of the new data proves that the efficiency of star formation in the dense molecular gas (SFE-dense) of extreme starbursts is a factor 3-4 higher compared to normal galaxies. We find a duality in Kennicutt-Schmidt (KS) laws that is reinforced if we account for the different conversion factor for HCN (alpha-HCN) in extreme starbursts and for the unobscured star formation rate in normal galaxies. This result extends to the higher molecular densities probed by HCN lines the more extreme bimodal behavior of star formation laws, derived from CO molecular lines by two recent surveys. We have confronted our observations with the predictions of theoretical models in which the efficiency of star formation is determined by the ratio of a constant star formation rate per free-fall time (SFR-ff) to the local free-fall time. We find that it is possible to fit the observed differences in the SFE-dense between normal galaxies and LIRGs/ULIRGs using a common constant SFR-ff and a set of physically acceptable HCN densities, but only if SFR-ff~0.005-0.01 and/or if alpha-HCN is a factor of a few lower than our favored values. Star formation recipes that explicitly depend on the galaxy global dynamical time scales do not significantly improve the fit to the new HCN data presented in this work.
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