  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.
