Abstract　 　　  

  Luminous and ultra-luminous infrared galaxies, (U)LIRGs, dominate the cosmic star formation (SF) at z > 1.  Local (U)LIRGs offer a effective way to study the SF process taking place in their distant counterparts.  I will present the results on (a) the mid-J CO rotational emission lines from warm and dense molecular gas and the fine-structure [NII] line at 205 micron from ionized gas from our Herschel/SPIRE 194-671 micron spectroscopic survey of a flux-limited sample of 125 local (U)LIRGs, and (b) the sub-100 pc distribution of the warm molecular gas in the nuclei of selected (U)LIRGs from our follow-up ALMA imaging in the CO(6-5) line and the dust continuum at 430 micron.  These results are used to study the interplay between the on-going SF and the underlying molecular gas.  I will show strong evidences that the on-going SF is directly related to the warm and dense molecular gas phase that emits its CO lines mainly in the mid-J regime (4 < J < 10), rather than to the general, more diffuse molecular gas dominating the total flux of the low-J CO lines, such as CO(1-0). I will also discuss its implication for the widely accepted bi-modal SF phenomenon.  Finally, in view of the ALMA's superior spectroscopic capability in the sub-mm, I will argue that the CO(7-6) line is an excellent SF tracer at high z, much better than the widely used [CII] line at 158 micron, and describe how one can characterize both the SF rate (SFR) band the effective SFR surface density (or its alternative measure, the rest-frame far-infrared color of the dust emission) in high-z galaxies by measuring only two spectral lines in the far-IR/sub-mm: CO\,(7-6) at 372 micron and the [NII] line at 205 micron.