AbstractResults from a large, multi-J CO, ¹³CO, and HCN line survey of Luminous Infrared Galaxies (LIRGs: L_IR≥ 10¹⁰ L_⊙) in the local Universe (z≤0.1), complemented by CO J=4–3 up to J=13–12 observations from the Herschel Space Observatory (HSO), paints a new picture for the average conditions of the molecular gas of the most luminous of these galaxies with turbulence and/or large cosmic ray (CR) energy densities U_CR rather than far-UV/optical photons from star-forming sites as the dominant heating sources. Especially in ULIRGs (L_IR>10¹² L_⊙) the Photon Dominated Regions (PDRs) can encompass at most a few % of their molecular gas mass while the large U_CR∼ 10³ U_{CR, Galaxy}, and the strong turbulence in these merger/starbursts, can volumetrically heat much of their molecular gas to T_kin∼ (100-200) K, unhindered by the high dust extinctions. Moreover the strong supersonic turbulence in ULIRGs relocates much of their molecular gas at much higher average densities (≥10⁴ cm⁻³) than in isolated spirals (∼ 10²–10³ cm⁻³). This renders low-J CO lines incapable of constraining the properties of the bulk of the molecular gas in ULIRGs, with substantial and systematic underestimates of its mass possible when only such lines are used. Finally a comparative study of multi-J HCN lines and CO SLEDs from J=1–0 up to J=13–12 of NGC 6240 and Arp 193 offers a clear example of two merger/starbursts whose similar low-J CO SLEDs, and L_IR/L_CO,1−0 and L_{HCN, 1−0}/L_CO,1-0 ratios (proxies of the so-called SF efficiency and dense gas mass fraction), yield no indications about their strongly diverging CO SLEDs beyond J=4–3, and ultimately the different physical conditions in their molecular ISM. The much larger sensitivity of ALMA and its excellent site in the Atacama desert now allows the observations necessary to assess the dominant energy sources of the molecular gas and its mass in LIRGs without depending on the low-J CO lines.