  Results from a large, multi-J CO, {13}CO, and HCN line survey of Luminous
Infrared Galaxies (L_{IR}>=10^{10} L_{\odot}) 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}>10^{12} L_{\odot}) the Photon Dominated Regions (PDRs) can
encompass at most \sim few% of their molecular gas mass while the large U_{CR}
and the strong turbulence in these merger/starbursts, can volumetrically heat
much of their molecular gas to T_{kin}\sim(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 than in isolated
spirals. 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. 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}, L_{HCN, 1-0}/L_{CO,1-0} ratios,
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 ....
