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We report single-dish multi-transition measurements of the 12^CO, HCN, andHCO^+ molecular line emission as well as HNC J=1-0 and HNCO in the twoultraluminous infra-red galaxies Arp220 and NGC6240. Using this new molecularline inventory, in conjunction with existing data in the literature, wecompiled the most extensive molecular line data sets to date for such galaxies.The many rotational transitions, with their different excitation requirements,allow the study of the molecular gas over a wide range of different densitiesand temperatures with significant redundancy, and thus allow good constraintson the properties of the dense gas in these two systems. The mass (~(1-2) x10^10 Msun) of dense gas (>10^5-6 cm^-3) found accounts for the bulk of theirmolecular gas mass, and is consistent with most of their IR luminositiespowered by intense star bursts while self-regulated by O,B star clusterradiative pressure onto the star-forming dense molecular gas. The highlyexcited HCN transitions trace a gas phase ~(10-100)x denser than that of thesub-thermally excited HCO^+ lines (for both galaxies). These two phases areconsistent with an underlying density-size power law found for Galactic GMCs(but with a steeper exponent), with HCN lines tracing denser and more compactregions than HCO^+. Whether this is true in IR-luminous, star forming galaxiesin general remains to be seen, and underlines the need for observations ofmolecular transitions with high critical densities for a sample of bright(U)LIRGs in the local Universe -- a task for which the HI-FI instrument onboard Herschel is ideally suited to do.Comment: 38 pages (preprint ApJ style), 3 figures, accepted for Ap

MOLECULAR GAS IN EXTREME STAR-FORMING ENVIRONMENTS: THE STARBURSTS Arp 220 AND NGC 6240 AS CASE STUDIES