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We present an extensive CO emission-line survey of the Planck's dusty GEMS, asmall set of 11 strongly lensed dusty star-forming galaxies at z = 2-4discovered with Planck and Herschel satellites, using EMIR on the IRAM 30-mtelescope. We detected a total of 45 CO rotational lines from Jup=3 to Jup=11,and up to eight transitions per source, allowing a detailed analysis of the gasexcitation and interstellar medium conditions within these extremely bright,vigorous starbursts. We applied radiative transfer models using the largevelocity gradient approach to infer the spatially-averaged molecular gasdensities, $n_{H_2}$~10$^{2.6-4.1}$ cm$^{-3}$, and kinetic temperatures,$T_k$~30-1000 K. In five sources, we find evidence of two distinct gas phaseswith different properties and model their CO ladder with two excitationcomponents. The warm (70-320 K) and dense gas reservoirs in these galaxies arehighly excited, while the cooler (15-60 K) and more extended low-excitationcomponents cover a range of gas densities. In two sources, the latter isassociated with diffuse Milky Way-like gas phases, which provides evidence thata significant fraction of the total gas masses of dusty starburst galaxies canbe embedded in cool, low-density reservoirs. Finally, we show that the CO lineluminosity ratios are consistent with those predicted by models ofphoton-dominated regions and disfavor scenarios of gas clouds irradiated byintense X-ray fields from active galactic nuclei. By combining CO, [CI] and[CII] line diagnostics, we obtain average PDR gas densities significantlyhigher than in normal star-forming galaxies at low-redshift, andfar-ultraviolet radiation fields at least 100 times more intense than in theMilky Way. These spatially-averaged conditions are consistent with those inhigh-redshift SMGs and in a range of low-redshift environments.

Planck's dusty GEMS. VI. Multi-J CO excitation and interstellar medium conditions in dusty starburst galaxies at z=2-4