  We study the Carbon Monoxide (CO) excitation, mean molecular gas density and
interstellar radiation field (ISRF) intensity in a comprehensive sample of 76
galaxies from local to high redshift (z~0-6), selected based on detections of
their CO transitions J=2-1 and 5-4 and their optical/infrared/(sub-)millimeter
spectral energy distributions (SEDs). We confirm the existence of a tight
correlation between CO excitation as traced by the CO(5-4)/(2-1) line ratio
(R52), and the mean ISRF intensity U as derived from infrared SED fitting using
dust SED templates. By modeling the molecular gas density probability
distribution function (PDF) in galaxies and predicting CO line ratios with
large velocity gradient radiative transfer calculations, we present a framework
linking global CO line ratios to the mean molecular hydrogen gas density nH2
and kinetic temperature Tkin. Mapping in this way observed R52 ratios to nH2
and Tkin probability distributions, we obtain positive U-nH2 and U-Tkin
correlations, which imply a scenario in which the ISRF in galaxies is mainly
regulated by Tkin and (non-linearly) by nH2. A small fraction of starburst
galaxies showing enhanced nH2 could be due to merger-driven compaction. Our
work demonstrates that ISRF and CO excitation are tightly coupled, and that
density-PDF modeling is a promising tool for probing detailed ISM properties
inside galaxies.
