The ATLAS 3D project - XVI. Physical parameters and spectral line energy distributions of the molecular gas in gas-rich early-type galaxies

[Abridged] We present a detailed study of the physical properties of the molecular gas in a sample of 18 molecular gas-rich early-type galaxies (ETGs) from the ATLAS$ 3D sample. Our goal is to better understand the star formation processes occurring in those galaxies, starting here with the dense star-forming gas. We use existing integrated $^{12}$CO(1-0, 2-1), $^{13}$CO(1-0, 2-1), HCN(1-0) and HCO$^{+}$(1-0) observations and present new $^{12}$CO(3-2) single-dish data. From these, we derive for the first time the average kinetic temperature, H$_{2}$ volume density and column density of the emitting gas, this using a non-LTE theoretical model. Since the CO lines trace different physical conditions than of those the HCN and HCO$^{+}$ lines, the two sets of lines are treated separately. We also compare for the first time the predicted CO spectral line energy distributions (SLEDs) and gas properties of our molecular gas-rich ETGs with those of a sample of nearby well-studied disc galaxies. The gas excitation conditions in 13 of our 18 ETGs appear analogous to those in the centre of the Milky Way. Such results have never been obtained before for ETGs and open a new window to explore further star-formation processes in the Universe. The conclusions drawn should nevertheless be considered carefully, as they are based on a limited number of observations and on a simple model. In the near future, with higher CO transition observations, it should be possible to better identify the various gas components present in ETGs, as well as more precisely determine their associated physical conditions. To achieve these goals, we show here from our theoretical study, that mid-J CO lines (such as the $^{12}$CO(6-5) line) are particularly useful.