The properties of the interstellar medium in dusty, star-forming galaxies at $z \sim 2-4$: The shape of the CO spectral line energy distributions

The molecular gas in the interstellar medium (ISM) of star-forming galaxy populations exhibits diverse physical properties. We investigate the $^{12}$CO excitation of twelve dusty, luminous star-forming galaxies at $z \sim 2-4$ by combining observations of the $^{12}$CO from $J_{\rm up} = 1$ to $J_{\rm up} = 8$. The spectral line energy distribution (SLED) has a similar shape to NGC 253, M82, and local ULIRGs, with much stronger excitation than the Milky Way inner disc. By combining with resolved dust continuum sizes from high-resolution $870$-$\mu$m ALMA observations and dust mass measurements determined from multi-wavelength SED fitting, we measure the relationship between the $^{12}$CO SLED and probable physical drivers of excitation: star-formation efficiency, the average intensity of the radiation field $\langle U\rangle$, and the star-formation rate surface density. The primary driver of high-$J_{\rm up}$ $^{12}$CO excitation in star-forming galaxies is star-formation rate surface density. We use the ratio of the CO($3-2$) and CO($6-5$) line fluxes to infer the CO excitation in each source and find that the average ratios for our sample are elevated compared to observations of low-redshift, less actively star-forming galaxies and agree well with predictions from numerical models that relate the ISM excitation to the star-formation rate surface density. The significant scatter in the line ratios of a factor $\approx 3$ within our sample likely reflects intrinsic variations in the ISM properties which may be caused by other effects on the excitation of the molecular gas, such as cosmic ray ionization rates and mechanical heating through turbulence dissipation.