Diagnosing the interstellar medium of galaxies with far-infrared emission lines I. The [C II] 158 microns line at z~0

Atomic fine structure lines have been detected in the local Universe and at high redshifts over the past decades. The [C II] emission line at 158 $\mu$m is an important observable as it provides constraints on the interstellar medium (ISM) cooling processes. We develop a physically motivated framework to simulate the production of far-infrared line emission from galaxies in a cosmological context. This first paper sets out our methodology and describes its first application, simulating the [C II] 158 $\mu$m line emission in the local Universe. We combine the output from EAGLE cosmological hydrodynamical simulations with a multi-phase model of the ISM. Gas particles are divided into three phases: dense molecular gas, neutral atomic gas and diffuse ionised gas (DIG). We estimate the [C II] line emission from the three phases using a set of Cloudy cooling tables. Our results agree with previous findings regarding the contribution of these three ISM phases to the [C II] emission. Our model shows good agreement with the observed ${\rm L_{[C II]}}$-star formation rate (SFR) relation in the local Universe within 0.4 dex scatter. The fractional contribution to the [C II] line from different ISM phases depends on the total SFR and metallicity. The neutral gas phase dominates the [C II] emission in galaxies with $\rm{SFR}\sim0.01$-$1\,\rm{M_{\odot}}\,\rm{yr^{-1}}$, but the ionised phase dominates at lower SFRs. Galaxies above solar metallicity exhibit lower ${\rm L_{[C II]}}$/SFR ratios for the neutral phase. In comparison, the ${\rm L_{[C II]}}$/SFR ratio in the DIG is stable when metallicity varies. We suggest that the reduced size of the neutral clouds, caused by increased SFRs, is the likely cause for the ${\rm L_{[C II]}}$ deficit at high infrared luminosities, although EAGLE simulations do not reach these luminosities at $z=0$....