A theoretical investigation of far-infrared fine structure lines at

The far-infrared (FIR) fine structure lines [C <ns0:sc>II</ns0:sc>]158 μm, [O <ns0:sc>III</ns0:sc>]88 μm, [N <ns0:sc>II</ns0:sc>]122 μm, and [N <ns0:sc>III</ns0:sc>]57 μm are excellent tools for probing the physical conditions of the interstellar medium (ISM). The [O <ns0:sc>III</ns0:sc>]88 μm/[C <ns0:sc>II</ns0:sc>]158 μm and [O <ns0:sc>III</ns0:sc>]88 μm/[N <ns0:sc>II</ns0:sc>]122 μm luminosity ratios have shown to be promising tracers of the ionisation state and gas-phase metallicity of the ISM. Observations of galaxies at redshift z > 6 show unusually high [O <ns0:sc>III</ns0:sc>]88 μm/[C <ns0:sc>II</ns0:sc>]158 μm luminosity ratios compared to local sources. The origin of the enhanced ratios has been investigated in the literature with different theoretical modelling approaches. However, no model has to date successfully managed to match the observed emission from both [O <ns0:sc>III</ns0:sc>]88 μm and [C <ns0:sc>II</ns0:sc>]158 μm, as well as their ratio. For this study we used C<ns0:sc>LOUDY</ns0:sc> to model the [C <ns0:sc>II</ns0:sc>]158 μm, [O <ns0:sc>III</ns0:sc>]88 μm, [N <ns0:sc>II</ns0:sc>]122 μm, and [N <ns0:sc>III</ns0:sc>]57 μm emission lines of P<ns0:sc>ONOS</ns0:sc>, a high-resolution (mgas = 883.4 M⊙) cosmological zoom-in simulation of a galaxy at redshift z = 6.5, which is post-processed using <ns0:sc>KRAMSES</ns0:sc>-<ns0:sc>RT</ns0:sc>. We modify carbon, nitrogen, and oxygen abundances in our C<ns0:sc>LOUDY</ns0:sc> models to obtain C/O and N/O abundance ratios respectively lower and higher than solar, more in line with recent high-z observational constraints. We find [O <ns0:sc>III</ns0:sc>]88 μm/[C <ns0:sc>II</ns0:sc>]158 μm luminosity ratios that are a factor of ∼5 higher compared to models assuming solar abundances. Additionally, we find an overall better agreement of the simulation with high-z observational constraints of the [C <ns0:sc>II</ns0:sc>]158 μm-SFR and [O <ns0:sc>III</ns0:sc>]88 μm-SFR relations. This shows that a lower C/O abundance ratio is essential to reproduce the enhanced [O <ns0:sc>III</ns0:sc>]88 μm/[C <ns0:sc>II</ns0:sc>]158 μm luminosity ratios observed at z > 6. By assuming a super-solar N/O ratio, motivated by recent z > 6 JWST observations, our models yield an [O <ns0:sc>III</ns0:sc>]88 μm/[N <ns0:sc>II</ns0:sc>]122 μm ratio of 1.3, which, according to current theoretical models, would be more appropriate for a galaxy with a lower ionisation parameter than the one we estimated for P<ns0:sc>ONOS</ns0:sc>. Most current simulations adopt solar abundance patterns that are not adequate for recently observed high-z predictions. Our results showcase the importance of theoretical modelling efforts, coupled with high-resolution zoom-in simulations, and with parallel multi-tracer observations to understand the physical and chemical conditions of the ISM at z > 6.