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In recent years, gas has been observed in an increasing number of debrisdiscs, though its nature remains to be determined. Here, we analyse COmolecular excitation in optically thin debris discs, and search ALMA Cycle-0data for CO J=3-2 emission in the Fomalhaut ring. No significant line emissionis observed; we set a 3-$\sigma$ upper limit on the integrated line flux of0.16 Jy km s$^{-1}$. We show a significant dependency of the CO excitation onthe density of collisional partners $n$, on the gas kinetic temperature $T_k$and on the ambient radiation field $J$, suggesting that assumptions widely usedfor protoplanetary discs (e.g. LTE) do not necessarily apply to their lowdensity debris counterparts. When applied to the Fomalhaut ring, we consider aprimordial origin scenario where H$_2$ dominates collisional excitation of CO,and a secondary origin scenario dominated by e$^-$ and H$_2$O. In eitherscenario, we obtain a strict upper limit on the CO mass of 4.9 $\times$10$^{-4}$ M$_{\oplus}$. This arises in the non-LTE regime, where the excitationof the molecule is determined solely by the well-known radiation field. In thesecondary scenario, assuming any CO present to be in steady state allows us toset an upper limit of $\sim$55% on the CO/H$_2$O ice ratio in the parentplanetesimals. This could drop to $\sim$3% if LTE applies, covering the rangeobserved in Solar System comets (0.4-30%). Finally, in light of our analysis,we present prospects for CO detection and characterisation in debris discs withALMA.

CO mass upper limits in the Fomalhaut ring - the importance of NLTE excitation in debris discs and future prospects with ALMA