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Line-intensity mapping (LIM) surveys will characterise the cosmologicallarge-scale structure of emissivity in a range of atomic and molecular spectrallines, but existing literature rarely considers whether these surveys canrecover excitation properties of the tracer gas species, such as the carbonmonoxide (CO) molecule. Combining basic empirical and physical assumptions withthe off-the-shelf Radex radiative transfer code or a Gaussian process emulatorof Radex outputs, we devise a basic dark matter halo model for CO emission bytying bulk CO properties to halo properties, exposing physical variablesgoverning CO excitation as free parameters. The CO Mapping Array Project(COMAP) is working towards a multi-band survey programme to observe bothCO(1-0) and CO(2-1) at $z\sim7$. We show that this programme, as well as afurther 'Triple Deluxe' extension to higher frequencies covering CO(3-2), isfundamentally capable of successfully recovering the connection between halomass and CO abundances, and constraining the molecular gas kinetic temperatureand density within the star-forming interstellar medium in ways thatsingle-transition CO LIM cannot. Given a fiducial thermal pressure of$\sim10^4$ K cm$^{-3}$ for molecular gas in halos of $\sim10^{10}\,M_\odot$,simulated multi-band COMAP surveys successfully recover the thermal pressurewithin 68% interval half-widths of 0.5--0.6 dex. Construction ofmulti-frequency LIM instrumentation to access multiple CO transitions iscrucial in harnessing this capability, as part of a cosmic statistical probe ofgas metallicity, dust chemistry, and other physical parameters in star-formingregions of the first galaxies and proto-galaxies out of reionisation.

Constraining the halo-ISM connection through multi-transition carbon monoxide line-intensity mapping