The redshifted spectral line radiation emitted from both atomic fine-structure and molecular rotational transitions in the interstellar medium (ISM) of high-redshift galaxies can be detected in the centimetre, millimetre and submillimetre wavebands. Here we predict the counts of galaxies detectable in an array of molecular and atomic lines. This calculation requires a reasonable knowledge of both the surface density of these galaxies on the sky, and the physical conditions in their ISM. The surface density is constrained using the results of submillimetre-wave continuum surveys. Follow-up OVRO Millimeter Array observations of two of the galaxies detected in the dust continuum have provided direct measurements of CO rotational line emission at redshifts of 2.56 and 2.81. Based on these direct high-redshift observations and on models of the ISM that are constrained by observations of low-redshift ultraluminous infrared galaxies, we predict the surface density of line-emitting galaxies as a function of line flux density and observing frequency. We incorporate the sensitivities and mapping speeds of existing and future millimetre/submillimetre-wave telescopes and spectrographs, and so assess the prospects for blank-field surveys to detect this line emission from gas-rich high-redshift galaxies.
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