Millimetre/submillimetre-wave emission-line searches for high-redshift galaxies

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. Key words: ISM: molecules ± galaxies: evolution ± galaxies: formation ± cosmology: observations ± infrared: galaxies ± radio lines: galaxies. 1 I N T R O D U C T I O N The redshifted far-infrared/submillimetre-wave line emission from the interstellar medium (ISM) in galaxies could be exploited to detect new samples of distant gas-rich galaxies and active galactic nuclei (AGN) (Loeb 1993; Blain 1996; van der Werf & Israel 1997; Silk & Spaans 1997; Stark 1997; Combes, Maoli & Omont 1999; van der Werf 1999). This emission is attributable both to molecular rotational transitions, in particular from carbon monoxide (CO), and to atomic fine-structure transitions, in particular from the singly ionized 158-mm carbon [C ii] line. Redshifted CO emission has been detected from a range of known high-redshift galaxies and quasars in the millimetre/ submillimetre waveband, as summarized by Frayer et al. (1998) and Combes et al. (1999). Many of these galaxies are known to be gravitationally lensed by a foreground galaxy, an effect which potentially complicates the interpretation of the results by altering the ratios of the inferred luminosities in the continuum and the detected lines (Eisenhardt et al. 1996). In only a small subsample of the detected galaxies (Solomon, Downes & Radford 1992; Barvainis et al. 1997; Downes et al. 1999) have multiple lines been detected, providing an opportunity to investigate the astrophysics of the ISM. So far there have been very few detections of redshifted finestructure lines, despite careful searches, for example, for both [C ii] (Isaak et al. 1994; Ivison, Harrison & Coulson 1998a; van der Werf 1999) and singly ionized 205-mm [N ii] emission (Ivison & Harrison 1996). Neutral carbon [C i] emission, which is considerably less intense than [C ii] and [N ii] emission in the Milky Way (Wright et al. 1991) and nearby galaxies (Stacey et al. 1991), has been detected from the gravitationally lensed Cloverleaf quasar (Barvainis et al. 1994). The most luminous high-redshift galaxies and quasars have necessarily been targeted in these searches. [C ii] fine-structure emission is powerful in both the Milky Way and in sub-L* galaxies, in which it accounts for about 0.5 per cent of the bolometric far-infrared luminosity (Nikola et al. 1998). However, based on observations of a limited number of lowredshift galaxies using the Infrared Space Observatory (ISO) (Malhotra et al. 1997; Luhman et al. 1998; Pierini et al. 1999), it q 2000 RAS w E-mail: [email protected] 560 A. W. Blain et al. appears that a systematically lesser fraction of the bolometric luminosity of more luminous galaxies appears as [C ii] emission, about 0.1 per cent (Luhman et al. 1998). As noted by Luhman et al. (1998) and van der Werf (1999), the results of these ISO observations are fully consistent with the non-detection of redshifted fine-structure emission from high-redshift galaxies using ground-based submillimetre-wave telescopes. The results of Kuiper Airborne Observatory (KAO) observations of the Galactic Centre (Erickson et al. 1991) indicate that 63and 146-mm neutral oxygen [O i] fine-structure emission becomes steadily more luminous as compared with that from [C ii] as the far-infrared luminosity of gas clouds increases. However, currently there are insufficient published data available to address this issue in external galaxies. We attempt here to predict the counts of distant gas-rich lineemitting galaxies that could be detected in the millimetre/ submillimetre waveband. There are two challenges to making reliable predictions. First, there are limited data available from which to construct a clear understanding of the astrophysics of the ISM in high-redshift galaxies. There are only a few tens of detections of line emission from these objects, the majority of which have been made in galaxies that are gravitationally lensed by foreground galaxies. Because of the potential for differential magnification across and within the lensed galaxy, neither the ratios of the line and continuum luminosities nor the excitation conditions in the ISM are known accurately in these cases. As shown by ISO [C ii] observations, extrapolation of the observed properties of low-redshift galaxies with relatively low luminosities to greater luminosities in high-redshift galaxies is not necessarily reliable. Secondly, the space density and form of evolution of gasrich galaxies at high redshifts has not been well determined. Thus the existing predictions of the observability of high-redshift submillimetre-wave line emission have concentrated on either discussing the potential observability of individual high-redshift galaxies (van der Werf & Israel 1997; Silk & Spaans 1997; Combes et al. 1999; van der Werf 1999), or have relied on extensive extrapolations, from the populations of low-redshift ultraluminous infrared galaxies (ULIRGs) (Blain 1996) and from low-redshift L* bulges to the properties of protoquasars at z , 10 (Loeb 1993). Both of these difficulties can be addressed by exploiting the results of deep 450and 850-mm dust continuum radiation surveys made using the Submillimetre Common-User Bolometer Array (SCUBA) camera (Holland et al. 1999) at the James Clerk Maxwell Telescope (JCMT) (Smail, Ivison & Blain 1997; Barger et al. 1998; Hughes et al. 1998; Barger, Cowie & Sanders 1999a; Blain et al. 1999b, 2000; Eales et al. 1999). These surveys are sensitive to galaxies at very high redshifts (Blain & Longair 1993), and have detected a considerable population of very luminous dust-enshrouded galaxies. The 15-arcsec angular resolution of the JCMT is rather coarse, but reliable identifications can be made by combining the SCUBA images with multiwaveband follow-up images and spectra (Ivison et al. 1998b, 2000; Smail et al. 1998, 2000; Barger et al. 1999b; Lilly et al. 1999), and crucially with observations of redshifted CO emission, which are currently available for two submillimetre-selected galaxies: SMM J0239920136 at z ˆ 2:81 and SMM J1401110252 at z ˆ 2:56 (Frayer et al. 1998, 1999; Ivison et al. 1998b, 2000). The bolometric and CO-line luminosities of these galaxies are reasonably well known, and because they are lensed by clusters rather than individual foreground galaxies, their inferred line ratios are not subject to modification by lensing. These observations thus provide a useful template with which to describe the properties of the ISM and line emission in high-redshift, dustenshrouded, gas-rich galaxies. In Section 2 we discuss the existing line observations, and summarize our current state of knowledge about the evolution and redshift distribution of galaxies that have been discovered in submillimetre-wave dust continuum surveys. In Section 3 we describe our model of line emission from these galaxies, and present the results, as based on our understanding of high-redshift continuum sources. In Section 4 we discuss the observability of this hypothetical population using existing and future millimetre/ submillimetre-wave spectrographs. Unless otherwise stated, we assume that H0 ˆ 50 km s Mpc; V0 ˆ 1 and VL ˆ 0. 2 B AC K G R O U N D I N F O R M AT I O N 2.1 Line observations Ground-based telescopes have detected molecular rotation lines and atomic fine-structure lines from low-redshift galaxies (e.g. Sanders et al. 1986; Wild et al. 1992; Devereux et al. 1994; Gerin & Phillips 1998; Mauersberger et al. 1999). Atomic fine-structure lines have also been observed from bright galactic star-forming regions and nearby galaxies using the KAO (Stacey et al. 1991; Nikola et al. 1998), COBE (Wright et al. 1991) and ISO (Malhotra et al. 1997; Luhman et al. 1998; Pierini et al. 1999). CO rotational line emission has been detected successfully from various high-redshift galaxies and quasars, including the first identified high-redshift ULIRG IRAS F1021414724 (Solomon et al. 1992), the gravitationally lensed Cloverleaf quasar H 14131117 (Barvainis et al. 1994; Kneib et al. 1998), various quasars at z . 4; including BR 120220725 (Ohta et al. 1996, 1998; Omont et al. 1996) and the extremely luminous APM 0827915255 (Lewis et al. 1998; Downes et al. 1999), and the submillimetre-selected galaxies SMM J0239920136 and SMM J1401110252 (Frayer et al. 1998, 1999). A significant fraction of the dynamical mass in many of these systems is inferred to be in the form of molecular gas, and it is plausible that they are observed in the process of forming the bulk of their stellar populations. 2.2 Atomic fine-structure lines Atomic fine-structure lines emitted at wavelengths longer than about 100mm ± [C ii] at 1900 GHz, [N ii] at 1460 and 2460 GHz, [O i] at 2060 GHz, and [C i] at 492 and 809 GHz ± are redshifted into atmospheric windows for galaxies at redshifts z & 5; the redshift range within which at least 80 per cent of dust-enshrouded galaxies detected by SCUBA appear to lie (Smail et al. 1998; Barger et al. 1999b; Lilly et al. 1999). There are many other midinfrared lines with shorter rest frame emission wavelengths (see, e.g., Lutz et al. 1998); however, unless massive galaxies exist at z , 10; these lines will not be redshifted into atmospheric windows accessible to ground-based telescopes. Here we assume that the line-to-bolometric luminosity ratio f line ˆ 10 for the [C ii] line in all high-redshift gas-rich galaxies, corresponding to the value observed in low-redshift ULIRGs. The equivalent value for ‰C iŠ492 GHz; f line ˆ 2:9 10; is chosen to match the value observed by Gerin & Phillips (1998) in Arp 220. The values of fline for other fine-structure lines listed in Table 1 are chosen by scaling the results of observations of the Milky Way and low-redshift galaxies (Genzel et al. 1990; Erickson et al. 1991; Stacey et al. 1991; Wright et al. 1991). q 2000 RAS, MNRAS 313, 559±570 Millimetre/submillimetre-wave line surveys 561 This approach should lead to a reliable estimate of the counts of [C i] and [C ii] lines, but greater uncertainty in the N and O line predictions. In the absence of more observational data, which would ideally allow luminosity functions to be derived for each line, we stress that the predictions of the observability of redshifted fine-structure lines made in Section 3 must be regarded as tentative and preliminary. 2.3 CO rotational transitions 2.3.1 CO line excitation Much more observational data are available about the properties of the ladder of CO rotational transitions in the ISM. The energy of the Jth level in the CO molecular rotation ladder EJ ˆ kBT; where T ˆ J…J 1 1†‰2:77 KŠ; and so the energy of a photon produced in the J 1 1! J transition is hn ˆ kB…J 1 1†‰5:54 KŠ: The population of the J states can be calculated by assuming a temperature and density for the emitting gas. The primary source of excitation is expected to be collisions with molecular hydrogen (H2), which dominates the mass of the ISM, with a role for radiative excitation, including that attributable to the cosmic microwave background radiation (CMBR). By taking into account the spontaneous emission rate, AJ11; J / n3…J 1 1†=…2J 1 3† with A1;0 ˆ 6 10 s; and details of the optical depth and geometry of gas and dust in the emitting region, the luminosities of the various J 1 1! J rotational transitions can be calculated. If the J state is to be thermally populated, then the rate of CO±H2 collisions in the ISM gas must be greater than about A J11; J : This condition will not generally be met for a temperature of 50 K in the CO…5! 4† transition unless the density of H2 molecules exceeds about 2 10 cm; which is many times denser than the 10 cm that appears to be typical of low-redshift ULIRGs (Downes & Solomon 1998). Radiative excitation and optical depth effects, perhaps in very non-isotropic geometries, with very pronounced substructure, will complicate the situation greatly in real galaxies. In general, calculations of level populations are very complex, and at high redshifts there are very few data with which to constrain models. Probably the best way to investigate the conditions in very luminous distant galaxies is to study their low-redshift ULIRG counterparts, and the rare examples of high-redshift galaxies and quasars for which more than one CO transition has been detected (e.g. Downes et al. 1999), bearing in mind the potential effects of gravitational lensing. In order to try and make reasonable predictions for the line ratios in high-redshift galaxies, we employed a standard large velocity gradient (LVG) analysis (e.g. de Jong, Dalgarno & Chu 1975) to estimate how the CO line ratios are affected by the temperature, density and finite spatial extent of the ISM, and by the radiative excitation caused by the CMBR. In the third column of Table 1 we show the results from this model, assuming a density of 10 cm, which is typical of the central regions of ULIRGs (Downes & Solomon 1998), and a standard value of X…CO†=…dv=dr† ˆ 3 10 …km s pc21†21: We assume a kinetic temperature of 53 K, which is the temperature of the dominant cool dust component in the SCUBA galaxy SMM J0239920136 (Ivison et al. 1998b). Higher dust temperatures of about 80 and 110 K are inferred for other well-studied high-redshift galaxies IRAS F1021414724 and APM 0827915255 respectively, but these are very exotic galaxies, and the results are potentially modified by the effects of differential gravitational lensing. We assume a background temperature of 10 K, the temperature of the CMBR at z ˆ 2:7; the mean redshift of the two SCUBA galaxies with CO detections. The line-to-continuum bolometric luminosity ratio f line ˆ LJ11!J=LFIR can be calculated if the bolometric continuum luminosity LFIR is known. There is a clear trend of a reduction in the CO line-to-bolometric luminosity ratio of luminous infrared galaxies as the bolometric luminosity increases, with a large scatter, which is consistent with f line / L FIR (Sanders et al. 1986). We normalize the results to the observed CO…3! 2† line luminosity in the LFIR . 10 L( SCUBA galaxies SMM J0239920136 and SMM J1401110252, in which the ratios of the luminosity in the CO…3! 2† line to LFIR is q 2000 RAS, MNRAS 313, 559±570 Table 1. The fraction of the bolometric luminosity of distant dusty galaxies that is assumed to be emitted in a variety of submillimetre-wave lines fline, their rest frame emission frequencies n rest, and the redshifts at which the lines would be detected in the important observing bands at 90, 230, 345 and 650 GHz: z90, z230, z345 and z650 respectively. The CO ratios fline listed in the third, fourth and fifth columns are calculated assuming the LVG model described in Section 2, and in two local thermal equilibrium models with kinetic temperatures of 38 and 53 K respectively. The fraction of the bolometric luminosity of the galaxies in all CO transitions, obtained by adding all values of fline is also shown. The values of fline listed for fine-structure transitions in the final six rows are derived from observations of low-redshift galaxies and the Milky Way; see Section 2.2. The line styles and thicknesses used to represent the various transitions are listed in the final column. Line n rest/GHz fline fline fline z90 z230 z345 z650 Line style (LVG) …T ˆ 38 K† …T ˆ 53 K† CO…1! 0† 115 1:8 10 3:4 10 2:8 10 0.28 N/A N/A N/A CO lines are represented CO…2! 1† 230 1:3 10 8:2 10 7:2 10 1.6 0.0 N/A N/A by solid lines whose CO…3! 2† 345 4:0 10 4:0 10 4:0 10 2.8 0.50 0.0 N/A thickness increases with CO…4! 3† 461 8:3 10 9:6 10 1:1 10 4.1 1.0 0.34 N/A the value of J CO…5! 4† 576 1:3 10 1:4 10 2:0 10 5.4 1.5 0.67 N/A CO…6! 5† 691 1:6 10 1:4 10 2:6 10 6.7 2.0 1.0 0.06 CO…7! 6† 806 1:4 10 1:1 10 2:8 10 8.0 2.5 1.4 0.24 CO…8! 7† 922 4:2 10 6:8 10 2:3 10 9.2 3.0 1.7 0.42 CO…9! 8† 1040 1:6 10 3:3 10 1:6 10 12 3.5 2.0 0.60 Total CO N/A 6:1 10 6:6 10 1:5 10 N/A N/A N/A N/A N/A [C ii] 1890 10 1⁄4 1⁄4 20 7.2 4.5 1.9 Thick dashed line [C i]809 GHz 809 2:9 10 1⁄4 1⁄4 8.0 2.5 1.3 0.24 Thick dot-dashed line [C i]492 GHz 492 2:9 10 1⁄4 1⁄4 4.5 1.1 0.43 N/A Thin dot-dashed line [N ii]2460 GHz 2460 6:4 10 1⁄4 1⁄4 26 9.6 6.1 2.8 Thick dotted line [N ii]1460 GHz 1460 4:0 10 1⁄4 1⁄4 15 5.3 4.2 1.2 Thin dotted line [O i]2060 GHz 2060 2:4 10 1⁄4 1⁄4 22 8.0 5.0 2.2 Thin dashed line 562 A. W. Blain et al. about 2:1 10 and 5:3 10 respectively (Frayer et al. 1998, 1999), with errors of order 50 per cent. The dominant source of error is the uncertainty in the bolometric luminosity. We compare the results obtained in a simple equilibrium case, in which the density of CO molecules, the spontaneous emission rates and the transition energies in different J states are multiplied to give the luminosity in each transition, LJ11!J / n3…J 1 1†2 exp{ 2 ‰2:77 KŠ…J 1 1†…J 1 2†=T}: …1† The results are shown in the fourth and fifth columns of Table 1 for J # 9; assuming kinetic temperatures of 38 and 53 K respectively, the temperature generated by simple fits to the observed counts of IRAS and ISO galaxies (Blain et al. 1999c), and the spectral energy distribution (SED) of SMM J0239920136. 2.3.2 The effects of different excitation conditions The values of fline listed in Table 1 for CO transitions in the LVG and the 38and 53-K thermal equilibrium models differ. However, in lines with J & 7 the differences are less than a factor of a few. Given the current level of uncertainty in the data that support these calculations, this is an acceptable level. The differences between the results are more marked at large values of J. The consequences of these differences for the key predictions of the source counts of line-emitting galaxies are discussed in Section 3.2. Throughout the paper we use the LVG model to describe the CO line emission of dusty galaxies. Observations of the CO line ratios in low-redshift dusty galaxies indicate a wide range of excitation conditions, Tb‰CO…3! 2†=CO…1! 0†Š , 0:2±1 (Mauersberger et al. 1999). In the central regions of starburst nuclei this temperature ratio tends to be systematically higher, with Tb‰CO…3! 2†=CO…1! 0†Š . 0:5±1 (Devereux et al. 1994). In our chosen LVG model, listed in Table 1, this ratio is .0.9, which is consistent with the observations of the central regions of M82 (Wild et al. 1992) and Arp 220 (Mauersberger et al. 1999). The value of the X parameter in the model has little effect on these ratios; however, reducing the density from 10 to 3300 and 1000 cm reduces the predicted ratio to 0.81 and 0.51 respectively. Our model looks reasonable in the light of these observations, as the high-redshift galaxies would typically be expected to be ULIRGs with high gas densities. There have been two recent discussions of the observability of CO line emission from high-redshift galaxies. Silk & Spaans (1997) describe the effect of the increasing radiative excitation of high-J lines at very high redshifts because of the increasing temperature of the CMBR. The median redshift of the SCUBA galaxies is likely to be about 2±3 (Barger et al. 1999b; Smail et al. 2000; Lilly et al. 1999), with perhaps 10±20 per cent at z * 5; and because there is currently no strong evidence for the existence of a large population of metal-rich galaxies at z * 10; this effect is unlikely to be very important. Combes et al. (1999) include a hot dense 90 K/10 cm component in the ISM of their model highredshift galaxies, in addition to the cooler, less dense component included in our models, and conduct LVG calculations to determine CO emission-line luminosities. Understandably, the luminosity of high-J CO lines is predicted to be greater in their models as compared with the values listed in Table 1. The continuum SED of the best studied SCUBA galaxy SMM J0239920136 certainly includes a contribution from dust at temperatures greater than 53 K, but here we avoid including additional hot dense phases of the ISM in our models in order to avoid complicating the models and to try and make conservative predictions for the observability of high-J CO lines at high redshift. Only additional observations of CO in high-redshift galaxies will allow us to improve the accuracy of the conditions in the ISM that are assumed in these models. The detection of the relative intensities of the CO…9! 8† and CO…5! 4† emission from APM 0827915255 (Downes et al. 1999) and the ratio of the intensities of the multiple CO lines detected in BR 120220725 at z ˆ 4:7 (Ohta et al. 1996, 1998; Omont et al. 1996) are broadly consistent with the values of fline listed in the 53-K thermal equilibrium model: see Table 1. The evolution of the abundance of CO and dust throughout an episode of star formation activity have been investigated by Frayer & Brown (1997), and the detailed appearance of the submillimetrewave emission-line spectrum of galaxies requires a careful treatment of the radiative transfer between stars, AGN, gas and dust in an appropriate geometry. However, given that the amount of information on the spectra of high-redshift galaxies is currently not very great, it seems sensible to base estimates of the properties of line emission on the template of the submillimetre-selected galaxies studied by Frayer et al. (1998, 1999). 2.3.3 Other molecular emission lines There could also be a contribution from rotational lines emitted by other species, such as NH3, CS, HCN, HCO 1 and H2O; however, it seems unlikely that these emission lines would dominate the energy emitted in CO unless the densities and excitation temperatures are very high. 2.4 High-redshift dusty galaxies The surface density of 850-mm SCUBA galaxies is now known reasonably accurately between flux densities of 1 and 10 mJy (Barger et al. 1999a; Blain et al. 1999b, 2000). By combining knowledge of the properties of the SCUBA galaxies, the lowredshift 60-mm IRAS galaxies (Saunders et al. 1990; Soifer & Neugebauer 1991), the 175-mm ISO galaxies (Kawara et al. 1998; Puget et al. 1999) and the intensity of far-infrared background radiation (Fixsen et al. 1998; Hauser et al. 1998; Schlegel, Finkbeiner & Davis 1998), it is possible to construct a series of self-consistent models that can account for all these data (Guiderdoni et al. 1998; Blain et al. 1999a,c), under various assumptions about the formation and evolution of galaxies. The `Gaussian' model, which is described by Blain et al. (1999c) and based on pure luminosity evolution of the low-redshift 60-mm luminosity function (Saunders et al. 1990), was modified slightly to take account of the tentative redshift distribution derived from the SCUBA lens survey by Barger et al. (1999b). This `modified Gaussian' model is used as a base for the predictions of line observability presented here. The evolution function increases as …1 1 z†g; with g . 4 at z & 1; has a 1.0-Gyr-long Gaussian burst of luminous activity centred at z ˆ 1:7; in which L* is 70 times greater than the value of L* at z ˆ 0 (Saunders et al. 1990), and then declines at z * 3. This model contains fewer parameters than there are constraining pieces of information, and so should provide a reasonable description of the properties of high-redshift dust-enshrouded galaxies, which are expected to be the most easily detectable sources of line emission. Future observations will inevitably provide more information and demand modifications to the model q 2000 RAS, MNRAS 313, 559±570 Millimetre/submillimetre-wave line surveys 563 of galaxy evolution discussed above; however, it is likely to provide a sound basis for the predictions below. 3 L I N E P R E D I C T I O N S In this paper we are concerned with the detectability of redshifted lines rather than with their resolution. As a result, we want to estimate the total luminosity of a line, and will not be concerned with details of its profile. The results are all presented as integrated flux densities, determined over the whole line profile. Where relevant, a linewidth of 300 km s is assumed. The evolution of the 60-mm luminosity function of dusty galaxies (Saunders et al. 1990) is assumed to be defined by the modified Gaussian model. The bolometric far-infrared continuum luminosity function Fbol(LFIR) can then be calculated by integrating over a template dusty galaxy SED (Blain et al. 1999b). Fbol can in turn be converted into a luminosity function for each line listed in Table 1, Fline(Lline, z), by evaluating Fbol at the bolometric luminosity LFIR that corresponds to the line luminosity Lline. Based on observations by Sanders et al. (1986), the ratio of the luminosity in the CO…1! 0† line, LCO, to LFIR is a function of LFIR, with LCO / L FIR: Hence, when making the transformation from Lline to LFIR for the CO lines listed in Table 1, we use the relationship LFIR ˆ Lline=f line; in which f line / L FIR and is normalized to the value listed in Table 1 at 10 L(, the luminosity of the submillimetre-selected galaxies SMM J0239920136 and SMM J1401110252. There is no evidence from ISO observations of any systematic luminosity dependence in the value of fline for [C ii] emission from ULIRGs, and so we assume that the values of fline listed in Table 1 describe the line-tobolometric luminosity relation in the fine-structure lines at all luminosities, that is LFIR ˆ Lline=f line; where fline is constant. q 2000 RAS, MNRAS 313, 559±570 Figure 1. Predicted counts of CO rotation and fine-structure lines detectable from galaxies in a 1-GHz band centred at 90 GHz (a), and in 8-GHz bands centred on (b) 230 GHz, (c) 345 GHz and (d) 650 GHz. The band in (a) is accessible to the existing BIMA, IRAM, Nobeyama and OVRO interferometer arrays. These arrays also operate with a narrower 1-GHz bandwidth around 230-GHz (b). ALMA will operate in all the bands shown in (b), (c) and (d), with a current goal bandwidth of 16 GHz (Wootten 2000). SPIFI and the HIFI and SPIRE-FTS instruments on the FIRST satellite will operate in the band shown in (d). The line styles and thicknesses correspond to each transition listed in Table 1. In (a) the CO rotational transitions from CO…1! 0† to CO…8! 7† and the [C i]492 GHz line are present, in (b) CO…2! 1† to CO(9!8) and all the fine-structure lines listed in Table 1 are detectable, in (c) CO…3! 2† to CO…9! 8† and all the fine-structure lines are present, and in (d) the CO transitions from CO…6! 5† to CO…9! 8† and all but the [C i]492 GHz fine-structure line are present. 10 W m is equivalent to 3.3, 1.3, 0.9 and 0.5 Jy km s at 90, 230, 345 and 650 GHz respectively. 564 A. W. Blain et al. The surface density of line-emitting galaxies N…. S† that can be detected at integrated flux densities brighter than S, as measured in Jy km s or W m, in an observing band spanning the frequency range between nobs and nobs 1 Dnobs can be calculated by integrating the luminosity function of a line Fline over the redshifts for which the line is in the observing band, and luminosities Lline greater than the detection limit Lmin(S, z). The count of galaxies is thus