Observing strategies for blank-field surveys in the submillimetre waveband

Transcription

1 Mon. Not. R. Astron. Soc. 279, (1996) Observing strategies for blank-field surveys in the submillimetre waveband A. W. Blain and M. S. Longair Cavendish Laboratory, Madingley Road, Cambridge CB3 ORE Accepted 1995 October 19. Received 1995 October 11; in original form 1995 August 23 ABSTRACT The coming generation of submillimetre bolometer array detectors, such as the Submillimetre Common-User Bolometer Array (SCUBA) instrument currently being constructed for the James Clerk Maxwell Telescope (JCMT), offer opportunities for searching for distant star-forming galaxies in the submillimetre waveband. The special features of observing in the sub millimetre waveband require a careful assessment of the optimum observing strategy, aimed both at maximizing the probabilities of detecting distant galaxies and at discriminating between different models of galaxy formation and evolution. The distinctive feature of faint sub millimetre sources is that their predicted source counts are inverted at low flux densities and this strongly influences the observing strategy. We illustrate how the strategy differs for distinct models of the evolution of the population of star-forming galaxies. In all cases the preferred observing wavelength is 85 11m. Plausible models of the evolution of lras galaxies suggest that the best strategy would be to observe an area of about deg 2 for an observing time of 3 x 5 s. In a hierarchical clustering picture, it would be preferable to observe a much smaller area of.1 deg 2 for the same time to a deeper limiting flux density of.3 mjy, at which the extremely steep source counts expected in these models should be detected. It is argued that both types of survey should be carried out. Key words: methods: observational - stars: formation - dust, extinction - galaxies: evolution - cosmology: observations - radio continuum: galaxies. 1 INTRODUCTION A large fraction of the bolometric luminosity of nearby starforming galaxies is emitted in the far-infrared waveband, and the equivalent emission of distant sources is redshifted into the submillimetre waveband. There are excellent prospects for detecting the continuum radiation from dust in star-forming galaxies at large redshifts in the submillimetre waveband (e.g. Dunlop et al. 1994; Isaak et al. 1994) because the modified blackbody emission spectra of dusty galaxies result in large negative K-corrections in the submillimetre waveband and flat flux density-redshift relations at redshifts between about 1 and (Franceschini et al. 1991; Blain & Longair 1993a). Sources at moderate and high red shifts can therefore be detected, provided the detectors are sufficiently sensitive to detect sources at redshifts 1. The advent of array bolometer detector systems for submillimetre telescopes, for example SCUBA for the JCMT (Gear & Cunningham 199; Cunningham et al. 1994), will increase the sensitivity and mapping speed of continuum observations by a sufficiently large factor to make a blank-field survey in the submillimetre waveband a practical possibility for the first time. In this paper we discuss observing strategies for the submillimetre waveband which are relevant to survey observations. We combine a simple model of the detector performance, based on the specifications of SCUBA, with new models of source counts in the submillimetre waveband developed from those presented by Blain & Longair (1993a). The improved models included in this paper describe two different schemes of galaxy formation, and are normalized to be consistent with the results of recent observations of the background radiation intensity in the millimetre waveband (Mather et al. 1994). The distinctive feature of all these computations is that, because of the redshifting of the far-infrared dust emission spectrum into the sub millimetre waveband, the source counts are expected to be inverted at sufficiently low flux densities. If the survey 1996 RAS

2 848 A. W. Blain and M. S. Longair extends to such flux densities, the steepness of the source counts means that more sources are detected by a modest increase in integration depth, as compared to surveying a larger area of sky in the same total observing time. The aim of this paper is to maximize the probability that a steep source count of submillimetre sources can be detected by a sub millimetre bolometer array detector such as SCUBA. The optimal frequency, depth and field area for blank-field extragalactic surveys in the submillimetre waveband are therefore discussed, and the prospects of using the results to discriminate between different models of galaxy formation and evolution are also investigated. Throughout this paper we assume that the density parameter n = 1 and that h =.5, where Hubble's constant Ho= h km S-I Mpc- I 2 OBSERVING IN THE SUBMILLIMETRE WAVEBAND It is difficult to observe faint point sources in the submillimetre waveband with current instrumentation, and mapping blank fields to search for extragalactic sources is not feasible at present. The emission of submillimetre radiation by molecules in the atmosphere limits ground-based observations to several narrow-wavelength windows, and within these windows careful subtraction of the rapidly varying atmospheric signal is essential (e.g. Matthews 1993). Careful calibration is also required to remove the effects of varying atmospheric absorption of astronomical signals. The technologies of both heterodyne and bolometer receiver systems are also pushed to their limits when operating in the submillimetre waveband. The current instrument for observing continuum radiation in the submillimetre waveband at the JCMT is the single-element bolometer UKT 14 (Duncan et al. 199). This receiver requires careful external calibration and very long integration times in good observing conditions in order to detect extragalactic objects (Dunlop et al. 1994; Isaak et al. 1994), and can only observe a single telescope beam area on the sky in each pointing, although the image plane of the JCMT covers a much larger area. Submillimetre bolometer arrays such as SCUBA can, however, fully exploit the imaging capabilities of the present generation of submillimetre telescopes. 2.1 Developments in instrumentation An instrument such as SCUBA with an array of detectors in the telescope image plane confers three significant dvantages over current receivers. (i) An array receiver greatly increases the area of the field that can be observed in a single telescope pointing, and immediately increases the mapping speed of the telescope by a factor equal to the number of array elements. The SCUBA bolometers are fed by a close-packed hexagonal array of circular horns, matched to the instrument optics at 45 and 85 with 91 and 37 bolometer elements respectively. Each array has a field of view of 2.3 arcmin. Additional filters are provided at the unmatched wavelengths of 35, 65 and 75 /lm to cover all the atmospheric windows in the submillimetre waveband. (ii) The SCUBA bolometers are cooled to - K, giving each element a sensitivity which is about times better than the UKT14 bolometer, and thus reducing the integration time required to detect a point source by a factor of about. Sensitivities of about and mjy Hz- l12 are expected at 35/45 and 75/85 /lm respectively, given as 'noise equivalent flux densities' (NEFDs), the flux densities of point sources that can be detected at a signal-to-noise ratio of unity in 1 s of integration. (iii) The design of SCUBA will allow much more reliable and accurate calibration of observations and reduce the dependence of the results upon favourable atmospheric conditions. Internal calibration sources will allow sky and instrument calibration to be performed independently, and narrow-band filters accurately aligned with atmospheric transmission windows will reduce atmospheric contamination of the signal. The potential also exists for real-time subtraction of most atmospheric noise by removing the correlated atmospheric signals in all the detector channels. These factors combine to predict an increase in mapping speed of up to 4 and a decrease in integration times for point-source photometry of about at the JCMT when SCUBA replaces UKT Observing a population of sources The performance of SCUBA will only be known after the instrument is commissioned; however, the specifications and some general features of observations in the submillimetre waveband can be used to estimate the number of sources which should be observed in a blank-field survey. The flux density of the faintest source that can be detected in a survey covering an areaa in a total integration time tis, ) I12 ( t )-112 Smin(t,A)=R.So -, a seconds where R. is the required signal-to-noise ratio,a a is the area subtended by the detector array in a single pointing, and So is the NEFD of the instrument, which incorporates all the factors of telescope and detector efficiency. In order to estimate the number of sources detectable in such a survey N s, a source countn(sv) is required to describe the surface density of sources per unit area of sky. The expression can then be used to determine the values of t and A which optimize the number of detected sources. Equations (1) and (2) provide an estimate ofthe number of sources expected; however, a differential observing technique is required in order to subtract the atmospheric signal llnd form an image of the survey field (Emerson, Klein & Haslam 1979), and so the performance of the instrument will probably not match these predictions over large fields. Equations (1) and (2) will only predict Ns accurately if the combined instrumental and atmospheric noise is Gaussian and integrates away smoothly with integration time t as t -1/2. If the noise has a non-gaussian component then there could be an effective limiting value of Smin, which cannot be (1) (2)

3 improved by increasing t further. We will return to this point in Sections 5.2 and 5.3. In order to allow predictions of N s, source counts have been derived in the submillimetre waveband for two different models of galaxy formation, developed from Blain & Longair (1993a), in Sections 3 and 4 below. The resulting counts are used in Sections 5 and 6 to investigate observing strategies for the deep submillimetre surveys. It is possible to progress straight to these sections without a detailed reading of the intervening material. 3 MODELLING THE SUBMILLIMETRE SKY 3. 1 Predicting source counts The derivation of source counts in the submillimetre waveband, required to predict the number of sources that can be detected in a survey (equation 2), was discussed in some detail by Blain & Longair (1993a). Generally, the source count N(S,), including all sources brighter than S, can be expressed in sources sr- 1 as, N(S,) = <I>(L, z) dld2(z) dz dz, IZO foo dr o L(S"z) which involves the comoving luminosity function <I>(L, z) describing an evolving population of sources at redshifts which extend to Zo, the comoving radial distance element r and the distance parameter D (z). The term D 2(Z) dr gives the comoving volume of a spherical shell of thickness dr at redshift z, within which only sources with luminosities exceeding Sf,. dv' L(S" z) = 41t(1 + Z)D2(Z)S, --, f'(i+z) are included in the count. f, is the spectral energy distribution of the sources, and can be adequately described by a blackbody spectrum and a frequency-dependent model of the dust grain emissivity function in the submillimetre waveband (e.g. Hildebrand 1983). The forms used by Blain & Longair (1993a) have again been adopted. All source counts take the Euclidean form NoocS,-3/2 at high flux densities; however, at fainter flux densities the effects of redshifting the spectral energy distributions of the sources and of the large-scale curvature of space-time both modify the slope of the count. In the optical and nearinfrared wavebands these effects result in flatter source counts, with a power-law index > - 3/2, for non-evolving galaxies; however, in the submillimetre waveband the K corrections are large and negative and so the counts 'invert' and rise more rapidly, with indices < - 3/2. Source evolution can change these results. Optical counts of galaxies are steeper than expected (see observations listed by Kauffmann & White 1993) and radio galaxies and radio quasars exhibit very marked evolution of their source populations with cosmic epoch (Dunlop & Peacock 1991). The derivative of the count dn/ds" is often used to describe the number of sources on the sky at flux density S,. The 'normalized differential count', I1N/I1No, is obtained by dividing dn/ds, by the derivative of the Euclidean count dno/ds,( ocs ;512), and normalizing the result to 1 at high (3) (4) Submillimetre SU1VeyS 849 flux densities. This is also a useful quantity, as it varies over a much smaller range than the count itself. The number of sources in a faint submillimetre count typically far exceeds the extrapolation of the Euclidean count to faint flux densities, and a large fraction of the excess sources is expected to be at moderate and high redshifts. This is the key to the feasibility and potential utility of survey observations in the submillimetre waveband. 3.2 Constraints on count models Any model used to predict source counts in the submillimetre waveband must be consistent with observations of the intensity of diffuse extragalactic background radiation, and with the density parameter of heavy elements at the present epoch. The diffuse background radiation intensity is produced by the integrated luminosity of all sources contributing to the luminosity function and heavy elements are created by nucleosynthesis in all sufficiently massive stars. The observations and the constraints that they impose on the models underlying the predictions of source counts in this paper are discussed by Blain & Longair (in preparation). The upper limit to the background intensity in the submillimetre waveband, determined by the FIRAS instrument on board the COBE satellite (Mather et al. 1994), imposes the more severe constraint on models of galaxy evolution than the requirement that the density parameter of heavy elements Om < - 3 at the present epoch. 3.3 The effects of source evolution The effects of galaxy evolution can be included in predictions of the source count by using a redshift-dependent luminosity function <I>(L, z) in equation (3). Evolution can be described by a combination of 'luminosity evolution' and 'density evolution'. Positive density evolution increases the number of sources by scaling the luminosity function, while positive luminosity evolution increases the luminosity of each source, shifting the whole function to higher luminosities. A common convention uses the functions n (z) and g(z), which describe density and luminosity evolution respectively, in order to modify the logarithmic luminosity function at zero redshift 'Po into the logarithmic luminosity function at redshift z through the relation 'P(L, z)=n(z)'po [J' g(z) (e.g. Condon 1984). 'II and 'II give the comoving space densities of sources per unit logarithmic luminosity interval to the base m, and are related to the luminosity function <I> in equation (3) by the expression, 1 'II(L) d(logml) = <I>(L) dl. Lin (m). In this paper, both n(z) and g(z) are described by parametric models of the form, (1 +zy, e(z) = { (1 +zmaxy, if z::::;;zmax if Zmax <z::::;;zo, (5) (6) (7)

4 85 A. W. Blain and M. S. Longair in which p describes the strength of evolution, Zmax designates the redshift at which evolution ceases, and Zo is the absolute maximum redshift of sources, beyond which the luminosity function is zero. If an identical form of the function e(z) is used to describe either density or luminosity evolution, then the total density of havy elements formed in star-forming galaxies is the same. Fig. 1 illustrates the effects of source evolution on the counts predicted at two different wavelengths in the submillimetre waveband using the lras luminosity function of Saunders et al. (199), and a fixed dust temperature of 6 K. The four models are: a non-evolving model, two models representing different forms of density evolution, and a model of luminosity evolution. The slopes of all the counts in Fig. 1 exceed the Euclidean prediction N ocs ;3/2 at flux densities less than mjy, and the onset of the enhancement is shifted to very much higher flux densities by evolution. Luminosity evolution produces the onset of the enhancement at considerably higher flux densities than density evolution, and produces greater counts at all but the lowest flux densities, at which the counts begin to saturate. The effects of evolution, especially the effects of luminosity evolution, are very pronounced in the submillimetre waveband because of the flat flux density-redshift relation, which allows very distant objects at z '" 5 - to be detected almost as easily as sources atz '" 1 (Blain & Longair 1993a). Only low-redshift galaxies and distant galaxies on the steep high-luminosity slope of the luminosity function contribute to the counts at high flux densities. The sharp enhancement of the counts occurs when large numbers of sources at redshifts greater than unity begin to contribute, at flux densities corresponding to those of distant galaxies with luminosities around the knee in the luminosity function. Positive luminosity evolution increases the luminosity at which the knee in the luminosity function occurs, also shifting the onset of the enhanced counts to higher flux densities. Density evolution simply increases the number of sources on the high-luminosity slope, producing a less dramatic effect. Near saturation the counts are higher in a model of density evolution, 6,, Solid line - i laoi'm, 5 '" I Dashed line - 45l'm - "... bd '" " '- <l u " "....1 rn " -" bo " -5 :E Limiting Flux Density / Jy Figure 1. Source counts expected from lras galaxies containing dust grains at 6 K in four evolution models. The counts at mjy increase from a non-evolving model, through two models of density evolution of the form (1 + Z)3 and (1 + Z)6, to a model of luminosity evoloution ofthe form (1 +Z)3, zo=5 and zmax=2 (equation 7). because almost all the luminous dusty galaxies in the universe are then detectable, and density evolution predicts the largest total number of sources. The predicted non-euclidean nature ofthe counts at faint flux densities in the submillimetre waveband provides a powerful motivation for a blank-field survey. A large fraction of the sources included in the enhanced regions should lie at moderate and high redshifts. 4 SOURCE COUNTS AND MODELS OF GALAXY FORMATION The source counts required to investigate the feasibility of surveys in the submillimetre waveband have been derived for two different schemes of galaxy formation. In one scheme the population of far-infrared luminous galaxies is described by a population of lras galaxies undergoing luminosity evolution and in the other by a population of merging galaxies based on a model of hierarchical clustering. We will refer to the first type of model as an 'IRASbased model', which can be thought of as representing the development of structure by the fragmentation of unstable supergalactic structures that form at redshifts of about 5. This form of structure evolution would be expected if the universal density field was dominated by hot dark matter (e.g. Zeldovich 1984). Fragmentation would lead to an initial burst of intense star formation activity followed by a decaying star formation rate, which should be naturally represented by luminosity evolution in a population ofiras galaxies (e.g. Bruzual 1983). The hierarchical model describes the development of structure by the progressive merger of dark matter clouds, whose average mass increases with cosmic epoch. This form of structure evolution would be expected if the density field was dominated by cold dark matter (e.g. Efstathiou 199). The merger rate in this model is derived by adopting the simple model of hierarchical clustering described by Blain & Longair (1993a,b). 4.1 Counts from evolving [BAS galaxies In Fig. 1 we derived source counts in the submillimetre waveband using equations (3) and (4) and thelras luminosity function. In the computations which follow, we adopt a dust temperature-luminosity relation which describes the observed range of dust temperatures in lras galaxies reasonably well. Dusty star-forming galaxies'are assumed to contain grains at a single temperature T related to the source luminosity L by ( ( L ).8 - =6. -. K L The resulting source counts are shown in Fig. 2 at each of the four SCUBA filter wavelengths, and for the five different models of luminosity evolution, listed in Table 1. Model 1 describes a population of non-evolving sources, and model 2 describes a population which includes the strongest form of smooth evolution permitted by the FIRAS limit. Models 3, 4 and 5 describe models in which evolution follows the (1 +Z)3 form that is required to account for the observed counts of both the lras galaxies at 6!lm (Oliver, Rowan-Robinson & Saunders 1992) and of powerful radio-' (8)

5 (a) Solid line - 45,um N I Dashed line - 35llID on "d "-..., u rn " on :9 5 '.1 (b) o :a " z <l Submillimetre surveys 851 Solid line - 45ILffi Dashed line - 35ILffi -3 -' -3.1 Limiting Flux Density / Jy Flux Density / Jy. (C) Solid line - 85,um N I Dashed line - 75,um 5 ' (d) Solid line - 85,um Dashed line - 75J1ID on "d "- " u Ul on : ',, Limiting Flux Density / Jy Flux Density / Jy Figure 2. Integral and differential counts derived from five models of evolving lras galaxy populations (Table 1) in all four SCUBA observing bands; (a) and (b) at 35 and 45 11m, (c) and (d) at 75 and 85 11m. In (a) and (b) the counts increase in the order of models 1, 2,5,4 and 3 at mjy in both bands. At the same flux density in (c) and (d) the counts at 75 11m increase in the order of models 1,5,2,4 and 3, and in the order 1, 5, 4, 2 and 3 at 85 11m. Table 1. Evolution parameters and densities of heavy elements in five lras-based models (1 to 5; equation 7) used to predict source counts in Figs 2 and 3 in the case of luminosity evolution, and an hierarchical model (H) used to predict source counts in Fig. 5. Model p Zmax Zo Metal density Om X X -' X -' X -' X -' X ' / Gyr Zo H 4 X X -' galaxies and quasars (Dunlop & Peacock 1991). Model 3 describes the form of evolution of the radio galaxies and quasars over the complete range of redshifts, and predicts a background radiation intensity in excess of the FIRAS limit. In models 4 and 5 this evolution function is modified at moderate and large redshifts to ensure that the FlRAS limit is not exceeded. In model 4 this is achieved by including a sharp cut-off in the evolution function at zo=1.7, while in model 5 active evolution stops at the lower redshift zmax=o.6, with no further change of the evolution function to Zo = 5. The density of heavy elements produced by nucleosynthesis up to the present epoch in these models is also given in Table Counts from the hierarchical model We have developed a semi-analytic model of hierarchical structure formation based on the Press-Schechter formalism (Press & Schechter 1974) which yields a rate of formation of galaxies with mass M at redshift Z of Nform(M, z) per unit comoving volume per unit galaxy mass. N form describes the development of structure in the Universe by the merger of galaxies and gas clouds of lower masses (Blain & Longair 1993a, b). Equations (3) and (4) cannot be used directly to predict source counts in this model, because the source population is described by a function of galaxy mass and not luminosity, and a different formulation is required. We assume that star formation activity and powerful dust emission are associated with actively merging galaxies and gas clouds, so that the merger rate Nform(M, z) can be combined with a model of the luminosity of galaxies during mergers to give a source count. This model requires two

6 852 A. W Blain and M. S. Longair parameters: (J, which determines the duration of the starbursts associated with merging objects; andx, the efficiency with which the mass of merging clouds is processed into high-mass stars. The time-evolution of the bolometric luminosity of a merging cloud of mass M around the merger epoch t merge is therefore taken to be [.7c 2 xm], L(t) = { 2(J, if It - tmerge I::; (J; otherwise. The term in square brackets gives the total energy emitted during the merger-induced starburst, which is correctly normalized if x is the fraction of the total mass of the merging cloud that is converted from hydrogen into heavy elements by nucleosynthesis (Blain & Longair 1993a). In a particular hierarchical model, the parameters x and (J are set to match the values of independent quantities. x is fixed in order to normalize the density of heavy elements at the present epoch (see Blain & Longair 1995), and (J is then chosen to fix the bolometric luminosity of a merging galaxy with the typical mass of an object in the Press-Schechter formalism M* at low redshift to the typical luminosity of a luminous lras galaxy, using equation (9). The bolometric luminosity of a merging cloud from equation (9) can be combined with a model of the spectral energy distribution of dust radiation!v to find the flux density of radiation from a merging cloud at redshift z. If It - t merge I ::; (J, 1 {1 }f, S = -.7c 2 xm v 41tD\z)(1 +z) 2(J [ ] Sfv' dv' (9) () In order to derive a source count in the hierarchical model, equation () can be used to find the minimum mass of a merging cloud M min(s v) that produces a flux density exceeding S v at any redshift. It may be necessary to use an iterative method to solve equation () because the dust temperature, implicitly included in!v> depends on the source luminosity (equation 8) and therefore on M (equation 9). A source count at flux density S v can then be produced by integrating the comoving space density of luminous merging objects, that is 2(JN form(m, z) per unit mass, over all redshifts and all masses exceeding M min(s v). The count takes the form of (J normalizes the hierarchical counts at high flux densities to the same number as the lras -based counts in Fig. 2, and gives a luminosity-to-mass ratio of merging clouds of -1.1 Lo Mol (equation 9), a reasonable value considering that the mass of a galaxy in this formalism includes the mass of dark matter driving the hierarchical clustering and that during this phase the galaxy is undergoing a very large starburst. The source counts illustrated in Fig. 3 are much steeper than those that we published earlier (Blain & Longair 1993a), and steeper than any of the models shown in Fig. 2. Several factors combine to produce these remarkable counts, the flux density-redshift relation for sources in the submillimetre waveband being crucial for explaining the effect. The steepness of a source count is very sensitive to the slope of the flux density-redshift relation at redshifts between 1 and. A flat slope indicates a strong negative K correction and results in a very sharp increase of the cosmological volume within which sources are observable as the flux density limit of the survey decreases, producing a very steep count. Fig. 4 shows four flux density-redshift relations for merging sources whose masses, M =M*, are character- (a) (b) '-. Z <l ' -3.1 Limiting Flux Density / Jy 6 5 la' loa.1-3 -' dr XD2(Z) -dz. dz (11) Counts derived using the hierarchical model are presented in Fig. 3, in the same form as the lras -based counts of Fig. 2. The values of the parameters x and (J and the associated metal density are listed in Table 1 (model H). The chosen value of x corresponds to a background radiation intensity that lies close to the FIRAS limit in the submillimetre waveband and a metal density safely below the limit am < - 3 (Blain & Longair 1995). The chosen value.1 Flux Density / Jy Figure 3. Source counts corresponding to a hierarchical clustering model of galaxy formation (equation 11) in the SCUBA observing bands at 85, 75, 45 and 35 11m, in order of increasing counts at mjy. Integral counts are presented in (a), differential counts in (b). There is a very sharp enhancement of the counts in the longwavelength filters at low flux densities.

7 i "- ; t- _ 5 ' 6 Solid lines: 85 J.lm Dashed lines: 45 /-tm Thin lines: Varying temperature Bold lines: Fixed temperature OL- --W c:i (a), " <- on '" '-. 1:i u " <- en '"... '" i Solid line: hierarchical model Submillimetre surveys Dolled lines: lras models. rising in the order of models 1, 2, 5, 4 and 3 at mjy Limiting Flux Density I Jy '.1-3 Redshift Figure 4. Flux density-redshift relations expected for typical merging sources at the frequencies of the principal SCUBA bands. The mass and bolo metric luminosity of the sources falls with redshift, while the K-correction boosts their flux densities at redshifts - 1 to 5. The effect of including a luminosity-dependent dust temperature is shown. (b) ' ' istic of Press-Schechter clouds at each redshift (Blain & Longair 1993a), the type of sources which dominate the counts of Fig. 3. The flux density-redshift relations are shown at frequencies of 45 and 85 Ilm for each of two models of the spectral energy distribution of the sources: one in which the dust temperature of the sources is fixed, and another in which the dust temperature is determined from the temperature-luminosity relation (equation 8). Since the masses and luminosities of the merger galaxies are smaller at large redshifts, the temperatures of the galaxies decrease systematically with increasing redshift. The relations at 85 J.lffi are considerably flatter than those at 45 Ilm, leading to steeper counts in Fig. 3. The relations which include a dust temperature-luminosity relation are flatter than those derived with a fixed temperature, explaining why the counts in Fig. 3 are steeper than those in our earlier paper. The steepening effect of varying the dust temperature on the counts is more pronounced at 85 than at 45 Ilm because the flux density-redshift relation is flatter at this frequency even with a fixed temperature, and so the relatively small effect of changing the dust temperature with redshift exerts a strong influence on the steepness of the resulting counts. 4.3 Comparing the counts in each model The counts expected in the hierarchical model and the five lras -based models listed in Table 1 are compared in the principal SCUBA filter bands at 45 and 85 J.lffi in Figs 5(a) and (b) respectively. At 45 J.lffi the predicted counts in the hierarchical model fall below those in the evolving lrasbased models at flux densities greater than about 1 mjy, rising quickly to match and exceed the counts in models 2, 4 and 5 at lower flux densities. At 85 Ilm the hierarchical counts follow closely the predictions from non-evolving lras galaxies at flux densities greater than about 1 mjy, with a very sharp rise at fainter flux densities. In general, the Solid line: hierarchical model Dolled lines: lras models, rising in the order of models 1, 5, 4, 2 and 3 at mjy.1 Limiting Flux Density I Jy Figure 5. A comparison of integral source counts from hierarchical and lras-based models, shown by solid and dotted lines respectively, at 45 J.lIll (a) and 85 J.lIll (b). counts based on the evolving lras luminosity function are greater than those in the hierarchical models unless very faint flux densities are reached. It is interesting to compare the counts at 6 Ilm predicted by the forms of evolution in Table 1 with the lras observations (Oliver, Rowan-Robinson & Saunders 1992 and references therein). These differential counts are presented in Fig. 6 ih the form of Oliver et al.'s fig. 1, for all six models. The sources contributing to these counts have mean redshifts less than, and so are only affected by the evolution occurring at low redshifts. The forms of density and luminosity evolution of the lras galaxies, which are well known to account for the observations, are shown by A and B, while the other models are seen to predict significantly fewer faint 6-J.lffi sources than observed. Hence, models 3, 4 and 5 in Table 1 are consistent with the observed lras counts, while models 1, 2 and H are not. Model 2 is. used because its evolution function is described by a single smooth function across the whole redshift range of interest, and produces submillimetre source counts that are always within a factor of 3, and normally much less, of those predicted by models 4 and 5 in the most important observing band at 85 J.lffi. Despite not reproducing the 6-llm counts, the hierarchical model (H) provides a very simple way of modelling a cold-dark-matter-

8 854 A. W Blain and M. S. Longair A 2 f (Non-evolving) o.5 loglo(flux Density / Jy) Figure 6. Observed IRAS counts at 6 m (Oliver, Rowan-Robinson & Saunders 1992), compared to counts predicted by the models above. The counts A and B include density evolution of the form (1 +Z)6, and luminosity evolution ofthe form (1 +Z)3 (models 3, 4 and 5 in Table 1) respectively. The counts predicted by models 1, 2 and H are labelled independently. type scheme of galaxy formation in our calculations. We therefore use these models to investigate observing strategies in the submillimetre waveband, and will investigate the 6-f.UIl counts in a future paper. Thble 2. Estimates of the limiting depth and the number of sources expected at a signal-to-noise ratio of 3 in 2 SCUBA surveys: one lasting 24 h and covering deg 2, and the other reaching a flux density limit 5 times deeper in 8 hover.13 deg 2 The results expected are calculated for the five IRASbased models and the hierarchical model in Table 1 (Figs 2 and 3). Filter / NEFD/ Smin/ Model Number of pm Jy &-1/2 mjy (Table 1) sources Ns ,2,3.2,15,4 4,5,H,5, ,2,3 1.3,68,64 4,5,H 43,36, ,2,3.4,3, SOO 4,5,H 3,13, ,2,3 3,,8 4,5,H 8,7, ,2,3.2,.5,25 4,5, H 3,.5, ,2,3.4,4, 4,5, H 2,5, to-i 16 1,2,3.6,1.5,5 4,5,H 13,3, ,2,3 1.6,8,13 4,5,H 44,16,8 5 SURVEY STRATEGIES FOR lras BASED MODELS Relations between the area of duration of a submillimetre waveband survey and its depth and between the underlying population of sources and the expected number of detections were given by equations (1) and (2). The source counts derived above (Figs 2 and 3) now allow us to use these expressions to investigate the most promising survey strategy in order to maximize the probability of detecting galaxies at large redshifts and successfully discriminating between different models of galaxy formation. We will discuss the choice of observing frequency, investigate the limits imposed on the area of a survey and then seek the optimal choice of this area for different underlying populations of sources and integration times. 5.1 The choice of frequency The numbers of sources expected in two SCUBA surveys differing by a factor of 5 in depth are compared in Table 2 using equations (1) and (2) in each of the SCUBA filter bands and in the five lras -based models and the hierarchical model described in Table 1. The NEFDs introduced in Section 2.1 are assumed, and only sources detected at a signal-to-noise ratio greater than three are included. The areaa. in equation (1) is assumed to be one-third of the total area subtended by the 2.3-arcmin SCUBA array, a value ofaa4 x - 4 deg 2, because the telescope has to be repainted twice in order to cover the interstices of the detector array and produce a fully sampled image. Table 2 shows that in all five lras models, the lower NEFD values at the longer wavelengths more than compen- sate for the smaller source counts expected and therefore that the prospects for executing a successful survey are best with the 75/85-f.UIl array. In the case of the hierarchical model, a useful number of detections is only predicted for the deeper survey, in which the 75/85-f.UIl array also predicts the best results. Better atmospheric transmission at 75/85 f.uil than at 35/45 f.uil should also be taken into account when deciding on the array for which survey observations should be designed. The table shows that in model 2, which predicts the most conservative count at 85 f,lm for an evolving population of lras galaxies, about 15 and 7 sources should be detected in the shallower and deeper surveys respectively, which produce Nyquist sampled images containing about 4 and 1.3 x 3 pixel. Many more sources should appear on the image at a lower significance level allowing information to be extracted from a confusion analysis of the noise (see, for example Barcons 1992; Franceschini et al. 1989). Observations in the 75-f.UIl band are expected to produce the best results; however, the SCUBA optics are matched to the detectors at 85 f.uil and the NEFD value at 85 f.uil will actually be better than at 75 f,lm, reducing the apparent advantage of observing at the higher frequency. For this reason, and in order to produce conservative estimates of the results, further detailed predictions of survey performance will be made for the 85-f.UIl band. 5.2 Constraints on the survey duration and area The integration time t and field area A are the key parameters for which the best strategy for a blank-field survey in the submillimetre waveband must be evaluated, and the ranges of usable values are limited by external factors. Most

9 "'io-2 Jy 1 Jy.1 Survey area / degree 2 Figure 7. A plot showing contours ofthe sensitivities reached (dotted lines) and the number of sources Ns expected at R;:c.3 (solid lines) in a SCUBA survey at a wavelength of 85 m. Contours of Ns are shown for both a Euclidean count (E) and for a count based on non-evolving lras galaxies (I). Regions of the diagram containing unrealistic survey parameters are lightly hatched. importantly, t cannot reasonably exceed about 3 x 5 s because of the pressure for telescope time. A longer integration should always predict better results, and so A must be chosen to maximize the number of detected sources, subject to some limit on t. The minimum possible values of A are restricted by the effects of source confusion owing both to the extragalactic point sources, which are sought in the programme, and to the structure in the diffuse background radiation intensity from galactic infrared cirrus clouds. The limits imposed by galactic cirrus confusion can be estimated from lras maps (Gautier et al. 1992). However, because the intensity of the galactic signal falls at longer wavelengths and on finer angular scales, observations with the 15-m JCMT at 85 Jlm should only be affected at sensitivities better than.2 mjy. Fig. 7 shows that only a very long integration in a very small field could achieve this sensitivity. Confusion owing to unresolved point sources should be dominated by sources with a density of about 1 per beam on the sky. At 85 Jlm, SCUBA has 37 resolution cells in each 1.2 x - 3 deg 2 field and so this condition is satisfied for a source count of about 3.1 x 4 deg- 2 Fig. 5(b) shows that for both plausible lras -based models and for the hierarchical model, source confusion should be associated with a flux density of about -.2 mjy. Franceschini et al. (1989) use a detailed model to estimate that source confusion is important at a flux density of about mjy. Neither galactic cirrus confusion nor point-source confusion noise should therefore introduce severe lower limits to A. At 45 Jlm cirrus confusion becomes important at flux densities of about a factor of 2 higher, while point-source confusion noise remains at about the same level. However, because the flux density limit at 45 Jlm is an order of magnitude higher than at 85 Jlm, confusion is less important. However, A must be large enough to include enough beam areas to ensure that the final survey image contains many more pixels than the number of extragalactic sources detected. In the 85-Jlm band the beam area is about 3 x - 5 deg 2, and at least three separate pointings are required for calibration, indicating a Submillimetre surveys 855 lower limit toa of about 4 x - 3 deg 2, corresponding to a Nyquist sampled map containing about 33 pixel. A larger area would be preferable if a reasonable number of 5 detected sources were sought, resulting in a rather flexible lower limit to A of about - 2 deg 2 Survey parameters which conflict with the results imposed by source confusion and array area lie in the lightly shaded areas of Figs 7, 8, 9 anadilipper limit to A is imposed by the desire to avoid fields containing bright point sources or strong cirrus emission. In a survey field away from the galactic plane these restrictions are easy to satisfy over areas of several square degrees, scales on which a survey would include 3 SCUBA fields, but an area over which secure calibration of the images would be difficult. The constraints imposed on the survey area by external factors thus require.1 A 1 deg 2 The lower end of this range is much preferred for obtaining uniform calibration of the images. 5.3 The choice of survey area Fig. 7 demonstrates the dependence both of the number of expected detections N s and of the sensitivity S min obtained in a survey, in the form of a contour plot for wide ranges of A and t, which include all reasonable values. The contours are drawn assuming the same instrumental parameters used in Table 2. This figure illustrates the results of equations (1) and (2), that S min and N s should depend on A -1/ 2 t 1 / 2 and A 1I4t 3/4 respectively in a Euclidean count model. In this case, a shallow large-area survey is clearly most efficient at detecting significant numbers of sources. However, including a non-euclidean source count in the calculations dramatically increases the effectiveness of a deep survey. At flux densities less than about mjy, the Ns contours of the nonevolving lras -based model separate from those for the Euclidean model, because the lras -based counts steepen sharply, to form a roughly s;- -26 (Fig. 2), while the Euclidean form remains S ;3/2. Equations (1) and (2) then predict N s cx:a t -1.3 in the lras -based model, implying that surveys covering the smallest fields should detect the largest number of sources in equal times. Fig. 7 demonstrates the key fact that SCUBA will provide sufficient sensitivity to exploit the unique enhanced faint counts in the submillimetre waveband. Even without source evolution, the N s = 1 contour labelled'!, on Fig. 7 can be reached in an integration time of about 5 s (28 h) in a small survey field with A =lsx - 2 deg 2 The survey area should be chosen to avoid the shaded region in Fig. 7, in which non-gaussian confusion noise owing to galactic cirrus and faint sources is expected to dominate the instrumental noise. An additional source of non-gaussian noise would increase the minimum flux density of a source visible in the survey, moving the boundary of the forbidden region down and to the right in Fig. 7. In this case, and if the limiting flux density attainable in the survey became larger than the flux density expected from a typical distant source, the optimum choice of survey area for a fixed integration time would be affected. The best strategy would then be to integrate down to reach the flux density limit over the largest area possible. A similar contour plot comparing the effects of the three 1996 RAS, MNRAS 279,

10 856 A. W. Blain and M. S. Longair lras -based count models of Table 1 is shown in Fig. 8. The number of detections predicted increases with the strength of evolution, confirming that significant numbers of sources can be detected in a practical survey covering deg 2 in a 24-h integration, assuming a modest amount of lras galaxy evolution. The slopes of the dashed contours flatten out at the smallest areas in Fig. 8, because the underlying counts begin to saturate at the corresponding flux density levels (Fig. 2). This effect is only significant in impractically small survey fields and does not affect our assessment of the optimal observing strategy. "- ad <1 'E l:. 7 " Summary The highest possible sensitivity is required for a successful survey, which is confirmed by the contours in Fig. 9, comparing the number of detections expected, assuming a source count from a modestly evolving lras -based model in each of the SCUBA filter bands. The best strategy is to undertake a deep survey covering a small field at a low frequency, in order to exploit the excellent NEFD values expected for the low-frequency array. A survey taking more than 5 hand covering '" deg 2 at 85 m in a region of the sky with low galactic cirrus emission should offer the best prospects for the detection of these source populations. It should be noted that the values expected for numbers of detected sources quoted in this section depend upon the final sensitivity achieved by SCUBA; however, the principles underlying the optimal choice of survey parameters would not be affected. 6 INVESTIGATING GALAXY FORMATION Fig. shows the number of detections expected in a survey at 85 m based on counts from three lras -based models (2, 4 and 5; Table 1), and from the hierarchical clustering model (H; Table 1). The contours confirm the suggestion of the counts in Fig. 5(b) that the lras -based model would predict more detections, until the source count in the hierarchical model rises very rapidly near the closely packed contours at A '".2 deg 2 and t '" 3 X 5 s. There is therefore a choice to be made in observing strategy. For a given amount of observing time, say, 3 x 5 s, surveying an area of -1 deg 2 would result in the detection of between about and 6 sources with flux densities exceeding between 1 and 3 mly, if one of the plausible models of lras galaxy evolution is correct. On the other hand, if a small area of, say,.1 deg 2 were surveyed, a lower flux density limit of about.3 mly would be achieved and the possibility of detecting directly the extremely steep source counts expected in the hierarchical clustering model might be realized. The discovery of such a steep source count, complemented by statistical analyses to the confusion limit of the survey, would be a definite signature of continuing star formation through the redshift interval 1::;; z ::;;. Hence, the number of sources detected in a submillimetre waveband survey should readily discriminate between a hierarchical, cold-dark-matter-type model and a hot-dark-matter-type model in which source luminosity increases with redshift. The redshift distributions of detected sources can be used to discriminate between these models more effectively, by comparing the resulting distributions with the predictions of Survey area / Ciegree 2 Figure 8. A contour plot of the number of sources Ns expected in a SCUBA survey field at 85 11m with R;:c. 3. Values of Ns = 1 and are shown by solid and dashed lines respectively for each of the lras-based models in Table 1 (1, 2 and 3) and the Euclidean model (E). Following Fig. 7 the sensitivities reached are also shown by dotted lines. 85 J1-m :,. '" ; J1-m Oul,O-1OL,lu.L-lLO Survey area / degree 2 Figure 9. Contours of the numbers of sources Ns expected with Ra:c. 3 in all four SCUBA filter bands in lras model 2 (Table 1). Ns values of 1 and are represented by solid and dashed lines respectively. Survey area / degree2 Figure. A contour plot of the numbers of sources expected in a SCUBA survey at 85 11m in both evolving lras models 2, 4 and 5 (dashed lines, in order of decreasing line weight) and the hierarchical model (solid lines). Three labelled contour levels are presented for each model.

11 Submillimetre SU1VeyS 857 h..d S z.1.1 Redshift Figure 11. Redshift distributions of sources expected in a 24-h.I-deg 2 SCUBA survey at 85 11m in the five IRAS-based models and the hierarchical clustering model in Table 1. The survey limit at Ra = 3 is 1.6 mjy. the integrands of equations (3) and (11), examples of which are shown in Fig. 11 for the six models in Table 1. The forms of the redshift distributions are similar in all three lrasbased models, which converge at low redshifts where evolution is insignificant. The hierarchical distribution does not converge to the same value because the hierarchicalluminosity function is not identical to that of the lras galaxies. The form of the distributions predicted by the hierarchical and lras-based models are markedly different, most significantly because the hierarchical distribution contains a clear redshift cut-off, while the distributions in all the lrasbased models extend to much higher redshifts, with a double-peaked form. The cut-off in the hierarchical model is due to the decline in the average luminosity of galaxies with increasing redshift, which counteracts the effect of the K-correction in the submillimetre waveband at the flux density limit in Fig. 11. In lras-based models there is no such effect, and sources at large redshifts should be readily detectable in a sensitive submillimetre survey. The distributions change as the sensitivity of the survey is improved, as shown by Fig. 12. In the lras -based model, the redshift distributions increase uniformly, maintaining their characteristic form. However, the extremely steep source counts predicted by the hierarchical model lead to a dramatic increase in the typical and maximum redshifts of detected sources as sensitivity improves in Fig. 12(b). If sufficiently high sensitivities can be obtained using SCUBA, then this effect would provide an extremely powerful test of hierarchical models of galaxy evolution. The differences in the calculated redshift distributions suggest that the redshifts of even a small number of sources detected in a submillimetre survey could be used to discriminate between the hierarchical and lras -based models. 7 CONCLUSION The advent of submillimetre detector arrays such as SCUBA offers the potential to search for distant galaxies in a waveband in which only the most luminous galaxies have been investigated to date. A blank-field survey could reach.1 Redshift N, 1: OJ).",, x.", e "- h a Ul a h Q) -3.D S Z.1 Redshift.1-4 Figure 12. The effects of survey depth Smin on the redshift distributions of sources expected in a submillimetre survey at 85 11m in both an IR4S-based model (a) and an hierarchical model (b). The redshift distribution curves corespond to Smin values of 1.6, 1.,.8,.5 and.2 mjy, in order of increasing source density. The form of evolution assumed in the IR4S model corresponds to model 2 in Table 1. directly to large redshifts and provide new insights into the processes involved in the formation and evolution of galaxies, offering excellent prospects for discriminating between different models of galaxy formation. We have investigated the factors affecting the optimal choice of observing strategy, and found that the depth of a survey is the most important, as a survey must exploit the strongly enhanced source counts in the submillimetre waveband. Contrary to what might have been intuitively expected, there are great advantages in surveying small fields to very low sensitivity limits. Our models illustrate what might be found in two quite different models of galaxy formation and evolution. In both cases, it seems that there are excellent prospects for detecting cosmologically distributed evolving galaxies. Given an observing time of 3 x 5 s, the best strategy for detecting an evolving lras population of sources would be to survey an area of about deg 2 in the expectation of detecting about 5 sources. To detect the extremely steep source count expected in the hierarchical clustering picture, it is preferable to survey a small area to very high sensitivity. For the same observing time, surveying an area of.1 deg 2 would result in the detection of about