Abstract

We present a quantitative analysis of the astrophysical and cosmological information that can be extracted from the many important wide-area, shallow surveys that will be carried out in the next few years. Our calculations combine the predictions of the physical model by Granato et al. for the formation and evolution of spheroidal galaxies with up-to-date phenomenological models for the evolution of starburst and normal late-type galaxies and of radio sources. We compute the expected number counts and the redshift distributions of these source populations separately and then focus on protospheroidal galaxies. For the latter objects, we predict the counts and redshift distributions of strongly lensed sources at 250, 350, 500 and 850 μm, the angular correlation function of sources detected in the surveys considered, and the angular power spectra due to clustering of sources below the detection limit in Herschel and Planck surveys. An optimal survey for selecting strongly lensed protospheroidal galaxies is described, and it is shown how they can be easily distinguished from the other source populations. We also discuss the detectability of the imprints of the one-halo and two-halo regimes on angular correlation functions and clustering power spectra, as well as the constraints on cosmological parameters that can be obtained from the determinations of these quantities. The novel data relevant to derive the first submillimetre estimates of the local luminosity functions of starburst and late-type galaxies, and the constraints on the properties of rare source populations, such as blazars, are also briefly described.

1 INTRODUCTION

So far, surveys at submillimetre (submm) wavelengths have covered only tiny fractions of the sky (less than 1 deg2) down to mJy levels (Mortier et al. 2005, and references therein). Forthcoming experiments, however, will enormously extend the sky coverage. The Legacy Surveys with SCUBA-21 include a Shallow Survey (SASSy) of ∼20 000 deg2 to a depth of 150 mJy (5σ) and a Wide-Area Extragalactic Survey2 of 5 deg2 to a 5σ flux limit of 2 mJy, both at 850 μm. The SPIRE instrument on ESA's Herschel satellite may be able to carry out a shallow survey of ∼400 deg2 to a 100 mJy level (5σ) in all three bands (250, 350 and 500 μm; Lagache, Dole & Puget 2003; Harwit 2004). The HFI instrument (Lamarre et al. 2003) on the Planck satellite will carry out all sky surveys in 6 bands, centred at 3000, 2100, 1380, 850, 550 and 350 μm; these surveys will be confusion limited to flux levels ranging from a few hundred mJy to ≃1.5 Jy (Vielva et al. 2003; Negrello et al. 2004; López-Caniego et al. 2006).

As first pointed out by Blain (1996), the submm region is exceptionally well suited to exploit strong gravitational lensing to detect large complete samples of high-redshift dusty galaxies. The reason resides in the combined effect of the large and negative K-correction, that makes the observed flux density of sources of given luminosity only weakly dependent on distance over a broad redshift range above z≳ 1 (Blain & Longair 1993), and of very steep evolution. These two ingredients cooperate in yielding large lensing optical depths and extremely steep counts, resulting in a strong magnification bias.

Strong lensing is a powerful tool for many interesting applications (see Kochanek, Schneider & Wambsganss 2004 for a comprehensive review). The statistics of lenses is a test of the cosmological model because the optical depth is roughly proportional to the volume to the source. If the redshifts of both the source and the lens are measured, the ratio between the lens–source and observer–source angular diameter distances, dls/dos, constitutes a sensitive probe of the geometry of the Universe (and in particular of ΩΛ, with weak dependence on Ωm) through the scaling of light ray deflections with the source redshift. To date, in the best-studied cases, the lens is a low-redshift cluster, such as A1689 or A2218, and this test is difficult to apply because, for high-redshift sources, dls/dos is always ≃1 and the effect of different cosmologies is very small. On the contrary, in the case of the submm-selected sources, the lenses are expected to be generally at substantial redshifts. The distribution of lens redshifts provides a further, potentially powerful, cosmological test since, in general, cosmologies with a large cosmological constant predict significantly higher lens redshifts than those without. Recent analyses of lens statistics have focused on the CLASS survey (see e.g. Mitchell et al. 2005) to derive constraints on cosmological parameters. Attempts to use the lens statistics to constrain dark energy have been worked out by Chae et al. (2004) and Kuhlen, Keeton & Madau (2004).

On the other hand, the probability that a source has an intervening lens also depends on the distribution of the lens galaxies, and the cross-section of the lens for producing a given effect depends on its velocity dispersion. Thus, if a cosmological model is adopted, the lens statistics is informative on the evolution of galaxy properties, and in particular of dark matter structures. Note that gravitational lenses are unique in providing a selection based on mass, rather than luminosity, so that the separation distribution of lenses is a direct mapping of the mass function of all haloes, luminous or dark.

Alexander et al. (2003, 2005) have found that a substantial fraction of submm bright galaxies harbour an active galactic nucleus (AGN). The magnification produced by gravitational lensing allows us to study fainter AGN host galaxies than otherwise possible. This will permit a better understanding of the growth of super-massive black holes and their relationships with their host galaxies.

Another important driver of large-area submm surveys is the investigation of the clustering properties of high-z galaxies, both in the one-halo and in the two-halo regimes, that can be performed more effectively than in other wavebands, and without being affected by obscuration effects. The large-scale angular correlation function of detected sources and the autocorrelation function of intensity fluctuations provide information on the masses of dark matter haloes, on the duration of the star formation process and on the evolution of the bias factor. They are also sensitive to, and will provide constraints on cosmological parameters.

Furthermore, these large-area surveys will yield complete samples of submm-selected low-z dusty galaxies, allowing, for the first time, direct estimates of the local luminosity functions (LFs). In addition, they may contain a number of extreme luminosity distant galaxies.

Other interesting, although only seldom mentioned, products of large-area submm surveys are complete samples of radio sources, selected in a spectral region almost unexplored so far, but where key information on their physics is expected to show up. For example, observations at these wavelengths may reveal the transition from optically thick to optically thin synchrotron emission in the most-compact regions, allowing the determination of the self-absorption (turnover) frequency which carries information on physical parameters. Also, the slope of the optically thin synchrotron emission steepens at high frequencies due to electron energy losses; the spectral break frequency νb is related to the strength of the magnetic field and to the ‘synchrotron age’ts by νb≃ 320(20 μ G/B)3 (ts/Myr)−2 GHz. Excess far-infrared (far-IR)/submm emission possibly due to dust, is often observed from radio galaxies (Knapp & Patten 1991). These surveys will allow us to assess whether this is a general property of these sources, with interesting implications for the presence of dust in the host galaxies, generally identified with giant ellipticals, usually devoid of interstellar matter.

The dominant radio source population at the relevant frequencies is made of flat-spectrum radio quasars and of BL Lac objects, collectively dubbed ‘blazars’. Multifrequency surveys in the far-IR/submm region will allow us to check, e.g. whether there are systematic differences in the turnover frequencies between BL Lacs and flat-spectrum quasars as expected if the BL Lac emission is angled closer to the line of sight, so that the turnovers are Doppler boosted to higher frequencies.

Correlations between turnover frequency and luminosity, which is also boosted by relativistic beaming effects, may be expected. On the other hand, Fossati et al. (1998) and Ghisellini et al. (1998) found an anti-correlation between the frequency of the synchrotron peak of blazars and their radio luminosity, referred to as ‘the blazar sequence’. The spectral energy distributions (SEDs) of the most-luminous blazars appear to peak at (rest-frame) mm wavelengths, while the peak moves towards optical wavelengths as the luminosity decreases. This sequence, if confirmed, is an important breakthrough in our understanding of the physical processes governing the blazar emission. On the other hand, Antón & Browne (2005) find that most of their low-luminosity flat-spectrum radio sources peak at much lower frequencies than expected from the Fossati et al. (1998) relationship. Also, Nieppola, Tornikoski & Valtaoja (2006) did not find any correlation between luminosity and frequency of the synchrotron peak for their sample of 300 blazars. However, only a limited region of the luminosity–peak frequency plane has been explored to date, mostly due to the dearth of data in the far-IR to mm region, and the samples have been collected from different surveys and are not complete.

In this paper we address all these important aspects. After an overview of the expected counts of extragalactic sources at submm wavelengths (Section 2), we present a comparative study of the potential of these surveys in relation to: the selection of strongly lensed, dusty primordial galaxies (Section 3), the clustering properties of submm sources at high redshifts (Section 4), and the selection of the first complete submm samples of local star-forming galaxies and of radio sources, mainly blazars (Section 5). In Section 6 we summarize and discuss our main conclusions. This investigation may help in guiding us on what to expect and in optimizing the survey programmes.

Throughout this paper we will adopt the ‘concordance model’, that is, a flat ΛCDM cosmology with Ωm= 0.27, Ωb= 0.045, and ΩΛ= 0.73, H0= 72 km s−1 Mpc−1, σ8= 0.8, and an index n= 1.0 for the power spectrum of primordial density fluctuations.

2 OVERVIEW OF THE MODELS AND SUBMILLIMETRE COUNTS

Submillimetre extragalactic sources are a mixed bag of various populations of dusty galaxies and of flat-spectrum radio sources. A summary of the models adopted for each population follows.

2.1 protospheroidal galaxies

Dusty galaxies detected at mJy levels by SCUBA and MAMBO surveys are modelled following Granato et al. (2004), who interpret them as protospheroidal galaxies in the process of forming most of their stars in a gigantic starburst lasting 0.3–1 Gyr (depending on redshift and on the halo mass). This physical model predicts the birth rate of galaxies as a function of cosmic time and the evolution with galactic age of their full SED, from radio to X-ray wavelengths, including the effect of the active nucleus, whose growth is closely tied to the evolution of the host galaxy. It is thus strongly constrained by, and indeed accounts for huge amounts of multifrequency data at different redshifts (Cirasuolo et al. 2005; Silva et al. 2005; Granato et al. 2006; Lapi et al. 2006; Shankar et al. 2006). In particular, it fits the 850-μm counts available at the time (Granato et al. 2004) and the data on the redshift distribution (Fig. 1).

Figure 1

Comparison of the redshift distribution of sources with S850 μm > 5 mJy yielded by the Granato et al. (2004) model (solid histogram) with the observational estimate by Chapman et al. (2005, dot–dashed line). The numbers of sources per bin of Chapman et al. (2005) have been rescaled to a surface density N(S850 μm > 5 mJy) = 620 deg−2 (Scott et al. 2002). As pointed out by Chapman et al. (2005), the dip at z≃ 1.5 and the high-z cut-off in the data are likely due to the radio and optical selections.

Figure 1

Comparison of the redshift distribution of sources with S850 μm > 5 mJy yielded by the Granato et al. (2004) model (solid histogram) with the observational estimate by Chapman et al. (2005, dot–dashed line). The numbers of sources per bin of Chapman et al. (2005) have been rescaled to a surface density N(S850 μm > 5 mJy) = 620 deg−2 (Scott et al. 2002). As pointed out by Chapman et al. (2005), the dip at z≃ 1.5 and the high-z cut-off in the data are likely due to the radio and optical selections.

However, the re-analysis of existing blank-field SCUBA surveys by Scott, Dunlop & Serjeant (2006) and the results of the SCUBA Half-Degree Extragalactic Survey (Coppin et al. 2006) converge in yielding lower counts than previous studies. To match these more recent results, we have scaled the 850-μm fluxes of Granato et al. (2004) by a factor of 0.68, thus preserving the agreement with the shape of the redshift distribution.

Most recently, Aretxaga et al. (2007) compared the redshift distribution of the SCUBA Half Degree Extragalactic Survey (SHADES), derived from rest-frame radio–mm–far-IR colours, with those yielded by the four different galaxy formation models studied by van Kampen et al. (2005) and with the semi-analytic models by Baugh et al. (2005) and by Granato et al. (2004). Only the latter model turned out to be compatible, within the uncertainties, with the data, although if sources detected only at 850 μm are introduced in the redshift probability distribution with other priors, two of the models considered by van Kampen et al. (2005) can come closer to acceptable.

2.2 Starburst and normal disc galaxies

The Granato et al. (2004) model is meant to take into account the star formation occurring within galactic dark matter haloes virialized at zvir≳ 1.5 and bigger than Mvir≃ 1011.6 M, which are, crudely, associated to spheroidal galaxies. We envisage discs (and irregulars) as associated primarily to haloes virializing at zvir≲ 1.5, which have incorporated, through merging processes, a large fraction of haloes less massive than 4 × 1011 M virializing at earlier times, which may become the bulges of late type galaxies.

The contributions of the latter galaxies to the number counts are estimated following, as usual, a phenomenological approach, which consists in simple analytic recipes to evolve their local LFs, and appropriate templates for their SEDs to compute K-corrections and to extrapolate the models to different wavelengths where direct determinations of the local LFs are not available. We have adopted the 60-μm local LFs by Saunders et al. (1990), that have been derived for ‘warm’ and ‘cold’ galaxies (based on IRAS colours) that we associate, respectively, to starburst and spiral galaxies. Transformations to longer wavelengths were done using the mean flux ratios determined by Serjeant & Harrison (2005) who extrapolated the IRAS detections using the submm colour temperature relations from the SCUBA Local Universe Survey (Dunne et al. 2000; Dunne & Eales 2001).

Rather detailed information on dusty galaxies at redshifts ≤1.5 came from the ISOCAM cosmological surveys performed with the LW3 filter, centred at 15 μm (see Lagache, Puget & Dole 2005 for a review). Various phenomenological models proposed to account for these data have been compared and discussed by Gruppioni et al. (2002), who showed that the data require a combination of luminosity and density evolution for starburst galaxies, stopping at some redshift zbreak∼ 1. Gruppioni et al. (2002) also allowed for non-evolving normal spirals and for type 1 AGN evolving in luminosity. The latter population, however, yields a very small contribution to the 15-μm counts (as well as to the submm counts) and can be neglected here. On the other hand, chemo/spectrophotometric evolution models for normal disc galaxies (Mazzei, Xu & de Zotti 1992; Matteucci 2006) indicate a mild increase in the star formation rate, hence of IR luminosity, with look-back time. In view of that, Silva et al. (2004) updated the Gruppioni et al. (2002) model by allowing for a moderate luminosity evolution of normal spiral galaxies. They obtained good fits to the observed 15-μm counts and redshift distributions with the following recipe. (i) The LF, Φsb, of the starburst population evolves both in density and in luminosity as Φsb[L(z), z]=Φsb[L(z)/(1 +z)2.5, z= 0]× (1 +z)3.5, with a high-luminosity cut-off of the local LF at L60 μm= 2 × 1032 erg s−1 Hz−1; and (ii) spirals evolve in luminosity as L(z) =L(0)(1 +z)1.5; (iii) the evolution of both starburst and spiral galaxies stops at zbreak= 1 and the LFs keep constant afterwards, up to a redshift cut-off zcut-off= 1.5, above which essentially all the massive haloes are associated, according to Granato et al. (2004), with spheroidal galaxies. We have adopted this model.

As shown by Silva et al. (2004, 2005), the model accounts for a broad variety of data, including optical, radio, near-, mid-, and far-IR counts, and redshift distributions. Moreover the model predicted, at the flux limit (F24 μm≥ 83 μJy) of the Chandra Deep Field South MIPS survey (Papovich et al. 2004), a surface density of sources at z≥ 1.5 of ≃1 arcmin2, that is, amounting to ≃22 per cent of the observed total surface density. This prediction was at variance with those of major phenomenological models which, for that flux limit, yielded (see fig. 2 of Pérez-González et al. 2005 and fig. 6 of Caputi et al. 2006) either very few (Chary et al. 2004) or almost 50 per cent (Lagache et al. 2004) sources at z≃ 1.5. The redshift distribution observationally determined by Pérez-González et al. (2005) and Caputi et al. (2006) for F24 μm≥ 83 μJy has 24–28 per cent of sources at z≥ 1.5, in nice agreement with the Silva et al. (2004) model. We add here (Fig. 2) a comparison with Spitzer counts at 160 μm (Dole et al. 2004; Frayer et al. 2006a), the wavelength closest to those relevant here, confirming once more the consistency of the model with the data.

Figure 2

Model 160 μm integral counts compared with the Spitzer counts by Dole et al. (2004, squares) and Frayer et al. (2006a, triangles). The heavy solid line on the lower left-hand corner shows the count estimate by Serjeant & Harrison (2005).

Figure 2

Model 160 μm integral counts compared with the Spitzer counts by Dole et al. (2004, squares) and Frayer et al. (2006a, triangles). The heavy solid line on the lower left-hand corner shows the count estimate by Serjeant & Harrison (2005).

2.3 Extragalactic radio sources

For radio sources we have adopted the model by De Zotti et al. (2005). In the wavelength range and at the relatively bright flux densities considered in this paper, the dominant population are blazars (flat-spectrum radio quasars and BL Lac objects). The usual power-law approximation of the spectra of these objects may break down at submm wavelengths, where the synchrotron emission of even the most-compact regions turns from the optically thick to the optically thin regime and is further steepened by electron ageing effects. To allow for the corresponding steepening of radio spectra, we have adopted the spectral shape given by Donato et al. (2001) and Costamante & Ghisellini (2002), including their (controversial, see Antón & Browne 2005 and Nieppola et al. 2006) anti-correlation between break frequency and radio luminosity (see Section 5).

2.4 Submm counts

Fig. 3 summarizes the expected contributions of protospheroids, of late-type (starburst plus spiral) galaxies, and of radio sources to the counts at 850, 550, 350, and 250 μm, yielded by the models described above. The 850 μm panel shows that the SCUBA counts are almost entirely accounted for by protospheroidal galaxies, with a minor contribution from starburst galaxies which, however, become increasingly important at low flux densities. At bright fluxes, the counts of dusty galaxies match the extrapolation of IRAS data by Serjeant & Harrison (2005). At the brightest levels, however, the counts may be dominated by flat-spectrum radio sources.

Figure 3

Predicted counts of high-z protospheroidal galaxies (solid line), late-type (starburst plus normal spiral) galaxies (dot–dashed line), and radio sources (dotted line). The dashed line shows the predicted counts of strongly lensed protospheroidal galaxies. The 850-μm data are from Coppin et al. (2006, circles) and Scott et al. (2006, squares). The heavy solid line on the lower right-hand corner of each panel shows the count estimates by Serjeant & Harrison (2005).

Figure 3

Predicted counts of high-z protospheroidal galaxies (solid line), late-type (starburst plus normal spiral) galaxies (dot–dashed line), and radio sources (dotted line). The dashed line shows the predicted counts of strongly lensed protospheroidal galaxies. The 850-μm data are from Coppin et al. (2006, circles) and Scott et al. (2006, squares). The heavy solid line on the lower right-hand corner of each panel shows the count estimates by Serjeant & Harrison (2005).

3 STRONGLY LENSED PROTOSPHEROIDAL GALAXIES

As already mentioned, the combination of the very steep counts (Fig. 3) and high redshifts (Fig. 1), hence large lensing optical depths, of sources detected by SCUBA surveys maximizes the fraction of strongly lensed sources.

Calculations have been carried out as in Perrotta et al. (2002, 2003), but with the evolutionary model for protospheroidal galaxies by Granato et al. (2004), updated as described above. The predicted counts of strongly lensed sources (defined as those with a gravitational amplification by a factor of ≥2) are displayed in Fig. 3. We find that the fraction of strongly lensed protospheroids can indeed be very high at relatively bright submm fluxes, where the surface density of unlensed protospheroidal galaxies sinks down rapidly. Dunlop et al. (2004) argue that the amplification distribution derived by Perrotta et al. (2003) is probably low because it neglects the effect of the halo substructure. If so, the fraction of strongly lensed sources in submm surveys is even higher than shown in Fig. 3.

Clearly, large-area submm surveys can provide far richer samples of strongly lensed sources than currently available. However, Planck/HFI is confusion limited at bright flux levels. Estimates of the 5σ detection limits have been obtained by López-Caniego et al. (2006), using up-to-date templates of the microwave/submm sky, but ignoring source clustering. The effect of latter on the confusion noise has been discussed by Negrello et al. (2004), using models fitting the earlier 850-μm counts. We have revised downwards the estimates by Negrello et al. (2004) based on the counts shown in Fig. 3. Combining these results with those by López-Caniego et al. (2006) we obtain 5σ detection limits of ≃335, 740 and 1290 mJy at 850, 550 and 350 μm, respectively. Note that the analysis by López-Caniego et al. (2006) refers to high Galactic latitude regions (|b| > 30°); at lower latitudes the detection limits increase because of the larger contributions of Galactic emissions to foreground fluctuations. Therefore, Planck/HFI can possibly pick up only the most extreme (very rare) high-z sources.

The situation is far better for Herschel/SPIRE. For example, the surface density of strongly lensed galaxies at a flux limit of 100 mJy is estimated to be ≃0.14 deg−2 at 500 μm. Thus a survey of, say, 400 deg2 to this flux limit will provide a complete sample of ≃60 strongly lensed galaxies. The SASSy will provide information on strongly lensed sources several times brighter (in terms of bolometric luminosity) and much rarer (surface density ∼0.002 deg−2) which, however, will be detected in a comparable number (≃40), thanks to the much larger area (20 000 deg2). For comparison, the radio Cosmic Lens All-Sky Survey (CLASS; Browne et al. 2003; Myers et al. 2003), that has been the basis for most analyses of lens statistics, has yielded a sample of 22 gravitational lens systems.

The other source populations contributing to the counts are easily distinguished from (lensed and unlensed) protospheroidal galaxies. Essentially all radio sources are flat-spectrum blazars, already detected by low-frequency surveys, like the 1.4-GHz NVSS (Condon et al. 1998) and FIRST (Becker, White & Helfand 1995; White et al. 1997), complete to a few mJy levels, the 0.84-GHz SUMSS (Mauch et al. 2003), with completeness limits ranging from 8 to 18 mJy, and the 5-GHz GB6 survey (Gregory et al. 1996), complete to 18 mJy, and PMN survey (Griffith & Wright 1993) with completeness limits ranging from 20 to 72 mJy. Altogether these surveys cover the full sky. The median spectral index of blazars between 5 and 18.5 GHz is (Ricci et al. 2006) α≃ 0.16 (Sν∝ν−α) and generally further steepens at higher frequencies. Therefore, the radio flux of blazars is significantly higher than at 500 μm, and thus well above the completeness limits of the quoted surveys; obviously, the low-frequency fluxes of steep-spectrum sources are even higher.

The starburst and normal late-type galaxies sampled at the relatively bright flux limits considered here are at z≪ 1 (see Fig. 4). For the median local ratios S60 μm/S250 μm≃ 1, S60 μm/S350 μm≃ 2.4, S60 μm/S550 μm≃ 9, and S60 μm/S850 μm≃ 41 (Serjeant & Harrison 2005, based on observational data by Dunne & Eales 2001) these sources have 60-μm fluxes above the completeness limit of the IRAS survey, S60 μm≃ 0.6 Jy, for λ≥ 500 μm, and well above that of the AKARI (formerly ASTRO-F) all-sky survey, S60 μm≃ 44 mJy (Pearson et al. 2004), at all wavelengths. Combining SPIRE and/or SCUBA-2 with IRAS and AKARI data, photometric redshift estimates can be obtained. On the contrary, lensed and unlensed protospheroidal galaxies, at typical z∼ 2 –3, have S60 μm/S550 μm≪ 1.

Figure 4

Redshift distribution of spiral (dotted) and starburst (dashed) galaxies brighter than 100 mJy at 550 μm, for Δz= 0.01, according to the model described in the text.

Figure 4

Redshift distribution of spiral (dotted) and starburst (dashed) galaxies brighter than 100 mJy at 550 μm, for Δz= 0.01, according to the model described in the text.

Also, starburst and normal late-type galaxies are expected to have optical counterparts in the available digitized sky surveys, such as SuperCOSMOS, APM, APS, DSS and PMM. In fact only 3.5 per cent of the galaxies in the IRAS PSCz sample (Saunders et al. 2000), complete to S60 μm= 0.6 mJy over 84 per cent of the sky had no identifications with bJ≤ 19.5 mag. On the other hand, star-forming protospheroidal galaxies are expected to be optically very faint (R > 25) because of their substantial redshifts bringing optical bands to the rest-frame ultraviolet where obscuration is very high.

At 500 and 850 μm, unlensed protospheroids have essentially disappeared above 100 mJy so that, after having removed radio sources and low-z galaxies as discussed above, we are left with a sample almost exclusively made of strongly lensed sources, that is, we expect an efficiency in the selection of such sources close to 100 per cent. For comparison, the CLASS survey found 22 gravitational lens systems out of over 16 000 radio sources.

The surface density of strongly lensed sources brighter than 100 mJy in the two other Herschel/SPIRE channels is higher than at 500 μm (≃0.6 deg−2 at both 250 μm and at 350 μm, corresponding to ≃240 strongly lensed galaxies over 400 deg2). It is, however, lower than that of unlensed protospheroids, so that singling out strongly lensed sources is not straightforward, although at 350 μm the ratio of lensed to unlensed sources is still high (≃0.4). Also, only a fraction of starburst/late type galaxies selected at these wavelengths are expected to show up in the IRAS catalogue. On the other hand, the different selection wavelengths provide complementary information, since, as illustrated by Fig. 5, they probe different redshift intervals, systematically shifted to higher values with increasing wavelength.

Figure 5

Predicted redshift distributions in bins of Δz= 0.36 of strongly lensed galaxies brighter than 100 mJy in the 3 Herschel/SPIRE channels (250, 350 and 550 μm) over an area of 400 deg2, and brighter that 150 mJy (the flux limit of the SCUBA-2 SASSy survey) at 850 μm over an area of 20 000 deg2, based on the Granato et al. (2004) model, updated as described in the text.

Figure 5

Predicted redshift distributions in bins of Δz= 0.36 of strongly lensed galaxies brighter than 100 mJy in the 3 Herschel/SPIRE channels (250, 350 and 550 μm) over an area of 400 deg2, and brighter that 150 mJy (the flux limit of the SCUBA-2 SASSy survey) at 850 μm over an area of 20 000 deg2, based on the Granato et al. (2004) model, updated as described in the text.

The SCUBA-2 ‘Wide area extragalactic survey’ should also detect hundreds of strongly lensed sources. In this case, however, they are only ≃2 per cent of the total number of high-z galaxies, and detailed follow up observations would be necessary to single them out. Herschel/SPIRE surveys going substantially deeper than 100 mJy will face a similar problem.

Clearly, the angular resolution of both Herschel/SPIRE and SCUBA-2 is far too poor to resolve the multiple images of lensed sources, so that optical/IR follow-up is necessary to see the multiple images. Note that the sources considered here are brighter, even by large factors, than the two highly magnified submm sources discovered by Kneib et al. (2004) and Borys et al. (2004), so that the follow-up measurements will be correspondingly easier.

4 CLUSTERING OF PROTOSPHEROIDAL GALAXIES

Both theoretical arguments and observational data indicate that the positions of powerful far-IR galaxies, detected by SCUBA surveys and interpreted as massive protospheroidal galaxies, are highly correlated (see e.g. Smail et al. 2003; Blain et al. 2004; Negrello et al. 2004; Scott et al. 2006). On the contrary, Poisson fluctuations dominate in the case of the other extragalactic source populations contributing to the submm counts, whose clustering is relatively weak, as in the case of spiral and starburst galaxies (cf. e.g. Madgwick et al. 2003) or highly diluted because of the very broad redshift distribution, as in the case of radio sources (Magliocchetti et al. 1999; Blake & Wall 2002a,b; Overzier et al. 2003; Blake, Mauch & Sadler 2004; Negrello, Magliocchetti & De Zotti 2006).

In the following, we will therefore consider only the clustering of dusty protospheroidal galaxies. As already mentioned, they are located at substantial (≳2) redshifts (Chapman et al. 2005) and display huge star formation rates (≳1000 M yr−1), allowing masses in stars of the order of 1011 M to be assembled in times shorter than 1 Gyr.

For dark matter to stellar mass ratios typical of massive ellipticals (see e.g. Marinoni & Hudson 2002; McKay et al. 2002), the dark matter haloes in which the submm galaxies reside have masses of ≥1013 M. Since such massive haloes sample the rare high-density peaks of the primordial dark matter distribution (Kaiser 1984; Mo & White 1996), these sources are expected to exhibit a strong spatial clustering, similar to that measured for extremely red objects (EROs; Daddi et al. 2001, 2003), for which similar masses have been inferred (Moustakas & Somerville 2002).

Direct measurements of clustering properties of SCUBA galaxies are made difficult by the poor statistics and by the fact that they are spread over a wide redshift range, so that their clustering signal is strongly diluted. However, tentative evidence of strong clustering with a comoving correlation length r0∼ 8 –13 h−1 Mpc, consistent with that of EROs, has been reported (Scott et al. 2002, 2006; Smail et al. 2003; Webb et al. 2003; Blain et al. 2005). A highly statistically significant detection of strong clustering for ultraluminous IR galaxies over 1.5 < z < 3 has been obtained by Farrah et al. (2006a,b) from an analysis of optically faint sources selected at 24 μm in three fields observed as a part of the Spitzer Wide-area Infrared Extragalactic (SWIRE) survey. They found r0= 14.40 ± 1.99 h−1 Mpc for 2 < z < 3 and r0= 9.40 ± 2.24 h−1 Mpc for 1.5 < z < 2. Magliocchetti et al. (2007) found that optically unseen (R > 25.5) 24-μm selected sources in the complete Spitzer FIRST Look Survey sample (Fadda et al. 2006) are highly clustered and pointed out several lines of circumstantial evidence consistently indicating that they are at z∼ 2. If so, their comoving clustering length is r0= 10.6+1.6−1.8 h−1 Mpc.

4.1 Formalism

We model the two-point spatial correlation function, ξ(r, z), of star-forming spheroids as in Negrello et al. (2004, their model 1):  

1
formula
where b(Meff, z) is the redshift-dependent (linear) bias factor (Sheth & Tormen 1999; Percival et al. 2003), Meff being the effective mass of the dark matter haloes in which the sources reside; ξDM is the linear two-point spatial correlation function of dark matter, determined by the power spectrum of primordial density perturbations as well as by the underlying cosmology. For the power spectrum of the primordial fluctuations we adopt the fitting relations by Eisenstein & Hu (1998) which account for the effect of baryons on the matter transfer function.

We adopt an effective halo mass of star-forming spheroids Meff= 1013 h−1 M, which yields a comoving clustering length, defined by ξ(r0, z) = 1, r0≃ 8 h−1 Mpc, consistent with the available observational indications for SCUBA galaxies (see Negrello et al. 2004), and somewhat below the estimates for Spitzer 24-μm sources at z≃ 2. Note that, on scales below a few Mpc, clustering enters the non-linear regime and the slope of ξ(r, z) becomes steeper than expected from the linear theory. In this range of scales, a power-law model provides a better description of the two-point spatial correlation function. Therefore, we assume  

2
formula
rvir being the comoving virial radius of the characteristic halo in which star-forming spheroids reside. In view of the tight connection between spheroidal galaxies and AGN at their centres entailed by the model of Granato et al. (2004), we assume the correlation length, r0, to be constant in comoving coordinates, as suggested by quasar data (Porciani, Magliocchetti & Norberg 2004; Croom et al. 2005).

The angular correlation function is related to the spatial one, ξ(r, z), by the relativistic Limber equation (Peebles 1980):  

3
formula
where x is the comoving radial coordinate, u is defined by r={u2+[xθ/(1 +z)]2}1/2, F(x) gives the correction for curvature, and the selection function Φ(x) satisfies the relation  
4
formula
in which forumla is the mean surface density, Ωs is the solid angle covered by the survey, and N(z) is the number of sources within the shell (z, z+ dz), given by the model.

The errors on w(θ) have been computed as  

5
formula
where Npair is the number of source pairs separated by an angular distance θ.

The angular correlation function of intensity fluctuations due to sources below the detection limit writes (Peebles 1980; De Zotti et al. 1996):  

6
formula
zmax being the redshift when the sources begin to shine, z(Sd, L) the redshift at which a source of luminosity L is seen with a flux equal to the detection limit Sd, and δz the redshift difference of sources separated by a comoving distance r. The effective volume emissivity jeff is expressed as  
7
formula
Φ(L, z) being the LF (per unit logarithmic luminosity interval), K(L, z) the K-correction and Lmax and Lmin the maximum and minimum local luminosity of the sources.

The angular power spectrum of the intensity fluctuations can then be obtained as  

8
formula
The fractional error on the power spectrum can be estimated summing in quadrature the instrumental, confusion and cosmic variances (Knox 1995):  
9
formula
where A (sr) is the surveyed area, σ is the rms noise per pixel (quadratic sum of instrumental and confusion noise), N is the number of pixels in the map, and W is the window function, describing the region of the ℓ-space observable with the considered instrument. We assume Gaussian beams with full width at half-maximum forumla, so that W= exp (−ℓ2σ2B).

In the dependence on the angular scale of both the angular correlation functions and the power spectra of background fluctuations due to clustering, three regimes can be distinguished (see Cooray & Sheth 2002 for a review). On small scales the power spectrum probes the statistics of the distribution of sources (subhaloes) within large-scale dark matter haloes, that is, the halo occupation function, determined by the non-linear evolution of large-scale structure. On intermediate scales, the amplitude of the signal is mostly determined by the bias parameter, which in turn depends on the effective mass of dark haloes associated to sources. On very large scales, the signal provides information on the cosmological evolution of primordial density fluctuations in the linear phase, and is thus sensitive to cosmological parameters, and in particular to the dark matter and baryon densities.

4.2 Predictions for wθ) and C

Before proceeding to describe our predictions it is interesting to compare our model with the four alternative models for the clustering of SCUBA galaxies investigated by van Kampen et al. (2005). These include a simple merger model, a hydrodynamical model, the stable clustering model by Hughes & Gaztañaga (2000), and a phenomenological model. For each of these models van Kampen et al. (2005) worked out estimates of the clustering properties, tailored for the SHADES survey. In Fig. 6, these estimates are compared with those of the model used here. There are clear differences among the models, so that the analysis of clustering will help to discriminate between them. The w(θ) predicted by our model is remarkably close to that of the hydrodynamical model, which assumes a redshift distribution (the analytical form of Baugh, Cole & Frenk 1996) close to being compatible with the results by Aretxaga et al. (2007).

Figure 6

Comparison of the angular correlation function of SHADES sources predicted by the model adopted in this paper (solid line with 1σ error bars) with the best fitting power laws of the sky-averaged correlation functions predicted by the four models studied by van Kampen et al. (2005).

Figure 6

Comparison of the angular correlation function of SHADES sources predicted by the model adopted in this paper (solid line with 1σ error bars) with the best fitting power laws of the sky-averaged correlation functions predicted by the four models studied by van Kampen et al. (2005).

Our predictions for the angular correlation function, w(θ), and/or the power spectrum of intensity fluctuations, C, for the large-area surveys defined in Section 1 are presented in Figs 7–10.

Figure 7

Predicted angular correlation function, with its uncertainties (3σ Poisson error bars), of protospheroids detected by 250-μm surveys to flux limits of 25, 60 and 100 mJy, over areas of 10, 50 and 100 deg2, respectively, and by a 350-μm survey of 10 deg2 to a flux limit of 25 mJy, averaged over bins Δ log (θ) = 0.4. The dilution of the signal due to the other, weakly clustered, source populations is taken into account. The corresponding predicted numbers of detected sources and their redshift distributions for redshift bins of Δz= 0.23 are also shown. The predicted redshift distributions of spiral and starburst galaxies (dotted and dashed histograms, respectively) are also shown. Surveys at 350 μm to flux limits of 60 and 100 mJy over areas of 50 and 400 deg2 would provide only very weak constraints on w(θ). Deep surveys, reaching the estimated confusion limits at 250 and 350 μm, can provide useful constraints on cosmological parameters, as illustrated by the top left-hand panels where, in addition to the predicted w(θ) for our reference cosmological model, with Ωm= 0.3 and Ωb= 0.047 (dashed line), we show the effect of different choices of these two parameters: Ωm= 0.15 and Ωb= 0.045 (dotted line); Ωm= 0.27 and Ωb= 0. (dot–short-dashed line); Ωm= 0.27 and Ωb= 0.09 (dot–long-dashed line).

Figure 7

Predicted angular correlation function, with its uncertainties (3σ Poisson error bars), of protospheroids detected by 250-μm surveys to flux limits of 25, 60 and 100 mJy, over areas of 10, 50 and 100 deg2, respectively, and by a 350-μm survey of 10 deg2 to a flux limit of 25 mJy, averaged over bins Δ log (θ) = 0.4. The dilution of the signal due to the other, weakly clustered, source populations is taken into account. The corresponding predicted numbers of detected sources and their redshift distributions for redshift bins of Δz= 0.23 are also shown. The predicted redshift distributions of spiral and starburst galaxies (dotted and dashed histograms, respectively) are also shown. Surveys at 350 μm to flux limits of 60 and 100 mJy over areas of 50 and 400 deg2 would provide only very weak constraints on w(θ). Deep surveys, reaching the estimated confusion limits at 250 and 350 μm, can provide useful constraints on cosmological parameters, as illustrated by the top left-hand panels where, in addition to the predicted w(θ) for our reference cosmological model, with Ωm= 0.3 and Ωb= 0.047 (dashed line), we show the effect of different choices of these two parameters: Ωm= 0.15 and Ωb= 0.045 (dotted line); Ωm= 0.27 and Ωb= 0. (dot–short-dashed line); Ωm= 0.27 and Ωb= 0.09 (dot–long-dashed line).

Figure 8

Predicted angular correlation function, with its uncertainties (3σ Poisson error bars), of protospheroids detected by the 850-μm SCUBA-2 survey of 2 deg2 to a detection limit of 2 mJy, and the corresponding redshift distribution. The dilution of the signal due to the other, weakly clustered, source populations is taken into account. The predicted redshift distributions of spiral and starburst galaxies (dotted and dashed histograms, respectively) are also shown. We have used bins of Δ log (θ) = 0.2 and Δz= 0.23. The different lines in the left-hand panel correspond to the same choices of the cosmological parameters Ωm and Ωb as in Fig. 7.

Figure 8

Predicted angular correlation function, with its uncertainties (3σ Poisson error bars), of protospheroids detected by the 850-μm SCUBA-2 survey of 2 deg2 to a detection limit of 2 mJy, and the corresponding redshift distribution. The dilution of the signal due to the other, weakly clustered, source populations is taken into account. The predicted redshift distributions of spiral and starburst galaxies (dotted and dashed histograms, respectively) are also shown. We have used bins of Δ log (θ) = 0.2 and Δz= 0.23. The different lines in the left-hand panel correspond to the same choices of the cosmological parameters Ωm and Ωb as in Fig. 7.

Figure 9

Power spectra of background fluctuations due to clustering of protospheroidal galaxies below the detection limit compared with the power spectra of the diffuse Galactic dust emission and of Poisson fluctuations. The ‘data’ points refer, from top to bottom panel, to Herschel/SPIRE surveys of 10, 50 and 400 deg2 with detection limits of 25, 60 and 100 mJy, respectively, for all channels (250, 350 and 550 μm). The error bars are estimated from equation (9), averaging over bins of Δℓ= 0.4ℓ. The broken lines show the power spectra of Galactic dust emission based on the Finkbeiner et al. (1999) model, averaged over a particularly dust-free 400-deg2 area centred at l≃ 245°, b≃−50°. The dashed lines show the Poisson contributions of extragalactic point sources below the detection limits. In the bottom right-hand panels, we show, in addition to the predicted power spectrum for our reference cosmological model, with Ωm= 0.3 and Ωb= 0.047 (dashed line), the effect of different choices of these 2 parameters: Ωm= 0.15 and Ωb= 0.045 (dotted line); Ωm= 0.27 and Ωb= 0. (dot-short dashed line); Ωm= 0.27 and Ωb= 0.09 (dot–long-dashed line).

Figure 9

Power spectra of background fluctuations due to clustering of protospheroidal galaxies below the detection limit compared with the power spectra of the diffuse Galactic dust emission and of Poisson fluctuations. The ‘data’ points refer, from top to bottom panel, to Herschel/SPIRE surveys of 10, 50 and 400 deg2 with detection limits of 25, 60 and 100 mJy, respectively, for all channels (250, 350 and 550 μm). The error bars are estimated from equation (9), averaging over bins of Δℓ= 0.4ℓ. The broken lines show the power spectra of Galactic dust emission based on the Finkbeiner et al. (1999) model, averaged over a particularly dust-free 400-deg2 area centred at l≃ 245°, b≃−50°. The dashed lines show the Poisson contributions of extragalactic point sources below the detection limits. In the bottom right-hand panels, we show, in addition to the predicted power spectrum for our reference cosmological model, with Ωm= 0.3 and Ωb= 0.047 (dashed line), the effect of different choices of these 2 parameters: Ωm= 0.15 and Ωb= 0.045 (dotted line); Ωm= 0.27 and Ωb= 0. (dot-short dashed line); Ωm= 0.27 and Ωb= 0.09 (dot–long-dashed line).

Figure 10

Power spectra of background fluctuations due to clustering of protospheroidal galaxies below the detection limit in the high frequency Planck/HFI surveys (points with error bars for bins Δℓ= 0.4ℓ) compared with the power spectra of the diffuse Galactic dust emission (broken lines), of Poisson fluctuations due to extragalactic point sources below the detection limits (dashed lines), and of the CMB (solid line, visible in the 850-μm panel, and in the bottom left-hand corner of the 550-μm panel). The dust power spectra are based on the Finkbeiner et al. (1999) model; the upper set of panels refer to the ≃9650 deg2 at |b| > 50°, the lower set refers to the 400-deg2 low-dust region centred at l≃ 245°, b≃−50°. In the bottom panels, we show, in addition to the predicted power spectrum for our reference cosmological model, with Ωm= 0.3 and Ωb= 0.047 (dashed line), the effect of different choices of these two parameters: Ωm= 0.15 and Ωb= 0.045 (dotted line); Ωm= 0.27 and Ωb= 0. (dot–short-dashed line); Ωm= 0.27 and Ωb= 0.09 (dot–long-dashed line).

Figure 10

Power spectra of background fluctuations due to clustering of protospheroidal galaxies below the detection limit in the high frequency Planck/HFI surveys (points with error bars for bins Δℓ= 0.4ℓ) compared with the power spectra of the diffuse Galactic dust emission (broken lines), of Poisson fluctuations due to extragalactic point sources below the detection limits (dashed lines), and of the CMB (solid line, visible in the 850-μm panel, and in the bottom left-hand corner of the 550-μm panel). The dust power spectra are based on the Finkbeiner et al. (1999) model; the upper set of panels refer to the ≃9650 deg2 at |b| > 50°, the lower set refers to the 400-deg2 low-dust region centred at l≃ 245°, b≃−50°. In the bottom panels, we show, in addition to the predicted power spectrum for our reference cosmological model, with Ωm= 0.3 and Ωb= 0.047 (dashed line), the effect of different choices of these two parameters: Ωm= 0.15 and Ωb= 0.045 (dotted line); Ωm= 0.27 and Ωb= 0. (dot–short-dashed line); Ωm= 0.27 and Ωb= 0.09 (dot–long-dashed line).

The steep counts of protospheroids shown by Fig. 3 imply that in the case of Planck/HFI and Herschel/SPIRE survey flux limited at 100 mJy at 500 μm the clustering signal comes entirely from sources below the detection limit. For the same flux limit, we expect significant signals both from sources above and below threshold at 350 and 250 μm. Only at the latter wavelength, however, a 400-deg2 survey would provide a large enough sample of detected spheroidal galaxies allowing a direct study of their angular correlation function (see Fig. 7). A deeper survey (flux limit of ≃25 mJy), on one hand, provides a substantially larger sample in a given observing time (thanks to the steepness of the counts of protospheroids in this flux density interval)but, on the other hand, contains a higher fraction of weakly clustered late-type (starburst and spiral), diluting the signal. The former effect, however, prevails, so that, as shown by Fig. 7, we expect a rather accurate determination of w(θ) particularly at 250 μm, where the data will effectively probe all the three clustering regimes mentioned above. The number of detected sources decreases rapidly, for given area and limiting flux, with increasing wavelength. Already at 350 μm the uncertainties are rather large. On the other hand, studies at longer wavelengths explore higher values of z (because of the effect of the strongly negative K-correction).

The SASSy survey will detect, apart from the strongly lensed sources discussed in Section 3, mostly blazars and low-z late-type/starburst galaxies. The size of the samples is unlikely to be sufficient to detect the weak clustering of these populations. On the other hand, the SCUBA-2 ‘Wide-Area Extragalactic Survey’ is well suited for the investigation of small- (non-linear), intermediate (sensitive to the bias parameter), and large-scale (sensitive to cosmological parameters) clustering, since it has a very good statistics and the clustering signal is less diluted than in the (already very fit) deepest 250- and 350-μm Herschel/SPIRE surveys considered in Fig. 7. It also covers a much wider redshift range (see Fig. 8).

Herschel/SPIRE surveys at 500 μm, which are expected not to provide large enough samples of individually detected high-z protospheroids to allow accurate estimates of their w(θ) are, however, better suited than surveys at shorter wavelengths to estimate the autocorrelation function of background fluctuations, from which the clustering power spectra can be derived. This is illustrated by Fig. 9 where the clustering signal is compared with competing astrophysical sources, namely the Poisson fluctuations due to the combination of all populations of extragalactic sources fainter than the detection limit, and to Galactic dust emission. To deal with the latter we have used the Finkbeiner, Davis & Schlegel (1999) model, built combining IRAS and COBE data. We have assumed that the Herschel survey is carried out in the low-dust region centred at l≃ 245°, b≃−50°. Because of the effect of redshift, the clustering power spectrum of high-z protospheroids dominates over that of dust (in the chosen region) over a multipole range increasing with increasing wavelength. At 500 μm, the ‘clean’ range is broad enough to provide information both on the bias factor and on cosmological parameters in each of the three considered cases, but the highest accuracy is achieved by the largest area, shallow survey. On the contrary, the possibility of taking full advantage of the Herschel/SPIRE resolution to learn about the non-linear clustering regime (whose onset is marked by the high-multipole inflection point) is seriously constrained by Poisson fluctuations.

While the much larger area covered by Planck allows, in principle, a much more accurate estimate of the power spectrum of the clustering signal, this advantage is largely impaired by the contamination by Galactic dust emission. For example, even if we restrict ourselves to the ≃9650 deg2 around the Galactic polar regions (|b| > 50°), the interstellar dust emission power spectrum blurs the clustering contribution for multipoles ℓ≲ 500 at 550 μm and for ℓ≲ 900 at 350 μm (Fig. 10). The effect of Galactic dust decreases substantially at 850 μm, but then the power spectrum is dominated by CMB anisotropies up to ℓ≃ 1000. On the other hand, it is possible to select regions of ∼400 deg2 where the Galactic dust contamination is much lower and below the clustering signal down to ℓ≲ 350 at 550 μm and to ℓ≲ 500 at 350 μm. Again, at high multipoles (ℓ≳ 2000), Poisson fluctuations dominate. An application of equation (9) shows that the uncertainties on the clustering power spectrum on scales much larger that the instrument FWHM are, at given wavelength, almost identical for Planck and Herschel surveys of the same area. The substantial increase in the error bars at ℓ≃ 2500 signals that we are approaching the limit set by Planck's resolution.

5 OTHER SOURCE POPULATIONS

5.1 Local late-type and starburst galaxies

According to the counts shown in Fig. 3, an Herschel/SPIRE survey of 400 deg2 to a flux limit of 100 mJy will yield ≃3000 low-z galaxies at 250 μm, ≃890 at 350 μm, and ≃190 at 500 μm. Low-z galaxies are expected to dominate the counts at these bright flux levels, except perhaps at 250 μm. In any case, they can be easily distinguished from the other populations as described in Section 3. The SCUBA-2 SASSy survey will yield about 350 low-z galaxies at 850 μm. Adopting the 5σ detection limits estimated in Section 3, we find that number of low-z galaxies detected by Planck/HFI in the 50 per cent of the sky at |b| > 30° will be of 110, 300 and 990 at 850, 550 and 350 μm, respectively.

Thus, these surveys will provide large enough samples of low-z galaxies to allow the first direct and accurate estimates of the local LFs of late-type and starburst galaxies at several submm wavelengths.

5.2 Radio sources

Based on the current understanding of the blazar spectra, the slope, β, of the integral counts (N(>S) ∝S−β) of blazars in the relevant flux density range cannot be much different from that measured by high-frequency radio surveys (Waldram et al. 2003; Ricci et al. 2004), that is, β≃ 1. Then, the number of detected sources in an observing time t increases as t1/2 if we use the time to go deeper over a fixed area, and increases as t if we cover a larger area to a fixed depth. Thus, a larger area shallow survey is substantially more productive than a deep survey over a small area. From the De Zotti et al. (2005) model we expect that the SASSy survey detects ≃1200 blazars, and that the Planck surveys detect ≃550, 130 and 50 sources at 850, 550 and 350 μm, respectively, for the flux limits given in Section 3 and at |b| > 30°. However, since a key scientific goal is the identification of the synchrotron peak frequency, a broad spectral coverage is essential and large-area surveys at Herschel/SPIRE wavelengths would also be extremely valuable. A 400-deg2 survey to a flux limit of 100 mJy is expected to provide a complete sample of ≃30 blazars detected at each SPIRE wavelength. As explained in Section 3, the radio source populations can be easily identified by a positional cross correlation with radio catalogues, covering the whole sky at at least two frequencies.

6 DISCUSSION AND CONCLUSIONS

Combining the predictions of the physical model by Granato et al. (2004) for the formation and evolution of spheroidal galaxies (updated to take into account the most recent 850-μm counts) with up-to-date phenomenological models for the evolution of starburst and normal late-type galaxies and of radio sources, we have worked out quantitative predictions for the many important wide-area, shallow surveys that will be carried out in the next few years.

We find that Planck/HFI surveys are confusion limited at bright flux levels, with 1 σ confusion noise increasing from ≃70 mJy at 850 μm to ≃260 mJy at 350 μm. These surveys will thus be not sufficiently deep to detect with good signal-to-noise ratio (S/N > 5) high-z galaxies, except in the case of extreme gravitational magnifications or exceptionally high luminosities. They will, however, provide complete samples of hundreds low-z star-forming (starburst and normal late-type) galaxies, enabling a direct, accurate, determination of their local LFs at submm wavelengths. In addition, they will detect, at least at 850 μm, several hundreds blazars, providing important information on their physical properties.

Although the number of submm sources of each population detected by Planck is too small to allow a measurement of their angular correlation function (unless their clustering is far stronger than indicated by current data), the Planck maps will allow accurate determination of the autocorrelation function of intensity fluctuations due to clustering of protospheroidal galaxies. It will, however, be difficult to take full advantage of Planck's sky coverage because of the contamination by Galactic dust. Even at |b| > 50°, the Galactic dust power spectrum is expected to overwhelm that of clustering for multipole numbers ℓ < 1000 in the 350-μm channel and for ℓ≤ 600 in the 500-μm channel (see Fig. 10). Nevertheless, the useful range of angular scales provides precise information on the bias parameter of haloes hosting the submm sources, hence on their masses.

There are, however, smaller low-dust regions where the clustering power spectrum may dominate over that of Galactic dust down to ℓ∼ 400 –500 at 550 and 350 μm, respectively. Such regions are large enough to allow us to investigate the regime where the shape of the clustering power spectrum is sensitive to the cosmological parameters (especially to the baryon density and to the matter density), so that its accurate determination sets interesting constraints on them. At 850 μm, the dominant signal up to ℓ≃ 1000 is the cosmic microwave background (CMB). For high multipoles, ℓ≳ 2500, Planck's ability to measure the power spectrum hits the limit set by angular resolution; however, already at somewhat lower multipoles (ℓ∼ 2000) Poisson fluctuations overcome the clustering ones.

One might expect that, thanks to its much higher angular resolution (compared to Planck/HFI) Herschel/SPIRE can remove much more efficiently point sources and thus substantially decrease the Poisson fluctuation level. However, while it is obviously true that Herschel/SPIRE can reach much fainter detection limits, the very steep submm counts of protospheroidal and starburst galaxies imply that the dominant contribution to Poisson fluctuations come from the still fainter fluxes where the slope of integral counts flattens below the critical value β= 2. Thus while, in principle, the Herschel/SPIRE resolution would allow us to reach very high multipoles (up to ℓ∼ 2 × 104), the Poisson fluctuations are expected to dominate for ℓ≳ 2 × 103, that is, over the multipole range where one-halo contributions (i.e. the clustering of sources within a single dark matter halo) come out, even for surveys reaching the confusion limits. On the whole, we do not expect that Herschel/SPIRE surveys will add much to clustering power spectrum determinations provided by Planck/HFI, except for the Herschel-only 250-μm channel. They will, however, provide independent estimates.

The great advantage of Herschel/SPIRE over Planck/HFI resides in its capability of detecting large enough source samples to allow a direct determination of the angular correlation function w(θ), at least at 250 and 350 μm (Fig. 7). In particular, a 250-μm survey of 10 deg2 reaching a flux limit of 25 mJy effectively probes all the three clustering regimes mentioned in Section 4.1. As illustrated by Fig. 6, measurements of the w(θ) are also useful to discriminate between galaxy formation models (van Kampen et al. 2005; Percival et al. 2003), although the error bars are expected to be rather large. Surveys of different depths probe different redshift intervals.

At 350 and 500 μm, Herschel/SPIRE surveys covering ∼400 deg2 are necessary to extend the counts up to flux density levels sampled by Planck which, at these wavelengths, is confusion limited at ∼1 Jy. A survey of a similar area, to a 5σ detection limit ≃100 mJy, would also provide sufficient samples of z≲ 0.1 late-type/starburst galaxies to estimate their local LFs.

At 500 μm, such a very large-area, shallow survey will also be ideal to select ≃60 strongly lensed high-z dusty galaxies with an essentially 100 per cent efficiency. As discussed in Section 3, the other source populations can be easily distinguished using only available photometric data. The simultaneous SPIRE surveys at shorter wavelengths with a similar detection limit will detect larger numbers of strongly lensed sources (e.g. we expect ≃240 strongly lensed galaxies at 350 μm), but singling them out will be a less easy and efficient process, although at 350 μm the ratio of lensed to unlensed protospheroidal galaxies is estimated to be still high (≃0.4). The redshift distributions of lensed sources is increasingly shifted to higher redshifts with increasing survey wavelength.

We have also estimated that the SASSy survey will detect ∼40 strongly lensed sources, substantially brighter (in terms of bolometric luminosity) and with more extreme amplifications than those detected by Herschel/SPIRE surveys with a flux limit of ∼100 mJy. These sources will be retrievable with an essentially 100 per cent efficiency, as in the case of the latter surveys.

On the other hand, in spite of its very large-area, the SASSy survey will probably not detect enough protospheroidal galaxies to obtain an estimate of their w(θ). The SCUBA-2 Wide-Area Extragalactic Survey will be much better suited for this purpose and may indeed yield more accurate information on all three clustering regimes than the other surveys discussed here. It will also detect hundreds of strongly lensed protospheroids; these, however, will be outnumbered by a factor of ≃100 by unlensed protospheroids, so that singling them out will require a toilsome effort.

The experience made in different contexts (e.g. analyses of Spitzer low-frequency data –Frayer et al. 2006a,b– or of WMAP data –López-Caniego et al. 2007) has demonstrated that source-detection algorithms, when applied to low-resolution maps, easily miss real sources or attribute to them wrong positions or misinterpret fluctuations of Galactic foregrounds or confusion noise in overdense regions as real sources. This implies that the subtraction of sources above the nominal detection limit leaves substantial residuals, that, in the case of CMB experiments, add to the contamination of CMB maps. The situation improves substantially if we can take advantage of the prior knowledge of source positions. This means that the SASSy and a very large area Herschel survey are extremely beneficial for detection and photometry of Planck sources.

For the purpose of cleaning Planck CMB maps from point source contamination, it should be noted that, as illustrated by the 850-μm panels of Fig. 10, point sources are important on small angular scales (large ℓ). Thus, although the SASSy survey will be an exceedingly important resource, a very large area Herschel/SPIRE survey will bring in key additional information. In fact, thanks to the steep rise in dust emission spectra with increasing frequency, a 250-μm survey to a flux limit of 100 mJy is deep enough to observationally characterize both Poisson and clustering fluctuations due to all the main populations of dusty galaxies, providing key input information for techniques, like the widely used Maximum Entropy Method, that need knowledge of the power spectra of the various components to allow an accurate subtraction of their contribution from CMB maps.

The considered surveys will also provide the first complete, submm-selected samples of rare objects, such as blazars. Owing to their much flatter counts (compared to those of dusty galaxies), blazars are prominent in shallow surveys, while they are swamped by dusty galaxies at faint flux levels. We thus expect that the richest blazar sample (∼1200 objects) will come from the SCUBA-2 SASSy survey. The Planck 850-μm survey should detect ∼550 blazars at |b| > 30°, while the number of detections is expected to rapidly decrease in shorter wavelength channels, as a consequence of the rapidly increasing detection limit. Although Herschel/SPIRE surveys of ∼400 deg2 to a flux limit of ∼100 mJy may detect only ≃30 blazars, the multi-wavelength information would be of great value to investigate the currently controversial issue of the ‘blazar sequence’.

2
After this paper was submitted, we learned that the Wide-Area Extragalactic Survey at 850 μm has been combined with the Deep 450-μm survey. The joint programme, known as the SCUBA-2 Cosmology Legacy Survey (http://www.strw.leidenuniv.nl/~pvdwerf/pdf/SCUBA2-CLS.pdf), has two elements: an 850-μm survey of ∼35 deg2 to a 5σ limit of 3.5 mJy and a deeper 450-μm survey of ∼1.5 deg2 to a 5σ limit of 2.5 mJy.

We are grateful to the referee for comments that helped substantially improving this paper. This work was supported in part by MIUR and ASI.

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