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Mon. Not. R. Astron. Soc. 000, 0–0 (2004) Printed 7 February 2020 (MN plain TEX macros v1.6)

The extragalactic sub-mm population: predictions for theSCUBA Half-Degree Extragalactic Survey (SHADES)

Eelco van Kampen1,2, Will J. Percival1, Miller Crawford1, James S. Dunlop1,Susie E. Scott1, Neil Bevis3, Seb Oliver3, Frazer Pearce4, Scott T. Kay3, EnriqueGaztanaga5,6, David H. Hughes5, Itziar Aretxaga51 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ2 Institute for Astrophysics, University of Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria3 Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH4 School of Physics & Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD5 Instituto Nacional de Astrofisica, Optica y Electronica, Apt. Postal 51 y 216, Puebla, Pue, Mexico6 Institut d’Estudis Espacials de Catalunya, Edifici Nexus, Gran Capita 2-4, desp. 201, 08034 Barcelona, Spain

Accepted ... Received ...; in original form ...

ABSTRACT

We present predictions for the angular correlation function and redshift distribution forSHADES, the SCUBA HAlf-Degree Extragalactic Survey, which will yield a sampleof around 300 sub-mm sources in the 850 micron waveband in two separate fields.Complete and unbiased photometric redshift information on these sub-mm sources willbe derived by combining the SCUBA data with i) deep radio imaging already obtainedwith the VLA, ii) guaranteed-time Spitzer data at mid-infrared wavelengths, and iii)far-infrared maps to be produced by BLAST , the Balloon-borne Large-Aperture Sub-millimeter Telescope. Predictions for the redshift distribution and clustering propertiesof the final anticipated SHADES sample have been computed for a wide variety ofmodels, each constrained to fit the observed number counts. Since we are dealing witharound 150 sources per field, we use the sky-averaged angular correlation function toproduce a more robust fit of a power-law shape w(θ) = (θ/A)−δ to the model data.Comparing the predicted distributions of redshift and of the clustering amplitude Aand slope δ, we find that models can be constrained from the combined final SHADES

dataset provided that redshifts, even with relatively large uncertainties, are availablefor the vast majority of sources.

Key words: galaxies: evolution

1 INTRODUCTION

An important problem with most current galaxy formationmodels is to establish whether a set of model parametersthat produces a good match to observations is unique. Themain reason for this is that, for most models, there are de-generacies amongst the various free parameters. As a fairfraction of useful observational data used to constrain themodel parameters are obtained from our local universe, this‘uniqueness problem’ can be resolved by comparing modelpredictions and observations at high redshift, which in manyrespects is independent from a comparison at low redshift.

For this purpose, highly valuable observational data willbe provided by the SCUBA HAlf-Degree Extragalactic Sur-vey (SHADES, see http://www.roe.ac.uk/ifa/shades andMortier et al. 2004 for details). This survey, which com-menced in December 2002, has been designed to cover 0.5sq. degrees to a 3.5-σ detection limit of S850µm = 8mJy,split between two 0.25 sq. degree fields. The two survey ar-

eas, the Lockman Hole East, and the Subaru-XMM DeepField (SXDF) field, have been selected on the basis of lowgalactic confusion at sub-mm wavelengths, and the wealthof existing or anticipated supporting multi-frequency datafrom radio to X-ray wavelengths.

In addition it is planned that the SCUBA data willbe combined with observations to be made by BLAST , a“Balloon-borne Large-Aperture Sub-millimeter Telescope”(see http://chile1.physics.upenn.edu/blastpublic andDevlin 2001 for details), which will undertake a series ofnested extragalactic surveys at 250, 350 and 500µm. Thisexperiment will significantly extend the wavelength range,sensitivity, and area of existing ground-based extragalac-tic sub-mm surveys (Hughes et al. 2002). The combinedSCUBA + BLAST survey provides the only means, priorto the launch of Herschel, to define the sub-mm–FIR spec-tral energy distributions (SEDs) for a substantial sampleof SCUBA sources. Moreover, the combination of SCUBA,VLA, Spitzer and BLAST data offers a uniquely powerful

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2 van Kampen et al.

way of providing the complete and unbiased redshift andSED information required to measure the cosmic evolutionand the clustering properties of the sub-mm sources.

It is anticipated that spectroscopic redshifts will ulti-mately be obtained for a substantial fraction of the SHADES

sources (e.g. Chapman et al. 2003). However, the key pointfor the work presented here is that, even where optical/near-infrared spectroscopy is impossible, the long-wavelengthdata provided by the combined SCUBA + VLA + Spitzer+ BLAST dataset will yield photometric redshifts for all

sources with uncertainties of ∆z ∼ ±0.3 − 0.4 (Hughes etal. 2002). This will allow the first unbiased measurement ofthe cosmic history of dust-enshrouded star formation thattakes place in very massive star-bursts with inferred star-formation rates of order 1000M⊙yr−1 (Scott et al. 2002).

These massive star-bursts could be associated with theformation of the progenitors of massive ellipticals if sus-tained for a significant amount of time (up to 1Gyr). How-ever, SCUBA sources could also be associated with bright,but short-lived bursts of intense star-formation occurring inmore modest galaxies drawn from the high-redshift galaxypopulation already discovered at optical/UV wavelengths(Adelberger & Steidel 2000 and many others). If the brightSCUBA sources are indeed the progenitors of massive el-lipticals then they are likely to be more strongly clusteredthan when drawn from the population of less massive galax-ies. This is an inevitable result of gravitational collapsefrom Gaussian initial density fluctuations: the rare high-mass peaks are strongly biased with respect to the mass(Kaiser 1984).

There is abundant evidence that this bias does occurat high redshift: the correlations of Lyman-break galaxiesat z ≃ 3 are almost identical to those of present-day fieldgalaxies, even though the mass must be much more uniformat early times. Moreover, the correlations increase with UVluminosity (Giavalisco & Dickinson 2001), reaching scale-lengths of r0 ≃ 7.5 h−1Mpc, 1.5 times the present-day value.Daddi et al. (2000) find a trend of clustering with colour forEROs, reaching r0 ≃ 11h−1Mpc for R−K > 5, which corre-sponds to fluctuations in projected number density that are∼unity on the scale of the SCUBA field-of-view, falling to10% rms on 1-degree scales. For SHADES to detect the clus-tering properties of bright sub-mm sources over co-movingscales reaching ≥10Mpc, the survey needs to cover a sig-nificant fraction of a square degree. At the time of writingthe survey is set to reach half a square degree within threeyears, and is making good progress towards achieving thatgoal.

The direct predecessor of SHADES was the 8-mJy sur-vey of Scott et al. (2002; see also Ivison et al. 2002).The correlation function for SCUBA sources derived fromthis survey alone did not yield a significant detection ofclustering, even though the large uncertainties meant itwas still consistent with the strong clustering displayed byEROs. There are nevertheless good reasons for believing theSCUBA source population to be highly clustered. In par-ticular, cross-correlation with X-ray sources (Almaini et al.2003) and Lyman-break galaxies (Webb et al. 2003) yieldclearly-significant detections of clustering. Scott et al. (inpreparation) have recently performed a combined cluster-ing analysis on the three main existing blank-field SCUBAsurveys (the 8-mJy survey, the CUDSS survey by Webb

et al. 2003, and the Hawaii survey by Barger et al. 1999)to determine whether the existing data are capable of re-vealing significant clustering within the sub-mm populationalone. Even though this analysis is based on combining datafrom several small fields, it has yielded the first significant(5-σ) measurement of sub-mm source clustering on scales≃ 0.5 − 2 arcmin, of a strength which does indeed appearcomparable to that found by Daddi et al. (2000) for EROs.Interestingly, if the integral constraint (see Section 4.2 forits definition) is varied as a free parameter, the inferred clus-tering in fact becomes stronger than that displayed by theERO population.

SHADES will improve on the Scott et al. (2004) analy-sis by yielding many more sources, of order 300 at the 3.5-σ level, while the large area of the survey, spread over twofields, ensures that uncertainty in the integral constraint willnot be an issue. Additionally, the redshift information pro-vided by the SCUBA + VLA + Spitzer + BLAST combina-tion will substantially improve on these existing clusteringmeasurements, which have lacked the crucial input of sub-stantial redshift information.

The study of the SHADES sample will help answer threefundamental questions about galaxy formation: What is thecosmic history of massive dust-enshrouded star-formationactivity? Are SCUBA sources the progenitors of present-day massive ellipticals? What fraction of SCUBA sourcesharbour a dust-obscured AGN? The aim of this paper isto review and compare the predictions of various existingmodels for the bright sub-mm population, and to considerhow they can best be tested and constrained by the finalSHADES dataset, and thus help answer the first two ques-tions. The third question will be addressed elsewhere, asit involves a detailed analysis of the combined radio, mid-infrared and X-ray properties of the SHADESsources.

The paper is organized as follows: Section 2 providesan overview of SHADES and its main aims, Section 3 de-scribes the various models used to make predictions, whichare compared to each other in Section 4. This section alsopresents the actual predictions for SHADES, and we discussthese in Section 5.

2 SHADES: A WIDE-AREA SUB-MM SURVEY

2.1 The unique power of SCUBA + BLAST

The combined SCUBA +BLAST approach enhances boththe SCUBA and BLAST observations. First, from theSCUBA perspective, the BLAST data will provide the vi-tal information for estimating the bolometric luminosity ofthe sources, and hence the cosmic history of energy out-put from dust-enshrouded star-formation activity, and ex-tend the source counts to shorter wavelengths. In return,from the BLAST perspective, the SCUBA maps will pro-vide the improved source positions which represent a cru-cial bridge en route to identifying BLAST detections withVLA or optical/infrared sources on arcsec scales. However,most importantly, both SCUBA and BLAST will benefitfrom the greatly-enhanced effectiveness of SED-based red-shift information which results from the combined 850, 500,350, 250µm database (Hughes et al. 2002).

The sensitivity and confusion limits of BLAST meanthat it should detect only 20% of SCUBA sources which

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predictions for SHADES 3

lie at z > 4. This means that a BLAST non-detection ofSCUBA sources offers a powerful method to isolate an ex-citing subset of the sub-mm population which potentiallylies at extreme redshifts, z > 4. Conversely, a 500µm BLASTsource at 15 mJy will not be detected by SCUBA if at z < 1.

2.2 Redshift estimation

Redshift information is not only crucial for deriving thecosmic history of dust-enshrouded star formation, but alsoholds the key to measuring the clustering properties ofthe sub-mm source population. Although precise spectro-scopic measurements of the redshift of a sample of SHADESsources will be possible if reliable radio/optical/IR counter-parts can be identified and readily followed-up with 10-mclass optical telescopes (e.g. Chapman et al. 2003), in prac-tice one will not be able to derive this information for themajority of sources in the survey. Sub-mm photometric red-shift techniques, although yielding crude estimates (∼ ±0.4)for individual sources, are, however, a cheap way to reliablycharacterize the properties of the whole population at noadditional survey cost.

A Monte-Carlo based photometric-redshift techniquehas been designed by Hughes et al. (2002) and Aretxaga etal. (2003), where sub-mm photometric information is com-bined with prior information on the population, such as thenumber counts and the likely evolution of the luminosityfunction of dust-enshrouded galaxies, to weigh the outputredshifts provided by a large sample of template SEDs thatrepresents the wide range of temperatures, dust emissivitiesand luminosities found in nearby IR-bright galaxies. Thecalculation of the redshift probability distribution for eachsource comes as one of the benefits of this technique.

Even though the distributions are wide (∼ ±0.4), thedetailed information on the shape of the distribution, com-bined with a large number of sources, provides a powerfulstatistical measurement of population properties such as theparent redshift-distribution and the global star-formationrate. Hughes et al. (2002) demonstrated that the combina-tion of three to four detections at the 250/350/500/850 µmbands for SHADES yields redshift distributions that havetypical widths of ±0.4, and that the combination of detec-tions in only two of these bands provides distributions withtypical widths ∼ ±1.0.

While, naively, these measurements might seem insuffi-ciently crude, the combination of the redshift distributionsof hundreds of sources can indeed measure the history ofstar formation of the galaxies detected in more than twoSHADES bands (LFIR > 2 × 1013 L⊙) with an accuracyof ∼20% (Hughes et al. 2002), and provide sufficient con-straints to tune the broad-band receivers of large mm-cmfacilities in a quest to detect the CO rotational transitionsof a selected sample of bright sources (Aretxaga et al. 2003).

Moreover, the inclusion of additional photometric infor-mation provided by detections or upper limits at 1.4 GHz(from the VLA), and at 70–3µm (from the Spitzer SpaceTelescope), can help break the remaining degeneracies inthe analysis undertaken by Hughes et al. (2002), further in-creasing the accuracy of the photometric redshifts.

3 FOUR ALTERNATIVE MODELS

Four different models for the clustering of SCUBA galaxiesare presented, some of which are designed especially withSHADES in mind, while for other models the SCUBA pre-dictions are just part of a range of predictions. The modelsalso vary in the level of complexity, and in the underlyingassumptions, including the choices for the cosmological pa-rameters, even though differences in the latter are minorcompared to the fundamental differences between the mod-els.

3.1 A simple merger model

This simple merger model is a very simple model includedin order to help determine the important processes at workin the creation of SCUBA galaxies. The underlying premiseof this model is that 8-mJy SCUBA galaxies are formed byobscured star formation driven by the violent merger be-tween two galaxy sized haloes. The emission is assumed tobe above the 8-mJy detection threshold for a lifetime tlifeafter the galaxy haloes have merged. No direct link is madebetween the luminosity of the SCUBA galaxy and the prop-erties of the merger except that a lower limit is placed on thefinal mass of haloes that contain a detectable 8-mJy SCUBAsource. In other words, a Poisson sampling of massive halomergers is assumed to form bright SCUBA galaxies. We haveadopted a mass limit of 1013 M⊙, corresponding to ‘radio-galaxy’ mass haloes.

Halo mergers were found in a 2563 N-body simulationrun within a co-moving (100h−1Mpc)3 box using gadget,a publicly available parallel tree code (Springel, Yoshida& White 2001). Cosmological parameters were assumedto have their concordance values (Ωm = 0.3, ΩΛ = 0.7,h = 0.70 & ns = 1), and the power spectrum normaliza-tion was set at σ8 = 0.9. Outputs from the simulation wereobtained at 434 epochs, separated approximately uniformlyin time, and halos were found at each epoch using a stan-dard friends-of-friends routine with linking length b = 0.2.New halos were defined to be halos with > 50% of the con-stituent particles not having previously been recorded in ahalo of equal, or greater mass. Of these, the halo was saidto have been created by major merger if there were two pro-genitors at the previous time output that had mass between25% and 75% of the final mass.

Obtaining the right number density of SCUBA sourcesis limited by the definition of merging used, the lifetimeof emission above the detection threshold, and the propor-tion of mergers that result in SCUBA sources. We thereforesimply assume that all of the mergers, defined as above, re-sult in a luminous SCUBA source, and allow tlife to vary togive ∼300 sources in 0.5 deg2. Because there are relativelyfew mergers > 1013 M⊙, obtaining the correct number den-sity of SCUBA sources required a relatively long lifetimetlife = 8× 108 years.

Mock SCUBA catalogues for a 0.5 deg2 survey were cal-culated by placing a (co-moving) light cone through an ar-ray of simulation boxes. This is done by selecting outputtime-steps such that corresponding redshifts are separatedby a box length in co-moving coordinates. Boxes are re-flected, rotated and translated randomly to reduce the ar-tificial correlation between neighbouring boxes inherent in

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4 van Kampen et al.

using a single simulation, this necessarily reduces real cor-relations due to structures that would cross boxes. Mergersthat occurred less than the model lifetime before the timecorresponding to their luminosity distance were flagged aspotential SCUBA sources, and their angular positions andredshifts were recorded in order to create mock catalogues.

Obviously, while this model does predict both thespatial distribution and redshift space distribution of theSCUBA sources, it does not predict the luminosity func-tion. In fact, we note that following successful comparisonbetween analytic theory and numerical simulations, both theredshift space distribution and the spatial distribution ofSCUBA galaxies in this model could have been accuratelyestimated analytically (Percival, Miller & Peacock 2000; Per-cival et al. 2003).

3.2 A hydrodynamical model

At the heart of the model is a simulation from Muan-wong et al. (2002) which is an adaptive particle-particle,particle-mesh code incorporating smoothed particle hydro-dynamics (SPH). The underlying code is HYDRA (Couch-man, Thomas & Pearce 1995) with the addition of a stan-dard pair-wise artificial viscosity (Thacker et al. 2000). Thecosmological model is Ωm = 0.35, ΩΛ = 0.65, h = 0.71,σ8 = 0.9, Ωb = 0.019h−2 . The simulation used here em-ploys a box of co-moving size (100h−1Mpc)3 with 1603 darkmatter particles and 1603 gas particles, and is evolved be-tween 50 > z > 0 in approximately 2000 time-steps. Thesimulations have various components: non-interacting darkmatter; gas; “star-like” (same as gas, but forming stars);“galaxy-fragments”, which are collisionless. Evolution of thevarious components is as follows: all particles evolve undergravity; gas can adiabatically heat and cool; gas can alsoradiatively cool; at ρ/ρ > 500, T < 12, 000K gas particlesbecome “star-like” (at this point all the mass is deemed tohave been converted into stars). An aggregation of 13 ormore close “star-like” particles become a “fragment”. Frag-ments may accrete more star-like particles but do not merge.

As a complete model of galaxy formation this simulationhas a number of strengths and weaknesses. It provides aself consistent treatment of large-scale-structure and galaxyevolution. However, the limited resolution and the arbitrarysolution to “cooling catastrophe” necessitated by this, limitits validity. For the present purpose the full power of thesimulation is not used; it serves as an ingredient to a morephenomenological model.

The first step is to construct a “galaxy-fragment” light-cone in the usual way (as described in Section 3.1). Withthe SHADES sample area it is not necessary to use morethan a single box transverse to the line-of-sight.

The redshift distribution of the fragments in this cone(dN/dz)frag is measured using redshift bins of uniformwidth ∆z = 0.7, however there are many more fragments ineach bin than would be detectable. Each fragment is treatedas the possible location of a SCUBA source and is selectedbased upon its star-formation rate (SFR), which is measuredas the mass per unit time of ”star-like” particles accreted tothis fragment (averaged over the last output time-step). Therequired number of fragments with the greatest SFR areselected from each bin such that the redshift distributionmatches a particular model: (dN/dz)SCUBA. For this pa-

per, the analytical form of Baugh et al. (1996) was adoptedwith a median redshift of 2.3 and the normalization suchto give 300 sources in the full 0.5 deg2 sample size. Hence,the redshift distribution produced is not derived from thehydrodynamical simulations and this model merely makesa reasonable choice as to which fragments SHADES will in-clude, and for these, encodes the positional information fromthe simulations.

3.3 A stable clustering model

This is the model of Hughes & Gaztanaga (2000), in whicha single output from a N-body simulation that fits wellthe local spatial correlation function as measured by APM(Gaztanaga & Baugh 1998) is used to generate a populationof SCUBA galaxies in a lightcone. This corresponds to as-suming stable clustering, i.e. a constant spatial correlationfunction in co-moving space. Fixing the spatial correlationfunction does not imply that we fix the angular one as well,as that depends on lightcone geometry, luminosity evolu-tion, and the redshift selection function. The prescriptionfor galaxy formation corresponds to the assumption thatthe probability for finding a galaxy somewhere within thelightcone is simply proportional to the local dark matterdensity, with the total number of galaxies normalized to thesurface-density required for a given flux limit. Although theredshift distribution is (exponentially) cut-off beyond z = 6,this model contains the highest redshift SCUBA galaxies ofall models considered in this paper.

So far this model has mainly been used to test the pho-tometric redshift estimation technique of Aretxaga et al.(2003), to constrain sample size and depth given the corre-lation length, and to test correlation function measurementsfrom surveys with relatively small sky coverage (Gaztanaga& Hughes 2001).

3.4 A phenomenological model

The phenomenological galaxy formation model of van Kam-pen, Rimes & Peacock (2004), a revised version of themodel of van Kampen, Jimenez & Peacock (1999), is semi-numerical, in the sense that the merging history of galaxyhaloes is taken directly from N-body simulations which in-clude special techniques to prevent galaxy-scale haloes un-dergoing ‘overmerging’ owing to inadequate numerical reso-lution. When haloes merge, a criterion based on dynamicalfriction is used to decide how many galaxies exist in thenewly merged halo. The most massive of those galaxies be-comes the single central galaxy to which gas can cool, whilethe others become its satellites.

When a halo first forms, it is assumed to have anisothermal-sphere density profile. A fraction Ωb/Ωm of thisis in the form of gas at the virial temperature, which cancool to form stars within a single galaxy at the centre of thehalo. Application of the standard radiative cooling curveshows the rate at which this hot gas cools and is able toform stars. Energy output from supernovae reheats some ofthe cooled gas back to the hot phase. When haloes merge,all hot gas is stripped and ends up in the new halo. Thus,each halo maintains an internal account of the amounts ofgas being transferred between the two phases, and consumedby the formation of stars.

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predictions for SHADES 5

Figure 1. Two simulated distributions of 8-mJy sources similarto those to be observed by SHADES, for the phenomenologicalmodel described in Section 3.4, with no redshift selection. Thesize of the fields are those of the two fields that will make upSHADES: a quarter square degree each. The diameter of eachdot is proportional to the sub-mm magnitude of the source itrepresents.

The model includes two modes of star formation: qui-escent star formation in disks, and star-bursts during majormerger events. Having formed stars, in order to predict theappearance of the resulting galaxy it is necessary to assumean IMF, which is generally taken to be Salpeter’s, and tohave a spectral synthesis code, for which we use the spec-tral models of Bruzual & Charlot (1993). The evolution ofthe metals is followed, because the cooling of the hot gasdepends on metal content, and a stellar population of highmetallicity will be much redder than a low metallicity oneof the same age. It is taken as established that the popula-tion of brown dwarfs makes a negligible contribution to thetotal stellar mass density, and the model does not allow an

adjustable M/L ratio for the stellar population. The cos-mological model adopted is Ωm = 0.3, ΩΛ = 0.7, h = 0.7,σ8 = 0.93, Ωb = 0.02h−2. The sub-mm flux is assumed tobe proportional to the star formation rate, with a randomterms of order 50 per cent added or subtracted to mimicthe uncertainty in dust temperature, grain sizes, and otherproperties which are not yet included in the modelling.

The model used in this paper has a mixture of burstingand quiescent star formation, with most of the recent starformation occurring in discs, following the Schmidt law witha threshold according to the Kennicutt criterion, and mostof the high-redshift star formation resulting from merger-driven star-bursts.

4 MODEL PREDICTIONS COMPARED

In Fig. 1 we show a complete simulation of the two 850µmdatasets that will comprise SHADES, produced using thephenomenological model of van Kampen et al. (2004). A sim-ple square geometry was chosen, although the actual surveygeometry of each of the two SHADES fields could be of asomewhat different shape. All sources with fluxes larger than8 mJy are shown, where the symbol size is proportional tothe logarithm of the flux, i.e. a sub-mm magnitude.

We now consider simulated SHADES datasets as pre-dicted from the various alternative models of the sub-mmsource population presented in Section 3. We do not con-sider the effects of noise and sidelobes, but we do take intoaccount the effects of the 15 arcsec SCUBA beam by merg-ing into single sub-mm sources anything closer together than7 arcsec. This reflects the resolution expected for the finalsource extraction from the SHADES images. After the re-moval of close pairs we assume the model sub-mm sourcesto reflect the final SHADES source list.

The final survey is expected to contain around 300sources, i.e. 150 sources per field. We produce, for eachmodel, 50 realizations of individual fields, i.e. 25 mockSHADES datasets of 300 sources each.

4.1 Predictions for the redshift distribution

For the four models, we show in Fig. 2 the redshift distri-butions expected after smoothing with a Gaussian filter ofwidth 0.8, which reflects, very crudely, the resolution achiev-able with photometric redshifts. The distributions are ob-tained by averaging over all realisations for each model, andare normalized to the total source count of 300.

Even after relatively heavy smoothing, we see that theredshift distributions are rather different, and are clearly dis-tinguishable from each other. This means that even crudebut complete redshift information will be of enormous bene-fit in differentiating between and constraining models. Obvi-ously, obtaining more accurate redshifts should help to tunethe models that survive this first test even further.

For the purpose of comparing clustering properties be-tween the models, note from Fig. 2 that the redshift range2 < z < 3 is the only range where all models have a rea-sonable number of sources to attempt a correlation func-tion analysis. The differences in the distribution stem fromthe different assumptions for each of the models: the sim-ple merger model assumes that only high-mass mergers can

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6 van Kampen et al.

Figure 2. Redshift distributions for all models with 300 sources,averaged over 25 mock SHADES datasets for each model.

form SCUBA sources, which, in a hierarchical structure for-mation scenarion, necessarily places them at lower redshiftsas compared to the other models, while the simple, unbiasedgalaxy formation prescription of the stable clustering modelplaces SCUBA sources at relatively high redshifts.

4.2 Clustering measures

The estimated redshifts have a predicted accuracy of δz ∼±0.4, which means that we cannot directly measure the 3Dspatial correlation function ξ(r), and have to restrict our-selves to measuring angular clustering. However, the photo-metric redshifts can still be used to boost the angular clus-tering signal-to-noise by splitting the sample in redshift bins,or by only considering pairs of galaxies that lie at similarredshifts.

Even so, the measured angular correlation function willbe noisy, so we use integrals of this function, as consideredin the early days of optical galaxy surveys when total sourcecounts were much lower than today (e.g. Davis & Peebles1983).

4.2.1 Estimating the angular correlation function

The method for modelling the clustering of sources proceedsas follows. From the data, the Landy & Szalay (1993) esti-mator 1 +wLS = 1+ (DD− 2DR+RR)/RR is calculated,where DD, DR and RR are the (normalized) galaxy-galaxy,galaxy-random and random-random pair counts at separa-tion θ, calculated from the galaxy sample and a large randomcatalogue containing 10000 points that Poisson samples thesurvey region. This estimator is then fitted by its expectedvalue

1 + 〈wLS〉 = [1 + w(θ)]/(1 + wΩ) , (1)

where wΩ is the integral of the model two-point correlationfunction over the sampling geometry:

wΩ =

∫

Ω

Gp(θ)w(θ)dΩ . (2)

The function Gp(θ) is the probability density function offinding two randomly placed points in the survey at a dis-tance θ. This “integral constraint” corrects for the effect ofnot knowing the true density of objects (Groth & Peebles1977; Landy & Szalay 1993) and stops the recovered correla-tion function being biased to low values compared with thetrue function. Note that eq. (1) implies that the true cor-relation function is biased low by a factor 1 + wΩ, whereasoften this is approximated by w(θ) = wLS−wΩ, i.e. ignoringthe term wLSwΩ.

We also introduce an alternative to the standard an-gular correlation function which takes redshift informationinto account in an unorthodox way. In the counting of DDpairs, we just consider those pairs which have a redshift sep-aration of at most 0.4, whereas the DR and RR counts arestill obtained for all galaxy pairs. This is equivalent to re-moving distant pairs that are expected to be unclusteredfrom an analysis of the angular correlation function of allof the objects in the survey. It is clear that this approachmust increase the signal-to-noise of the recovered correlationfunction.

4.2.2 The sky-averaged angular correlation function

For the relatively small number of sources being detected inSHADES, fitting to an integral of this estimator should bebetter behaved. Such an approach has previously been usedto analyse clustering within early galaxy redshift surveys(eg. the CfA survey; Davis & Peebles 1983). The statisticthat was often obtained in these analyses was the integratedquantity J3, defined as

J3(r) ≡

∫ r

0

ξ(y)y2dy . (3)

The dimensionless analogue of J3 is called the volume-averaged correlation function:

ξ(r) ≡3

r3

∫ r

0

ξ(y)y2dy = 3J3(r)

r3. (4)

This measures the fluctuation power up to the scale r, andis therefore a useful measure for a survey which is limitedin object numbers. For reference, ξ(10h−1Mpc) = 0.83 wasfound for the optical CfA survey (Davis & Peebles 1983; noerror given).

As we cannot measure the spatial clustering functionξ(r), as the redshift determinations are very uncertain, weuse the two-dimensional version of ξ, the sky-averaged an-gular correlation function

w(θ) ≡2

θ2

∫ θ

0

w(φ)φdφ , (5)

where w(θ) is the angular correlation function, which is theprojection of ξ(r) along the line-of-sight. Our estimator ofthis statistic was calculated by numerically integrating theangular correlation function (calculated using the Landy &Szalay 1993 estimator), in the form of logarithmically binnedestimates wi, up to the angle θi using eq. (5):

wi =2

θ2i

∑

j≤i

wjθ2j∆ , (6)

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where ∆ is the logarithmic binsize. The errors on wi areobtained by propagating the errors on wi through this sum-mation. This estimate for the true sky-averaged angular cor-relation function is also biased, and has its own integralconstraint, similar to the one for w(θ). For a power-law cor-relation function wpl(θ) = (θ/A)−δ

wpl(θ) =2

2− δ

(

θ

A

)−δ

. (7)

Thus, the sky-averaged correlation function is also a power-law, with the same slope and slightly different amplitude(except for δ ≈ 2), and the integral constraint wΩ is scaledby the same factor 2/(2 − δ) with respect to wΩ (see eq.2). We can therefore fit power-law models to either w(θ) orw(θ), and constrain the same parameters.

4.2.3 Fitting to the model correlation functions

Traditionally, χ2 minimization is used to fit a power-lawfunction to the estimated correlation function. This typeof minimization is, strictly speaking, only valid for binneddata which is uncorrelated and has Gaussian errors. Ourestimates of w(θ) and w(θ) are correlated for different max-imum separations, and have errors that are non-Gaussian (itis easy to see that, in the absence of clustering, the errorsare strictly Poisson, as shown by Landy & Szalay (1993) forthe estimator of w(θ)). In order to constrain models of thecorrelation function using our binned estimates, we shouldtherefore perform a full likelihood calculation taking intoaccount the potentially complex shape of the likelihood. Agood example of such a method was published by Fisher etal. (1994). Both w(θ) and w(θ) obviously contain the sameinformation, and should therefore result in the same likeli-hood surface for a given model.

The reason for fitting w(θ) rather than w(θ) lies in theapproximations that are made in estimating the likelihood.For the sky-averaged angular correlation function, the binsare dependent on more pairs of galaxies than the correspond-ing direct estimator of the correlation function, so that thesky-averaged statistic will have a distribution closer to aGaussian form. Of course, switching from a direct estimateto a sky-averaged estimate increases correlations betweendata points. However, we only wish to find the likelihoodmaximum, rather than using the likelihood surface to pro-vide errors on the fitted parameters so, for a small numberof galaxies, the Gaussian behaviour is more important, andwe expect fitting to wi to be better behaved than fitting towi.

In fact, as we simply wish to observe basic trends be-tween models by reducing each measured correlation func-tion to a few model parameters, then we simply assume thatthe bins are independent and use the χ2 statistic to approxi-mately fit models to the data, instead of adopting a methodlike the one used by Fisher et al. (1994).

A single parameter fit to the correlation function is of-ten adopted assuming that w(θ) = (θ/A)−0.8 (e.g. Roche etal. 1993; Daddi et al. 2000). It is not at all clear what theslope for high-redshift sub-mm sources is going to be, but itis easier to fit a one-parameter function for a small number ofgalaxies. In the following we consider both a one-parametermodel with constrained power-law slope δ = 0.8, and a two-parameter fit for the generic power-law w(θ) = (θ/A)−δ to

both w(θ) and w(θ). We use non-linear χ2-fitting for bothfunctions, using the Levenberg-Marquardt method (Press etal. 1988), ;which also produces the covariance matrix of thefitted parameters, ;and thus a good estimate on their errors.which allows to easily take into account the multiplicativeintegral constraint. For each fit the χ2 probability Q is cal-culated using the incomplete gamma function, and any fitswith Q < 0.1 are discarded. Remember that because theconditions for using χ2 minimization are not met, the pa-rameter Q should be used with care, hence the somewhatliberal acceptance criterion.

4.2.4 Examples

In Figs. 3, 4 and 5 we show examples of w(θ) and w(θ) forrealisations of all models, for the case where we have noredshift information (Fig. 3) and for the case where we have(Fig. 4 and 5). The same realisations have been used forall three figures. Round symbols show the angular correla-tion function w(θ), where open symbols indicate negativevalues, and error bars indicate Poisson errors. The best fitpower-law is shown as a solid line, if a fit was possible. Ifnot, the line is simply omitted. The sky-averaged angularcorrelation function w(θ) is shown using stars, and the best-fitting power-law is shown as a dashed line (again, if a fitwas successful).

Before continuing, please note that these examples areby no means meant to be representative, they are merelyshown to demonstrate the difference between using w(θ) andw(θ) for fitting, and to show the effect of the additionalredshift information. The examples should not be used tocompare the differences in clustering strength between themodels; this will be covered in the next section.

First focusing on Fig. 3, we see that the correlationfunctions for the complete line-of-sight are noisy, and fortwo of the models a fit to w(θ) fails completely. The figuredemonstrates the use of sky-averaging, as w(θ), plotted usingstars, is better behaved. This is perhaps best illustrated inthe first panel, but also in the third panel, where it has thedirect consequence that a fit to w(θ) is possible where a fitto w(θ) failed (third panel of Fig. 3). In general, for mostrealisations a simple χ2 fit to w(θ) turns out to be easierthan a fit to the angular correlation function itself. Thisis very helpful for our purpose of comparing models, as itis important that fitting is possible for a large number ofrealisations, which needs to be largely automatic.

A stronger clustering signal is expected for sources se-lected in a redshift range, as the signal is less polluted byuncorrelated sources at very different redshifts. Indeed, Fig.4 shows that for all realisations the selection of sources inthe redshift interval 2 < z < 3 boosts the clustering signal.Even though the errors are larger due to the small number ofsources, the stronger signal means that a fit is possible in allcases shown, even for w(θ) itself, although the sky-averagedcorrelation function is to be preferred nevertheless. However,a fraction of realizations still produce unacceptable fits.

An alternative to redshift intervals is to only countgalaxies that are paired in redshift space, e.g. have |zi−zj | <0.4, as described in Section 4.2.1. The result for the samerealizations as used for Figs. 3 and 4 is shown in Fig. 5,and again much better results are obtained as compared tothe estimates without any redshift information (Fig. 3). The

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Figure 3. Bias-corrected angular correlation function w(θ)(round symbols, open meaning negative) and its sky-averagedcounterpart w(θ) (stars, negative values not plotted) for a sin-

gle realisation of each model, with best fit power-law functionsoverplotted for both, and no redshift selection. See text for fulldetails.

Figure 4. Same as Fig. 3, but with sources selected to be in theredshift range 2 < z < 3, which is the range where all modelshave a reasonable amount of sources in this redshift bin (see Fig.

2). All realisations are exactly the same as in Fig. 3, in order todemonstrate what redshift availability can achieve.

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Figure 5. Same as Fig. 3, but with sources selected to be inredshift pairs with δz < 0.4. All realisations are exactly the sameas in Figs. 3 and 4.

binned data looks cleaner than those for the redshift inter-vals (Fig. 4), which is due to more galaxies being used in theDR counts. The fitting is therefore somewhat more reliable,as demonstrated by the small difference between fits to w(θ)

Figure 6a. Scatter plot for the two fitting parameters of thesky-averaged angular correlation function, Asky and δsky, for 50fields of 150 sources. Only fits of sufficient quality, i.e. those with aχ2 probability larger than 0.1, are included (see text for details).

and w(θ). However, the clustering amplitudes are generallylower than for the redshift intervals.

4.2.5 Distribution over fitting parameters

So far we have considered single realisations for each model,which should really be treated as examples of how the finalSHADES dataset might appear. In order to be able to makea quantitative comparison possible between the actual finalSHADES dataset and all models, we need to find the proba-bility distribution over the fitting parameters for each modelgiven the survey constraints (area, flux limit, etc.).

We therefore produced 50 realizations for each model,and fitted a power-law correlation function to all of these.The resulting amplitudes Asky and slopes δsky of the sky-averaged correlation function w(θ) are shown as scatter plotsin Fig. 6a, which shows the case where no redshift informa-tion is available, and in Figs. 6b and 6c, where we are ableto split up the sample in redshift intervals (three of theseare shown), or select pairs of galaxies in redshift space.

In the case of no redshift information, a significant frac-tion of the mock fields do not produce a correlation functionwhich can be fitted by a power-law, and the number of es-timates in Fig. 6a is therefore less than 50 for each model.However, for each model still more than half the realisa-tion allow a good fit, so crudely speaking one would expectthat at least one of the two observed SHADES fields shouldproduce a good fit. All models spread out over a relativelylarge region of parameter space, and seem to overlap witheach other for most of that region. This merely reflects thefact that whatever intrinsic correlation exists in the under-lying sub-mm population is weakened by projection, whichproduces this large spread in fitting parameters and the sig-nificant overlap between the models. Only the high-massmerger model shows a larger clustering amplitude overall,which also results in a larger fraction of the realisations pro-ducing good fits, and a smaller spread in the slope δ.

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Figure 6b. As Fig. 6a, but for three different redshift bins. Againonly parameters from ‘good’ fits are shown, i.e. those with a χ2

probability larger than 0.1.

Figure 6c. As Fig. 6a, but for close redshift pairs with δz < 0.4.Again only parameters from ‘good’ fits are shown, i.e. those fitswhich have a χ2 probability larger than 0.1.

Let us now use the redshift information to split upthe mock samples into redshift intervals, which should showstronger clustering but larger errors (due to the smaller num-ber of sources). Three intervals are shown in Fig. 6b, wherethe 2 < z < 3 case is the most relevant one as it has the mostsources for all models. The other two intervals both haveat least one model where the number of sources is signifi-cantly lacking, which means that only the remaining modelscan reasonably be compared. The first thing to notice in allthree panels of Fig. 6b is that the clouds of fitted parameterpairs start to separate out somewhat, reducing the overlapbetween the models.

Various interesting effects can be seen for the differentmodels. The stable clustering model (open squares) showsfairly strong clustering in the 1 < z < 2 redshift interval, butbecause of the low number of sources in this redshift range(see Fig. 2), few of the realisations actually produce a goodfit. The high mass merger model (open diamonds) shows rel-atively strong clustering for 1 < z < 3, but of course lacknumbers in the highest redshift interval. The phenomeno-logical model (crosses) shows strongest clustering in thatsame 3 < z < 4 interval, but this is also somewhat troubledby low source counts. It also shows the weakest clusteringfor the lowest redshift interval. The clustering strength ofthe hydrodynamical model (open triangles) is pretty muchindependent of redshift.

In Fig. 6c we show the results for the close pairs, i.e.galaxies with |δz| < 0.4. The points scatter in a similarfashion to the 2 < z < 3 interval, with some differences,and the separating of model point clouds is comparably, ex-cept for the simple merger model, which can quite clearlybe distinguished. This diagram should thus be a good testfor high mass merging versus the other models consideredin this paper.

If we concentrate on the 2 < z < 3 interval, which al-lows the cleanest comparison between the four models con-sidered here, and on the close pairs, we do see that the clouds

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Figure 7. Distribution of clustering amplitude Asky over fifty re-alisation for each model, for the redshift bin 2 < z < 3, where each

model has a sufficient number of sources available (top panel), andfor close redshift pairs with δz < 0.4 (bottom panel).

of points are overlapping significantly, but the distributionsare elongated somewhat along the δ axis, and seem to havedifferent mean clustering amplitudes A. Also, the mean ofthe distribution over δ is near the δ = 0.8 slope which isoften found for a whole range of galaxy populations, whichis perhaps surprising.

This leads us to finally consider the traditional one-parameter fit to the data, assuming δ = 0.8. This producesa single clustering amplitude Asky for each mock field, anda distribution over Asky for each model. These distributionsare plotted in Fig. 7, for the redshift interval 2 < z < 3(top panel), and for the close pairs. Interestingly, for theredshift interval all distributions are different, and althoughthere is significant overlap, having two fields in the finalSHADES dataset means that one should be able to distin-guish between the models, especially in combination withthe different redshift distributions (see Fig. 2). For the closepairs, the result of Fig. 6c is made more apparent, in thatthe high-mass merger model is clearly different from the restof the models, which show almost identical distributions.

Figure 8. Distribution of the angular correlation measure w(1′)over fifty realisation for each model, again for the redshift bin

2 < z < 3 (top panel), and the close redshift pairs with δz < 0.4(bottom panel).

Another measure of clustering is the sky-averaged angu-lar correlation function as it was originally intended: just asa measurement. Therefore we also plot, in Fig. 8, the distri-bution over a particular w(θi), which we choose to be w(1′),i.e. the sky-averaged correlation function within 1 arcmin,for the same redshift selected data as used for Fig. 7. Theresult is a similar, although the distributions overlap morethan those for Asky (as seen in Fig. 7). However, the majoradvantage with the measure w(1′), or one at a different an-gle, is that we do not need to assume a model for the formof the correlation function.

5 CONCLUSIONS

One of the primary science drivers for the SHADES projectis to place strong constraints on galaxy formation modelsby observations of luminous sub-mm galaxies in the highredshift Universe. In order to achieve this, it is worth con-sidering how models are best constrained by the data, andexamine the range of possible predictions. With this aim

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in mind, we have presented 4 different models of the sub-mm galaxy population, selected to be as widely varying inconcept as possible, without worrying about every aspect ofeach model.

For each model, the redshift distribution and clusteringproperties of the sub-mm population were predicted for thecurrently on-going SHADES project. These models rangefrom simple ones built specifically for this purpose to moreelaborate ones for which the sub-mm properties are part of asuite of predictions, often for many wavebands and redshiftranges. For each of these models, 50 realisations were pro-duced, each comprising around 150 sources. Thus, 25 mockSHADES datasets have been produced for each model.

These simulated SHADES catalogues were used to in-vestigate the ability of the clustering statistics of the finalSHADES dataset to constrain the various models collectedhere. Direct and sky-averaged estimators of the correlationfunction have been considered and their relative merits dis-cussed. We have argued that power-law fits are best per-formed on the sky-averaged angular correlation function,and that a relatively good fit is possible in most cases. Mostof the models predict sufficiently strong clustering, so thatwe expect to detect clustering within the SCUBA popula-tion when SHADES is complete. Although cosmic varianceremains a concern, we can certainly quantify the probabil-ity of a given model to produce the observed dataset, andthis is expected to reject some of the models included in thispaper.

The observed redshift distribution will provide an com-plementary strong test of models, even with relatively coarsephotometric redshift information. In fact, the combintaion ofclustering and redshift data offers the best discriminator be-tween the different models that we have considered: modelswith similar redshift distributions have different clusteringstrengths, while models with similar clustering propertieshave different redshift distributions.

The primary conclusion from our analysis is that, withthe area coverage and the expected number of sources, andparticularly with the expected photometric redshift infor-mation, SHADES is capable of distinguishing between cur-rently widely varying scenarios for the production of thebright sub-mm population.

ACKNOWLEDGEMENTS

This research was supported in part by the Austrian Sci-ence Foundation FWF under grant P15868, and by twoPPARC rolling grants. EvK thanks Duncan Farrah and TomBabbedge for many useful comments and suggestions. DHHand IA are partly supported by CONACyT grants 39953-Fand 39548-F.

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