An ALMA survey of the SCUBA-2 Cosmology Legacy SurveyUKIDSS/UDS Field: Halo Masses for Submillimetre Galaxies
S. M. Stach, I. Smail, A. Amvrosiadis, A. M. Swinbank, U. Dudzevi?iūt?, J. E. Geach, O. Almaini, J. E. Birkin, Chian-Chou Chen, C. J. Conselice, E. A. Cooke, K. E. K. Coppin, J. S. Dunlop, D. Farrah, S. Ikarashi, R. J. Ivison, J. L. Wardlow
MMNRAS , 1–13 (2020) Preprint 26 February 2021 Compiled using MNRAS L A TEX style file v3.0
An ALMA survey of the SCUBA-2 Cosmology Legacy SurveyUKIDSS/UDS field: Halo Masses for Submillimetre Galaxies
S. M. Stach, ★ I. Smail, A. Amvrosiadis, A. M. Swinbank, U. Dudzeviči ¯ut˙e, J. E. Geach, O. Almaini, J. E. Birkin, Chian-Chou Chen, C. J. Conselice, E. A. Cooke, , K. E. K. Coppin, J. S. Dunlop, D. Farrah, , S. Ikarashi, R. J. Ivison, J. L. Wardlow Centre for Extragalactic Astronomy, Department of Physics, Durham University, Durham, DH1 3LE, UK Centre for Astrophysics Research, School of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield AL10 9AB, UK School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD, UK Academia Sinica Institute of Astronomy and Astrophysics, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK Department of Physics and Astronomy, University of Hawaii, 2505 Correa Road, Honolulu, HI 96822, USA Institute for Astronomy, 2680 Woodlawn Drive, University of Hawaii, Honolulu, HI 96822, USA European Southern Observatory, Karl Schwarzschild Strasse 2, Garching, Germany Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
Accepted —. Received —; in original form —
ABSTRACT
We present an analysis of the spatial clustering of a large sample of high-resolution, inter-ferometically identified, submillimetre galaxies (SMGs). We measure the projected cross-correlation function of ∼
350 SMGs in the UKIDSS Ultra Deep-Survey Field across a redshiftrange of 𝑧 = ( 𝑀 halo [ ℎ − M (cid:12) ]) ∼ ( 𝑀 halo [ ℎ − M (cid:12) ]) ∼ 𝑧 = 𝑧 = Key words: galaxies:starburst – galaxies:high-redshift – submillimetre:galaxies
Submillimetre galaxies (SMGs) are a population of high-redshiftdusty galaxies (typically 𝑧 ∼ 𝐿 IR ) ∼ − L (cid:12) (see: Casey et al. 2014, for review). It isbelieved that the majority of this far-infrared emission corresponds ★ E-mail: [email protected] to dust-reprocessed radiation from recent star formation, with theluminosity of this emission implying high dust masses ( (cid:38) M (cid:12) ),and high star-formation rates ( >
100 M (cid:12) yr − ), and thus SMGs aresome of the most massive and rapidly star-forming galaxies inthe Universe. In addition, their selection at submillimetre wave-lengths ( ∼ 𝜇 m) corresponds to the Rayleigh-Jeans tailof the galaxy spectral energy distribution (SED) in high-redshiftgalaxies, which results in a strongly negative 𝑘 -correction (figure4: Blain et al. 2002). As a result, at a fixed observed wavelength inthe submillimetre, as the redshift of an SMG is increased the SED © a r X i v : . [ a s t r o - ph . GA ] F e b Stach et al. is sampled along the rising Rayleigh-Jeans tail and this increasingbrightness approximately cancels out the luminosity dimming fromthe increasing distance out to 𝑧 ∼
7. Therefore, submillimetre sur-veys for SMGs provide effectively volume-limited probes of stronglystar-forming galaxies with high dust mass and by implication highgas mass, in the high-redshift Universe.Given the estimated gas masses and star-formation rates ofSMGS (e.g. Bothwell et al. 2013; Dudzeviči¯ut˙e et al. 2020), theirextreme star formation rates can only be a relatively short lived( ∼
200 Myr e.g. Birkin et al. 2021) and much work has been un-dertaken to understand where this infrared-bright phase fits intoa larger evolutionary pathway for SMGs. One suggestion is theSanders et al. (1988) scenario for the local Universe analogues:ultra-luminous infrared galaxies (ULIRGs, with 𝐿 IR ≥ L (cid:12) ),which places the strongly star-forming ULIRGs as an intermediatephase following a galaxy merger and preceding a resultant quasarphase, with the present-day descendant being a massive passivespheroidal galaxy. There is circumstantial evidence for this linkfrom the redshift distributions of SMGs and quasars which peak atsimilar redshifts (Chapman et al. 2005; Wardlow et al. 2011; As-sef et al. 2011; Dudzeviči¯ut˙e et al. 2020). In addition a number ofother observational tests are claimed to support this evolutionarylink (e.g. Swinbank et al. 2006; Tacconi et al. 2008; Hainline et al.2011; Simpson et al. 2014, 2017; Hodge et al. 2016; Stach et al.2019; Dudzeviči¯ut˙e et al. 2020). However, these tests are uncertainas they rely on measurements and models that are poorly constrainede.g. stellar masses and star-formation histories (e.g. Hainline et al.2011; Michałowski et al. 2012).Another method for contextualising the evolution of a galaxypopulation is through measurements of their spatial clustering,which is linked to the masses of their dark matter haloes (Peebles1980). With inferred dark matter halo masses, the present-day halomasses for a galaxy population can be estimated based on the darkmatter mass assembly histories from N-body simulations (Fakhouriet al. 2010). Using these, comparisons can then be made with clus-tering measurements of the proposed evolutionary descendants inthe local Universe. This spatial clustering method has been appliedto SMGs (e.g. Blain et al. 2004; Weiß et al. 2009; Cooray et al.2010; Lindner et al. 2011; Hickox et al. 2012; Chen et al. 2016b;Wilkinson et al. 2017; Amvrosiadis et al. 2019) and where cluster-ing signals could be detected there is general agreement that SMGsreside in massive dark matter haloes of mass 𝑀 halo ∼ − M (cid:12) .This halo mass is broadly in agreement with those expected foran evolutionary track connecting QSOs (Croom et al. 2005; Myerset al. 2006; Hickox et al. 2011) and local massive spheroidal galax-ies (Quadri et al. 2007; Zehavi et al. 2011), supporting a Sanderset al. (1988)-like evolutionary model.The major difficulties with measuring the clustering signal forSMGs are the relative low number densities of SMGs, resulting insmall sample sizes, and the reliance on uncertain identification andsimilarly uncertain photometric redshifts, due to the challenges andexpense of obtaining spectroscopic redshifts for SMGs. Hickox et al.(2012) attempted to minimise these issues by cross-correlating asmall sample of probable SMGs (some with spectroscopic redshifts)in the Extended Chandra
Deep Field-South, with a larger galaxysample in the same field from the
Spitzer
InfraRed Array Camera(IRAC). In addition they adopted the Myers et al. (2009) methodfor incorporating the information in the full probability distributionfunction (PDF) for the photometric redshift estimations for the IRACgalaxies to improve the resulting clustering signal. With this methodthey derived an auto-correlation length for ∼
50 SMGs in the redshift range 𝑧 = 𝑟 = + . − . ℎ − Mpc which corresponded to adark matter halo mass of log ( 𝑀 halo [ ℎ − M (cid:12) ]) = + . − . .More recently, larger submillimetre samples have becomeavailable, as degree-scale extra-galactic fields have been mappedat submillimetre wavelengths with single-dish facilities, increas-ing the precision of clustering measurements (Chen et al. 2016b;Wilkinson et al. 2017; Amvrosiadis et al. 2019; An et al. 2019; Limet al. 2020). These larger surveys allow the samples to be split byredshift to measure the evolution in clustering strength as a functionof redshift, however currently there are disagreements about thetrends found. Wilkinson et al. (2017) found redshift evolution suchthat SMG activity occurs in more massive dark matter halo masses( 𝑀 halo ∼ M (cid:12) ) at higher redshifts ( 𝑧 >
2) and in lower masshaloes ( 𝑀 halo ∼ M (cid:12) ) at 𝑧 <
2. Contrary to this, the observedclustering measurements from Chen et al. (2016a), Amvrosiadiset al. (2019), and An et al. (2019) (as well as results from semi-analytical models of Cowley et al. 2016) suggest SMGs inhabithaloes of 𝑀 halo ∼ M (cid:12) at all redshifts.Finding the source of this disagreement is complicated by thediffering methods used to identify SMGs from the low-resolutionsingle-dish maps, which are known to suffer from source blending(e.g. Karim et al. 2013; Stach et al. 2018). All of these studies relyon probabilistic radio, mid-infrared and colour-selection for identi-fications that are known to be incomplete and contaminated (Hodgeet al. 2013). Any mis-identification of the SMGs can have dramaticeffects on the resulting clustering measurements and could be re-sponsible for the conflicting claims about the halo mass evolution.For example, with the availability of robust identifications for sam-ples of SMGs using the Atacama Large Millimetre/submillimetreArray (ALMA), García-Vergara et al. (2020) has suggested thatsingle-dish clustering studies could be overestimating the SMG halomasses by as much as 3.8 + . − . times their true value. Therefore, toobtain robust results such an analysis needs to be based on a largesample of SMGs across a contiguous field that are accurately iden-tified through submillimetre interferometry at sub-arcsecond reso-lution to yield a precise and accurate measurement of SMG halomasses.We have recently completed an ALMA follow-up survey of the ∼
700 submillimetre sources in the 850- 𝜇 m map of the UKIDSSUltra Deep Survey (UDS) field obtained by the SCUBA-2 Cosmol-ogy Legacy Survey (S2CLS, Geach et al. 2017). The S2CLS UDSmap reached a median sensitivity of 𝜎 = and all 716 single-dishes sources detected at ≥ 𝜎 significance were imaged at 870 𝜇 m with ALMA as the ALMASCUBA-2 UDS Survey (AS2UDS) (Simpson et al. 2015; Stachet al. 2018, 2019). This resulted in the largest, homogeneously-selected sample of SMGs to date across a contiguous field withexcellent multi-wavelength coverage from which robust photomet-ric redshifts could be derived both for the ALMA-detected SMGsand the > 𝐾 -detected galaxies in this field (Dudzeviči¯ut˙eet al. 2020). In this paper we present the results of the projectedtwo-point cross-correlation analysis of the SMGs with the 𝐾 -banddetected field galaxy sample (Almaini et al. in prep.) utilising thefull redshift PDFs from Dudzeviči¯ut˙e et al. (2020) to constrain SMGclustering at redshifts 𝑧 = MNRAS , 1–13 (2020)
S2UDS: Clustering of SMGs SMGs. In §3 we present the results and discussion of our clusteringanalysis, including the dark matter halo masses as a function ofredshift for our sample and the comparisons with previous SMGclustering results. §4 presents our main conclusions. Throughoutthis paper we assume a
Planck
XIII cosmology with Ω m = = − Mpc − (and using the standard definition for ℎ from H = ℎ km s − Mpc − ), and for the amplitude of the mat-ter power spectrum we use 𝜎 = Our clustering analysis employs a similar cross-correlation methodas used by Hickox et al. (2012). We focus on this methodology asit allows the inclusion of spectroscopic and photometric redshiftinformation in the clustering analysis and hence it is likely to beincreasingly adopted in future studies as the available redshift infor-mation expands on submillimetre galaxy samples. We therefore startby defining four catalogues: an SMG catalogue (‘SMGs’), a compar-ison population within the same volume as the SMGs (‘Galaxies’),and randomised distributions for both SMGs and the comparisonsample (‘Random’). For the SMG catalogue we use as a basis the707 SMGs in the catalogue from Stach et al. (2019)’s AS2UDSALMA survey in the UKIRT Infrared Deep Sky Survey (UKIDSS,Lawrence et al. 2007) Ultra-Deep Survey field (UDS, Almaini etal. in prep.), which we briefly discuss here (for a full descriptionsee: Stach et al. 2019; Dudzeviči¯ut˙e et al. 2020). The AS2UDS sur-vey obtained ALMA Band 7 (870 𝜇 m) continuum observations ofall > 𝜎 sources from the S2CLS SCUBA-2 850- 𝜇 m map of theUDS region (Geach et al. 2017). This SCUBA-2 map also formedthe basis for the earlier Wilkinson et al. (2017) clustering analy-sis, that relied upon probabilistically-identified radio, mid-infraredand colour-selected counterparts to the single-dish submillimetresources (some of which were subsequently shown to be incorrect,see An et al. (2018)). In contrast, we now have robust ALMA inter-ferometric identifications, at 870 𝜇 m and ∼ (cid:48)(cid:48) resolution, of thetrue counterparts to the single-dish sources. The S2CLS UDS mapreached a median depth of 0.9 mJy beam − across the 0.96 deg (Geach et al. 2017). ALMA continuum mapping of 716 single-dishsources located 708 submillimetre galaxies at > 𝜎 significance(spanning a flux range of 𝑆 = 𝑢𝑣 -tapered to 0.5 (cid:48)(cid:48) resolution. In ad-dition, previously discovered very bright, 𝑧 = Figure 1.
The redshift distributions for the SMG sample from the AS2UDSsurvey in comparison to the probabilistically-identified SMGs in the sameredshift range from the LESS survey analysed by Hickox et al. (2012)(hatched histogram). This illustrates the ∼ × increase in sample size thatcomes from our analysis of the deep ∼ UDS field compared tothe somewhat shallower ∼ LESS survey. In addition, our anal-ysis benefits from robust and unambiguous ALMA-identified counterparts,compared to the earlier study. The redshifts for the AS2UDS sample arethe photometric redshifts reported for these ALMA-identified SMGs fromDudzeviči¯ut˙e et al. (2020), whilst the LESS sample are a combination ofspectroscopic redshifts and photometric redshifts from Wardlow et al. (2011)for the probabilistic radio/mid-infrared-identified counterparts (subsequentALMA studies are reported by Hodge et al. 2013; Danielson et al. 2017).Also shown is the redshift distribution for the 𝐾 -selected field galaxies thathave been redshift-matched to the SMG sample as discussed in §2.1, that areused for the projected cross-correlation analysis with the SMGs, the scalefor this sample is shown on the right-hand ordinate. in the vicinity of brighter sources, rather than due to intrinsicallyclustered sources, a point we return to in our analysis. Finally, toassess the influence of any inhomogeneity in the properties of theoriginal SCUBA-2 catalogue which was the basis for our ALMAfollow-up observations, we also investigate the effect of applyinga 𝑆 𝜇𝑚 ≥ 𝜎 in theSCUBA-2 map) cut on our ALMA sample, as the parent single-dish catalogue is expected to be close to ∼
100 per cent completefor sources brighter than this limit.
Photometric redshifts and other physical parameters (as well as theirassociated probability density functions) were estimated by Dudze-viči¯ut˙e et al. (2020) through spectral energy distribution (SED)fitting of the multi-wavelength ultra-violet–to–radio coverage in theUDS field using the magphys modelling code (da Cunha et al. 2008,2015; Battisti et al. 2019). For full details of the results from themagphys analysis and the extensive testing of these, see Dudze-viči¯ut˙e et al. (2020). Here we provide a brief description of thetesting of the photometric redshifts that is relevant to this work.The uncertainties on the redshifts (and thus the broadnessof their PDFs) is reliant in part on the number of constraints tothe SED, i.e. the number of photometric bands with detections orlimits for each galaxy. For the subset of optically-bright SMGswith detections in all 22 photometric bands available, the mag-
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MNRAS000 , 1–13 (2020)
Stach et al. phys PDFs are narrow with a 16–84 th percentile range of Δ 𝑧 ∼ Δ 𝑧 ∼ Δ 𝑧 /(1+ 𝑧 spec ) = − ± Δ 𝑧 /(1+ 𝑧 spec ) = − ± 𝑧 𝑠 ) we measure how many lie within their expected confidenceinterval of their respective predicted photometric redshift PDFs, i.e.do 1 per cent of the 𝑧 𝑠 lie within the 1 per cent confidence interval,10 per cent within the 10 per cent confidence interval, etc. This isachieved by measuring the fraction of the PDF that lies within theredshift intervals where the distribution is greater than the value ofthe PDF at the spectroscopic redshift 𝑝 ( 𝑧 𝑠 ) , i.e.: 𝑐 𝑖 = ∑︁ 𝑧 ∈ 𝑝 𝑖 ( 𝑧 ) ≥ 𝑝 𝑖 ( 𝑧 𝑠,𝑖 ) 𝑝 𝑖 ( 𝑧 ) (1)where 𝑝 𝑖 ( 𝑧 ) is the PDF for the 𝑖 th SMG with a spectroscopicredshift and 𝑐 𝑖 is the resulting threshold credibility. The empiri-cal cumulative distribution function of these threshold credibilities,ˆ 𝐹 ( 𝑐 ) , should then follow a one-to-one relation with 𝑐 if the redshiftPDFs were accurately measuring the uncertainties in the photomet-ric redshifts. If the cumulative distribution falls below the unityrelation then it suggests the PDFs are underestimating the uncer-tainties (the peaks are too narrow) and likewise if the distribution isabove the line then this suggests the PDFs are over-estimating theuncertainties. In Figure 2 we show the cumulative distribution plotfor our AS2UDS SMGs which shows the hint of an underestimationin the redshift uncertainties but a Kolmogorov-Smirnov test of theSMGs against the ideal one-to-one relation finds a probability ofthis occuring by chance of 12 per cent suggesting that this is nota statistically significant deviation. Therefore, from the sample ofSMGs for which we have spectroscopic redshifts, we conclude thatmagphys returns both accurate photometric redshifts and represen-tative uncertainties. In addition, we tested the potential impact ofan underestimation or overestimation of the broadness of the mag-phys derived redshift PDFs by replacing the PDFs of sources in ouranalysis with Gaussians of varying widths, greater and smaller than Δ 𝑧 ∼ K -band Galaxies For the cross-correlation analysis we make use of the UKIDSS UDSDR11 catalogue (Almaini et al. in prep.), a 𝐾 -band selected cata- Figure 2.
The empirical cumulative distribution function of the thresholdcredibilities for those AS2UDS SMGs with spectroscopic redshifts. Theinterpretation of the trend in this plot is that a sample that falls belowthe one-to-one relation, shown in black, results from photometric redshiftPDFs that are on average narrower than expected given their spectroscopic tophotometric redshifts offsets, whilst samples with lines above the one-to-onerelation have overly broader PDFs and are thus on average overestimatingthe photometric redshift uncertainties. For our magphys PDFs we find thatthe trend is consistent with the one-to-one relation and hence that the PDFsappear to accurately reflect the uncertainties expected from the observedoffsets of the photometric redshifts compared to the available spectroscopicredshifts. logue that covers the majority of the S2CLS map of the UDS field(634/707 SMGs covered) which has been matched to up-to 21 otherphotometric bands from the ultra-violet to radio (these same photo-metric data are used to model the SMG sample, see An et al. 2018;Dudzeviči¯ut˙e et al. 2020). The DR11 catalogue contains 296,007sources extracted from the 𝐾 -band image with a median 5- 𝜎 depthof 𝐾 = 𝐾 -band‘Galaxy’ catalogue and the AS2UDS ‘SMG’ catalogue and the twoassociated ‘Random’ catalogues. These ‘Random’ catalogues weremade by randomly assigning spatial positions for galaxies withinthese unmasked regions at approximately ten times the density ofSMGs to create the Random SMG catalogue and, again, approx-imately ten times the density of the 𝐾 -band galaxies to form aRandom Gal . Then each source in these catalogues was assigned aredshift by sampling the associated mean photometric redshift PDFsfor the ‘real’ SMGs and galaxies.To better define our ‘Galaxy’ sample we first note that thefraction of the 𝐾 -band selected galaxy sample at redshifts 𝑧 > 𝑧 <
1, thus toensure a statistically robust Galaxy sample to cross-correlate with
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S2UDS: Clustering of SMGs our SMGs we restrict our clustering measurements to 𝑧 = 𝐾 -selected galaxies (Figure 1). Next, we followprevious clustering studies in the UDS field (Hartley et al. 2013;Wilkinson et al. 2017) and apply a 90 per cent mass completenesslimit using our new magphys masses. These previous studies haveapplied a redshift dependant mass limit, but from our experimen-tation the resulting clustering results are insensitive to the minorevolution in the mass completeness limits with redshift across theredshift range of interest. We therefore apply a uniform 90 per centmass completeness limit for our 𝑧 = 𝑀 lim >10 . M (cid:12) to all redshift bins, removing the lowest stellarmass galaxies that have no analogues in the SMG sample. Finally,because the redshift distribution for the mass-limited galaxy sam-ple peaks at a significantly lower redshift ( 𝑧 = 𝑧 = ± 𝐾 -band galaxy catalogue such that their resultingredshift distribution approximately matches the distribution of theSMGs (Figure 1). To derive the halo masses of our sample of SMGs we have toestimate their two-point correlation function 𝜉 ( 𝑟 ) . This function isdefined as the probability 𝑃 above Poisson of finding two galaxiesphysically separated a distance 𝑟 in a volume element 𝑑𝑉 , i.e.: 𝑑𝑃 = 𝑛 [ + 𝜉 ( 𝑟 )] 𝑑𝑉, (2)where 𝑛 is the mean space density of the galaxies. In the projectedtwo-point correlation function we project the 𝑟 separation into twocomponents, perpendicular ( 𝑟 𝑝 ) and parallel ( 𝜋 ) to the line-of-sight.The projected correlation function 𝑤 𝑝 ( 𝑟 𝑝 ) is then defined as theintegral of the correlation function, 𝜉 ( 𝑟 ) , over the line-of-sight: 𝑤 𝑝 ( 𝑟 𝑝 ) = ∫ 𝜋 max 𝜉 ( 𝑟 𝑝 , 𝜋 ) 𝑑𝜋. (3)Following Davis & Peebles (1983) we measure the line-of-sightseparations from the co-moving radial distances, 𝐷 , derived fromtheir redshifts ( 𝜋 = 𝐷 − 𝐷 ). The perpendicular components foreach galaxy pair can then be calculated from the on-sky separations( 𝜃 ) and radial co-moving distances simply using the cosine rule: 𝑟 𝑝 = [ 𝐷 𝐷 ( − cos 𝜃 )] / (4)By integrating the correlation function by a suitable distancealong the line-of-sight the issues of redshift space distortions (Kaiser1987) owing to the peculiar velocities of the galaxies are removed. 𝜉 ( 𝑟 ) can be approximated by a simple power-law in the form 𝜉 ( 𝑟 ) = ( 𝑟 / 𝑟 ) − 𝛾 and we choose to fix 𝛾 = 𝜉 ( 𝑟 ) then, from Peebles (1980), this can berelated to the projected cross correlation function 𝑤 𝑝 ( 𝑟 𝑝 ) using: 𝑤 𝑝 ( 𝑟 𝑝 ) = 𝑟 𝑝 (cid:18) 𝑟 𝑟 𝑝 (cid:19) 𝛾 Γ ( / ) Γ [( 𝛾 − )/ ] Γ ( 𝛾 / ) , (5)where Γ is the gamma function. Therefore from a simple power-law fit to the projected correlation function we can directly estimate thecorrelation length 𝑟 for the corresponding galaxies. The power-lawparameterisation does assume we integrate to 𝜋 𝑚𝑎𝑥 = ∞ , howeverthe integral in Equation 2 is in practice limited to a set co-movingdistance which has to be large enough to recover all the clusteringsignal, but small enough to reduce the noise from including uncor-related pairs at larger separations. For this work we chose a valueof 𝜋 𝑚𝑎𝑥 = ℎ − Mpc that is consistent with Hickox et al. (2011,2012) who also fitted projected cross-correlation functions usingphotometric redshifts and their PDFs in a similar redshift rangeto this work. There are however other studies with projected cor-relation functions using photometric redshifts which apply larger 𝜋 𝑚𝑎𝑥 values of ∼
400 Mpc h − (Georgakakis et al. 2014), to tryand encapsulate the larger uncertainties inherent with photometricredshifts. To test the impact of increasing 𝜋 𝑚𝑎𝑥 we estimated 𝑟 val-ues for our full redshift sample described below with the increased 𝜋 𝑚𝑎𝑥 =
400 Mpc h − . We found that the 𝑟 value was largely in-sensitive to this increase, showing only an ∼
10 per cent increasein 𝑟 from 𝜋 𝑚𝑎𝑥 =
100 Mpc h − , which is well encapsulated by thesubstantial uncertainties in our derived 𝑟 values, thus we retain 𝜋 𝑚𝑎𝑥 =
100 Mpc h − for consistency with Hickox et al. (2012).To estimate the correlation function we use the Landy & Szalay(1993) estimator for cross-correlation: 𝜉 (cid:0) 𝑟 𝑝 , 𝜋 (cid:1) = 𝐷 SMG 𝐷 Gal − 𝐷 SMG 𝑅 SMG − 𝑅 Gal 𝐷 Gal + 𝑅 SMG 𝑅 Gal 𝑅 SMG 𝑅 Gal (6)where 𝐷 SMG 𝐷 Gal is the normalised number of SMG–Galaxy pairs, 𝐷 SMG 𝑅 SMG
SMG–Random
SMG , 𝑅 Gal 𝐷 Gal
Galaxy–Random
Gal and 𝑅 SMG 𝑅 Gal the Random
SMG –Random
Gal pairs at separations 𝑟 𝑝 ± Δ 𝑟 𝑝 and 𝜋 ± Δ 𝜋 . For our cross-correlation analysis we calculatedpair counts in logarithmic 𝑟 𝑝 bins in the range 0.05–14 ℎ − Mpc,which at the median redshift of the SMG sample ( 𝑧 ∼ ∼ (cid:48)(cid:48) .To incorporate the photometric redshift PDFs we measure theprojected correlation function with a Monte Carlo method by repeat-ing the projected correlation as a function of 𝑟 𝑝 bins whilst samplingthe redshifts of every SMG and galaxy by randomly selecting fromtheir respective PDFs. We set the contribution to the uncertaintiesfor the final estimation of the projected correlation function from thesampling as the 16th and 84th percentile of the 𝑤 𝑝 ( 𝑟 𝑝 ) distributionin each 𝑟 𝑝 bin from the resulting 3,000 redshift-sampling iterationswhich are combined with the poisson uncertainties estimated fromthe median pair counts for each bin. For the small subset of SMGswith archival spectroscopic redshifts (37 SMGs with 𝑧 = The projected cross-correlation of the SMGs with the 𝐾 -band fieldgalaxies provides us with the relative bias between SMGs and thesegalaxies as described below. However, to estimate the character-istic halo mass of the SMGs we need to determine the absolutebias of the SMGs relative to the dark matter and thus we need toestimate the absolute bias of the galaxies with respect to the darkmatter. To determine this we measure the auto-correlation functionfor the comparison 𝐾 -band galaxies that were redshift-matched tothe SMGs. This redshift-matched galaxy sample is large enough( 𝑁 ∼ angular auto-correlation MNRAS000
Gal pairs at separations 𝑟 𝑝 ± Δ 𝑟 𝑝 and 𝜋 ± Δ 𝜋 . For our cross-correlation analysis we calculatedpair counts in logarithmic 𝑟 𝑝 bins in the range 0.05–14 ℎ − Mpc,which at the median redshift of the SMG sample ( 𝑧 ∼ ∼ (cid:48)(cid:48) .To incorporate the photometric redshift PDFs we measure theprojected correlation function with a Monte Carlo method by repeat-ing the projected correlation as a function of 𝑟 𝑝 bins whilst samplingthe redshifts of every SMG and galaxy by randomly selecting fromtheir respective PDFs. We set the contribution to the uncertaintiesfor the final estimation of the projected correlation function from thesampling as the 16th and 84th percentile of the 𝑤 𝑝 ( 𝑟 𝑝 ) distributionin each 𝑟 𝑝 bin from the resulting 3,000 redshift-sampling iterationswhich are combined with the poisson uncertainties estimated fromthe median pair counts for each bin. For the small subset of SMGswith archival spectroscopic redshifts (37 SMGs with 𝑧 = The projected cross-correlation of the SMGs with the 𝐾 -band fieldgalaxies provides us with the relative bias between SMGs and thesegalaxies as described below. However, to estimate the character-istic halo mass of the SMGs we need to determine the absolutebias of the SMGs relative to the dark matter and thus we need toestimate the absolute bias of the galaxies with respect to the darkmatter. To determine this we measure the auto-correlation functionfor the comparison 𝐾 -band galaxies that were redshift-matched tothe SMGs. This redshift-matched galaxy sample is large enough( 𝑁 ∼ angular auto-correlation MNRAS000 , 1–13 (2020)
Stach et al. function for this sample. We measure the angular auto-correlationusing the same Landy & Szalay (1993) estimator, but now modifiedfor auto-correlation: 𝜔 ( 𝜃 ) = 𝑅𝑅 ( 𝐷𝐷 − 𝐷𝑅 + 𝑅𝑅 ) (7)where 𝐷𝐷 , 𝐷𝑅 , and 𝑅𝑅 are the normalised number of Galaxy–Galaxy, Galaxy–Random, and Random–Random galaxy pairs, re-spectively, at an angular separation 𝜃 . The errors for the auto-correlation function are calculated by dividing the field into nineroughly equal sized sub-fields and using the ‘delete one jackknife’method (Shao 1986; Norberg et al. 2009).As with the projected correlation function, we fit a power-lawto the galaxy auto-correlation function of the form 𝑤 ( 𝜃 ) = 𝐴𝜃 − 𝛿 . Tomeasure the absolute correlation length for the SMGs we convert 𝐴 and 𝛿 to the projected correlation function equivalent 𝑟 and 𝛾 following Peebles (1980) by deprojecting the auto-correlationfunction through the Limber equation (Limber 1954): 𝛿 = 𝛾 − , (8)and 𝐴 = 𝐻 𝛾 ∫ ∞ ( 𝑑𝑁 / 𝑑𝑧 ) ( 𝑑𝑁 / 𝑑𝑧 ) 𝐸 𝑧 𝜒 − 𝛾 𝑑𝑧 (cid:104)∫ ∞ ( 𝑑𝑁 / 𝑑𝑧 ) 𝑑𝑧 (cid:105) (cid:104)∫ ∞ ( 𝑑𝑁 / 𝑑𝑧 ) 𝑑𝑧 (cid:105) 𝑟 𝛾 (9)where 𝐻 𝛾 = Γ ( . ) Γ ( . [ 𝛾 − ]) Γ ( . 𝛾 ) , (10)and 𝑑𝑁 / 𝑑𝑧 and 𝑑𝑁 / 𝑑𝑧 are the redshift distribution for the samples,where for our auto-correlation 𝑑𝑁 / 𝑑𝑧 = 𝑑𝑁 / 𝑑𝑧 and 𝐸 𝑧 is 𝐻 𝑧 / 𝑐 where 𝐻 𝑧 is the Hubble parameter. With an 𝑟 for the SMG–Galaxycross-correlation and the Galaxy–Galaxy auto-correlation we canestimate the correlation length for the auto-correlation function ofSMGs, using a fixed 𝛾 = 𝜉 SMG = 𝜉 − Gal / 𝜉 Gal (Coil et al. 2009).
As described above, to estimate the dark matter halo masses for theSMGs we first must measure the absolute bias of the 𝐾 -band Galaxysample from their auto-correlation. A measurement of bias relieson an estimation of the dark matter angular correlation function 𝑤 𝑑𝑚 . To estimate 𝑤 𝑑𝑚 we use the HaloFit code (Smith et al.2003), with the updated Halofit fitting parameters from Takahashiet al. (2012), to calculate the non-linear dark matter power spectrum 𝑃 ( 𝑘, 𝑧 ) assuming the slope of the initial fluctuation power spectrum Γ = 𝑤 ( 𝜃 ) = 𝑐 ∫ (cid:18) 𝑑𝑁𝑑𝑧 (cid:19) 𝐻 ( 𝑧 ) ∫ 𝑘 𝜋 𝑃 ( 𝑘, 𝑧 ) 𝐽 [ 𝑘𝜃 𝜒 ( 𝑧 )] 𝑑𝑘𝑑𝑧 (11)where 𝐽 is the zeroth order Bessel function, 𝜒 is the radial co-moving distance, and 𝑑𝑁 / 𝑑𝑧 is the stacked redshift PDF normalisedsuch that ∫ ∞ [ 𝑑𝑁 / 𝑑𝑧 ] 𝑑𝑧 = 𝑏 𝑔 ) of the auto-correlated galaxiesby scaling the dark matter angular correlation function to the galaxycorrelation function: 𝑤 ( 𝜃 ) = 𝑏 𝑔𝑎𝑙 𝑤 𝑑𝑚 ( 𝜃 ) . (12)For the relative SMG–Galaxy bias we calculate the dark matter projected correlation function from the same power spectrum, butnow Fourier transformed to give 𝜉 ( 𝑟 ) which is then integrated usingEquation 3.The projected dark matter correlation function is scaled tothe SMG–Galaxy projected cross-correlation function in the samemanner as for the galaxy auto-correlation, but with the linear scalingnow equal to 𝑏 smg 𝑏 gal and thus the absolute SMG bias can becalculated. We convert the absolute bias into a dark matter halomass by assuming a Tinker et al. (2010) bias function.From recent studies using Halo Occupation Distribution(HOD) models (Peacock & Smith 2000; Benson et al. 2000), thegalaxy correlation function can be broken down into the sum oftwo components, a ‘one-halo’ term that dominates at smaller spatialscales and measures the contributions from pairs of galaxies withina single dark matter halo, and the ‘two-halo’ term dominating atlarger scales involving clustering of galaxies in separate haloes.Whilst we lack the signal-to-noise in our SMG correlation func-tions to constrain the increased number of parameters involved infitting HOD models, we still test the potential influence of a contri-bution from a one-halo component on our results by restricting theminimum spatial scale where we apply both the power-law and darkmatter correlation function fits to minimise contributions from theintra-halo pairs. From the HOD fitting of 250 𝜇 m selected SMGsin Amvrosiadis et al. (2019) we set the minimum spatial scale to 𝑟 𝑝 > ℎ − Mpc, this value chosen to be roughly consistent withthe region in which the ‘one-halo’ term dominates their galaxy clus-tering function. We indicate this in Fig. 3 by showing the regionused in the fitting as a solid line.
As our clustering measurements are for a sample in a finite-areafield we check the magnitude of the correction to the measuredcorrelation function from the absence of information on densityfluctuations on the scale of the field from the ‘integral constraint’(IC): 𝑤 𝑝 = 𝑤 obs 𝑝 + IC (13)As we are constraining our fits to relatively small scales( < ℎ − Mpc) in comparison to the degree-scale UDS field weexpect this offset to be negligible in comparison to the measuredclustering (e.g. Kashino et al. 2017). We estimate the integral con-straint for the projected correlation function using our ‘Random’catalogues by following the iterative method of Roche & Eales(1999) and Kashino et al. (2017):IC = ∫ 𝜋 max (cid:205) 𝑖 𝑅𝑅 ( 𝑟 𝑖 ) 𝜉 mod ( 𝑟 𝑖 ) (cid:205) 𝑖 𝑅𝑅 ( 𝑟 𝑖 ) 𝑑𝜋 (14)where 𝜉 mod is the bias-scaled correlation function of the darkmatter: 𝜉 mod = 𝑏 𝜉 d 𝑚 where 𝑏 is initially set to the SMG–Galaxyrelative bias as measured in §2.6. The integral constraint is thencalculated and applied and the process is repeated with the updatedrelative bias values incorporating the estimated integral constraintoffsets until convergence. We find a final integral constraint cor-rection for our field of IC = ℎ − Mpc, which is an insignificantcorrection to the observed correlation function at the separationscales we consider. Nevertheless, we still correct our observed cor-relation function amplitudes for this integral constraint. The galaxyauto-correlation amplitudes were similarly corrected for the integralconstraint, estimated with:IC = (cid:205) 𝑖 𝑅𝑅 ( 𝜃 𝑖 ) 𝑤 ( 𝜃 𝑖 ) (cid:205) 𝑖 𝑅𝑅 ( 𝜃 𝑖 ) (15) MNRAS , 1–13 (2020)
S2UDS: Clustering of SMGs where, following Hartley et al. (2013), for 𝑤 ( 𝜃 ) we use the angularcorrelation function of the dark matter correlation function traced byour galaxy sample, scaled by the absolute galaxy bias, as measuredin §2.6. There is a complexity in measuring the clustering of SMGs that aredetected from follow-up surveys of single-dish observations, arisingfrom a potential bias due to the low resolution of the parent survey,e.g. ‘blending bias’ Cowley et al. (2016), that could increase themeasured clustering. These issues arise because the low-resolutionsingle-dish observations not only detect individual galaxies, butalso can uncover groups of faint SMGs (either physically associatedor simply seen in projection) that are separated on the sky by lessthan the single-dish beam ( ∼
30 % for S2CLS sources: Stach et al.2018). As expected, below the flux limit of the single-dish parentsurvey the ALMA follow-up survey is incomplete to fainter SMGs,but some faint galaxies at these flux limits are included due to suchblending (the bulk arise due to noise-boosting), including those intrue SMG groups, whose summed flux density raise them abovethe single-dish flux density detection threshold. Missing isolatedexamples of such faint galaxies, but detecting those preferentiallylying in small separation groups could result in an overestimation ofthe true SMG clustering. In addition, if angular correlation functionsare used to derive the clustering measurements, then these groupsof SMGs, even if just projected systems with a wide spread inredshift between the components, rather than physically associated,can become correlated if the redshift bins are coarse enough (Cowleyet al. 2016).A recent test of the potential significance of blending bias wasshown by García-Vergara et al. (2020) who assessed the strengthof the effect for the ALMA follow-up of the single-dish LABOCAsources used in Hickox et al. (2012), by applying a complex forwardmodelling technique to attempt to both account for incompletenessand assess the clustering of the SMGs. They suggested that suchan approach is necessary to correctly return the true characteristichalo masses and that by just measuring clustering from the single-dish sources alone can result in a halo mass 3.8 + . − . times higherthan the true mass. We expect that our sample is less sensitive tothis bias than that used in the García-Vergara et al. (2020) analy-sis, as the AS2UDS SMGs are follow-up of SCUBA-2 single dishobservations with SCUBA-2 having a smaller beam size than thebeam-convolved LABOCA map on which their analysis is based(14.6 (cid:48)(cid:48) , c.f ∼ (cid:48)(cid:48) ), as well as the higher significance cut used asthe basis for the subsequent ALMA follow-up observations (4.0 𝜎 ,c.f 3.75 𝜎 ). We also note that for our AS2UDS survey, of the 440SCUBA-2 sources lying within the unmasked regions described in§2.1, only 57 of these SCUBA-2 sources contain more than a singleSMG and thus are potentially introducing some ‘blending bias’. Asonly ∼
13 per cent of our SCUBA-2 sources being considered herecontain multiple SMGs (whether physically associated or not), andof those we only expect ∼
30 per cent to be physically associated(Stach et al. 2018; Simpson et al. 2020), we therefore do not ex-pect a significant overestimation of the clustering from these biases.Nevertheless, with our increased sample size we can test for themagnitude of the blending bias (and other inhomogeneities in theparent SCUBA-2 catalogue) by measuring the clustering for boththe ‘full’ SMG sample and for a subset of SMGs by applying a cut atthe flux density where we are close to 100 per cent complete in thesingle-dish survey (ALMA 𝑆 ≥ Figure 3.
The projected cross-correlation function for the AS2UDS SMGsacross the redshift range 𝑧 = 𝑆 ≥ > ℎ − Mpc used in the fitting), forwhich we derive correlation lengths of 𝑟 = + . − . ℎ − Mpc for the ‘full’and 𝑟 = + . − . ℎ − Mpc for the 𝑆 ≥ 𝑟 for clarity). This confirms that the clustering measured from thetwo samples are statistically indistinguishable. The dotted line shows theprojected correlation of the underlying dark matter which is then linearlyscaled to the samples to derive their relative bias measurements. both the lower significance sources and any groups of faint SMGswithin a single-dish beam.In Figure 3 we show the projected cross-correlation functionfor the AS2UDS SMGs with the 𝐾 -band galaxies, both with andwithout the ALMA 870- 𝜇 m flux cut mentioned above, across theredshift range of 𝑧 = ℎ − Mpc (toreduce the influence of the one-halo term), using a maximum like-lihood estimator with a single parameter, power-law model given inEquation 5, where we fix 𝛾 = 𝐾 -band galaxy samples to return the estimated SMGauto-correlation lengths 𝑟 = + . − . ℎ − Mpc for the ‘full’ sampleand 𝑟 = + . − . ℎ − Mpc for the 𝑆 ≥ 𝑆 < ∼
19 such SMGs to our analysis if we do not apply the flux cut,
MNRAS , 1–13 (2020)
Stach et al. this modest change is unsurprising and therefore for the remainderof the analysis we have therefore chosen to use the ‘full’ sample.
The correlation lengths are a useful measure for comparisons be-tween different clustering studies as they are not dependent on dif-ferent bias models, that can alter halo masses derived from the linearbias fitting. The weakest clustering (corresponding to the shortestcorrelation length) found for SMGs in the 𝑧 = 𝑟 = + . − . ℎ − Mpc from their angular correlation func-tions which is ∼ 𝜎 below our value for the same redshift range.As noted earlier, Wilkinson et al. (2017) had to rely on radio, mid-infrared and colour selection to identify likely counterparts to thesingle-dish submillimetre sources, as opposed to high-resolutionALMA imaging. Therefore contamination from mis-identifications(Hodge et al. 2013; An et al. 2018) is a likely cause for their lowercorrelation lengths; as they note, if they limit their analysis to themore robust (but less complete) radio-identified counterparts theywould estimate a longer correlation length: 𝑟 = + . − . ℎ − Mpc,in better agreement with our measurements.Comparing to clustering estimates in other fields from the liter-ature, we find reasonable agreement with our measurements, e.g. inthe Extended
Chandra
Deep Field South, Hickox et al. (2012) alsomeasured the projected correlation functions of probabalistically-identified counterparts to single-dish detected sources and founda correlation length of 𝑟 = + . − . ℎ − Mpc for sources at 𝑧 = 𝑟 < ℎ − Mpc in Williams et al. (2011) for SMGs selected at1.1 mm, and 𝑟 = ± ℎ − Mpc in Blain et al. (2004). All of theseprevious studies have suffered from modest sample sizes (
𝑁 < 𝑏 𝑠 = ± 𝑆 ≥ 𝑏 𝑠 = ± ( 𝑀 halo [ ℎ − M (cid:12) ]) = + . − . for both samples. As with the correlation lengths, mostprevious studies are consistent with our estimates, e.g.log ( 𝑀 halo [ ℎ − M (cid:12) ]) = + . − . by Hickox et al. (2012);log ( 𝑀 halo [ ℎ − M (cid:12) ]) = + . − . by Chen et al. (2016b); andlog ( 𝑀 halo [ ℎ − M (cid:12) ]) ∼
12 from Wilkinson et al. (2017). Our di-rect estimate of the halo mass for SMGs in the AS2UDS surveyalso agrees well with that inferred from the redshift distributionof the AS2UDS SMG population by Dudzeviči¯ut˙e et al. (2020):log ( 𝑀 halo [ ℎ − M (cid:12) ]) ∼ upper limit for a > ( 𝑀 halo [ ℎ − M (cid:12) ]) < 𝜎 error range of our characteristic mass. We returnto discuss the connection between halo mass and redshift for theSMG population in §3.2 and 3.3.In comparison to theoretical simulations of SMGs, our me-dian halo mass lies between the results from the semi-analyticmodel galform finding SMGs inhabiting haloes with masseslog ( 𝑀 halo [ ℎ − M (cid:12) ]) = 𝑆 > ( 𝑀 halo [ ℎ − M (cid:12) ]) = + . − . (McAlpine et al. 2019). We can exploit our relatively large sample size of ∼
400 galaxies( ∼ × larger than similar previous studies) to split the sampleinto independent redshift bins to test any potential evolution in thehalo masses. We split our sample into three redshift bins with equal Δ 𝑧 = 𝑧 = 𝑧 = 𝑧 = 𝑟 ∼ ℎ − Mpc, and inferred halo masses,log ( 𝑀 cen [ M (cid:12) ]) ∼ 𝜎 uncertainties.We note that our bias measurements do not show evidence thatSMGs at 𝑧 = 𝑧 > 𝑧 < 𝑧 > ( 𝑀 halo [ ℎ − M (cid:12) ]) > 𝑧 = 𝑁 = 𝑧 > ( 𝑀 halo [ ℎ − M (cid:12) ]) = + . − . , that is just 0.1 dex above our es-timate for the 𝑧 = 𝜎 ). Whilst this hintsto a potential marginal increase in halo mass for 𝑧 > MNRAS , 1–13 (2020)
S2UDS: Clustering of SMGs Table 1.
Clustering results for each redshift bin considered. The 𝑧 = 𝜎 errors.Redshift < 𝑁 smg > 𝑟 𝑏 smg log ( 𝑀 halo ) ( ℎ − Mpc) (log ( ℎ − M (cid:12) ) )1.5–3.0 329 6 . + , − . . ± . . + . [ . ]− . [ . ] . + . − . . ± . . + . [ . ]− . [ . ] . + . − . . ± . . + . [ . ]− . [ . ] . + . − . . ± . . + . [ . ]− . [ . ] − r p ( h − Mpc) w p ( r p )( h − M p c ) . < z < . AS2UDS - SMGsPower-Law Fit − r p ( h − Mpc)2 . < z < . − r p ( h − Mpc)2 . < z < . Figure 4.
The two-point cross-correlation functions of submillimetre galaxies identified in the AS2UDS survey with redshift-matched 𝐾 -band selected fieldgalaxies from the UKIDSS UDS catalogue in three redshift bins ( 𝑧 = 𝛾 = 𝐾 -band galaxy auto-correlations, the dark matter halo masses for the submillimetre galaxies are derived and reportedin Table 1). Figure 5.
Left:
The predicted redshift evolution of the galaxy bias for the AS2UDS submillimetre galaxies. The dotted lines show the expected bias evolutionfor dark matter haloes at their labelled masses. For comparison we show similar measurements from the Hickox et al. (2012), Chen et al. (2016a) and Wilkinsonet al. (2017) SMG samples. Contrary to Wilkinson et al. (2017), but in agreement with Hickox et al. (2012) and Chen et al. (2016a), we do not see statisticallysignificant evolution in the bias with redshift compared to that expected for a constant dark matter halo mass with a mass of ∼ M (cid:12) . The red dashed linesshow the predicted halo mass growth rates from Fakhouri et al. (2010) for our three redshift bins. These converge towards bias values for the descendantgalaxy population at 𝑧 ∼ ∼ 𝐿 ★ galaxies, a populationdominated by massive, passive spheroidal galaxies. Right:
The auto-correlation lengths for our AS2UDS SMGs in three redshift bins compared to observationalestimates for a range of galaxy populations. The solid black curves show the expected correlation lengths for dark matter haloes of various masses. The AS2UDSSMGs at 𝑧 =000
The auto-correlation lengths for our AS2UDS SMGs in three redshift bins compared to observationalestimates for a range of galaxy populations. The solid black curves show the expected correlation lengths for dark matter haloes of various masses. The AS2UDSSMGs at 𝑧 =000 , 1–13 (2020) Stach et al. likely source of this discrepancy comes from the mis-identificationof the SMGs.We estimate this contamination in the Wilkinson et al. (2017)sample by taking their parent SMG catalogue from Chen et al.(2016a) and applying their same ‘Class 1’ SMG selection, definedas SMGs in regions of the UDS map with both optical and radiocoverage (similar to our own selection described above). Of the 645Chen et al. (2016a) SMGs used by Wilkinson et al. (2017), just392 match to an ALMA AS2UDS SMG to within a 1.0 arcsecondsmatching radius, corresponding to a 42 % contamination rate. Thecontamination rate is highest at the lower redshift end ( 𝑧 = ∼
52 per cent of the probabilistically-identified SMGs havingno ALMA detection. As the genuine ALMA-detected SMGs areexpected to be on average more massive than contaminant mis-identified galaxies, the significantly lower characteristic halo massespredicted by Wilkinson et al. (2017) at this redshift range wouldbe a natural consequence of this level of contamination. In theredshift range 𝑧 = ∼
30 per cent. A similar contamination rate of ∼
30 percent applies at the highest redshift bin of 𝑧 = ( 𝑀 halo [ ℎ − M (cid:12) ]) ∼ 𝑧 ∼
0. The evolved 𝑧 = 𝐿 ★ galax-ies, a population that is dominated by passive spheroidal systems(Zehavi et al. 2011).The high present-day halo masses and the properties of galax-ies normally found to populate such haloes, are consistent with arange of other circumstantial evidence linking SMGs with the for-mation of spheroidal galaxies. Recent examples include the broadagreement between SMG properties and the scaling relations seenin local spheroids between baryonic surface density and total stellarmass, Σ bar – 𝑀 ∗ , by Hodge et al. (2016) (see also Franco et al. 2020),and between velocity dispersion and total baryonic mass, 𝜎 – 𝑀 bar ,by Birkin et al. (2021), as well as their general sizes (e.g. Fujimotoet al. 2017; Ikarashi et al. 2017) and environmental trends (e.g.Zavala et al. 2019). Similarly, Dudzeviči¯ut˙e et al. (2020) demon-strate that the space density of massive SMGs roughly matches thatfor the most massive, evolved galaxies (in general agreement withsome theoretical simulations, McAlpine et al. 2019). Our resultson the halo masses of SMGs provide further support for the sim-ple empirical model linking these massive intensely star-formingand metal rich galaxies at high redshift to the early formation andevolution of local, passive spheroids.For an empirical comparison to other high-redshift galaxy pop-ulations we show in Figure 5 the estimated correlation lengths fora range of previous studies collected from Hickox et al. (2012) e.g.luminous QSOs (Myers et al. 2006; Ross et al. 2009), Lyman-breakgalaxies (LBGs) (Adelberger & Steidel 2005), Spitzer
MIPS 24- 𝜇 m selected star-forming galaxies (Gilli et al. 2007), and opticallyselected clusters (Estrada et al. 2009). We also show the predic-tions for the correlation length evolution with redshift at varyinghalo masses using the Peebles (1980) formalism. Our results areagain, roughly consistent with Hickox et al. (2012), showing across Figure 6.
The estimated halo masses for the AS2UDS SMGs compared totwo recent theoretical models of SMGs. The two lines show the medianvalues for each model, with the shaded region representing the 16–84 thpercentile range for the semi-analytic model shark (Lagos et al. 2020), andthe 10–90 th percentile range in the case of the N-body hydrodynamicalsimulation eagle (McAlpine et al. 2019). Whilst we see general agreementto our observational results, we note that the AS2UDS SMGs are typicallybrighter 𝑆 (cid:38) 𝑧 = 𝑆 > 𝑆 > 𝑆 ∼ 𝜇 m fluxes reside in haloes of lower masses,with 𝑆 > ∼ 𝑆 > Finally, we investigate the stellar-to-halo mass ratio (SHMR) forthe AS2UDS SMGs as a function of their halo masses. SHMR is ameasure of the efficiency with which these galaxies form stars. Toestimate the SHMR we take the stellar mass estimates for the SMGsfrom Dudzeviči¯ut˙e et al. (2020) that are estimated using SED fittingwith magphys (da Cunha et al. 2008, 2015; Battisti et al. 2019).
MNRAS , 1–13 (2020)
S2UDS: Clustering of SMGs Figure 7.
Left:
The stellar mass-halo mass ratio for the AS2UDS SMGs as a function of halo mass. For comparison we show a number of empirical modelpredictions for central galaxies at 𝑧 = 𝜎 uncertainties. Observational results for 𝑧 ∼ 𝑔𝑧𝐾 𝑠 -selected quiescent galaxies (Cheema et al. 2020) are shown by the diamond points. The models show that SMGs, with halo masseslog ( 𝑀 halo [ M (cid:12) ]) ∼ Right:
The redshift-binned SMG characteristic halo masses in comparison to the Dekel & Birnboim (2006) schematic for the thermal properties of gas flowingonto galaxies. Below the almost horizontal orange line the galaxy discs are fed by cold streams conducive to future star formation. The diagonal solid blue lineis the upper limit for a galaxy’s mass at the critical redshifts where the cold streams can still penetrate the hot shock heated halos and thus star formation canstill occur. Our estimated halo masses for the SMGs lie around this boundary, suggesting that they may represent the most massive, common haloes onto whichgas can cool to fuel star formation. The dotted lines are from the Press–Schechter estimates for halo formation masses and show the estimated percentage ofthe total halo mass at a given redshift that resides in haloes of mass greater than 𝑀 halo (i.e. 1 𝜎 =
22 per cent, 2 𝜎 = 𝜎 = To estimate the SHMR we first separate our SMGs intotwo bins of stellar mass (to minimise variations in the halomass estimates), split at the median stellar mass of the sam-ple, log ( 𝑀 ★ [ M (cid:12) ]) ∼ ( 𝑀 stellar / 𝑀 halo ) = − + . − . for our lower mass bin andlog ( 𝑀 stellar / 𝑀 halo ) = − + . − . for the higher mass, both estimatesbeing consistent within the significant uncertainties.For comparison, we also show observational estimates of theSHMR for 𝑔𝑧𝐾 -selected quiescent galaxies at 𝑧 ∼ ( 𝑀 halo [ M (cid:12) ]) ∼ ( 𝑀 halo / 𝑀 (cid:12) ) ∼ ( 𝑀 halo / 𝑀 (cid:12) ) ∼
12, gas cancool from the intragalactic medium onto the central galaxy at allredshifts. However, for more massive haloes a shock forms in thehalo that increasingly limits the ability of streams of cold gas to beaccreted onto the central galaxy at lower redshifts. This behaviouris illustrated in Figure 7, that shows the boundaries of the variousregimes as well as the collapse redshifts for haloes of differentmasses as indicated by the rarity of the fluctuations they representbased on the Press-Schecter formalism (which gives some indicationof the likely rarity of haloes with a given mass as a function ofredshift). In Dekel & Birnboim (2006) the disruption of the coldstreams in massive haloes occurs at a mass scale that is a multipleof the characteristic halo mass at that epoch, reflecting the influenceof the local environment and growing large-scale structures on thehalo accretion. This results in the diagonal boundary line shown inFigure 7 – between regimes at high and low redshifts where the coldstreams can or cannot feed the central galaxy.
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We indicate on Figure 7 the halo masses and median redshiftsfor SMGs in the three redshift ranges analysed in § 3.2. Thesemeasurements are broadly consistent with these SMGs lying nearthe boundary defining the most massive galaxies where the coldstreams can still feed the star-formation activity in galaxies. Thesegalaxies thus represent the most massive galaxies that can continueto support significant star-formation rates fueled by the accretionof gas supplies from the surrounding intragalactic medium. Thismodel also naturally explains the peak in the redshift distribution ofdust-mass-selected samples of SMGs at 𝑧 ∼ 𝑧 ≥ We have measured the clustering strength of the largest sample ofinterferometrically identified SMGs in a single contiguous field. Weuse Monte Carlo methods to incorporate the complete photometricredshift PDFs for both the main SMG samples and the 𝐾 -selectedfield galaxy sample, into the calculation of the projected cross-correlation functions. The main results of our clustering analysisare as follows: • Across the entire redshift range considered ( 𝑧 = 𝑟 = + . − . ℎ − Mpc. This isconsistent with previous studies of smaller samples of single-dishdetected SMGs that show SMGs to be more strongly clustered thantypical star-forming galaxies at their redshift and more similar to theclustering strength seen for luminous QSOs (or ‘bright quasars’).From linearly scaling the dark matter projected correlation func-tion we derive dark matter halo masses for 𝑧 = ( 𝑀 halo [ ℎ − M (cid:12) ]) = + . − . . • We split our sample into three redshift bins 𝑧 = ( 𝑀 halo [ ℎ − M (cid:12) ]) ∼ 𝑧 = ( 𝑀 halo [ ℎ − M (cid:12) ]) ∼ • Exploiting the stellar mass estimates for the SMGs fromDudzeviči¯ut˙e et al. (2020) we split the AS2UDS sample into twostellar mass bins and calculated their respective characteristic halomasses. The stellar to halo mass ratio for these sub-samples are con-sistent with the theoretical models, with the SMGs lying at the peakof the stellar-to-halo mass ratio for the models. This suggests thatSMGs are amongst the most efficient galaxy populations in termsof the conversion of baryons into stellar mass. • We compare the estimates of the halo masses for SMGs as afunction of redshift to a simple model that describes the bimodalityin the local galaxy population through a dichotomy in the modeof gas accretion, driven by the presence of a stable shock in gasaccreting in more massive halos. We show that the SMGs fall nearthe boundary where cold gas streams can still be accreted onto thecentral galaxies in the most massive haloes. This would naturallyexplain several characteristics of the SMG population, includingtheir intense star-formation rates, masses and redshift distribution,as they represent the most massive galaxies that can still support their star-formation activity through accretion of gas from the intra-galactic medium.
ACKNOWLEDGEMENTS
All Durham co-authors acknowledge financial support from STFC(ST/T000244/1). AA is supported by ERC Advanced Investiga-tor grant, DMIDAS [GA 786910], to C.S. Frenk. CCC acknowl-edges support from the Ministry of Science and Technology ofTaiwan (MOST 109-2112-M-001-016-MY3). KEKC acknowledgesupport from the UK Science and Technology Facilities Council(STFC) (grant number ST/R000905/1) and a Royal Society Lev-erhulme Trust Senior Research Fellowship (grant number RSLTSRF/R1/191013). JLW acknowledges support from an STFC ErnestRutherford Fellowship (ST/P004784/1 and ST/P004784/2).
The data underlying this article are available in the JCMT, ALMAand ESO archives.
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