Estimating sizes of faint, distant galaxies in the submillimetre regime
L. Lindroos, K. K. Knudsen, L. Fan, J. Conway, K. Coppin, R. Decarli, G. Drouart, J. A. Hodge, A. Karim, J. M. Simpson, J. Wardlow
MMon. Not. R. Astron. Soc. , 1–11 (2016) Printed October 9, 2018 (MN L A TEX style file v2.2)
Estimating sizes of faint, distant galaxies in the submillimetreregime
L. Lindroos (cid:63) , K. K. Knudsen , L. Fan , J. Conway , K. Coppin , R. Decarli , G.Drouart , J. A. Hodge , A. Karim , J. M. Simpson , and J. Wardlow , Department of Earth and Space Sciences, Chalmers University of Technology, Onsala Space Observatory, SE-439 92 Onsala, Sweden Shandong Provincial Key Lab of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Science,Shandong University, Weihai, 264209, China Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield, AL10 9AB, UK Max-Planck-Institut für Astronomie, Köonigstuhl 17, D-69117 Heidelberg, Germany International Centre for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark
Accepted 2016 July 05. Received 2016 June 25; in original form 2016 December 21
ABSTRACT
We measure the sizes of redshift ∼ star-forming galaxies by stacking data fromthe Atacama Large Millimeter/submillimeter Array (ALMA). We use a uv -stackingalgorithm in combination with model fitting in the uv -domain and show that thisallows for robust measures of the sizes of marginally resolved sources. The analysisis primarily based on the 344 GHz ALMA continuum observations centred on 88sub-millimeter galaxies in the LABOCA ECDFS Submillimeter Survey (ALESS). Westudy several samples of galaxies at z ≈ with M ∗ ≈ × M (cid:12) , selected usingnear-infrared photometry (distant red galaxies, extremely red objects, sBzK-galaxies,and galaxies selected on photometric redshift). We find that the typical sizes of thesegalaxies are ∼ . (cid:48)(cid:48) which corresponds to ∼ kpc at z = 2 , this agrees well with themedian sizes measured in the near-infrared z -band ( ∼ . (cid:48)(cid:48) ).We find errors on our size estimates of ∼ . (cid:48)(cid:48) − . (cid:48)(cid:48) , which agree well with the ex-pected errors for model fitting at the given signal-to-noise ratio. With the uv -coverageof our observations (18-160 m), the size and flux density measurements are sensitiveto scales out to 2 (cid:48)(cid:48) . We compare this to a simulated ALMA Cycle 3 dataset withintermediate length baseline coverage, and we find that, using only these baselines,the measured stacked flux density would be an order of magnitude fainter. This high-lights the importance of short baselines to recover the full flux density of high-redshiftgalaxies. Key words: techniques: interferometric – galaxies: high-redshift – galaxies: structure– sub-millimetre: galaxies
The star-formation rate density in the universe peaks at z ∼ (e.g. Madau & Dickinson 2014), making this a veryimportant epoch in the formation of galaxies. For galaxies atthese redshifts submillimeter (sub-mm) emission is a com-monly used tracer of star formation (e.g. Daddi et al. 2010b),often used in combination with ultraviolet and optical mea-surements to allow reliable star-formation rate (SFR) esti- (cid:63) E-mail: [email protected] mates for galaxies with very different dust properties (e.g.Tacconi et al. 2013; da Cunha et al. 2015). The AtacamaLarge millimeter/submillimeter Array (ALMA) and IRAMNOrthern Extended Millimeter Array (NOEMA) are cur-rently producing a large wealth of data at frequencies of − GHz, allowing us to measure the sub-mm emis-sion from high-redshift galaxies previously to faint to study.Observing at these frequencies is efficient for high redshifts,as the flux density for galaxies at a given SFR is expected tobe almost constant for redshift z ∼ − due to the negative K -correction (e.g. Blain et al. 2002; Casey et al. 2014). c (cid:13) a r X i v : . [ a s t r o - ph . GA ] A ug L. Lindroos et al.
Current observations with ALMA and NOEMA primar-ily focus on the galaxies with high SFR, >
100 M (cid:12) yr − ,however, these galaxies constitute a small fraction of the to-tal star formation (e.g. Bouwens et al. 2011; Rodighiero et al.2011). It is possible to study single sources from much faintergalaxy populations, e.g., with 50 ALMA antennas and ∼ hour integration we can reach a depth of 20 µ Jy/beam at 345GHz, which corresponds to 1 σ uncertainty of ∼ (cid:12) yr − at z = 2 . However, to obtain large samples of galaxies forstatistical studies is very expensive. An alternate approach isto study galaxies that are amplified by gravitational lensing.By using lensing it is possible to detect very faint sourceswith shorter observations, e.g., Watson et al. (2015) detecteda z ∼ galaxy with a SFR of 9 M (cid:12) yr − and a flux den-sity of 0.61 mJy at 220 GHz, which would require only a ∼
30 s integration for a 5 σ detection with 50 ALMA anten-nas. However, it can be difficult to obtain large samples ofsuch galaxies as high magnifications are rare. A third ap-proach is stacking, which uses shallower surveys to studystatistical properties of large samples galaxies which havepreviously been detected at other wavelengths. Stacking is acommon technique used across many different wavelength: γ -rays (e.g. Aleksić et al. 2011), X-rays (Chaudhary et al.2012; George et al. 2012), optical/near infrared (Zibetti et al.2007; Matsuda et al. 2012; González et al. 2012), mid/far in-frared (e.g. Dole et al. 2006), and radio (Boyle et al. 2007;Ivison et al. 2007; Hodge et al. 2008, 2009; Dunne et al. 2009;Karim et al. 2011).Looking specifically at sub-mm emission, stacking hasbeen applied to data from James Clerk Maxwell Telescope(JCMT) and Atacama Pathfinder EXperiment (APEX),using several different samples of high-redshift galaxies,(e.g. Webb et al. 2004; Knudsen et al. 2005; Greve et al.2010). Compared to these surveys, ALMA can achieve sub-arcsecond resolution, which is orders of magnitude betterthan the . (cid:48)(cid:48) and . (cid:48)(cid:48) at GHz of APEX and JCMT re-spectively. Firstly, this allows us to measure the flux densityof the sources without being affected by confusion, whichis believed to impact the result of stacking at JCMT andAPEX resolutions (e.g. Webb et al. 2004). Secondly, we canstudy the structure of our stacked source. Several studieshave found star-forming galaxies at redshifts of z ∼ havelarge sizes, e.g. Daddi et al. (2010b) found sizes up to 1 . (cid:48)(cid:48) z ∼ galaxies.Decarli et al. (2014) used stacking to measure the sub-mm flux density of star-forming galaxies in the ExtendedChandra Deep Field South (ECDFS) with data from theALMA. In this paper we will build on the work by Decarliet al. (2014), using the same data, but extending the analy-sis to focus on the sizes of the stacked sources. Decarli et al.(2014) performed stacking on the imaged pointings, analo-gous to how stacking is done at other wavelengths. However,as seen in Lindroos et al. (2015), this may not be ideal forinterferometric data. In this paper we instead adopt the uv -stacking approach described in Lindroos et al. (2015), whichperforms the stacking directly on the visibility data. Whenusing image stacking in mosaiced data sets, it is necessary tocombine data from pointings imaged with different restoringbeams. Because of this, it is very difficult to deconvolve thesource structure from the beam in the final stacked image.Using uv -stacking, we combine the data in the uv -domain,and the beam can be directly calculated from the new uv - coverage. Therefore, using the uv -stacking algorithm is es-pecially important for measuring the sizes of the stackedsources.While the work in this paper is primarily focused onstacking high-redshift galaxies, the stacking techniques ap-plied are quite general. Many of the lessons learned apply toany ALMA stacking of marginally extended sources.The paper is structured as following. In §2, we describethe ALMA data we use and in §3 we describe the sample, aswell as the photometric near infrared and optical catalogue.In §4 we describe a set of simulations performed to testvarious aspect of the stacking result and in §5, we describeour uv -stacking procedure. In §6, we summarise our results,including the typical galaxy sizes for each sample. Finally,in §7 we discuss the implications of the results both for starformation at z ∼ , and for general stacking of ALMA data.In this paper we use a standard cosmology with H =67 . − Mpc − , Ω Λ = 0 . , and Ω m = 0 . (PlanckCollaboration et al. 2014). All magnitudes are in AB (Oke1974) unless otherwise specified. Our analysis is based on data from the ALMA survey of thesubmillimetre galaxies (SMGs) detected in LESS (ALESS,Hodge et al. 2013), where LESS is the LABOCA ECDFSSubmm Survey, LABOCA is the Large Apex BOlometerCAmera mounted on APEX, and ECDFS is the ExtendedChandra Deep Field South. The ALESS survey is com-posed of 122 pointings across the ECDFS, centred on 122SMGs, observed during ALMA Cycle 0 between Octoberand November 2011. The observations are tuned to a fre-quency of 344 GHz and have a typical resolution around1 . (cid:48)(cid:48) × . (cid:48)(cid:48)
2. The median value of the noise (standard devia-tion) in the centre of each pointing is ∼ . mJy/beam. Allpointings with central noise > . mJy/beam or beam axisratio > are excluded from the analysis, see Hodge et al.(2013) for more details. As such our data consist of 88 “goodquality” pointings, each with a field of view (full width athalf power of the ALMA primary beam) of 17 . (cid:48)(cid:48) . In this paper we extend the analysis of Decarli et al. (2014),using the same sample selection. The selection is based onthe photometric catalogue of the ECDFS assembled usingthe same procedure as Simpson et al. (2014), using pri-marily data from the Wide MUlti-wavelength Survey byYale-Chile (MUSYC; Taylor et al. 2009). The MUSYC cat-alogue is a K -band flux limited sample, covering a (cid:48) × (cid:48) area of the ECDFS, with photometry for the sources in thebands UBV RIzJHK . At K AB = 22 mag the sample is100 per cent complete for point sources, and 96 per centcomplete for extended sources with a scale radius of 0 . (cid:48)(cid:48) J and K band catalogue Zibetti et. al (in prepara-tion), the Taiwan ECDFS NIR survey (Hsieh et al. 2012),and Spitzer/IRAC 3.6, 4.5, 5.8, and 8.0 µ m images from theSpitzer IRAC/MUSYC Public Legacy Survey (Damen et al. c (cid:13) , 1–11 stimating sizes of faint, distant galaxies in the submillimetre regime Figure 1.
Distribution of stellar masses for each sample. Thestellar masses are estimated by SED fitting to optical and near-infrared band using PEGASE 2 (Fioc & Rocca-Volmerange 1997). K Vega < , z > , and further limitthe samples as follows:(i) All sources with ( z − K − . > . B − z + 0 . − . which separate the galaxies from the stars (Daddi et al.2004). This sample was refered to the K Vega < samplein Decarli et al. (2014) and will be refered to as the K20sample in this paper.(ii) Actively star-forming galaxies selected using the sBzKcriteria by Daddi et al. (2004), i.e., ( z − K − . − ( B − z + 0 . > − . .(iii) Distant Red Galaxies (DRGs) selected using J − K > . (Franx et al. 2003).(iv) Extremely Red Objects (EROs) selected using ( R − K ) > . and ( J − K ) > . (Elston, Rieke & Rieke 1988).This results in our samples being the same as the z > samples in Decarli et al. (2014).Using the MUSYC photometry we also estimate thestellar mass ( M ∗ ) of our selected galaxies. The stellar-mass estimates are done using PEGASE 2 (Fioc & Rocca-Volmerange 1997). For each galaxy we use all availablebands, i.e., U , B , V , R , I , z , J , H , and K . Using four dif-ferent galaxy templates (elliptical, spiral Sa, spiral Sd, andstarburst), all assuming Kroupa IMF, we fit for stellar mass.The redshift is not fitted directly, instead we use the photo-metric estimates from Taylor et al. (2009). For each sourcewe choose the model with the lowest χ , with more than 90per cent of sources best fitted by the elliptical or the star-burst model. The distributions of stellar masses for thesesamples are shown in Fig. 1. The samples are stacked using the uv -stacking algorithmdescribed in Lindroos et al. (2015). The algorithm performsthe stacking operation directly on the visibility data. We usemodel fitting in uv -domain to estimate the flux densities andsizes of our stacked sources. For comparison with previousimage stacking results we also use a simpler flux density es-timate which assumes a point source, where the flux densityis estimated using the weighted average of all non-flaggedvisiblities. We refer to this estimate at the point-source es-timate . Prior to stacking each sample, all bright sources not part ofthe sample are subtracted from the visibility data.The modelling and subtraction was performed as fol-lows. The data is imaged and cleaned using Common As-tronomy Software Applications package (CASA) version4.4. Each pointing is imaged separately with a cell size of0 . (cid:48)(cid:48) (cid:48)(cid:48) of a stacking position. The model issubtracted from the uv -data, to produce a residual data set.To ensure that the visibility weights are accurate after thesubtraction, they weights are recalculated using the scatterof the visibilities in each baseline and time bin.Note that the aim of the bright source subtraction isto remove bright sources that are unrelated to the targetstacking sources, not to remove those bright in the targetsample. As such, this subtraction is performed separately foreach sample. We also note that the bright source subtractionis based on the clean models, which while not fully removingthe sources, is found to be sufficient for stacking, see section6.2. The uv -stacking method prescribed in Lindroos et al. (2015)uses a weighted average. We calculate the stacking weightsfor each position from the primary beam attenuation. Noisevariations between pointings are included in the visibilityweights, and are thus not included in the stacking weights.To ensure that the visibility weights are accurate, they arerecalculated prior to stacking from the scatter of each base-line and integration,The primary beam attenuation ( A N ) is estimated us-ing the ALMA model present in CASA version 4.4, i.e., anAiry pattern with a full width at half maximum (FWHM)of 1.17 λD ≈ . (cid:48)(cid:48) . This results in stacking weights calculatedas W k = (cid:104) A N ( ˆ S k ) (cid:105) , (1)where W k and ˆ S k are the weight and position of source k . http://casa.nrao.educ (cid:13) , 1–11 L. Lindroos et al.
Two different models are used to characterise the stackedsources.The first: a point source model defined by V ps ( u, v ) = Φ e πi ul + vmλ (2)where ( u, v ) are the projected baselines, λ is the wavelength, ( l, m ) are the direction cosines relative to the phase centre,and Φ is the flux density of the source.The second: a Gaussian model defined by V ( u, v ) = e (cid:18) π × ( u + v )2 λ (cid:19) V ps ( u, v ) (3)where V ps is defined according to Equation 2, and Θ is thesource size (FWHM) in radians.The models are fitted in the uv -domain to our stackedsources using the least square minimizer package Ceres . The model fitting is done to all non-flagged visibilities, andincludes the visibility weights in the χ minimization. We use two different methods to estimate the uncertaintiesof our size and flux density estimates: a Monte Carlo methodwhere random sources are inserted into the data and stacked,and a bootstrapping method.The Monte Carlo simulation for a given sample andmodel is performed as follows: a set of Monte Carlo sourcesis generated with the same number of sources as the givensample. The position for each source is randomized, however,always within the same pointing as their corresponding ac-tual source. Each source is modelled as the fit for the givenmodel to the stacked sources of the given sample. The set ofMonte Carlo sources are introduced into the residual dataset for the given sample and stacked using the same pro-cedure as for the actual samples. Finally the flux densityand size of the stacked Monte Carlo sources are estimatedusing the given model. This procedure is repeated a 100times for each sample and model to produce a distributionof estimated Monte Carlo flux densities and sizes. The un-certainties are calculated as the standard deviation of ourMonte Carlo estimates.The bootstrapping method is performed by resamplingthe galaxies in each sample allowing replacements, e.g., pick-ing galaxy 1 two times and galaxy 2 one time, and galaxy4 one time from a sample of 4 galaxies. We stack the newsample, and estimate the flux density and size using modelfitting. By studying the distribution of the parameters in dif-ferent resamples we can measure the influence of noise andunderlying sample variance on the result. To fully exhaustall possible resamplings would require (cid:0) N × − N (cid:1) resamplings Ceres (Agarwal, Mierle et al. 2015) uses a Levenberg-Marquardt algorithm (Levenberg 1944) for non-linear least squareminimization. It supports several different solvers for the linearstep. We use the solver based on Cholesky decomposition, whichfor our data set typically run 2 times faster compared to a stan-dard QR factorisation. The fit is terminated at the first to occurwithin 50 iterations, a parameter change in the last step of lessthan − , or a relative χ change less than − . All fluxdensities are constrained to be positive. where N is the number of galaxies in the sample. This isapproximately for the sBzK sample, however, we canget a good estimate using a much smaller number of resam-plings. As such we resample 1000 times for each target sam-ple. The error on each paramater is reported as where themeasured cummulative distribution function (CDF) crosses0.159 and 0.841, equivalent to ± σ of a Normal distribution.The estimated parameters are also recentered on where themeasured CDF crosses 0.5, thereby reducing the influenceof outliers on the result.We choose to refer to the first method as the MonteCarlo method as this is the same as the Monte Carlo methodused in Decarli et al. (2014). However, it is worth noting thatthe bootstrap method is also a Monte Carlo method as wedo not fully exhaust all possible resamples, however, in thiswork we will refer to it as bootstrapping. The bootstrapping described in §4.4 uses resampling of thegalaxies to estimate the uncertainties of stacking. Usingbootstrapping we can also estimate the uncertainty of themodel fitting, by resampling the visibilities of the uv -data.This method is not used for the stacked results as it will notestimate uncertainty from variance within the sample, how-ever, it is powerful for model-fitting of individual sources.We will refer to this method as visibility bootstrapping. The model fitting described in section 4.3 allows us to esti-mate the total flux densities and typical sizes of our stackedsources. The uv -models used aim to simulate the behaviourof the averages of our samples. They are not based on theunderlying morphologies of our samples. However, lookingat the data in the uv -domain we can obtain hints on the un-derlying structures of our sources. We have simulated severalpossible morphologies for the galaxies of our samples, to testif they produce different signatures in the stacked data, andto be able to compare them with our actual stacked data.For each simulation we generate a model of fake sourcesand then simulate an ALMA data set with the followingprocedure. We take the raw ALESS data set and set allvisibilities to zero, then we add the model and noise to thedata set. The noise is added using the simulator tool ( sm )in CASA, using the default parameters which produces arealistic noise for the ALMA site. After this the visibilityweights are recalculated by using the scatter in each baselineand time bin.This simulated data set is then stacked using the sameprocedure as for our real data sets (section 4.) Observations of high-z star-forming galaxies at rest-framewavelengths of ∼
200 nm indicate that they are more clumpycompared to their counterparts at lower redshifts (e.g. Imet al. 1999; Förster Schreiber et al. 2009). Based on this wehave generated a model where all the sub-mm flux is comingfrom a few clumps. c (cid:13) , 1–11 stimating sizes of faint, distant galaxies in the submillimetre regime For each source in the sample we generate 3 clumps, i.e.,3 point sources. The clumps are scattered uniformly aroundthe source position, with a maximal distance of 0 . (cid:48)(cid:48) pc, resulting in a total flux of 2.1 mJy for each simulatedgalaxy.We simulate two different uv -coverages. Firstly, thesame as our ALESS observations, with a similar level ofnoise added using the standard sm parameters. Secondly, anintermediate length baseline array with 36 antennas takenfrom ALMA Cycle 3: the C36-5 configuration described inthe ALMA Cycle 3 technical handbook , with baselinesfrom 45 m to 1.4 km. The total observation time is scaleddown to achieve a similar noise, i.e. 1 h spread evenly overthe 122 pointings. The inner parts of the ECDFS are covered by the GOODS-Ssurvey (Giavalisco et al. 2004), with Hubble Space Telescope(
HST ) observations in z -band (900nm) with a point-sourcesensitivity of 27.4 mag. The wider field of ECDFS is ob-served in the Galaxy Evolution from Morphology and SEDs(GEMS, Rix et al. 2004), with HST data in the F606Wand F850LP filters, however, at a shallower depth comparedto the GOODS-S observation: 2000 s typical integration ascompared to 6000 s. At z ∼ the z -band observed corre-sponds to a rest-frame wavelength of approximately nm,where the emission is dominated by light from intermediatemass stars (Bruzual & Charlot 2003).In contrast the sub-mm emission observed by ALMA at344 GHz will primarily trace star-formation surface density(Leroy et al. 2012). We can use this to test whether the starformation follows a significantly different morphology com-pared to the stellar population. Since we are working withstacking we can not study individual galaxies, however, wecan say something about average properties. As such, weproduce a simulated dataset where the star formation hasthe same surface density as the stellar mass traced by theHST z -band. This simulated dataset can be directly com-pared with the actual stacked data.The simulated dataset is produced as follows: we selectall sources which are part of the K20 and have at least a 5 σ detection in either GEMS (band F850LP) or GOODS-S, atotal of 32 sources. These galaxies are stacked using the samemethod as for the other samples. For each source we takethe HST image, mask all pixels below 5 times the noise, andscale to the same total flux density as the stacked averagefor the sample, i.e., 1.4 mJy. These images are then used asinput model for a simulation, following the same method asdescribed for the clumpy model. As part of the stacking process, we re-align the astrome-try from our optical catalogue with our ALMA astrometry. https://almascience.eso.org/documents-and-tools/cycle3/alma-technical-handbook From model fitting with a point source we find an offsetin declination of approximately 0 . (cid:48)(cid:48)
3, with small variations( < . (cid:48)(cid:48) ) between different samples. We also fit the positionusing the disk and Gaussian models, finding a variation of ∼ . (cid:48)(cid:48)
02 between the different models. This is consistent withthe offset found by Simpson et al. (2014) for the bright galax-ies in the same data. Based on this, all stacked datasets werephase rotated with 0 . (cid:48)(cid:48) K -band detections. Of the 100sources in the K20 11 galaxies are detected at a peak SNR > . For these galaxies we estimate the peak position using apoint-source model, and the errors of the fitted positions us-ing visibility bootstrapping (see §4.5). We find that weightedmeans of offset between the optical positions and submm po-sitions are . (cid:48)(cid:48) ± . (cid:48)(cid:48) in right ascension and . (cid:48)(cid:48) ± . (cid:48)(cid:48) indeclination. The errors on the averaged offsets are estimatedusing bootstrapping, where the 11 galaxies are resampled1000 times. We model the offsets between the sources as thesystematic offset, combined with one random offset for eachsource between the optical and submm position, plus theerror of the position measurement for the submm position.We find that the offset between the submm and measuredoptical position can be modelled as a circular Gaussian witha FWHM of . +0 . − . . Again the errors are estimated basedon bootstrapping, where the 11 galaxies are resampled 1000times. To ensure robustness of our new results based on uv -stacking, we perform several test on the stacked data andmethod. By inserting and stacking point sources in theALESS data, using the method described in section 4.4,we evaluate biases in the stacking result. We find that theflux density agrees with the expected values, except for thevery shortest baselines, where the flux density is approxi-mately 20 per cent too high. The results for the sBzK sam-ple is shown in Fig. 2, however, the other samples showvery similar structure in the uv -plane. Lindroos et al. (2015)found similar biases on the shortest baselines for simulateddatasets. In Lindroos et al. (2015) this could be shown tobe due to nearby bright sources which were not fully sub-tracted. This is consistent with our data, as the bright sourcesubtraction is based on the clean models, which may notfully subtract the sources. Based on this we flag all baselineshorter than . for the sBzK and DRG samples ( ∼ m forthe K20 and ERO samples ( ∼ ∼ . The stacked data are well fitted by a Gaussian,as shown in Fig. 3, with a flux density . ± . mJy and . (cid:48)(cid:48) ± . (cid:48)(cid:48) . This agrees well with Simpson et al. (2015),which found typical sizes (FWHM) of SMGs between 0 . (cid:48)(cid:48) . (cid:48)(cid:48) c (cid:13) , 1–11 L. Lindroos et al.
Figure 2.
Stacked flux densities for a simulated dataset, pro-duced by inserting point sources into the ALESS data. Flux den-sities averaged over 100 simulated datasets accurately estimatesystematic biases. The noise is estimated as the standard devia-tion between the different simulations. The red line indicates theexpected flux density for the stacked point sources. The shortestbaseline is higher than the expected flux density due to contribu-tions from residuals of bright sources, see Lindroos et al. (2015)for more discussion of such effects.
Figure 3.
Flux densities for the stacked visibilities of the SMGsample. The visibilities are binned by baseline length. The redline indicates a Gaussian fit. The errors are estimated from thestandard deviation of the real part of the visibilities within eachbin. The horizontal error is estimated from the standard deviationof the uv -distance within each bin. Fig. 4 shows flux density as a function of baseline length foreach sample. In the plot is shown the fit from the Gaussianmodel, with two free parameters: the total flux density andthe FWHM size.The typical sources in all of our samples are found to beextended, with stacked sizes between . (cid:48)(cid:48) and . (cid:48)(cid:48) (see Ta- Figure 4.
Stacked visibilities for each sample binned by baselinelength. The errors are estimated from the standard deviation forthe real part of the visibilities within each bin. The horizontalerror is estimated from the standard deviation of the uv -distancewithin each bin. The lines show uv -models that are fitted to thefull uv -data. The blue dash-dotted line is a Gaussian, the solidgreen line is a Gaussian plus a point source, and the black dashedline is a disk plus a point source. Note that no Gaussian model isvisible for the DRG sample, as it is identical to the Gaussian +point source model for this sample. ble 1). The measured stacked sizes are broadened by randomoffsets between the measured K -band positions and submmpositions. Accounting for this effect, we find deconvolvedsizes for our samples between . (cid:48)(cid:48) and . (cid:48)(cid:48) . The uncer-tainties are estimated by using the bootstrap and MonteCarlo methods described in section 4.4. The bootstrappingerrors are larger as they account for variance within the se-lected sample as well as observational uncertainties, whilethe Monte Carlo only accounts for observational uncertain-ties. For the deconvolved sizes, the reported errors are thecombination of the Monte-Carlo errors and the errors onrandom offset measurements, assuming that these two er-rors are independent.Roughly half the galaxies in our samples are detected inthe HST z -band observations from GOODS-S and GEMS.By fitting a Sérsic distribution to these sources we can esti-mate the sizes at z -band wavelength. We find a median sizeof 0 . (cid:48)(cid:48)
46 for the K20 sample and 0 . (cid:48)(cid:48)
52 for the other samples.The median Sérscic index n is around 1.33 for each sample,although slightly lower for the sBzK sample at 0.94.Compared to the results from Decarli et al. (2014), wefind flux densities which are 20 to 40 per cent higher. Thisis expected as the image stacking method in Decarli et al.(2014) uses the peak flux density in the stacked stamp, whichassumes that the sources are unresolved at the image res-olution of 1 . (cid:48)(cid:48)
6. When fitting a point source model to our uv -stacked data, the measured flux densities deviate fromthe Decarli et al. (2014) measurements by less than a fewper cent. c (cid:13) , 1–11 stimating sizes of faint, distant galaxies in the submillimetre regime Table 1.
Flux density estimates with uv -stacking. The flux density in uv -stacking is estimated using twodifferent methods. Method one (model): the flux density is estimated as the best fit Gaussian model. Methodtwo (point source): the flux density is estimated as the weighted average of all unflagged visibilities. Thesetwo estimates would coincide for point sources. We also present the fitted size of the Gaussian model, as wellas fitted size deconvolved from the random offsets between opical and sub-mm positions. For comparison thetable also shows the image stacking results from Decarli et al. (2014). The errors are estimated by stackingfake sources introduced into the data. uv -stacking Image stackingGaussian Point sourceSample N.gal flux density Size Deconvolved flux density Peak flux density[mJy] size [mJy] [mJy]K20 52 . ± .
30 0 . (cid:48)(cid:48) ± . (cid:48)(cid:48)
14 0 . (cid:48)(cid:48) ± . (cid:48)(cid:48)
16 1 . ± .
07 1 . ± . sBzK 22 . ± .
32 0 . (cid:48)(cid:48) ± . (cid:48)(cid:48)
15 0 . (cid:48)(cid:48) ± . (cid:48)(cid:48)
17 1 . ± .
10 1 . ± . ERO 25 . ± .
22 0 . (cid:48)(cid:48) ± . (cid:48)(cid:48)
17 0 . (cid:48)(cid:48) ± . (cid:48)(cid:48)
19 1 . ± .
10 1 . ± . DRG 19 . ± .
28 0 . (cid:48)(cid:48) ± . (cid:48)(cid:48)
14 0 . (cid:48)(cid:48) ± . (cid:48)(cid:48)
16 1 . ± .
11 1 . ± . Table 2.
Distributions of stacked parameters as estimated from bootstrapping, resampling the galaxieswithin each sample 1000 times. These distributions include both errors from measurement uncertainties andvariance within the samples. The presented range of 15.9 per cent to 84.1 per cent corresponds to the ± σ range for a Gaussian distribution. The distributions are also presented as histograms in A.Sample Gaussian flux [mJy] Size Point source flux [mJy]15.9% 50% 84.1% 15.9% 50% 84.1% 15.9% 50% 84.1%K20 1.33 1.90 2.58 0 . (cid:48)(cid:48)
63 0 . (cid:48)(cid:48)
94 1 . (cid:48)(cid:48)
38 0.95 1.25 1.61sBzK 1.62 2.38 3.14 0 . (cid:48)(cid:48)
54 0 . (cid:48)(cid:48)
74 0 . (cid:48)(cid:48)
91 1.31 1.86 2.33ERO 1.03 1.56 2.20 0 . (cid:48)(cid:48)
48 0 . (cid:48)(cid:48)
76 1 . (cid:48)(cid:48)
05 0.80 1.14 1.56DRG 1.81 2.43 3.16 0 . (cid:48)(cid:48)
54 0 . (cid:48)(cid:48)
72 0 . (cid:48)(cid:48)
85 1.49 1.91 2.32
To study the effect of substructure, we perform a simulationin which the emission originates from kpc-scale clumps inthe galaxies, described in more detail in section 5. At base-lines shorter than ∼ m the stacked visibilities are wellfitted by a Gaussian model, as is shown in Fig. 5. The blacksquares indicate the ALESS baselines. The simulation alsoinclude a set of longer baselines modelled on a intermedi-ate length baseline configuration from ALMA Cycle 3, withbaselines from 45 m to 1400 m, shown in Fig. 5 as red cir-cles. The Gaussian model recovers an average flux densityfor the stacked sources of . ± . mJy, compared to theinput flux density for the simulation of 2.1 mJy per source.The flux density is primarily recovered by using the ALESSbaselines, using the long baselines from the ALMA Cycle 3configuration, we measure an average flux density of only 90 µ Jy. When fitting to the data from both baseline configu-rations, the size measured for the Gaussian is 0 . (cid:48)(cid:48) ± . (cid:48)(cid:48) .This agrees well with the distribution of the positions forthe clumps, which are spread in a disk with a diameter of1 . (cid:48)(cid:48)
2. For the
HST z -band detected galaxies, we measure andcompare the HST sizes to our stacked ALMA sizes, and findthe values to be consistent with uncertainties for all samples.However, for those sources with a strong detection we canperform a more in-depth comparison. We select all sourcesfrom the K20 sample with peak SNR > in z -band, a totalof 32 sources. Stacking these sources in the ALESS data wemeasure an average size of 0 . (cid:48)(cid:48) ± z -band (0 . (cid:48)(cid:48) z -band morphology, described in detail in section 5.2. Fig. Baseline length [m] − . . . . . . . F l u x den s i t y [ m Jy ] Figure 5.
Stacked flux densities for simulated dataset. Eachgalaxy is simulated as a combination of three clumps, scat-tered within a radius of 5 kpc from the centre position for thegalaxy. The errors are estimate from the standard deviations ofthe visibilities in each bin. The plot combines data from simula-tions with two different baseline configuration, The shorter base-lines, marked with black squares, are simulated with the same uv -coverage as the ALESS observations. The longer baselines,marked with red circles, are simulated using an ALMA Cycle3 configuration with baselines from 45 m to 1.4 km. uv -plane, indicating that the z -band and the sub-mm emissiontrace a similar radial morphology. c (cid:13)000
Stacked flux densities for simulated dataset. Eachgalaxy is simulated as a combination of three clumps, scat-tered within a radius of 5 kpc from the centre position for thegalaxy. The errors are estimate from the standard deviations ofthe visibilities in each bin. The plot combines data from simula-tions with two different baseline configuration, The shorter base-lines, marked with black squares, are simulated with the same uv -coverage as the ALESS observations. The longer baselines,marked with red circles, are simulated using an ALMA Cycle3 configuration with baselines from 45 m to 1.4 km. uv -plane, indicating that the z -band and the sub-mm emissiontrace a similar radial morphology. c (cid:13)000 , 1–11 L. Lindroos et al.
Figure 6.
Simulation of stacked flux densities based on HST z -band emission maps shown in black, binned by baseline length.The errors are estimated from the standard deviation for the visi-bilities within each bin. For comparison the stacked flux densitiesof the z -band detected galaxies of our sample, using the samebinning. Note that for the middle bin the simulated and real dataare very close, and as such the simulated data point is hiddenbehind in the plot. Decarli et al. (2014) stacked each of the four samples in thethree
Herschel
SPIRE bands. Using data from the
Herschel
Multi-tiered Extragalactic Survey (Oliver et al. 2012). Wecombine these values with our stacked ALESS flux densitiesto better constrain the dust spectral energy distributions(SED) of our samples. The dust emission is modelled as amodified black body: S ν ∝ ν β B ν ( T ) where S ν is the dustSED, B ν ( T ) is the Planck function, T is the dust temper-ature (typically T ≈ β describes the effect ofdust opacity (typically β ≈ . − ) (e.g. Kelly et al. 2012).The total IR luminosity ( L IR ) is calculated between 8 µ mand 1000 µ m (e.g. Sanders et al. 2003). The dust emissionis fitted using a χ minimization, with two free parameters, T and L IR . The value of β is fixed to 1.6. Each data pointis weighted by σ − . Data and fitted SEDs are shown in Fig.7, and results are summarised in Table 3.The SFRs are calculated from L IR assuming a Chabrier(2003) initial mass function (Genzel et al. 2010) SFR = 1 . × − M (cid:12) yr − L IR L (cid:12) . (4)We find that the SFRs are similar for all samples at ∼ (cid:12) yr − , with the DRG sample showing a ∼
20 per centlarger star-formation rate compared to the other samples.In Fig. 8 we show SFR as a function of stellar mass for eachsample. The measured values fall close to the best-fit “mainsequence” for star-forming galaxies at similar redshifts, (e.g.the Tacconi et al. (2013) parametrization for comparison).We also split the sBzK sample into two subsets based onstellar mass, estimating the flux density of the stacked data
Table 3.
Infrared luminosity and SFR estimates for the stackedsamples, using a combination of
Herschel and the new stackedALMA results. We also show the average stellar mass for eachsample. The errors are estimated from χ when varying both T and L FIR simultaneously.Sample L FIR T dust SFR M ∗ [ L (cid:12) ] [K] [ M (cid:12) yr − ] [ M (cid:12) ]K20 . ± . ± ±
18 5 . × sBzK . ± . ± ±
14 5 . × ERO . ± . ± ±
22 4 . × DRG . ± . ± ±
20 6 . × sBzK(high mass) . ± . ± ±
20 2 . × sBzK(low mass) . ± . ± ±
20 8 . × ERO DRGK20 sBzK F l u x d e n s i t y [ m J y ] Wavelength [ µ m] Figure 7.
Stacked flux densities for the samples and fitted dust-emission SEDs. Combines the three wavelengths from the
Her-schel /SPIRE with our new ALMA estimates. The parameters ofthe fitted models can be found in Table 3. with a Gaussian. The star-formation rate is calculated usingthe same dust temperature as for the full sBzK sample.
Our stacked results show that the stacked sources have ex-tended emission with typical sizes ∼ . (cid:48)(cid:48)
7. Assuming that thetarget sources are compact or unresolved, as was done inDecarli et al. (2014), the flux density is systematically un-derestimated. For the samples in this study with between30 and 40 per cent. For the SMGs, where we measure thestacked size to be 0 . (cid:48)(cid:48)
4, this effect is smaller with the peakbrightness only ∼ uv -domain we can effectively re-cover the full flux density. This does, however, rely on havingaccess sufficient sensitivity on short baselines. The ALESS c (cid:13) , 1–11 stimating sizes of faint, distant galaxies in the submillimetre regime × Stellar mass [M ⊙ ] S t a r -f o r m a t i o n r a t e [ M ⊙ y r − ] Figure 8.
Average star-formation rate and stellar mass for theeach sample shown as blue triangles (see Table 3). The sBzKsample is also split into two sub-sample based on stellar mass,shown as black circles. The red line indicates the best-fit “mainsequence” for star-forming galaxies at z ∼ , using the Tacconi etal. (2013) parametrization. data were observed in a very compact ALMA configuration,with most baselines shorter than 100 m, or 115 k λ . This re-sults in a naturally weighted beam size of ∼ . (cid:48)(cid:48) , i.e., theobservations are sensitive to scales of 1 (cid:48)(cid:48) -2 (cid:48)(cid:48) .The filtering of spatial scales is a well known effectwithin interferometry, however, the results of this studyshow that the effect is especially pronounced for stacking.For the mapping of individual galaxies, most of the fluxdensity will originate from smaller scales, allowing it tobe resolved with higher resolutions. Only emission which issmooth over larger scales is filtered. In the case of stacking,the averaging of multiple galaxies smooth out substructure.As such, having access to sufficiently short baselines is essen-tial to measure the total flux density of the stacked sources.Emission at larger scales, at sizes larger than approximately2-3 (cid:48)(cid:48) , would be similarly suppressed in the ALESS data.However, HST data at z -band set an upper limit for oursamples at around 2 (cid:48)(cid:48) , as the dust-emission is unlikely toextend much beyond the stellar region. Our simulations show that with stacking, we can efficientlyestimate the total flux density and the radial distribution ofthe emission. Using Gaussian models, we find sizes around0 . (cid:48)(cid:48) . (cid:48)(cid:48) . (cid:48)(cid:48)
17. This means thatall samples are extended at a greater than σ significance.Martí-Vidal, Pérez-Torres & Lobanov (2012) calculate thelimitation of model fitting of detected sources in a interfer-ometric data set and find that the minimal size that can bemeasured is given by Θ min = β (cid:18) λ c (cid:19) (cid:18) S/N (cid:19) × Θ beam (5) where S/N is the SNR of the averaged visibilities, β is a pa-rameter that depends on the array configuration (typicallybetween 0.5 and 1.0), Θ beam is the FWHM of the beam us-ing natural weighting, and λ c depends on the probabilitycut-off for false detection (3.84 for σ ). Using this formulawe find our size error to be consistent with a β between 0.4and 0.5. This both indicates that the sizes of . (cid:48)(cid:48) are veryrobust, and also shows that model fitting of stacked sourceshas similar noise to individual sources with similar SNR. Forcomparison we also stacked the SMGs in our data, and findan average size of . (cid:48)(cid:48) ± . (cid:48)(cid:48) . This is marginally larger thanthe median size measured by Simpson et al. (2015) of 0 . (cid:48)(cid:48) . (cid:48)(cid:48) > σ , we find that the typical offsets are . (cid:48)(cid:48) ± . (cid:48)(cid:48) . If we deconvolve this from the measured sizeswe find that the sizes the actual galaxies are . (cid:48)(cid:48) − . (cid:48)(cid:48) .We also estimate the variance of the target samples us-ing bootstrapping. This indicate larger errors on our esti-mated parameters due to the sample sizes, with size errorsincreasing to 0 . (cid:48)(cid:48)
20 - 0 . (cid:48)(cid:48)
35. Larger samples of star-forminggalaxies have been studied using
HST , e.g. van der Wel et al.(2014) measured the sizes of ∼ star-forming galaxiesat z > . Based on this sample they find that the opticalsizes follow a log-normal distribution. Looking at the sBzKgalaxies, if we assume that the sub-mm sizes of our samplesfollow a similar distributions, we would expect this to con-tribute 0 . (cid:48)(cid:48)
04 to error of our stacked size assuming we sample22 random galaxies. This effect is similar for the other sam-ples, getting smaller the larger the sample is. Looking atresults from bootstrapping, we find that the results are con-sistent for the sBzK and DRG samples. For the K20 thebootstrap estimated error is larger than expected from theoptical sizes of star-forming galaxies, however, this sample isnot selective to star-forming galaxies leading probably lead-ing to a more heterogeneous sample. For the flux densities ofour stacked sample, the bootstrap errors are larger than themeasurement errors. This is consistent with the large varia-tion seen for star-forming galaxies, where the SFR can varywith more than an order of magnitude within a sample. Wenote that this indicates the error on the SFRs measured forour samples are dominated by sample variance. This wouldbe true even if each galaxy was individually detected, indi-cating the importance of large samples to accurately esti-mate the typical SFR for a population of galaxies.
Looking at the sizes of the galaxies with a detection in the
HST z -band data (peak SNR > ), we can estimate thesize of the stellar component of the galaxies. Using a Sersicdistribution, we find an median effective radius ( r e ) of 0 . (cid:48)(cid:48) n of 1.33. The sizes measured at sub-mmwavelengths for our stacked sources are based on a Gaussianprofile in place of a Sersic profile. For comparison we fitour stacked sources using a Sersic profile, with n fixed to c (cid:13) , 1–11 L. Lindroos et al.
HST observations arefrom the GEMS survey. The GEMS z -band observations arenot as deep as the GOODS-S z -band observations. As suchis possible that we are missing low flux surface density emis-sion, and underestimating the size of these galaxies. How-ever, as this primarily affects half the sample, the impact onthe median value is not expected to be very large.Another limitation of the z -band measurements is dustobscuration. The measured submm continuum emission in-dicates that dust is abundant in all samples. We can com-pare to the shallower HST H -band observations from GEMSand GOODS-S, which are less affected by dust absorption.However, only 16 galaxies are detected in H -band. For thesegalaxies we measure a median size of 0 . (cid:48)(cid:48)
6, which agrees wellwith the sizes measured in z -band.The size of . (cid:48)(cid:48) corresponds to a physical size of 6 kpc atthe average redshift of the sBzK sample. For SMGs severalmeasurements of the sizes at sub-mm wavelengths exist, e.g.,Simpson et al. (2015) find a median size of 2.4 ± ± ∼ × Focusing on the sBzK sample, the total SFR is estimatedto be 100 M (cid:12) yr − , over a size of 10 kpc, or a SFR surfacedensity ( Σ SFR ) of 1 M (cid:12) yr − kpc − . This value is consistentwith other measurements of sBzK galaxies, e.g., Daddi et al.(2010b) which found values for 0.1 to 30 M (cid:12) yr − kpc − . Ofthis, 40 per cent originates in the centre. This corresponds to Σ SFR ≈
13 M (cid:12) yr − kpc − in the inner 1 kpc of the galax-ies. While this is higher than the corresponding value forthe DRGs ( ∼ M (cid:12) yr − kpc − ), it is a very small valuecompared to LIRGs at lower redshift. E.g., in Arp 220 witha similar SFR (Anantharamaiah et al. 2000), the major-ity of the star formation occurs inside 1 kpc of the centre(Scoville, Yun & Bryant 1997), resulting in an average Σ SFR of approximately 70 M (cid:12) yr − kpc − (Anantharamaiah et al.2000). We can also compare this to SMGs, e.g., Hodge et al.(2015) measured Σ SFR in the centre of a z = 4 SMG to be ∼ M (cid:12) yr − kpc − , which is similar to Arp 220, but muchhigher than our sBzK galaxies.As noted, Σ SFR in the centre of the DRG sample isvery low, at 2 M (cid:12) yr − kpc − it is only a factor 4 above thesame value for the Milky Way (Robitaille & Whitney 2010),despite a factor 100 difference in SFR. In Decarli et al. (2014), all samples were found to have anexcess of star formation compared to the similar samples inother fields. Our updated flux-density estimate are ∼
30 - 40per cent higher than those found by Decarli et al. (2014).However, after fitting the SED of the dust emission, thefitted dust temperatures are typically lower. For the sBzKand DRG samples, this results in SFRs which are consistentwith the Decarli et al. (2014) measurements within the un-certainties. However, for the K20 and ERO sample the SFRdrops with ∼ per cent compared to Decarli et al. (2014).This results in the K20, ERO and sBzK samples having verysimilar star-formation rates, at ∼
90 M (cid:12) yr − .We also compare the measured star-formation rates tothe stellar masses, and find them to be consistent with Tac-coni et al. (2013) for star-forming galaxies at z ∼ . We alsosplit the sBzK sample, the sample with highest SNR, by stel-lar mass. Both the low- and high-mass samples fall close tothe best-fit “main sequence” using the Tacconi et al. (2013)parametrization. This indicates, that while these galaxies aretypically more massive compared to other similar samples,the star formation is driven by the same mechanics. In this paper we use stacking to measure the average mor-phologies and sizes of samples of galaxies using ALMA. Weuse a uv -stacking algorithm combined with model fitting inthe uv -domain. We select star-forming galaxies at z ∼ us-ing four different criteria: K VEGA < , ERO, DRG, andsBzK. The samples are stacked in the ALMA 344 GHz con-tinuum observations from the ALESS survey. We find thatall samples are extended, with FWHM sizes of ∼ . (cid:48)(cid:48) ± . (cid:48)(cid:48) estimated using a Gaussian model. Accounting for randomoffsets between optical catalogue positions and submm po-sitions in the data, we find that the actual average sizes aresomewhat smaller at ∼ . (cid:48)(cid:48) ± . (cid:48)(cid:48) .The uv -model fitting results in flux densities that are ∼ per cent higher than if the sources are assumed to bepoint sources. Furthermore, assuming that the dust emissionmeasured at 344 GHz is primarily heated by star formation,we find that the majority of the star formation is takingplace outside the inner kpc of the galaxy. We compare thisto the stellar distribution in the same galaxies, using HST z -band data. The median effective radius is measured to 0 . (cid:48)(cid:48) z -band maps as inputmodel for each galaxy. The distribution are found to agreewell, indicating no systematic difference in size or radial dis-tributions between the stellar and star-forming component.Using a Monte Carlo method to estimate the robust-ness of the result, we find the measured sizes to be robust at > σ for all samples. The measured difference between the c (cid:13) , 1–11 stimating sizes of faint, distant galaxies in the submillimetre regime sBzK and DRG sample, is larger than the uncertainties witha statistical significance of σ . We find that the measuredaccuracy of the sizes is comparable to the theoretical limitsfor individual sources (e.g. Martí-Vidal et al. 2014). As in allcases with stacking we do not measure the properties of theindividual galaxies, but the average properties of the sam-ples, and this smoothing effect can simplify the modelling ofthe stacked source. However, it also increase the interfero-metric effect of filtering of large spatial scale, making shortspacings very important to recover the full flux density.We can conclude that for the stacking of any sourcesthat may be marginally extended, using uv -stacking withmodel fitting can provide a flux-density estimate that is sig-nificantly more robust and valuable additional informationsuch as the typical sizes of the sources of the stacked sam-ple. This is also important for future facilities such as theSquare Kilometer Array (SKA), showing that having accessto uv -data in stacking is invaluable. LL thanks Robert Beswick for useful discussion, IvanMartí-Vidal for helpful input on the model fitting, andIan Smail for useful discussion. We thank an anony-mous referee for helpful suggestions and useful comments.This paper makes use of the following ALMA data:ADS/JAO.ALMA
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Figure A1.
Distribution of stacked size for the K20 sample asestimated through bootstrapping.
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APPENDIX A: FITTED MODELS
In this appendix we present the distributions determined forthe fitted sizes using bootstrapping on the stacking samples.The method for the bootstrapping is described in §4.4, andthe plotted distribution indicate the probability of possiblesizes for the population of each sample. The bootstrappingmethod approximate errors from observational noise as wellas sample variance.
Figure A2.
Distribution of stacked size for the sBzK sample asestimated through bootstrapping.
Figure A3.
Distribution of stacked size for the ERO sample asestimated through bootstrapping.c (cid:13) , 1–11 stimating sizes of faint, distant galaxies in the submillimetre regime Figure A4.
Distribution of stacked size for the DRG sample asestimated through bootstrapping.c (cid:13)000