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5 1 0 HI galaxy simulations for the SKA: number counts 2 n and bias a J 6 1 ] O C Mario G.Santos1,2,DavidAlonso3, PhilipBull4, Marta Silva5,1,SahbaYahya1 . h 1 DepartmentofPhysics,UniversityofWesternCape,CapeTown7535,SouthAfrica p 2 SKASA,4rdFloor,ThePark,ParkRoad,Pinelands,7405,SouthAfrica - o 3Astrophysics,UniversityofOxford,DWB,KebleRoad,OxfordOX13RH,UK r t 4InstituteofTheoreticalAstrophysics,UniversityofOslo,POBoks1029Blindern,0315Oslo, s a Norway [ 5 CENTRA,InstitutoSuperiorTécnico,UniversidadedeLisboa,Lisboa1049-001,Portugal 1 E-mail: [email protected] v 0 9 This chapter describes the assumed specifications and sensitivities for HI galaxy surveys with 9 SKA1 and SKA2. It addresses the expected galaxy number densities based on available simu- 3 0 lations as well as the clustering bias over the underlyingdark matter. It is shown that a SKA1 . 1 HIgalaxysurveyshouldbeabletofindaround5 106 galaxiesover5,000deg2(uptoz 0.8), 0 × ∼ whileSKA2shouldfind 109galaxiesover30,000deg2(uptoz 2.5).Thenumberspresented 5 ∼ ∼ 1 herehavebeenusedthroughoutthecosmologychaptersforforecasting. : v i X r a AdvancingAstrophysicswiththeSquareKilometreArray June8-13,2014 GiardiniNaxos,Italy (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ HIgalaxysimulations 1. Introduction Largescalestructuresurveyshavesofarreliedonimagingofalargenumberofgalaxiesatop- tical ornear-infrared wavelengths combined withredshift information toobtain the3-dimensional galaxy distribution (e.g. DES,BOSS,Euclid). Hydrogen isthe mostabundant baryon inthe Uni- verse, making it a prime candidate for a tracer of the underlying dark matter distribution and an invaluable probe of the energy content of the Universe and the evolution of large-scale structure. In order to use hydrogen as a tracer at radio wavelengths, we need to rely on radio observations of the neutral hydrogen (HI) 21cm line. With a rest frequency of 1420 MHz, telescopes probing the sky between this and 250 MHz would be able to detect galaxies up to redshift 5. Although we expect most galaxies to contain HI, since the spin-flip transition responsible for the 21cm signal is highly forbidden, the emission is quite weak: at z=1.5 most galaxies with a HI mass of 109 M will be observed with a flux density of 1 m Jy. This implies that for low sensitivity ⊙ ∼ surveys, onlytheclosest orHI-richest galaxies willbeobserved, andtherefore massive telescopes willberequired inordertodetectsufficientnumbersofHIgalaxiesatalevelcapableofproviding competitive constraints in Cosmology. This chapter analyses the expected number density of HI galaxies as a function of redshift as well as its clustering bias taking into account the flux sensitivities forHIgalaxy surveysusingtheplanned SKAtelescopes. Wediscuss theprocess bywhich we obtain these numbers -using a set ofstate of the art simulations -identify inherent limitations in the current set up and suggest a series of improvements. Although we focus on SKA1 and 2, the sensitivity calculations presented here allow toeasily consider other configurations such asan earlydeployment ofSKA1(corresponding toabout50%ofitsexpected sensitivity). 2. Sensitivity calculations Thenoiseassociatedtothefluxdensity(fluxperunitfrequency)measuredbyaninterferometer canbeassumedtobeGaussianwitharmsgivenapproximately by 2k T s B sys , (2.1) S≈ A 2dn t eff p p for anarray withtotal effective collecting areaA , frequency resolution dn andobservation time eff perpointingt (k istheBoltzmannconstant). Theextrafactorof1/√2comesfromassumingdual p B polarisedsystems. Telescopesensitivitiesaresometimesquotedintermsofthe"SystemEquivalent Flux Density": SEFD 2k T /A or just "A over T": A /T . The effective collecting area B sys eff eft sys ≡ depends on the efficiency of the system which is expected to be the in the range of 70 to 80% for theSKA1dishes. Notethattheexpressionabovegivesthefluxdensitysensitivityperresolutionbeam(nottobe confused with the dish primary beam or telescope field of view). Moreover, this is the sensitivity if the full array is used without constraints on the psf (point spread function or resolution beam). This is what is usually called the natural array sensitivity. Although angular resolution is not a crucialissueinaHIgalaxysurvey,someweightingofthevisibilities mightberequiredinorderto obtain a better behaved psf. In that case, the sensitivity of the telescopes will be further reduced. The required shape and resolution of the psf will depend on the source detection algorithm for 2 HIgalaxysimulations HI galaxies which is still a subject of active research. For SKA1, using uniform weighting plus Gaussiantaperingofthevisibilitiescanproduce"wellbehaved"Gaussianbeamsat10arc-seconds resolution with flux sensitivities that are 30% to 80% worse than what is quoted here (see Braun 2013). However, new source extraction algorithms might be able to deal with more complex psfs (forinstance, byworkinginthevisibilityspacedirectly)andapproachthenaturalarraysensitivity. Thereforeinthischapterwedecidedtoquotevaluesusingthefullarray,underlining thatinreality thesituation mightbelessoptimistic. Theequivalent brightness temperature uncertainty is s c2 s = S , (2.2) T 2k n 2(dq )2 B wheredq istheangularresolution oftheinterferometer. Thetotaltemperature is T =T +T (2.3) sys inst sky with T 60 300MHz 2.55 K and T the instrument temperature, which is usually higher than sky n inst ≈ theskytempera(cid:0)tureabo(cid:1)ve300MHz. Fortypicalinstrumentspecifications,thearraynoisefluxrms canbewrittenas: T 25,000m2 0.01MHz 1/2 1h 1/2 s =260m Jy sys . (2.4) S (cid:18)20K(cid:19)(cid:18) A (cid:19)(cid:18) dn (cid:19) (cid:18)t (cid:19) eff p For a given survey area, S we will need approximately S /(q )2 pointings where (q )2 area area B B isthetelescope primarybeam(instantaneous fieldofview)withthefullwidthathalfmaximumof thebeam,givenby(inradians) q 1.22l / A , (2.5) B dish ≈ p where A is the effective area of each dish. The time per pointing t is then related to the total dish p integration timet through tot (q )2 B t =t . (2.6) p tot S area This will increase the time per pointing at the lowest frequencies (for a fixed t ). Note however tot thatwhendoing asurveysomeoverlap ofthebeamsisexpected inorder toachieve uniformity on the noise across the sky map. Following the SKA1 imaging performance memo, we assume that themosaicked beamswillcorrespond toasizeof: p 1.3l 2 q 2 [sr]. (2.7) B≈ 8 (cid:18) D (cid:19) ThesituationisslightlymorecomplicatedforPAFs(PhasedArrayFeeds)asbelowacertaincritical frequency(usuallytakenasthemiddleoftheband),thebeamswillremainconstant. Therefore,for a PAF with a certain number of beams, N , we will assume that above the critical frequency, the b fullbeam willbeN SPF,whereSPFisthesinglepixel feedbeamasabove(eq. 2.7)andbelow b × thatcriticalfrequency, itwillremainconstant. Following the current baseline design (Dewdney et al. 2013), table 1 summarises the specifications for each telescope, while table 2 contains the specifications assumed for the target HI 3 HIgalaxysimulations Telescope Band[MHz] z T [K] N D [m] A [m2] beam[deg2]a Fluxrms[m Jy]b inst dish dish eff SKA1-MID 350-1050 0.35-3.06 28 190 15 21,824 1.78 417 SKA1-MID 950-1760 (0)-0.50 20 190 15 26,189 0.48 247 MeerKAT 580-1015 0.40-1.45 29 64 13.5 6,413 1.68 1015 MeerKAT 900-1670 (0)-0.58 20 64 13.5 5,955 0.64 1093 MID+MK 580-1015 0.40-1.45 28.5 254 – 28,237 1.37 328 MID+MK 950-1670 (0)-0.50 20 254 – 32,144 0.51 202 SKA1-SUR 350-900c 0.58-3.06 50 60 15.0 8,482 61(710)d 1918 SKA1-SUR 650-1670c (0)-1.19 30 60 15.0 8,482 18(1300)d 1151 ASKAP 700-1800 (0)-1.03 50 36 12.0 3,257 30(1250)d 4996 SUR+ASKAP 650-1670e,f (0)-1.19 30e 96 – 11,740 18(1300)d 832 SKA2g 480-1290 0.1-2.0 20 250h 50 400,000 30 16 Table1:Telescopeconfigurations. Notes: a Thisistheprimarybeam(FoV)calculatedatthecenterofthebandunlessstatedotherwise. Itis proportionaltol 2 (exceptforPAFs). Forthecombinedtelescopes,thesmallestbeamofthetwotelescopes isused.bFluxdensityrmsatthecentreoftheband(orthefrequencyshowninparenthesis)forafrequency intervalof0.01MHzand1hourintegrationusingEq.2.4.Thisisthenaturalarraysensitivity.Requirements on the shape of the psf can increase this value by about50%. c Only 500MHz instantaneousbandwidth. d PAF beams assumed constantbelow the critical frequency(shown in parenthesisin MHz) and going as 1/n 2abovethat.eAssumingthatallASKAPPAFswillbereplacedtomeettheSKA1-SURfrequencyband andinstrumenttemperatureof30K.f Assumingthatonlyband2willbe initiallydeployed. g Valuesonly indicative-canbechanged. Bothbeamandfluxrmsareassumedconstantacrosstheband. h Theseshould bestations(denseaperturearrays). galaxy surveys. Note that, in order for the sensitivities to be proportional to n , we assume that the mosaicking is done at the highest frequency used for the HI survey (i.e. telescope pointings are packed side by side at the highest frequency). This way, at lower frequencies, the beams will overlap, since their size varies as 1/n 2, which will increase the time per pointing with the corresponding improvement in sensitivity. Of course, for PAFs, this sensitivity will remain constant below the critical frequency. Table 2 shows that both SKA1 instruments (MeerKAT+MID and ASKAP+SUR)havecomparableperformanceforaHIgalaxysurveyatthehighestfrequencyband considered here (band 2). Note however that this assumes that all ASKAP PAFs will have to be replaced withnewgeneration receiversatsomepoint. Ontheotherhand,atthelowestfrequencies (band1),SKA1-SURseemstoperformbetter. 3. HIgalaxysurveys TocalculatetheHIgalaxynumberdensityandbiasasafunctionoffluxrms,weusedtheSAX- sky simulation1. The SAX-Sky simulation consists of a galaxy catalogue containing the position andseveralastrophysical properties foreachobject inamockobserving cone. Itwasproduced by Obreschkow et al. (2009) by adding HI and CO properties to the galaxies obtained by De Lucia 1http://s-cubed.physics.ox.ac.uk/s3_sax 4 HIgalaxysimulations Telescope Band[MHz] beam[deg2] Surveyarea[deg2] t [hours] S(ref) [m Jy] p rms SKA1-MIDa (Band1) 350-1050 0.88 5,000 1.76 315 SKA1-MIDa (Band2) 950-1760 0.88 5,000 1.76 187 MeerKATa (Band1) 580-1015 1.06 5,000 2.13 696 MeerKATa (Band2) 900-1670 1.06 5,000 2.13 750 SKA1-MID+MeerKATa (Band1) 580-1015 0.88 5,000 1.76 247 SKA1-MID+MeerKATa (Band2) 950-1670 0.88 5,000 1.76 152 SKA1-SURb(Band1) 350-900 61(710) 5,000 122 174 SKA1-SURb(Band2) 650-1670 18(1300) 5,000 36 192 ASKAPb 700-1800 30(1250) 5,000 6 645 SKA1-SUR+ASKAPb 700-1670 18(1300) 5,000 36 140 SKA2c 480-1290 30 30,000 10 5.14 Table2:Surveyspecifications.Weassumeatotalobservationtimeof10,000hours.Thecorrespondingflux (ref) densityrms(S )iscalculatedforafrequencyintervalof0.01MHzusingthenaturalarraysensitivity. rms Notes: a Valuescalculatedatthetargetfrequencyof1.0GHzunlessstatedotherwise. Thebeamandtime per pointing(t ) are assumed to changeas (1.0GHz)2 acrossthe band. Theflux density rmsis assumed to p n changeas n acrosstheband.bValuescalculatedatthePAFcriticalfrequency(inparentheses,inMHz). 1.0GHz Belowthatfrequency,valuesareassumedconstant. Aboveit, thebeamandt areassumedtogoas1/n 2, p and the flux density rms as n . c Indicative values. The beam and flux density rms are assumed constant acrosstheband. & Blaizot (2007) through the post processing of the Millennium dark matter simulation (Springel etal.2005). TheMillennium simulation isalarge dark matter simulation witha box size of500 h 1 Mpc − and a spatial resolution of 5 h 1 Kpc. It was run in order to study the formation of dark matter − halos and their evolution with time and therefore comoving snapshots are available at 64 fixed time steps from redshifts 127 to 0. The simulation was run using a WMAP1-compatible cosmology and a mass resolution of 8.6 108h 1M . The halos were identified as Friends-of-Friends − × ⊙ groups containing more than 20 particles, and merger trees were built to link each halo with its substructures throughout time. In De Lucia & Blaizot (2007) the simulation was post-processed in order to populate the halos with simple models of galaxies and used a semi-analytical scheme to evolve the galaxies properties independently unless a merger occurs. The properties of each of these galaxies includes themassofthedifferent constituents, such asthemassofthecold andhot gascomponents, aswellasthegalaxystarformation rate(SFR),Hubbletype,etc. TheatomicandmoleculargascomponentsofeachgalaxywerethenobtainedbyObreschkow et al. (2009) from the cold gas component based on properties of nearby regular spiral galaxies. Assuming that the cold gas of these galaxies resides in flat, approximately symmetric disks, it followsthatthelocalgaspressureisconnectedtothemoleculargasfractionbyaknownphysically based prescription, which is also compatible with observations (at least for the more HI massive galaxies). The hydrogen mass of the cold gas component was assumed to be 0.74% of the total 5 HIgalaxysimulations mass. In order to properly emulate the light cone, the SAX-sky simulation used only part of each snapshot oftheMillennium simulation asisdescribed inObreschkow etal.(2009). Therefore, the boxesatfixedredshiftswithHIpropertieswhichcanbeobtainedfromtheSAX-skysimulationare considerably smaller than 500h 1Mpc along the line of sight. Thiscan undermine the estimation − of statistical quantities such as the galaxy bias (the number densities, on the other hand, can be easilyobtained fromthequeries inthes-cubedwebsite). 3.1 Galaxynumbercounts Forthedetection ofagalaxy,werequired thatatleasttwopointsontheHIlinearemeasured, that is, the width of the line has to be larger than twice the assumed frequency resolution of the survey. TheideaistoobtaininformationonthetypicallinedoublepeakexpectedfromHIgalaxies duetotheirrotation. Thisinprinciple willremoveanygalaxythatisseen“faceon”sinceitwould showjustanarrowpeak. Moreevolvedsourcedetection methodscan(andshould) beexplored, to avoid spurious detections due to RFI, but we will keep this simple approach for now. A signal to noise of 10is then usually required for the detection of agalaxy. Weused the following variables andprescription todetectHIgalaxies intheSAXdatabase: zapparent–Apparentredshift (including Dopplercorrection). • Experiment spectral resolution set to dV =2.1(1+zapparent) at the galaxy rest frame • (inKm/s). Thiscorresponds toanobservedfrequencyresolutionof0.01MHzwhichiswhat hasbeenassumedforthesensitivity calculations. hiwidthpeak [Km/s] – Line width between the two horns of the HI-line profile in the • galaxyrestframe(alreadycorrected forthegalaxyinclination). Followingwhathasbeendiscussed, wetakeonlygalaxies wherehiwidthpeak/2>dV. • hiintflux[JyKm/s]–Velocity-integratedlinefluxoftheHI-line(intheobserversframe). • Foreachgalaxytheobserved sensitivity intermsoffluxdensity (fluxperfrequency) isthen • flux =hiintflux/hiwidthpeak. Notethatthisalreadyfactorsintherelationbetween d velocity integrated fluxandthefrequency integrated fluxaswellastherelation between the velocity linewidthintherestframeandthecorresponding observed frequency. Finally, we take only galaxies where flux > N S / (hiwidthpeak/dV) (we took cut rms • N =10fora"10-sigma"cut). p cut In order to be as general as possible we did not try to match completely the flux rms used in the query to the survey specifications given in table 2. Instead wegive results for different values so that a simple interpolation can be used for different surveys. In particular, if we decide to use a 5s cut instead of 10s , it isjust a question of looking for the numbers corresponding to the flux rmsthatishalfthesurveyvalue(sincetheobtained numbercountsallassumeda10-sigmacut). 6 HIgalaxysimulations 105 δ rms=0.0 104 z=1.0 rms=1.0 rms=3.0 103 rms=5.0 rms=10.0 rms=23.0 2π 102 2 / k) (HI P 101 3k 100 10-1 10-2 10-1 100 101 k[h/Mpc] Figure 1: The dimensionless HI galaxy power spectrum for differentflux cuts using the prescription de- scribedaboveforeachfluxrms. BlacksolidlineshowsthedarkmatterpowerspectrumfromCAMB.The straightlinesathighkindicateshotnoise. 3.2 Galaxybias To obtain the bias, the “detected” galaxies were put in a box, for which the power spectrum of the number counts was calculated. The bias squared was then taken as the ratio of that power spectrum to the dark matter one at k = 0.2 h/Mpc. To perform this study we used boxes with different sizes from L = 58h 1 Mpc and L = 100h 1 Mpc for redshift 0.06 to L = 162h 1 − − − Mpc and L =398h 1k Mpc for redshift 2.07⊥. This makes the bias extraction problemkatic, since − ⊥ we cannot efficiently probe modes below k 0.1 h/Mpc, where we can safely neglect non-linear ∼ effects. For high flux rms, the number of galaxies is low and the shot noise dominates up to very smallwavenumbers. Theconclusionisthatforhighflux/redshiftvaluestheresultsshouldbetaken as only an indication, in particular for bias above 20m Jy flux rms. As an example we show in ∼ Figure 3.1 the dimensionless HI galaxy power spectrum at z=1 for different sensitivities. Also notethattheMillenniumsimulationonlyhasgalaxieswithmasses&2 1010M andthereforethe × ⊙ biasandnumberdensityforrms=1mightbeaffectedbythelackofsmallergalaxies. Howeverthis should not affect the statistics for higher rms values. The cosmological analysis should compare results fordifferent fiducialvaluesandfullymarginalise overthebiasandnumbercounts. 4. Fitting functions forgalaxynumber counts and bias In this section we describe fitting functions for the galaxy number counts, dn/dz, and bias, b(z), as a function of redshift. At a constant flux rms, these are well described by the functions 7 HIgalaxysimulations (Yahyaetal.2014) dn/dz = 10c1zc2exp( c z) (4.1) 3 − b(z) = c exp(c z), (4.2) 4 5 wherec arefreeparameters. dn/dzisthenumberofgalaxies persquare degree, perunit redshift. i Best-fit values of the parameters are given in Table 3, and Fig. 2 compares the fitted curves with datapointsfromthesimulations. Note that for a redshift bin of width D z, the observed number of galaxies in a 1 deg2 patch is dnD z,andthecomovinggalaxynumberdensityisn(z)= dn 1 ,where dV = dr(p /180)2r(z)2, ≈ dz dzdV/dz dz dz r(z)isthecomingdistance toredshift zanddr/dz=c/H(z). As explained above, the flux sensitivities for the telescopes listed in Table 2 are taken to ei- ther evolve with frequency, S (cid:181) n , or are assumed constant across the band. For the non-PAF rms receivers (SKA1-MIDandMeerKAT),thefluxrmsasafunction ofredshiftis N n S (z)=S(ref) cut 21 (1+z) 1, (4.3) rms rms − (cid:18) 10 (cid:19)1GHz (ref) where S is the reference flux rms at 1 GHz listed in Table 2, N is the detection threshold in rms cut multiples of the flux rms, and n 1.42 GHz is the rest frame frequency of the HI line. For the 21 ≃ PAFreceivers(SKA1-SURandASKAP),thefluxrmsis 1 n n (ref) Ncut ≤ crit Srms(z)=Srms  , (4.4) (cid:18) 10 (cid:19)× n21(1+z) 1 n >n n − crit crit  where n is the PAF critical frequency listed in Table 2, and S(ref) is the reference flux at the crit rms critical frequency. ThefullSKAhasS (z)=const. WetakeN =5forallbut thefullSKA,to rms cut whichweapply amorestringent 10s threshold. Toobtain thenumbercounts andbiasforagiven telescope, weinterpolate Eqs. 4.1and4.2asafunction offluxrmsusingthevaluesgiveninTable 2,andevaluatethemasafunctionofredshiftusingtheappropriateS (z)function. Wethenfitthe rms same fitting functions to the results – best-fit parameters are given in Table 4. Note that while the frequency-corrected numbercountsareallsatisfactorilyfittedbyEq. 4.1,fourofthebiasfunctions arenotwelldescribed byEq. 4.2;theseareflaggedinthetable. 5. Limitations ofcurrent simulationsandfuture improvements AlthoughtheSAX-Skyisastate-of-the-art simulationintermsofcosmologicalradiosurveys, itsuffers from anumber ofshortcomings thatwould bedesirable toaddress inthefuture, inorder toimprovethequalityoftheforecasts fortheSKA. Probably the most challenging problem for this type of simulations is finding a compromise between theenormous volumethat experiments like theSKAareexpected toprobe, andthecom- plexity necessary to simulate the type of objects that will be observed with sufficient precision. In our case, we would ideally need a simulation with a box of size L 15h 1Gpc and resolv- box − ∼ ing haloes down to 109h 1M , which is simply out of the question given the present typical − ∼ ⊙ 8 HIgalaxysimulations 108 0µJy 1µJy 107 3µJy 5µJy 106 6µJy 105 7.3µJy 10µJy 2− eg 104 23µJy d dN/dz 40µJy 103 70µJy 100µJy 102 150µJy 200µJy 101 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 redshift (z) 10 0µJy 1µJy 3µJy 8 7.3µJy 23µJy 70µJy 6 100µJy z) b( 4 2 0 0.0 0.5 1.0 1.5 2.0 2.5 redshift (z) Figure2: Upperpanel: DependenceoftheHIgalaxyredshiftdistributiondn/dz(units: deg 2). Notethat − thenumbersarefordifferentfluxrmswhichwillinturncorrespondtoagivengalaxyfluxcutaccordingto thesourcedetectionproceduredescribedinthetext(whichassumeda10-sigmacut). Curvesfollowthefits accordingto Eq.4.1anddotsare fromthe S3-SAX simulation. LowerPanel: HIgalaxybiasfordifferent S . Note that above 70 m Jy values for high redshifts are purely extrapolations. However, this has little rms impactasathighz,shotnoisewilldominateforthesesensitivities. 9 HIgalaxysimulations S [m Jy] c c c c c rms 1 2 3 4 5 0 6.23 1.82 0.98 0.8695 0.2338 1 7.33 3.02 5.34 0.5863 0.6410 3 6.91 2.38 5.84 0.4780 0.9181 5 6.77 2.17 6.63 0.5884 0.8076 6 6.84 2.23 7.13 0.5908 0.8455 7.3 6.76 2.14 7.36 0.5088 1.0222 10 6.64 2.01 7.95 0.4489 1.2069 23 6.02 1.43 9.03 0.5751 0.9993 40 5.74 1.22 10.58 0.5125 1.1842 70 5.62 1.11 13.03 0.6193 1.0179 100 5.63 1.41 15.49 0.6212 1.0759 150 5.48 1.33 16.62 – – 200 5.00 1.04 17.52 – – Table3: ValuesofthefittedparametersforEqs.4.1and4.2. Forhigherfluxcutswesuggesttakingb(z)=1, asonlygalaxiesbelowz 0.5arerelevantanyway. ∼ Telescope c c c c c N 1 2 3 4 5 gal SKA1-MID(Band1) 4.6399 0.8577 13.402 0.8675* 0.4767* 7.40 104 × SKA1-MID(Band2) 5.3662 1.2611 13.979 0.7038* 0.8616* 3.39 106 × MeerKAT(Band1) 4.9849 1.8010 17.359 1.0000 0.0000 6.27 103 × MeerKAT(Band2) 7.6813 4.5740 28.067 1.0000 0.0000 1.22 105 × SKA1-MID+MeerKAT(Band1) 5.1667 1.1585 14.170 0.8116* 0.7163* 7.76 104 × SKA1-MID+MeerKAT(Band2) 5.4381 1.3315 11.836 0.6343 0.9708 4.96 106 × SKA1-SUR(Band1) 5.1814 1.1668 14.208 0.8108* 0.7174* 9.95 103 × SKA1-SUR(Band2) 5.9023 1.5879 16.149 0.6232 1.0636 4.25 106 × ASKAP 4.8577 1.1551 17.499 1.0000 0.0000 8.08 105 × SKA1-SUR+ASKAP 5.8611 1.4742 15.344 0.6206 1.0559 5.55 106 × SKA2 6.7767 2.1757 6.6874 0.5887 0.8130 9.12 108 × Table 4: Valuesofthebest-fitnumbercount/biasparameters(Eqs. 4.1and4.2)forindividualtelescopes, afterscalingthefluxrmswithfrequencyandinterpolating. Someoftheresultingbiasfunctionsarenotfit verywellbyEq.4.2;theseareflagged.Thetotalnumberofgalaxiesexpectedforeachsurveyisalsoshown (with5,000deg2exceptforSKA2). 10