HI galaxy simulations for the SKA: number counts and bias

0.22 MB·English

by  Mario G. Santos

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0 HI galaxy simulations for the SKA: number counts
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n and bias
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J

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C Mario G.Santos1,2,DavidAlonso3, PhilipBull4, Marta Silva5,1,SahbaYahya1
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h 1 DepartmentofPhysics,UniversityofWesternCape,CapeTown7535,SouthAfrica
p
2 SKASA,4rdFloor,ThePark,ParkRoad,Pinelands,7405,SouthAfrica
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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
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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
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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 sig-
nal 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 sensi-
tivities 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 spec-
ifications 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 cor-
responding 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 cosmol-
ogy 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