Relations Between the Sizes of Galaxies and their Dark Matter Halos at Redshifts $0 < z < 3$

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by  Kuang-Han Huang

DRAFTVERSIONMARCH24,2017
PreprinttypesetusingLATEXstyleAASTeX6v.1.0
RELATIONSBETWEENTHESIZESOFGALAXIESANDTHEIRDARKMATTERHALOSATREDSHIFTS0<z <3
KUANG-HANHUANG1,S.MICHAELFALL2,HENRYC.FERGUSON2,ARJENVANDERWEL3,NORMANGROGIN2,ANTON
KOEKEMOER2,SEONG-KOOKLEE4,PABLOG.PE´REZ-GONZA´LEZ5,ANDSTIJNWUYTS6
1UniversityofCaliforniaDavis,1ShieldsAvenue,Davis,CA95616,USA;[email protected]
2SpaceTelescopeScienceInstitute,3700SanMartinDrive,Baltimore,MD21218,USA
7 3MaxPlanckInstituteforAstronomy,Koenigstuhl17,D-69117Heidelberg,Germany
1 4CenterfortheExplorationoftheOriginoftheUniverse,DepartmentofPhysicsandAstronomy,SeoulNationalUniversity,Seoul,Korea
0
2 5DepartamentodeAstrof´ısica,FacultaddeCC.F´ısica,UniversidadComplutensedeMadrid,E-28040,Madrid,Spain
r 6DepartmentofPhysics,UniversityofBath,ClavertonDown,Bath,BA27AY,UK
a
M ABSTRACT
WederiverelationsbetweentheeffectiveradiiR ofgalaxiesandthevirialradiiR oftheirdarkmatter
2 eff 200c
2 halos over the redshift range 0 < z < 3. For galaxies, we use the measured sizes from deep images taken
withHubbleSpaceTelescopefortheCosmicAssemblyNear-infraredDeepExtragalacticLegacySurvey;for

] halos,weusetheinferredsizesfromabundancematchingtocosmologicaldarkmattersimulationsviaastellar
A
mass–halo mass (SMHM) relation. For this purpose, we derive a new SMHM relation based on the same
G
selectioncriteriaandotherassumptionsasforoursampleofgalaxieswithsizemeasurements. Asacheckon
.
h therobustnessofourresults,wealsoderiveR –R relationsforthreeindependentSMHMrelationsfrom
eff 200c
p theliterature. WefindthatgalaxyR isproportionalonaveragetohaloR ,confirmingandextendingto
eff 200c
-
o highredshiftsthez = 0resultsofKravtsov. Late-typegalaxies(withlowSe´rsicindexandhighspecificstar
r formationrate[sSFR])followalinearR –R relation,witheffectiveradiiat0.5 < z < 3closetothose
t eff 200c
s predictedbysimplemodelsofdiskformation;atz < 0.5,thesizesoflate-typegalaxiesappeartobeslightly
a
below this prediction. Early-type galaxies (with high Se´rsic index and low sSFR) follow a roughly parallel
[
Reff–R200c relation,∼0.2–0.3dexbelowtheoneforlate-typegalaxies. Ourobservationalresults,reinforced
3
byrecenthydrodynamicalsimulations,indicatethatgalaxiesgrowquasi-homologouslywiththeirdarkmatter
v
1 halos.
0 Keywords:galaxies: evolution—galaxies: high-redshift—galaxies: structure—methods: dataanalysis
0
4
0 1. INTRODUCTION galaxiesare,onaverage,proportionaltothesizesoftheirsur-
. rounding dark matter halos. For galactic disks, this propor-
1 The size of a galaxy, as measured by its half-mass radius
0 R, for example, is among the most basic of its properties. tionality in sizes follows directly from the assumed propor-
7 tionality of the specific angular momentum of baryons and
Together with the mass M, the size R determines the bind-
1
: ing energy, −E ≈ GM2/(4R), and hence the energy radi- darkmatterresultingfromtidaltorquesintheearlystagesof
v galaxy formation (Fall & Efstathiou 1980; Mo et al. 1998).
ated away during the formation of the galaxy. For galactic
i
X disks, with stars and gas on nearly circular orbits with ro- Thisassumptionunderliespracticallyallofthesemianalyti-
cal models of galaxy formation in current use (e.g., Cole et
r tation velocity V , the size R is determined by the angu-
rot
a al.2000;Crotonetal.2016). Recenthydrodynamicalsimu-
lar momentum J ≈ MRV , which in turn determines the
rot
energy E = −1/2MVr2ot ≈ −G2M5/(8J2). The basic de- lationsofgalaxyformationconfirmtheapproximatepropor-
tionalitybetweenthespecificangularmomentumofgalaxies
scriptionofgalaxiesingeneralconsistsofM,R,andV ,or
rot
and their dark matter halos (Genel et al. 2015; Pedrosa &
equivalently M, E, and J, while for disk-dominated galax-
Tissera2015;Tekluetal.2015;Zavalaetal.2016).
ies,anytwoofthesequantitiessuffice.
There have been numerous searches for the expected de-
As a result of the hierarchical growth of galaxies, we ex-
creaseingalacticsizeswithredshiftbasedonmeasurements
pecttheirmassesandradiitoincreasewithcosmictimeand
ofdeepimagestakenwiththeHubbleSpaceTelescope(HST)
thus to decrease with redshift. In the simplest models of
over the past dozen years (e.g., Ferguson et al. 2004; Hathi
galaxy formation, the sizes of the baryonic components of
etal.2008;Moslehetal.2012). Thesesearchesallfindthat
galaxies were smaller in the past, by roughly the predicted
amount,althoughtherearesignificantdifferencesinthepre-
E-mail:[email protected]
2
cisedeclineofgalacticsizeswithredshiftamongthesestud- logicalparameters: h=0.7,Ω =0.27,andΩ =0.73.
m Λ
ies (compare, e.g. Shibuya et al. 2015 and Curtis-Lake et
al.2016). Partofthediscrepancyamongtheseresultsstems 2. OBSERVATIONS
fromthefactthattheapparentevolutioninsizesdependson Forthisstudy,weneedagalaxysamplewithhomogeneous
howgalaxiesatdifferentredshiftsarecompared,whetherat data quality that enables accurate size measurements. HST
fixedstellarmassorluminosityoratvariablestellarmassor images are required because galaxies at z > 1 are gener-
luminosity. ally smaller than 1(cid:48)(cid:48). We also need a galaxy sample with
Kravtsov(2013)usedstellarmass–halomass(SMHM)re- good constraints on redshifts, stellar masses, and star for-
lations derived via the technique of abundance matching to mationrates,sothatwecanconnectgalaxiestodarkmatter
comparetheobservedsizesofpresent-daygalaxieswiththe halos and distinguish star forming galaxies from quiescent
sizesoftheirmatcheddarkmatterhalosincosmologicalN- galaxies. The Cosmic Assembly Near-infrared Deep Extra-
bodysimulations.Hefoundthatthesizesofgalaxiesatz =0 galacticLegacySurvey(CANDELS)isthebestdatasetcur-
are proportional on average to the sizes of their halos. Fur- rentlyavailableforthisstudy:allfiveCANDELSfields,cov-
thermore,thecoefficientofproportionalityisconsistentwith ering ≈ 800 arcmin2 in total, have HST images at optical
a simple model in which galactic disks grow with approxi- andnear-IRwavelengthswithuniformquality(Groginetal.
mately the same specific angular momentum as their halos 2011; Koekemoer et al. 2011). The high angular resolution
until z ∼ 2 and then stop growing after that. The question ofHST((cid:46)0(cid:48).(cid:48)15inthenear-IR)isabletoresolvemostgalax-
immediately arises whether the same or a different relation iesatz ≤ 3. Inaddition, ancillaryspectroscopicandimag-
holds between the sizes of galaxies and their halos at high ingdatacombinewithHST datatoprovidetightconstraints
redshifts. The purpose of this paper is to answer this ques- ongalaxyredshifts, stellarmasses, andstarformationrates.
tion. CANDELShasthreetiersofdepth. TheWideregioncovers
Theadvantageofcomparingthesizesofgalaxiesatmulti- ∼675arcmin2toa5σlimitingmagnitudeH ∼27.3mag
160
pleredshiftswiththesizesoftheirmatchedhalosatthesame ina0(cid:48).(cid:48)17aperture. TheDeepregioncovers∼ 125arcmin2
redshifts,aswedohere,isthattheresultsarethenexpressed toH ∼28.1mag. ThesurveyalsoencompassestheHub-
160
directlyinsimple,physicallymeaningfulterms. Thisframe- ble Ultra-Deep Field (HUDF)—the HUDF09 (Bouwens et
workalsohelpstoclarifytheresultsofprevioussearchesfor al. 2010) and HUDF12 (Ellis et al. 2013; Koekemoer et al.
theevolutionofgalacticsizes. 2013;seealsoIllingworthetal.2013)—covers∼ 5arcmin2
Therearealreadyacoupleofindicationsthatthesizesof toH ∼29.7mag.
160
galaxies and their halos evolve in lockstep. First, semiem- We take the photometry, spectroscopic and photometric
pirical models of galaxy formation that make this assump- redshifts, and stellar-mass estimates from the CANDELS-
tionagreebetterwithdeepHST imagesthanthesamemod- team catalogs (Guo et al. 2013; Galametz et al. 2013; San-
elswithdifferentassumptionsabouttheevolutionofgalactic tinietal.2015;Nayyerietal.2016;G.Barroetal. 2017,in
sizes (Taghizadeh-Popp et al. 2015). Second, recent mea- preparation; M. Stefanon et al. 2017, in preparation). The
surementsofthesizesandrotationvelocitiesofgalacticdisks sizeestimatesaretakenfromvanderWeletal.(2012).
at1 < z < 3and0.2 < z < 1.4indicatethattheyhaveap- WeselectgalaxiesintheCANDELSsurveyat0 < z < 3
proximatelythesamespecificangularmomentaastheirdark for this study. We cap our galaxy redshifts at z = 3 be-
matterhalos(Burkertetal.2016;Continietal.2016). While cause this is the highest redshift that HST still samples red-
these results are suggestive, it is still important to make a ward of rest-frame 4000A˚, and because selection biases in-
direct, independent comparison of the sizes of high-redshift duced by cosmological surface brightness dimming are ex-
galaxieswiththesizesoftheirmatchedhalos,theinvestiga- pected to be relatively mild for z ≤ 3 (Taghizadeh-Popp et
tionwedescribehere. al. 2015). Sources are detected using SExtractor (Bertin &
The plan for the remainder of this paper is the following. Arnouts1996)inH . Roughly10%ofthesesourceshave
160
In Section 2, we describe our sample of galaxies and mea- high-quality spectroscopic redshifts, which are used in cali-
surements of their sizes and other properties. In Section 3, bratingthephotometricredshiftsfortheremainingsources.
we discuss the abundance-matching method and its imple- Galaxy sizes are measured in H and J by fitting a
160 125
mentationwithfourdifferentSMHMrelations. InSection4, single Se´rsic profile to each galaxy using GALFIT (Peng et
wepresenttheresultsofourcomparisonofgalaxyandhalo al. 2010). We define galaxy sizes as effective radii (R )
eff
sizes,andinSection5,wediscusstheuncertaintiesinthese along the major axis, the radii within which Se´rsic profiles
results. WediscusssomeimplicationsofourresultsinSec- containhalfofthetotalintegratedlight. Wediscussthede-
tion6.Weshowtheconnectionbetweenthegalaxysize–halo projectionfrom2Dto3Dlaterwhencomparingwiththeoret-
size relation and the more familiar galaxy size–stellar mass ical expectations. Our overall sample is dominated by late-
relationinanappendix. Allmagnitudesquotedinthispaper type galaxies at all redshifts, whose disk components have
areintheABsystem, andweassumethefollowingcosmo- thesame2Dand3Dhalf-lightradii.
Usingsimulationswithartificialgalaxiesandcomparisons
3
Table1.GalaxySampleSizes
Redshift Wide Deep HUDF Total z M a
med ∗,low
(M )
(cid:12)
0.0<z<0.5 4388 923 50 5361 0.34 1.0×107
0.5<z<1.0 9706 2435 116 12257 0.73 5.0×107
1.0<z<1.5 6666 1395 113 8174 1.23 8.2×107
1.5<z<2.0 5152 1224 90 6466 1.70 1.7×108
2.0<z<2.5 2580 727 47 3354 2.23 2.1×108
2.5<z<3.0 1483 497 54 2034 2.69 3.8×108
AllRedshifts 29975 7201 470 37646 ··· ···
a
TypicalstellarmassofthegalaxiesfromHUDFwith26.6mag<H <
160
26.8magandnearthemedianofeachredshiftbin.Inthelowestredshift
bin,weimposeahardcutinstellarmassat107M .
(cid:12)
of measurements in different imaging depths, van der Wel upmanymorelowsurfacebrightnessobjects,wewouldhave
etal.(2012)concludedthatbrighterthanH = 24.5mag expected to see them show up in the tail of the distribution.
160
in the Wide region, the systematic (random) errors of R Instead,weseemorelarge-radiusobjectsintheWidesample,
eff
measurementsarebelow∼20%(30%). Meanwhile,thesys- mostofwhichareprunedawayasbadfits,butwithouthaving
tematic(random)errorsofSe´rsicindexnmeasurementsare much impact on the median R . A Kolmogorov-Smirnov
eff
below∼50%(60%). Thequotederrorshereareforgalaxies testyieldspvaluesconsistentwiththesamplesbeingdrawn
with n > 3, which tend to have larger errors than galax- from the same underlying distribution. The bottom panels
ies with n < 3. Therefore, we select all galaxies brighter of Figure 1 show the same comparison for the Deep region
than H = 24.5 mag in the Wide region, H = 25.2 inthemagnituderange24.2mag< H < 25.2mag. We
160 160 160
magintheDeepregion,andH =26.7magintheHUDF madeasimilarcomparisonforthestellarmassdistributions,
160
(SExtractor-measured magnitudes). These magnitude limits alsofindingnostatisticallysignificantdifferencebetweenthe
correspondtosimilarsignal-to-noiselimits. HUDFandtheDeepandWidesamples.
Inadditiontomagnitudecuts,weprunethesampleasfol- We have also estimated the completeness of our sample
lows. We reject all sources that have problematic photom- fromthedetectionefficienciesfortheCANDELSsurveyde-
etry (generally those at the borders of the image or falling rived by Guo et al. (2013). They inserted artificial galaxies
on stellar diffraction spikes). We eliminate sources that are into images from the Wide, Deep, and HUDF regions and
identified as active galactic nuclei (AGNs) via X-ray or IR analyzed them with SExtractor in the same way as the real
spectral energy distributions (SEDs). We discard as point surveytodeterminethedetectionefficiencyasafunctionof
sources all objects that have half-light radii (measured by apparent magnitude H , effective radius R , and Se´rsic
160 eff
SExtractor) smaller than 2.6 pixels. We enforce the follow- index n (see their Fig. 5). From these results, we estimate
ingcriteriatoeliminategalaxieswithpoorGALFITfits: (1) thatoursampleasawholeismorethan85%complete. This
theGALFITmeasurementisflaggedaspoorinthecatalogs highlevelofcompletenesshelpstoensurethatselectionbi-
from van der Wel et al. (2012); (2) the error in the mea- aseshaverelativelylittleimpactonourgalaxysize–halosize
sured R exceeds 0.3R ; (3) the measured n lies outside relations(estimatedinSection5).
eff eff
the range 0.1 < n < 8, which usually signals problematic Studyinggalaxysizeevolutiondemandsthatwecompare
fits. TheGALFIT,AGN,andpoint-sourcecriteriacombined R valuesatasimilarrest-framewavelengthacrossredshift
eff
rejectroughlyone-fourthofthesourcesthatsatisfythemag- bins,sothatwecaneliminatethecontributionsfromdustor
nitude cuts. The numbers of sources that pass all the cuts stellar age gradient to the observed size evolution. We fol-
abovearelistedinTable1. lowtheprocedureinvanderWeletal.(2014)tocorrectfor
TheexistenceoftheverydeepHUDFdataallowsustotest galaxy color gradients and place galaxy sizes on the same
whetherselectioneffects,measurementbiases,orthepruning rest-framewavelength. Todothis,weusegalaxysizesmea-
procedurearebiasingoursamplesneartheirfaintlimits. In suredinH forgalaxiesatz > 1.5andusethesizesmea-
160
thetoppanelsofFigure1,wecomparethesizedistributions suredinJ atz < 1.5. Colorgradientsthatleadtodiffer-
125
in the Wide region and the HUDF for the magnitude range ent galaxy sizes at different wavelengths are accounted for
23.5 mag < H < 24.5 mag before and after pruning, by a correction factor that is a function of galaxy redshift,
160
findingnosignificantdifference. IftheHUDFwerepicking stellar mass, and galaxy type (late-type or early-type). As
4
Raw sample HUDF Pruned sample
2.0 2.0
WIDE
s s
nt nt
u1.5 = 0.29 u1.5 = 0.27
o o
c = 0.28 c = 0.28
d d
e = 0.83 e = 0.35
z z
ali1.0 ali1.0
m m
or or
n n
0.5 0.5
0.0 0.0
1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.0 0.5 0.0 0.5 1.0
log(R /arcsec) log(R /arcsec)
Raw sample HUDF Pruned sample
2.0 2.0
DEEP
s s
nt nt
u1.5 = 0.25 u1.5 = 0.23
o o
c = 0.26 c = 0.24
d d
e = 0.59 e = 0.53
z z
ali1.0 ali1.0
m m
or or
n n
0.5 0.5
0.0 0.0
1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.0 0.5 0.0 0.5 1.0
log(R /arcsec) log(R /arcsec)
Figure1. HistogramsofeffectiveradiusReff forgalaxiesinnarrowmagnituderangesintheWide,Deep,andHUDFregionsofoursample.
ThetoppanelscomparethedistributionsofR intheWideandHUDFregionsinthemagnituderange23.5mag<H <24.5mag,while
eff 160
thebottompanelscomparethedistributionsofR intheDeepandHUDFregionsinthemagnituderange24.2mag<H <25.2mag.For
eff 160
reference,theselectionlimitsofoursampleintheseregionsareH =24.5(Wide),25.2(Deep),and26.7mag(HUDF).Theleftandright
160
panelscomparethedistributionsbeforeandafterthesamplepruningdescribedinSection2.Thelegendsinthepanelslistthemedianvaluesof
R inthefourhistograms,andtheKolmogorov-Smirnovprobabilitiesthatthehistogramsaredrawnfromthesameunderlyingdistribution.
eff
Theconsistencyofthehistogramsinregionswithdifferentdepths,beforeandafterpruning,indicatesthatthedistributionofgalacticsizesin
oursampleisunbiasedevenneartheselectionlimits.
theresultofthiscolorgradientcorrection,themeasurements than the magnitude limits mentioned above, where we are
are converted into the R near rest-frame 5000A˚. The size confidentthatourmeasurementsarerobustandunaffectedby
eff
correction is typically only a few percent, but it does reach size-dependentbiases.Foreachredshiftinterval,weestimate
∼60% in some cases. For more details about the color gra- thetypicalstellarmassofthefaintestgalaxiesM bytak-
∗,low
dient correction, we refer the readers to van der Wel et al. ing the median SED-fitted stellar mass estimate of galaxies
(2014),Section2.2,andtheirequations(1)and(2). within0.1magoftheHUDFmagnitudelimit. Thevaluesof
Stellar masses and star formation rates are estimated by M arelistedinTable1andshownasthicktickmarksat
∗,low
comparing our photometry with model SEDs, adopting a thebottomsofFigures5–9. SED-basedstarformationrates
Chabrier (2003) initial mass function (IMF). Here the stel- can be uncertain by ∼ 0.4 dex (Salmon et al. 2015); there-
lar masses of galaxies include all luminous stars and dark fore,theuncertaintiesinthespecificstarformationrates(sS-
remnants at the time of observation (but not stellar ejecta). FRs) are roughly (cid:46) 0.6 dex for our galaxy sample. In this
This method of estimating stellar masses has been exten- paper,weselectsubsamplesintheupperandlower20%tails
sively tested in Mobasher et al. (2015), and they found that of the sSFR distribution. Because we are making a differ-
typicalstellarmassuncertaintiesare∼0.25dexforthemag- entialcomparisonbetweentherelativelylargepopulationsin
nitude limits adopted here. The primary sources of system- thesetails,ourresultsarenotsensitivetothesSFRuncertain-
atic uncertainties are IMF and stellar evolution models; for ties.
galaxies with strong nebular emission lines, systematic un-
certaintiesforstellarmasscanbeupto∼0.4dex. 3. ABUNDANCEMATCHING
WerestrictthisstudytogalaxieswithstellarmassesM∗ > In this study, we employ the technique of abundance
107M(cid:12). Above this limit, we include all galaxies brighter matchingtoestimatethemassandhencethesizeofthedark
5
matter halo associated with each galaxy in our sample. In results for the four bins of width ∆z = 0.25 covering the
essence, this technique compares the measured sizes of ob- range 0.5 < z < 1.5 into two bins of width ∆z = 0.5.
served galaxies with the inferred sizes of matched halos in In this step, we weight the observed comoving densities of
cosmologicaldarkmattersimulations. Thebasicassumption galaxiesbythecomovingvolumeineach∆z =0.25binand
isthattherankorderingofgalaxy(stellar)massesM reflects thenfitadoubleSchechterfunctiontothecombinedcomov-
∗
on average the rank ordering of halo (virial) masses M , ingdensitiesineach∆z = 0.5bin. Forourlowestredshift
200c
i.e., that the cumulative number densities of galaxy masses bin, 0 < z < 0.5, weadopttheTomczaketal. stellarmass
andhalomassesareequal:n (>M )=n (>M ).This function in their lowest redshift bin, 0.2 < z < 0.5, be-
g ∗ h 200c
ansatz leads directly to a correspondence between M and causeitagreeswellwiththeoneat< z >= 0.1derivedby
∗
M knownasthestellarmass–halomassrelation. While Moustakas et al. (2013). Finally, we have derived the halo
200c
the assumption that galaxy masses and halo masses follow massfunctionn (>M )fromtheMillennium-IIsimula-
h 200c
thesamerankorderingisareasonableapproximationforsta- tion (Boylan-Kolchin et al. 2009) at the snapshot closest to
tistical studies based on large samples such as ours, it can- themiddleofeachredshiftbinandthenmatchedthistothe
notbeexactlytrueforindividualgalaxies,whichexperience stellarmassfunctionasdescribedabovetoobtaintheSMHM
stochastic events such as mergers and starbursts throughout relation.
theirhistories. As a check on this procedure, we have independently de-
Given an SMHM relation, we compute the halo mass rived our own stellar mass function from scratch by the
M ofeachgalaxyinoursamplefromitsstellarmassM . 1/V methodforthegalaxiesinallfiveCANDELSfields
200c ∗ max
WethencomputethevirialhaloradiusR usingthestan- in the six ∆z = 0.5 bins (albeit with approximate K-
200c
dardformula corrections in our estimates of V ). The resulting stellar
max
mass function is nearly identical to the rebinned one from
(cid:20) 3M (cid:21)1/3
R = 200c , (1) Tomczaketal.(2014). Thisaddstoourconfidenceintheva-
200c 4π·200ρ (z)
crit lidity of SMHM relation 1, which we regard as the primary
SMHMrelationinthisstudy.
where ρ (z) is the critical density of the universe at red-
crit
Becauseourgalaxysamplecoversawiderrangeinstellar
shiftz. Inordertoassesshowsensitiveourresultsaretothe
mass than the Tomczak et al. sample, we linearly extrapo-
choiceofSMHMrelation,weperformallofourcalculations
late the SMHM relation in log–log space to both lower and
withfourdifferentSMHMrelations. AlloftheseSMHMre-
highermasses. ThesolidlinesinFigure2showtheSMHM
lationsarebasedontheChabrier(2003)stellarIMFandthe
relationderiveddirectlyfromtheTomczaketal. data,while
samehalomassdefinitionM .TheyareplottedinFigures
200c
the dashed lines show the extrapolated parts of the SMHM
2,3,and4anddiscussedbelow.
relation.
SMHM relation 1. We have derived this new SMHM re-
SMHM relation 2. Behroozi et al. (2013) derived this
lation specifically for this study so that it is as consistent as
SMHM relation from published stellar mass and halo mass
possiblewiththeCANDELSdataset,selectioncriteria,and
functionsoverawiderangeofredshifts(0<z <8). Thisis
SED fitting procedure for our sample of galaxies with size
probablythemostprevalentSMHMrelationintheliterature.
measurements. In particular, we combine the stellar mass
However, sinceitisbasedonstellarmassfunctionsthatare
functionn (>M )fromTomczaketal.(2014)withourde-
g ∗
quitedifferentfromthosederivedusingCANDELSdata,itis
terminationofthehalomassfunctionn (>M )fromthe
h 200c
notidealforthepresentstudy. Weuseitmainlytogaugethe
Millennium-IIsimulation(Boylan-Kolchinetal.2009).
sensitivity of our results to different SMHM relations. For
Tomczaketal.(2014)derivedthestellarmassfunctionof
consistency, we convert their halo mass M , defined us-
galaxiesat0.2<z <3inthreeofthefiveCANDELSfields, vir
ing a redshift-dependent overdensity factor ∆ (z) (Bryan
usingselectioncriteriaandproceduresforestimatingstellar vir
& Norman 1998), to our halo mass definition M . The
massessimilartothoseforoursample,asdescribedinSec- 200c
conversionassumesanNFWhalomassprofileandthehalo
tion2. Wehavecomparedourstellarmasseswiththosede-
rivedbyTomczaketal.(2014)1andfindnosystematicoffset mass–concentrationmodelcalibratedinDiemer&Kravtsov
(2015). The corrections are very small in general (< 0.1
andonlyasmallscatter(∼ 0.1dex). Tomczaketal. fitteda
dex).
doubleSchechterfunctiontotheobservedstellarmassfunc-
SMHM relation 3. This is the same SMHM relation
tion in differential form dn (> M )/dM in each of eight
g ∗ ∗
adopted by Kravtsov (2013). He derived his own SMHM
redshift bins. We adopt the Tomczak et al. results directly
relation out of concerns that previous relations used stellar
for the three bins of width ∆z = 0.5 covering the range
massfunctionsthatarebiasedatboththehigh-massandlow-
1.5 < z < 3.0. However, for simplicity, we combine their
mass ends. By using the same SMHM relation as Kravtsov
(2013), we can directly compare our galaxy size–halo size
1ThesestellarmassesarepublishedbytheZFOURGEteam(Straatman relationwithhisatz =0.
etal.2016)andcanbedownloadedfromhttp://zfourge.tamu.edu. SMHMrelation4. ThereareseveralSMHMrelationssep-
6
aratedbygalaxytypeatz < 0.5intheliterature, whichwe 0.5
plotinFigure3. Theserelationsusedifferentapproachesto
1.0
deriving the ratio between stellar masses and halo masses,
1.5
ranging from abundance matching (Rodr´ıguez-Puebla et al. )
2015) to weak lensing (Hudson et al. 2015; Mandelbaum 2.0
/
et al. 2016) to a mixture of the two methods (Dutton et * 2.5
al. 2010). We adopt the SMHM relation from Rodr´ıguez- (
3.0
Pueblaetal.(2015)becauseithasthelargestdynamicrange
Dutton+10 (late-type) Hudson+15 (blue, z=0.5)
in halo mass and is in the middle of the range spanned by 3.5 Dutton+10 (early-type) Hudson+15 (red, z=0.5)
Rodriguez+15 (blue) Mandelbaum+16 (blue)
the other type-dependent relations from the literature. We 4.0 Rodriguez+15 (red) Mandelbaum+16 (red)
use the Rodr´ıguez-Puebla et al. SMHM relations for blue 10 11 12 13 14 15
( / )
and red central galaxies at z = 0 for galaxies in our sam-
ple with Se´rsic index n below and above 2.5, respectively. Figure3.RatioofgalaxystellarmassM∗tohalovirialmassM200c
SinceRodr´ıguez-Pueblaetal. definedtheirhalomassusing plottedagainstM200c forfourlow-redshiftSMHMrelationsfrom
theliteraturethatdependongalaxycolorortype. Thesewerede-
∆ (z), we have applied the same conversion to M as
vir 200c rivedbyabundancematching(Rodr´ıguez-Pueblaetal.2015),weak
wedidforSMHMrelation2. lensing(Hudsonetal.2015;Mandelbaumetal.2016),oracombi-
We compare the four SMHM relations in Figure 4. Ev- nationofbothtechniques(Duttonetal.2010).ThreeoftheSMHM
relationspertaintoz =0andonetoz =0.5(Hudsonetal.2015).
idently, there are significant discrepancies among these
Notethelargediscrepanciesamongthesecolor-andtype-dependent
SMHM relations, especially the first and second, for which SMHMrelations.
thedifferencescanbeupto∼0.5dexatz ∼3. OurSMHM
relation1, derivedspecificallyfortheCANDELSsampleat
1.0
0 < z < 3, shows stronger redshift evolution than SMHM
relation 2 from Behroozi et al. (2013). As already noted, 1.5
thisdifferencecomesmainlyfromthedifferentstellarmass ) 2.0
functions used as input to these SMHM relations. Fortu-
2.5
nately, as we show in Sections 4 and 5, our main scientific /
*
resultsarerelativelyinsensitivetotheadoptedSMHMrela- 3.0
(
tion,largelyduetotheweakdependenceofhalosizeonhalo 3.5
SMHM 1 (. < < . ) SMHM 3 (= )
mass(R200c ∝M210/03c). 4.0 SSMMHHMM 12 ((.=<. )< . ) SSMMHHMM 44 ((brelude, , ==))
SMHM 2 (= . )
4.5
10 11 12 13 14 15
SMHM Relation 1
1.0 ( / )
1.5
Figure4.RatioofgalaxystellarmassM∗tohalovirialmassM200c
2.0 plottedagainstM200cforthefourSMHMrelationsadoptedinthis
) work. SMHMrelation1: derivedasdescribedinSection3forall
2.5 galaxies at 0 < z < 3 and displayed here at 0 < z < 0.5 and
/* 2.5<z<3.0,whichbrackettherelationatintermediateredshifts.
g( 3.0 SMHMrelation2: derivedbyBehroozietal.(2013)forallgalax-
o
l 3.5 ies at 0 < z < 8 and displayed here at z = 0.1 and z = 3.0.
0.0 < z < 0.5 1.5 < z < 2.0 SMHMrelation3:derivedbyKravtsov(2013)forallgalaxiesonly
4.0 0.5 < z < 1.0 2.0 < z < 2.5 at z = 0. SMHM relation 4: derived by Rodr´ıguez-Puebla et al.
1.0 < z < 1.5 2.5 < z < 3.0 (2015)separatelyforblueandredgalaxiesonlyatz=0.Notethat
4.5
10 11 12 13 14 15 therearesignificantdifferencesamongtheseSMHMrelations,but
log( / ) becausehalosizedependsweaklyonhalomass(R ∝M1/3),
200c 200c
ourmainresultsarenotsensitivetothesedifferences.
Figure2.RatioofgalaxystellarmassM∗tohalovirialmassM200c
plottedagainstM forourprimarySMHMrelationinsixred-
200c
shiftbinscoveringtherange0 < z < 3. WederivedthisSMHM ThemainresultsofthispaperaredisplayedinFigures5–
relationbyabundancematchingfromanevolvingstellarmassfunc-
9 and described in this section. The uncertainties in these
tionappropriatefortheCANDELSsample(Tomczaketal.2014)
results,mostlystemmingfromtheSMHMrelationandmor-
and the evolving halo mass function in the Millennium-II simula-
tion(Boylan-Kolchinetal.2009)asdescribedinSection3. Solid phologicalclassification,arediscussedinSection5.
linesarebaseddirectlyonthestellarmassfunctionfromTomczak Ourfirstmainresultisthatgalaxysizesareproportionalto
etal.(2014);welinearlyextrapolatetheSMHMrelationinlog–log
halosizesoverawiderangeofsizeandmass.Figure5shows
spacetocoverthestellarmassrangeofoursample(dashedlines).
galaxyR plottedagainsthaloR at0<z <0.5forthe
eff 200c
fourdifferentSMHMrelations.Ineachpanel,themediansof
logR inbinsofwidth∆logR = 0.15dexareplotted
eff 200c
4. RESULTS
aspentagons,andthe16th–84thpercentilerangesasvertical
7
bars; only the bins with more than five galaxies are shown. This helps explain why our overall relation in Figure 5 is
The halo radius limit corresponding to the reference stellar higher than Kravtsov’s at intermediate masses. Our sample
mass M from Table 1 is shown as a thick tick mark at is dominated by late-type galaxies (∼90% have n < 2.5),
∗,low
thebottomofeachpanel. Thecoefficientofproportionality whileKravtsov’ssampleisdominatedbyearly-typegalaxies
αintherelationR =αR isnearlythesameinallfour (∼ 80% by number). He noted that late-type galaxies are
eff 200c
cases; the median values of α are 0.021, 0.025, 0.023, and systematically larger in R than early-type galaxies at in-
1/2
0.024 for SMHM relations 1–4, respectively. These R – termediatestellarmasses,whichiswhereweseethelargest
eff
R relationsareapproximatelylinear,butwithsomesub- offset between these sequences in Figure 5. The changing
200c
tledifferencesdependingontheadoptedSMHMrelation. morphological mix as a function of mass also helps explain
Kravtsov (2013) also found a linear relation, using com- theapparentcurvatureoftheoverallrelationinFigure5,be-
pletely independent samples of galaxies at z = 0 and de- cause early-type galaxies dominate the high- and low-mass
projected 3D half-mass radii R rather than the projected endsoftherelation.
1/2
2Dhalf-lightradiiR . ThesolidlineinFigure5showshis Our third main result is that the R –R relation for
eff eff 200c
derivedrelationR =α(cid:48)R withα(cid:48) =0.015,assuming late-typegalaxiesisclosetothepredictionsofthesimplean-
1/2 200c
R = R for pure-disk galaxies. The bulk of our sam- alyticmodelofdiskformation.Thescaleradiusandeffective
eff 1/2
plebynumberliesabovethisrelationby∼0.2dex,agreeing radiusofanexponentialdiskembeddedinadarkmatterhalo
better at the high- and low-mass ends. There are a number withavirial(outer)radiusR andaspinparameterλare
200c
of possible explanations for this offset, one of them being givenby
thedifferencebetween2Dhalf-light(effective)and2Dhalf- λ
R = √ R (2)
d 200c
massradii. Szomoruetal.(2013)notedthatforthegalaxies 2
moremassivethan5×1010M at0<z <2.5,rest-frameg-
(cid:12) and
band2Dhalf-lightradiiareonaverage∼25%largerthan2D
R =1.68R , (3)
half-massradii(presumablyduetotheinfluenceofbulges), eff d
whichcouldaccountfor∼0.1dexoftheoffset. Wewillad- when the disk and halo have the same specific angular mo-
dress other explanations below in connection with morpho- mentum (J/M). Equation (2) is exact for isothermal halos
logicaltypes,deprojectioneffects,andtheredshiftevolution. (Fall&Efstathiou1980;seetheirFigure3andequation42;
Our second main result is that the R –R relations Fall 1983, see his equation 4) and is approximate for NFW
eff 200c
are offset for late-type and early-type galaxies. To sepa- haloswithtypicalconcentrations(Moetal.1998;Burkertet
rate morphological types, we split our sample in two dif- al. 2016). This prediction is shown as the dashed lines in
ferent ways: (1) high-n (early-type) and low-n (late-type) Figures6to9forλ = 0.035, thepeakoftheuniversalspin
subsamples, and (2) low-sSFR (early-type) and high-sSFR parameterdistribution(Bullocketal.2001;Bettetal.2007).
(late-type)subsamples.Weonlyincludethehighestandlow- We find that late-type galaxies at 0 < z < 0.5 lie ∼ 0.2
est 20% of the sample in either n or sSFR in the hope that dex below the J/M equality line; in other words, our late-
this procedure will isolate disk-dominated from spheroid- type galaxies have slightly less specific angular momentum
dominated galaxies. The resulting R –R relations for thantheirdarkmatterhalos. Thisoffsetisconsistentwithdi-
eff 200c
late-andearly-typegalaxiesusingallfourSMHMrelations rectmeasurementsofspecificangularmomentumatz = 0,
areshowninFigures6and7. whichindicateJ/M retentionfactorsη ∼ 80%±20%for
j
We see in both Figures 6 and 7 that galaxies of different galacticdisks(Fall&Romanowsky2013).
typesfollowsequencesroughlyparalleltotheR ∝ R Ourfourthmainresultisthatthereisremarkablylittleevo-
eff 200c
linewithanoffsetof∼ 0.2dexat0 < z < 0.5. Thisresult lution in the R –R relation from z = 3 to z = 0.
eff 200c
isrelativelyrobustagainstSMHMrelationandmorphologi- This is shown in Figures 8 and 9. As in the previous dia-
cal classification method: early-type (high-n or low-sSFR) grams, we select the highest and lowest 20% tails of the n
galaxies have smaller R than late-type (low-n or high- andsSFRdistributions. WeonlyshowresultsforSMHMre-
eff
sSFR)galaxiesatthesamehalomasses. Theeffectpersists lation 1, but we have checked that they are similar for the
even if we compare 3D half-light radii rather than 2D half- other SMHM relations. Figures 8 and 9 show again that in
light radii R , although with a smaller separation between allredshiftbins,late-typegalaxiesfollowanearlylinearre-
eff
thesequences. Theparallelsequencesofearly-andlate-type lation: R = αR . At 0.5 < z < 3, late-type galaxies
eff 200c
galaxies in the R –R diagram are reminiscent of the haveα≈0.034inFigure8(α≈0.029inFigure9)andlie
eff 200c
parallelsequencesofspheroid-anddisk-dominatedgalaxies close to the J/M equality line (within (cid:46) 0.1–0.2 dex) with
intheJ/M vs. M diagram(Fall1983;Romanowsky&Fall no discernible evolution. (There is a slight offset to smaller
2012;Fall&Romanowsky2013). Thelatterisduetoacom- sizes in the late-type sample when selected by sSFR rather
bination of different sizes (by a factor of ∼2) and different thanSe´rsicindex.) Thisresultagreeswithrecentdirectmea-
rotation velocities (also by a factor of ∼2–3) of spheroid- surements of specific angular momentum at 0.2 < z < 1.4
anddisk-dominatedgalaxiesofthesamestellarmass. (Continietal.2016)andat1 < z < 3(Burkertetal.2016),
8
SMHM Relation 1 SMHM Relation 2 SMHM Relation 3 SMHM Relation 4
. < < . . < < . . < < . . < < .
c
p
k
/
Kravtsov13
/ kpc / kpc / kpc / kpc
Figure5.GalaxyeffectiveradiusReffplottedagainsthalovirialradiusR200cinthelowestredshiftinterval(0<z<0.5)forthefullsampleofgalaxies.The
fourpanelsshowresultsforSMHMrelations1,2,3,and4asindicated.Thefaintgraydotsrepresentindividualgalaxies,whilethefilledpentagonsandvertical
barsindicatethemedianvaluesand16th–84thpercentilerangesofReff inbinsofwidth0.15inlogR200c.ThediagonallinesshowtheR1/2–R200crelation
atz =0fromKravtsov(2013)assumingReff =R1/2. Thethicktickmarkatthebottomofeachpanelindicatesthehalosizecorrespondingtothereference
stellarmassM∗,low listedinTable1. NotethattheReff–R200c relationsaresimilarforthefourdifferentSMHMrelationsandareroughlyconsistentwith
Kravtsov’sresults. TheReff–R200crelationsarelinearinafirstapproximationbutexhibitsomecurvatureathighandlowmassesasaresultofthechanging
mixofgalaxymorphologies.ComparewithFigures6and7.
SMHM relation 1 SMHM relation 2 SMHM relation 3 SMHM relation 4
. < < . . < < . . < < . . < < .
c
p
k
/
J/M equality
Kravtsov13
Lowest 20% in
Highest 20% in
/ kpc / kpc / kpc / kpc
Figure6.GalaxyeffectiveradiusReffplottedagainsthalovirialradiusR200cinthelowestredshiftinterval(0<z<0.5)forsubsamplesofgalaxieswiththe
lowestandhighest20%ofthemeasuredSe´rsicindexnasproxiesforlate-andearly-typegalaxies,respectively.ThefourpanelsshowresultsforSMHMrelations
1,2,3,and4asindicated.Thefaintblueandreddotsrepresentindividuallow-nandhigh-ngalaxies,respectively,whilethefilledbluesquares,openredcircles,
andverticalbarsindicatethecorrespondingmedianvaluesand16th–84thpercentilerangesofReff inbinsofwidth0.15inlogR200c.Thediagonalsolidlines
showtheR1/2–R200crelationatz =0fromKravtsov(2013)assumingReff =R1/2,whilethediagonaldashedlinesshowthepredictionforgalacticdisks
withthesameJ/Mastheirsurroundinghalos.Thethicktickmarkatthebottomofeachpanelindicatesthehalosizecorrespondingtothereferencestellarmass
M∗,lowlistedinTable1.NotethattheReff–R200crelationforlow-ngalaxiesissystematicallyabove,androughlyparallelto,therelationforhigh-ngalaxies.
TheReff–R200crelationsforbothsubsamplesofgalaxiesaremorelinearthantherelationsforthefullsample.ComparewithFigures5and7.
whichshowthatJ/M ingalacticdisksisnearlythesameas could be due to either size-measurement biases (due to dif-
intheirdarkmatterhalos. fuse outer halos surrounding central galaxies in groups and
Kravtsov(2013)speculatedthatthesizesofgalaxiesgrew clusters) or the breakdown of abundance matching for the
inproportiontothesizesoftheirhalosuntilz ∼ 2andthen group-orcluster-masshalos.
stopped,whiletheirhaloscontinuedtogrowinmassandsize.
5. UNCERTAINTIES
We find instead that the R –R relations at z < 2 are
eff 200c
very similar to those at z > 2. Our R –R relations How robust are these results? The uncertainties in this
eff 200c
forthelate-typegalaxiesatz <0.5havesmalleramplitudes study potentially include measurement and statistical errors
thanthoseatz > 0.5,indicatingapossibleslowdowninthe internal to the CANDELS data set, as well as external sys-
growthofdisks,butthisdeviationismild(∼0.2dex)andnot tematicerrorsfromtheadoptedSMHMrelationsandstellar
establishedbeyondalldoubt(seebelow). population models. Here we provide a brief assessment of
The R –R relation for early-type galaxies is also theseuncertainties.
eff 200c
nearly constant. We see in Figures 8 and 9 that the trend As noted in Section 2, errors in the measurements of ef-
for early-type galaxies at all redshifts roughly parallels that fective radii Reff (from fits to Se´rsic profiles) are relatively
for late-type galaxies, but shifted down by ∼ 0.2 dex at small: < 20%(systematic)to30%(random). Evenifthese
0 < z < 0.5 and by ∼ 0.2–0.3 dex at 0.5 < z < 3. There errorswereattheupperendofthisrangeforallgalaxiesand
is a slight hint of a “turnover” at the most massive end at varied systematically with galactic masses and sizes, they
0 < z < 0.5 (see Figures 8 and 9). This turnover, if real, wouldhaveanegligibleinfluenceonthecoefficientandex-
ponentofthegalaxysize–halosizerelation: R = αRβ
eff 200c
9
SMHM relation 1 SMHM relation 2 SMHM relation 3 SMHM relation 4
. < < . . < < . . < < . . < < .
c
p
k
/
J/M equality
Kravtsov13
Highest 20% in sSFR
Lowest 20% in sSFR
/ kpc / kpc / kpc / kpc
Figure7.GalaxyeffectiveradiusReffplottedagainsthalovirialradiusR200cinthelowestredshiftinterval(0<z<0.5)forsubsamplesofgalaxieswiththe
highestandlowest20%ofthemeasuredsSFRasproxiesforlate-andearly-typegalaxies,respectively.ThefourpanelsshowresultsforSMHMrelations1,2,3,
and4asindicated.Thefaintblueandreddotsrepresentindividualhigh-sSFRandlow-sSFRgalaxies,respectively,whilethefilledbluesquares,openredcircles,
andverticalbarsindicatethecorrespondingmedianvaluesand16th–84thpercentilerangesofReff inbinsofwidth0.15inlogR200c.Thediagonalsolidlines
showtheR1/2–R200crelationatz =0fromKravtsov(2013)assumingReff =R1/2,whilethediagonaldashedlinesshowthepredictionforgalacticdisks
withthesameJ/Mastheirsurroundinghalos.Thethicktickmarkatthebottomofeachpanelindicatesthehalosizecorrespondingtothereferencestellarmass
M∗,lowlistedinTable1. NotethattheReff–R200crelationforhigh-sSFRgalaxiesissystematicallyabove,androughlyparallelto,therelationforlow-sSFR
galaxies.TheReff–R200crelationsforbothsubsamplesofgalaxiesaremorelinearthantherelationsforthefullsample.ComparewithFigures5and6.
with|∆α/α| (cid:46) 0.02and|∆β| (cid:46) 0.08(assuminga∼ 20% size–halo size relations from the detection efficiencies for
or smaller systematic deviation in R for a factor of 10 or the CANDELS survey derived by Guo et al. (2013) as fol-
eff
more variation in R ). Because the sample size in this lows. TheydividetheR –H planeintoregionsthatare
200c eff 160
studyissolarge(N ∼ 38000),theeffectsofrandomerrors 0–50%,50–90%,and90–100%complete. Mostofoursam-
in the size measurements on the mean R –R relations ple(88%)liesintheregionof90–100%completeness,while
eff 200c
areevensmaller.Inasituationlikethis,withnegligibleinter- theremainder(12%)liesintheregionof50–90%complete-
nalerrors,formaltestsofgoodnessoffitarenotinformative, ness. To place an upper limit on the impact of selection bi-
andwedonotattemptthem. ases,weadoptthelowerlimitsof90%and50%onthecom-
Thedominantuncertaintiesinourgalaxysize–halosizere- pletenessinthesetworegionsoftheR –H plane,assign
eff 160
lationsaremostlikelycausedbypossiblesystematicerrorsin weights 2.0 (i.e., 1/0.5) and 1.1 (i.e., 1/0.9) to the galax-
our adopted SMHM relations. We can judge the magnitude ies in our sample in these regions, and then recompute the
of these errors by comparing the R –R relations plot- R –R relations. ForR (cid:38)100kpc,wefindnegligi-
eff 200c eff 200c 200c
tedinFigures5to7forthefourdifferentSMHMrelations. blecorrectionstothemedianR –R relations,whilefor
eff 200c
This comparison indicates that the SMHM relation may be R (cid:46) 100kpc, wefindcorrectionsbelow0.1dexforall
200c
responsible for systematic errors at the level of ∼ 0.1–0.2 galaxy types and redshifts 0 < z < 3. We conclude from
dex,perhapsalittlelessforthecombinedsampleofgalaxies, this exercise that selection biases are likely to be subdomi-
perhapsalittlemoreforthesubsamplessplitbymorpholog- nantsourcesofuncertaintyinourR –R relations.
eff 200c
icaltype. Quantitativemeasuresofthedeviationsamongthe Basedonthisassessmentofuncertainties, mostofthere-
R –R relations at 0 < z < 0.5 confirm these impres- sultsofthispaperappeartoberobust. Inparticular,thereis
eff 200c
sions. a strong, approximately linear correlation between the sizes
Thecontributionstotheerrorbudgetfromtheadoptedstel- of galaxies and their dark matter halos over the full range
lar population models, which determine the stellar masses of redshifts examined here, 0 < z < 3. The coefficient of
andspecificstarformationrates,aresmallerthanthosefrom proportionalityislargerforlate-typegalaxiesthanforearly-
the adopted SMHM relations. Systematic errors in stellar type galaxies, which follow roughly parallel sequences, ex-
masses could affect the R –R relations at about the ceptpossiblyatthehighestredshifts. Forlate-typegalaxies,
eff 200c
samelevelassystematicerrorsinR . Theclassificationof theobservedR –R relationisgenerallyconsistentwith
eff eff 200c
the3Dshapesofgalaxies(i.e.,flatdisksvs.roundspheroids) simple models in which galactic disks grow with the same
bySe´rsicindexisanothersourceofuncertainty,becauseitis specificangularmomentumastheirdarkmatterhalos. There
basedonlyontheradialdeclineoftheprojected2Dsurface issomeevidenceforaslowdownindiskgrowthatz < 0.5,
brightnessprofiles. FittingasingleSe´rsicprofileinsteadofa buttheapparentdeviationfromtheJ/M equalitylineisonly
detaileddisk/bulgedecompositionpossiblyaddsfurtherun- ∼0.2dex.
certainty. Nevertheless, the R –R relations we obtain
eff 200c
fromsubsamplessplitbySe´rsicindexagreeatthe(cid:46)0.1dex
levelwiththosefromsubsamplessplitbyspecificstarforma-
tionrate.
We estimate the impact of selection biases on our galaxy
10
SMHM Relation 1
. < < . . < < . . < < .
c
p
k
/
. < < . . < < . . < < .
c
p
k
/
J/M equality
Kravtsov13
Lowest 20% in
Highest 20% in
/ kpc / kpc / kpc
Figure8. GalaxyeffectiveradiusReff plottedagainsthalovirialradiusR200c atdifferentredshiftsforsubsamplesofgalaxieswiththelowestandhighest
20%ofthemeasuredSe´rsicindexnasproxiesforlate-andearly-typegalaxies,respectively. ThesixpanelsshowresultscomputedfromSMHMrelation1in
redshiftintervalsof∆z =0.5coveringtherange0<z <3. Thefaintblueandreddotsrepresentindividuallow-nandhigh-ngalaxies,respectively,while
thefilledbluesquares,openredcircles,andverticalbarsindicatethecorrespondingmedianvaluesand16th–84thpercentilerangesofReffinbinsofwidth0.15
inlogR200c.ThediagonalsolidlinesshowtheR1/2–R200crelationatz=0fromKravtsov(2013)assumingReff =R1/2,whilethediagonaldashedlines
showthepredictionforgalacticdiskswiththesameJ/M astheirsurroundinghalos. Thethicktickmarkatthebottomofeachpanelindicatesthehalosize
correspondingtothereferencestellarmassM∗,lowlistedinTable1.NotethattheReff–R200crelationsforbothlow-nandhigh-ngalaxiesarenearlyconstant
withredshift,andthattheoneforlow-ngalaxiesisclosetothepredictedrelationforequalityofJ/Mindisksandhalos.ComparewithFigure9.
Table2.VerificationofMainResults
SMHM1 SMHM2 SMHM3 SMHM4
1.TheR –R relationsareroughlylinearinallredshiftbins. T T T T
eff 200c
2.TheR –R relationsareoffsetforearly-andlate-typegalaxies. T T T T
eff 200c
3.TheR –R relationforlate-typegalaxiesareclosetotheJ/M equalityline. T T T T
eff 200c
4.TheR –R relationshowslittleevolutionbetweenz=0andz=3. T T T T
eff 200c
WehaveplottedandexaminedtheR –R relationsat scientific results of this study are robust relative to discrep-
eff 200c
all redshifts (0 < z < 3) for all four SMHM relations to anciesamongtheSMHMrelations(becauseoftheweakde-
determinewhetherornottheysupportthefourmainresults pendenceofR onM ).
200c 200c
discussedinSection4. Theoutcomeofthistestisrecorded
in Table 2 by a T (for true) or F (for false) for each combi- 6. DISCUSSION
nation of SMHM relation and result. All of the entries are
We have found that the sizes of galaxies are proportional
Ts. Table2thereforereinforcesourconclusionthatthemain
on average to the sizes of their dark matter halos over a