Which galaxy property is the best indicator of its host dark matter halo properties?

11.5 MB·English

by  David A. Wake

Toappear in ApJ.
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
WHICH GALAXY PROPERTY IS THE BEST INDICATOR OF ITS HOST DARK MATTER HALO
PROPERTIES?
David A. Wake1, Marijn Franx2, Pieter G. van Dokkum1
Toappear inApJ.
ABSTRACT
In this work we investigate the link between galaxy velocity dispersion, mass and other properties
2 (color, morphology) with the properties of dark matter halos by comparing the clustering of galaxies
1 atbothfixedmassandvelocitydispersion(σ). WeusetheSloanDigitalSkySurveytodefineavolume
0 limited sample of massive galaxies complete in both stellar mass (M ) and σ with M >6×1010
star star
2 h−1M⊙ and σ > 75 km/s. Using this sample we show that at fixed σ there is no dependence of

n the clustering amplitude on stellar or dynamical mass (Mdyn). Conversely when Mstar or Mdyn are
a fixed there is a clear dependence of the clustering amplitude on σ with higher σ galaxies showing a
J higher clustering amplitude. We also show that whilst when M or M are fixed there remains
star dyn

9 a dependence of clustering amplitude on morphology, there is no such dependency when σ is fixed.
However,wedoseeadependenceoftheclusteringamplitudeong−r colorwhenbothmassandσ are

] fixed. Despitethis,evenwhenwerestrictoursamplestoonlyellipticalsorredgalaxiestherelationship
O
betweenσ andclusteringamplitudeatfixedmassremains. Itseemslikelythattheresidualcorrelation
C withcolorisdrivenbysatellitegalaxiesinmassivehalosbeingredderatfixedσ. Thelackofasimilar
residual morphology dependence implies that the mechanism turning satellites red is not changing
.
h their morphology. Our central result is that σ is more closely related to the clustering amplitude
p of galaxies than either M or M . This implies that σ is more tightly correlated than M or
star dyn star
- M with the halo properties that determine clustering, either halo mass or age, and supports the
o dyn
notion that the star formation history of a galaxy is more closely related to its halo properties than
r
t its overallmass.
s
a Subject headings: galaxies: evolution—galaxies: formation—galaxies: halos—large-scalestructure
[ of universe

1
v 1. INTRODUCTION Wake et al. 2008b; Mandelbaum et al. 2009) and halo
3 mass. Perhaps unsurprisingly the strongest relationship
In cosmological models of galaxy formation the for-
1 occurs with stellar mass, which for central galaxies in
mation history of galaxies, and hence their present day
9 halos show a tight correlation with a relatively small
properties, are intimately linked with the formation his-
1 scatter of 0.17 dex (Yang et al. 2011). At fixed stellar
tory and properties of the dark matter (DM) haloes in
1. which they reside. Therefore, if we wish to understand mass properties that indicate the star formation history
0 the process of galaxy formation it is important to ob- of a galaxy then show little dependence on halo mass
2 servationally establish a link between the properties of (More et al. 2011), when only considering central galax-
1 galaxies and those of the halos in which they reside. ies.
: In this work we focus on investigating how the dy-
v In recent years many studies have linked galaxy prop-
namical properties of galaxies, such as velocity disper-
i ertiestohalomassusingclustering,gravitationallensing,
X sion (σ) or surface mass density (Σ), in addition to
satellite kinematics, group catalogues or subhalo abun-
stellar mass are related to the clustering amplitude of
r dance matching, with the main focus being on linking
a the stellar properties (stellar luminosity, stellar mass, galaxies and hence their host dark matter halo prop-
erties. This is a particularly relevant question as it
color, SFR) or those of the central black hole (lumi-
is becoming increasingly clear that many galaxy prop-
nosity, black hole mass) to the galaxy host dark matter
erties such as their current star-formation rates, star-
haloproperties. Thisworkhasshownstrongcorrelations
formation histories, metalicities, and black hole masses
between halo mass and stellar mass, color, star forma-
are more closely related to these dynamical proper-
tion rate, or morphology, such that more massive (lu-
ties than their total stellar mass (e.g. Bender et al.
minous), redder (less star-forming), or more spheroidal
1993; Ferrarese & Merritt 2000; Gebhardt et al. 2000;
galaxies live in more massive halos (e.g. Li et al. 2006;
Kauffmann et al. 2003; Franx et al. 2008). If the forma-
Yang et al. 2008, 2009; Zehavi et al. 2011; More et al.
tion history and properties of the host dark matter halo
2011; Mandelbaum et al. 2006; Leauthaud et al. 2011).
are the major underlying cause of these galaxy proper-
For AGN the relationships are less clear although cor-
ties, we might expect that σ or Σ are better indicators
relations do exist between narrow line AGN luminos-
than stellar mass of these halo properties.
ity or radio luminosity (Wake et al. 2004; Li et al. 2006;
In this paper we make use of the very large sample
of galaxies with velocity dispersion, stellar mass, size,
1DepartmentofAstronomy,YaleUniversity,NewHaven,CT
color and morphology measurements from the seventh
06520
2SterrewachtLeiden,LeidenUniversity,NL-2300RALeiden, datareleaseoftheSloanDigitalSkySurvey(SDSSDR7;
TheNetherlands. Abazajian et al. 2009). We split these galaxies into nar-
2 Wake et al.
row bins of one parameter, in particular mass or σ, and
then see if there is any dependence of the clustering am-
plitude onanother galaxyparameterwithin thatnarrow
bin. Inthiswaywecaninvestigateifthereareanyresid-
ualdependenciesonhalopropertiesmanifestedbyhigher
orlowerclusteringamplitudesasoneparameterisvaried
whilst another remains fixed.
Throughoutthispaper,weassumeaflatΛ–dominated
CDM cosmology with Ω = 0.27, H = 73km
m 0
s−1Mpc−1, and σ =0.8 unless otherwise stated.
8
2. DATA
Thegalaxydatausedinthisanalysisaregatheredfrom
the seventh data release of the SDSS (Abazajian et al.
2009). We begin with the Large Scale Structure sam-
ples of the NYU Value Added Galaxy catalog (VAGC;
Blanton et al. 2005). These samples have been carefully
constructedformeasurementsoflargescalestructureand
include corrections for fiber collisions, the tracking of
spatially dependent completeness and appropriate ran-
domcatalogsallofwhicharerequiredtoaccuratelymea-
sure clustering. The sample we use has an r band mag-
nitude range of 14.5 < r < 17.6. In addition the NYU
VAGC gives k-corrected(to z=0.1)absolute magnitudes
(Blanton et al. 2003), velocity dispersion measurements
from the Princeton Spectroscopic pipeline, and circu-
larised sersic fits for each galaxy, all of which we make
use of in this analysis.
For estimates of the stellar mass we make use of the
MPA-JHU value added catalog which provides stellar
mass estimates based on stellar population fits to the Figure 1. Thedistributionofgalaxiesfromtheparentsample
SDSS photometry (Kauffmann et al. 2003; Salim et al. withMstar >6×1010M⊙ intheMstar –σ plane. Thecolored
regions in thetop panel show thehigh and low σ samples at
2007). The overlap between the MPA-JHU and NYU
VAGCs is close to but to quite 100% and so we remove fixed Mstar and in the bottom panel they show the high and
the regions where they do not match from the analy- low Mstar samples at fixedσ.
sisandadjustthecompletenesscorrectionsappropriately To estimate dynamical mass we follow Taylor et al.
(see section 4). We also remove any region of the sur- (2010) where
veythathasaspectroscopiccompletenesslessthan70%.
M =K σ2r /G (1)
This leaves an area of 7640 deg2 and a total sample of dyn V 0 e
521,313 galaxies. with
The SDSS velocity dispersions are measured within 73.32
K = +0.954 (2)
the 3” diameter SDSS fiber. We correct to a com- V 10.465+(n−0.95)2
mon aperture of one eighth of an effective radius (r ),
e
the central velocity dispersion, using the relation σ = where n is the sersic n parameter and the gravitational
σap(8rap/re)0.066 where rap = 1.”5 (Cappellari et0al. constant G=4.3×10−3 pc M⊙−1 km2 s−2.
2006). r istakenfromthebestfittingcircularisedsersic Finally we will make use of both galaxy color and
e
profile fit. morphologyto further parameterize our galaxy samples.
Inadditiontoinvestigatinghowclusteringdepends on Throughout we use g − r colors from the K-corrected
stellar mass and velocity dispersion, we also include the NYU VAGC absolute magnitudes. As already stated
dynamicalmass(M )asanadditiongalaxymassindi- these are corrected to z = 0.1; although we will refer to
cator. Estimates ofdygnalaxy stellar masses are uncertain themasg−r colorstheyarenotquite g−r atrest. The
due to the complex nature of the star formation history, morphological classifications we use come from Galaxy
stellarpopulationsynthesismodeling,extinctionlawand Zoo (Lintott et al. 2011) which provides multiple visual
initial mass function (e.g. see Marchesini et al. 2009; classifications for each galaxy in the SDSS spectroscopic
Muzzin et al. 2009; Conroy et al. 2009b). The dynam- sample. The parameter we use is the probability that
ical mass, which is estimated purely from the velocity a galaxy is an elliptical (Pel) corrected for redshift bias
dispersionandthe sizeofthe galaxy,willnotbe affected whereby higher redshift galaxies are more likely to be
by these same systematic uncertainties and will likely classifiedasellipticalsduetotheirsmallerapparentsizes
have errors more similar to those of the velocity disper- (see Lintott et al. 2011, for details).
sion. In addition there is a strong and fairly tight cor-
relationbetweenstellar massanddynamicalmassin the 3. SAMPLES
SDSS (Taylor et al. 2010, see also Figure 7) and so both We define a parent sample with stellar mass > 6 ×
masses are largely tracing the same physical property of 1010M⊙and0.04<z <0.113. Thestellarmasslimitand
a galaxy. redshift cuts are chosen to yield the largest stellar mass
3
Figure 2. The projected correlation functions of high and low σ samples at fixed Mstar (left). The projected correlation
functions of high and low Mstar at fixed σ (right). The correlation function measurements are colored to match the selection
regions shown in Figure 1.
limited sample of galaxies that can be defined from the and allowed σ to vary and visa-versa. We define a se-
NYU SDSS VAGC galaxy catalog, thus maximizing the ries of samples with a narrow range in one parameter
numberofgalaxiesavailableforclusteringmeasurements. and then take the upper and lower quartile of the sec-
Thisrelativelyhighstellarmasscutisalsohelpfulasthe ond parameter. We begin by defining the first velocity
SDSSvelocitydispersionsareonly reliableat>75km/s dispersionrange,startingatthevelocitydispersioncom-
and thus become incomplete in σ at low stellar masses. pleteness limit (210km/s)andcutting at adispersionof
By inspecting the distribution of velocity dispersion in 242.3 km/s, the limit that contains 20% of the galaxies
narrowstellarmassbinsweestimatethat98%ofgalaxies with σ larger than 210 km/s. We then split that sam-
with stellar mass >6×1010M⊙ have velocity dispersion ple by either stellar mass or dynamical mass into upper
in excess of 75 km/s. Finally we remove galaxies with and lower quartiles. In a similar manner we construct
unreliableσ measurementsbyrequiringthatthe errorin two more samples with the same interval in σ with suc-
σ be less than 10%. cessively higher σ limits again splitting into highest and
Since there is a significant scatter in the relation be- lowestquartilesineitherstellarmassordynamicalmass.
tweenstellarmassandvelocitydispersionwemustmake Details of these samples are given in Tables 3 and 5 in
further cuts if we wish to define samples that are com- Appendix A. To make the reciprocal samples, i.e. high-
plete in both. To define the velocity dispersion com- est and lowest quartiles in σ at fixed stellar mass or dy-
pleteness limit we investigate the distribution of stellar namicalmass,wefindtheappropriatestartingminimum
mass in narrow velocity dispersion bins for the full DR7 mass and interval that produces three samples with the
sample. Using these measurements we find the velocity same number of galaxies as in the σ range samples. We
dispersion that defines a complete sample of galaxies at then spit these into the highest and lowest quartiles of
thestellarmassof6×1010M⊙. Wefindthatforgalaxies σ. Details of these samples are given in Tables 4 and
with a velocity dispersion of 210 km/s 85% have stellar 6. Figure 1 shows the distribution of galaxies in the σ -
massesgreaterthan6×1010M⊙ andthuswerestrictour Mstar plane for the parent sample as well as the samples
sample to have σ grater than this limit when defining at fixed σ and Mstar.
σ limited samples. Wealsodefineaseriesofsamplesinnarrowstellarmass
The aim of this work is to investigate which is more rangeswhere we take the highest and lowestquartiles in
fundamental in determining the clustering amplitude, a stellarsurfacemassdensity(Σ)andtheirreciprocals(Ta-
galaxy’s mass, either stellar or dynamical, or its cen- bles 8 and 7) again ensuring that they are complete in
tral velocity dispersion. We approach this question by both stellar mass and surface mass density. This is an
measuring the clustering where we have fixed the mass important consistency check for our measurements. It is
4 Wake et al.
Figure 3. Theratiooftheprojectedcorrelationfunctionsbetweenhighandlowvelocitydispersionsamplesatfixedstellarmass
(left). The ratio of the projected correlation functions between high and low stellar mass samples at fixed velocity dispersion
(right). The best fit ratio on scales 1.6<rp <25.1 h−1Mpc is shown as thedashed line.
possible that the scatter introduced by larger errors on 4. CLUSTERINGMEASUREMENTTECHNIQUES
M could reduce any differences in relative clustering
star The two-point correlation function, ξ(r), is defined as
amplitudecomparedtoσ,somethingwhichshouldbere-
the excessprobabilityabovePoissonoffinding anobject
ducedby using M . Whilst anyerrorsonM will be
dyn dyn at a physical separation r from another object. This is
differentandlikelytobesmallerthanthoseonM they
star calculated by comparing the number of pairs as a func-
will still be larger than the error on σ due to the ad-
tion of r in our galaxy catalogs with the number in a
ditional uncertainties in fitting the profile to determine
randomcatalogthatcoversthesamevolumeasourdata.
R andthesersic-nparameter. However,Σ,whichshould
e Since the space density of galaxies in our subsamples
show similar trends to σ, contains both the errors on
isverylowshotnoisewouldbe asignificantsourceofer-
M and on R . Therefore, any reduction in clustering
star e rorfor measurementsoftheir auto-correlationfunctions.
trends due to scatterinthe measurementofthe physical
However,wecanovercomethisproblembyusingamuch
parameters should be largest for this sample.
denser sample of galaxies to trace the underlying den-
For each pair of samples (e.g. high and low stellar
sity field and measure the clustering of our sub-samples
mass at fixed dispersion) we find that the mean of the
relative to this larger sample using the cross-correlation
fixed parameter changes very little between the samples
function. The obviouschoicefor the largersample is the
split into high and low quartiles. For a fair comparison
parent sample we defined above containing all galaxies
we compare the mass ratio to the square of the σ ratio
passingourbasicselectioncriteria. Thissamplecontains
since mass is proportional to σ2. For all samples the
almost65,000galaxiesandsohasaspacedensityalmost
ratio of the mean of the fixed parameter varies by less
20 times higher than our largestsub-sample. Whilst the
than 5% and so should have a negligible effect on the
individual cross-correlation functions will reflect the in-
clustering amplitude. Conversely the ratio of the means
trinsic clustering amplitude of both the sub- and parent
of the varying parameter covers a factor of 2 to 3.5.
samplestheratiobetweenanytwoofthesecrosscorrela-
In the Appendix A we plot redshift, velocity, mass,
tions will just reflect the ratio between the clustering of
mass density and morphology distributions for each
the two sub-samples in question, with the clustering of
sample where appropriate. As expected, since we have
the parent sample in effect being ’canceled out’.
defined a volume limited parent sample and have only
Measuringthe line-of-sightdistance using redshifts in-
made cuts where we are complete in σ, M and mass,
den troduces distortions into the correlation function. The
the redshift distributions for all samples are the same.
effect of these distortions can be overcome by separat-
ing the clustering signal into contributions perpendicu-
lar (r ) and parallel (π) to the line-of-sight (ξ(r ,π)).
p p
5
Figure 4. Theratiooftheprojectedcorrelation functionsbetweenhighandlowvelocitydispersionsamplesatfixeddynamical
mass (left). The ratio of the projected correlation functions between high and low dynamical mass samples at fixed velocity
dispersion (right). The best fit ratio on scales 1.6<rp <25.1 h−1Mpc is shown as the dashed line.
One canthen integrateoverthe line ofsightdirectionto pair counts in a grid which is logarithmically spaced in
estimate the projected correlation function r of width 0.2 log(h−1Mpc) and linearly spaced in the
p
∞ ∞ rdrξ(r) π directionofwidth2h−1Mpc withallseparationsinco-
w (r )=2 dπξ(r ,π)=2 . (3) moving coordinates. The random catalogue is based on
p p Z0 p Zrp (r2−rp2)1/2 theoneprovidedbytheNYUVAGCwithadditionalcuts
to match the exact spatial coverage of the MPA/JHU
The final expression only involves the real-space corre-
stellar masses and redshifts matching the redshift distri-
lation function ξ(r) showing that w (r ) is not com-
p p bution of our samples. The resulting random catalogue
promised by redshift space distortions (Davis & Peebles
containsalmost1.9millionrandoms,closeto29timesas
1983). In practice it is only possible to integrate out
many as in our parent galaxy catalog, which is a suffi-
tosomemaximumπ becauseξ(r ,π)ispoorlyknownon
p cient number such that our errors are never dominated
verylargescalesresultinginadditionalnoisebeingintro-
duced to the measurement. We integrate to 60h−1Mpc by shot noise in the random pair counts.
Sincetheindividualbinsinacorrelationfunctionmea-
whichissufficientlylargetoincludemostcorrelatedpairs
surementarehighlycorrelatedweneedtogenerateaccu-
and gives stable results.
rate covariance matrices if we wish to compare our clus-
We make this measurement using the Landy & Szalay
tering measurements in a meaningful way. The simplest
(1993) estimator in the cross-correlationform, with
approach and the one we chose to follow is to use jack-
ξ(r ,π)= knife resampling (Scranton et al. 2002). There is some
p
debate about the accuracy of covariance matrices gen-
DD1 n2R −DR nR −D1R nR +1 (4) erated in this way; for instance Norberg et al. (2009)
RR (cid:18)n n (cid:19) RR (cid:18)n (cid:19) RR (cid:18)n (cid:19) suggest that the structure of jackknife covariance ma-
D D1 D D1
trices may not be entirely accurate for w (r ) mea-
p p
where DD1 are the pair counts between the parent and
surements, although the variance is reliable. Conversely
sub-sample, DR are the pair counts between parent and
Zehavi et al.(2002,2004,2005)findthattheycanclosely
random sample, D1R are the pair counts between sub-
reproducethestructureandamplitudeofcovariancema-
sample and random sample and RR are the pair counts
tricesgeneratedbymockcatalogsusingjackkniferesam-
between the random sample, all split into the r and π
p pling. Because of this uncertainty, it might be prefer-
directions. We are able to use the same random sample
able to generate covariancematrices fromlargenumbers
in each term as all samples are complete throughout the
ofmockgalaxycataloguesgeneratedby populating dark
samevolume,i.e. theycoverthesamespatialextentand
matterhalocatalogssoastoreproducetheclusteringand
have the same redshift distributions. We calculate the
6 Wake et al.
Figure 5. The ratio of the projected correlation functions between high and low surface mass density samples at fixed stellar
mass(left)andhighandlowstellarmasssamplesatfixedsurfacemassdensity(right). Thebestfitratioonscales1.6<rp <25.1
h−1Mpc is shown as the dashed line.
density of eachsample. However,since we have so many the clustering amplitude on M . This is the central
star
sub-samples, each with a complex selection, it would be result of this paper: clustering amplitude depends on
enormouslychallengingto undertake sucha scheme. We σ atfixedmass,butdoesnotdependonmassatfixedσ.
have therefore chosen to use jackknife resampling which ThisresultisillustratedmoreclearlyinFigure3which
should be sufficient for our purposes. We split the SDSS shows the ratio of w (r ) between the high and low
p p
areainto146equalarearegionsandthenrepeatedlycal- σ samples at fixed M (left) and the high and low
star
culate w (r ) removing one area at a time. These 146 M samples at fixed σ (right). Figure 4 is similar to
p p star
w (r ) measurements are then used to generate a full Figure 3, except M is replaced by M and shows
p p star dyn
covariance matrix. exactly the same trends as with M . We also show
star
in Figure 5 the ratio of w (r ) between high and low
p p
5. RESULTS ΣsamplesatfixedMstar andhighandlowMstar samples
at fixed Σ. This is, in effect, the same as splitting by
5.1. Is galaxy mass or σ more closely related to
size at fixed mass or mass at fixed size. Since M and
clustering amplitude? dyn
M are highly correlated one would expect galaxies
star
Figure 2 shows the wp(rp) measurements for the high with smaller Re at a fixed Mstar, thus higher Σ, to have
and low σ samples at fixed Mstar (left) and the high and a higher σ. These samples should then be analogous to
low Mstar samples at fixed σ (right). We find that when thosewithvaryingσ atfixedMstar andindeedtheyshow
the mass is held fixed the high σ samples have a higher the same trend of higher clustering amplitude at higher
clusteringamplitudethanthelowσsamples. Thisistrue Σ when M is fixed and very little dependence with
star
onallscalesandforallmassranges. However,whenσ is M whenΣisfixed. AsdiscussedinSection3thisalso
star
held fixed there appears to be little or no dependence of
7
Figure 6. The best fit large scale bias ratios for thesamples plotted in Figures 3, 4 and 5. The top panels show thebias ratio
for high and low Mstar at fixed σ and high and low σ at fixed Mstar. The middle panels show the bias ratio for high and low
M at fixed σ and high and low σ at fixed M . The bottom panels show the bias ratio for high and low Σ at fixed σ and
dyn dyn
high and low σ at fixed Σ. There is a significant dependence of the bias on σ or Σ at fixed mass, whereas there is little or no
dependenceon mass at fixedσ or Σ.
implies that these observed trends are not the result of with respect to the dark matter clustering and give an
largermeasurementerrorsonM orM comparedto indication of the relative linear bias of the two samples.
star dyn
σ, which could have reduced the clustering dependence For the w (r ) ratios shown in Figures 3, 4 and 5 this
p p
on mass compared to σ. appears to be the case and indeed a scale independent
To quantify these ratiosin w (r ), whichconstitutes a ratio is an acceptable fit for all but one of the samples.
p p
ratio in galaxy bias, we find the best fitting scale inde- These fits are made using the full covariance matrices
pendent amplitude on scales 1.6 < r < 25.1 h−1Mpc. for the w (r ) ratio, and are shown as the dashed lines
p p p
These scales are sufficiently large as to be dominated in Figures 3, 4 and 5. Details of the fits are given in
by pair counts between DM halos, rather than galaxies Table 1 and the best fit bias ratios and errors are plot-
within halos, and so should show a scale independence ted for all these samples in Figure 6. In addition to the
8 Wake et al.
Table 1
Fitstocorrelationfuntionratios
Sample Bestfitratio Bestfitχ2 Ratio=1χ2 Ratio=1prob
Mstar atσ 1 0.92±0.06 1.69 3.26 0.661
Mstar atσ 2 1.01±0.07 2.07 2.07 0.839
Mstar atσ 3 1.15±0.12 5.52 6.97 0.223
Mstar atσ All - - 12.30 0.66
σ atMstar 1 1.17±0.07 7.51 12.10 0.033
σ atMstar 2 1.23±0.10 2.24 7.31 0.198
σ atMstar 3 1.32±0.12 4.88 11.54 0.042
σ atMstar All - - 30.95 0.0089
Mdyn atσ 1 0.97±0.05 0.40 0.74 0.981
Mdyn atσ 2 1.11±0.07 3.15 5.27 0.384
Mdyn atσ 3 1.13±0.12 2.13 3.18 0.673
Mdyn atσ All - - 9.18 0.87
σ atMdyn 1 1.18±0.05 6.81 14.87 0.011
σ atMdyn 2 1.26±0.09 4.62 12.80 0.025
σ atMdyn 3 1.56±0.16 3.06 13.89 0.016
σ atMdyn All - - 41.57 0.00026
Mstar atΣ1 1.09±0.05 0.22 2.60 0.762
Mstar atΣ2 1.04±0.16 2.98 3.05 0.692
Mstar atΣ3 1.00±0.14 0.46 0.46 0.994
Mstar atΣ4 1.17±0.07 5.88 10.73 0.057
Mstar atΣAll - - 16.84 0.66
ΣatMstar 1 1.23±0.09 4.24 10.31 0.067
ΣatMstar 2 1.12±0.05 6.55 10.69 0.058
ΣatMstar 3 1.22±0.09 1.96 6.54 0.257
ΣatMstar 4 1.11±0.08 9.47 11.23 0.047
ΣatMstar All - - 38.77 0.0071
best fit we also calculate the χ2 for a w (r ) ratio ofone
p p
overthesamescales,andgivetheχ2 valueaswellasthe
probability of an acceptable fit in Table 1. In all cases
the best fit ratio is higher when σ or Σ is varied at fixed
massthanwhenσ orΣarefixedandthemassallowedto
vary. Similarly when σ or Σ are fixed the w (r ) ratios
p p
between the high and low mass samples are consistent
with one in all but one case whereas the varying σ or
Σ samples at fixed mass are nearly all inconsistent with
a w (r ) ratio of one. For each parameter pair we can
p p
combine the samples and determine the probability that
allare consistentwith a w (r )ratioof1. Againwefind
p p
that at fixed σ or Σ the samples are indeed consistent
with showing no dependence of the clustering amplitude
on mass at the 66%, 87% and 66% levels, whereas there
is only a 0.9%, 0.03% and 0.7% chance that all three of
the varying σ or Σ samples at fixed mass are consistent
withnovariationinclusteringamplitudeonlargescales.
On small scales the differences are just as clear with
thevaryingσ orΣsamplestypicallyshowingevenlarger
w (r ) ratios than at large scales. The samples with
p p
varying mass at fixed σ or Σ occasionally show moder-
Figure 7. The relationship between dynamical and stellar ately larger w (r ) ratios on small scales than on large
p p
mass for all DR7 galaxies with 0.04 < z < 0.113 and σ > scales,butoftenthereisnodifferencewiththew (r )ra-
p p
75 km/s (black contours) and those galaxies classified as ei-
tio remaining consistent with one on all scales.
therellipticals (red contours) orspirals (bluecontours). The
Overall these measurements make it clear that veloc-
points show the median Mstar in bins of Mdyn and the solid ity dispersion or stellar mass surface density are more
lines thebiweight best fit. Thedotted lineshows theone-to-
closely related to galaxy bias than either stellar mass or
one relation. This relationship is almost identical indepen-
dent of morphology over the whole mass range covered by dynamical mass.
this plot, and becomes indistinguishable for the mass range
of interested in thispaper.
9
5.2. The Importance of Morphology and Color evidence for any dependence of the clustering amplitude
onmorphologyatanyscale. Atfixedmasstheratiosare
We have included the distribution in elliptical proba-
all non-zero with galaxies with a higher P showing a
bility for each of the samples in the plots in Appendix el
higher clustering amplitude than those with a low P .
A as it demonstrates one area that is both potentially a el
However, the significance of these positive w (r ) ratios
concern but also scientifically important for this analy- p p
is low with all being consistent with no difference in the
sis. σismuchbettercorrelatedwithmorphologyorcolor
clusteringamplitude. Thisimpliesthatthecorrelationof
thanstellarmassis. Σisalsoabetterdiscriminatorthan
morphologywithdispersionisnotthedeterminingfactor
mass, but not as good as σ (see Wake et al. 2012 for a
inthestrongcorrelationofclusteringamplitudewithσat
detailed study of these trends). Because of this when
fixed mass.
we fix σ we come pretty close to fixing morphology re-
Whilstthestrongcorrelationofcolorwithσisunlikely
gardlessofthemassandwhenwesplitintohighandlow
to introduce any systematics in our measurements it is
σ samples at fixed mass we are pretty close to splitting
stillinformativetoinvestigatewhetherthereisanyresid-
into disks and spheriods (we also see a splitting by color
ual clustering dependence on color at either fixed mass
in the same way). This is of course the heart of our in-
or σ. Since color is better correlated with σ than mass
vestigation: is a tighter correlation between σ and halo
one might imagine that any residual clustering depen-
properties (compared to the correlation between mass
dence on color would be lower at fixed σ than at fixed
and halo properties) the explanation as to why σ is a
mass if the halo properties are key to determining the
better indicator than stellar mass of a galaxy’s stellar
color, much as is seen with morphology above. To test
population? This does however raise a potential worry
this we show in Figure 8 the best fit large scale correla-
regardingsystematics: are we measuring the same thing
tion function ratios of high and low g−r color samples
when we measure σ for a disk galaxy as compared to a
at fixed mass and dispersion and list the fitting results
spheroidal galaxy?
in Table 2 (the full w (r ) ratios are shown in Figure
We tackle this question in several ways. Firstly we p p
18 in Appendix A). Since the color evolves with redshift
show in Figure 7 the observed relationship between
we have again had to split the samples in a similar way
M and M for all the galaxies in SDSS with 0.04<
star dyn to the elliptical probability samples such that we have
z <0.113andσ >75km/s. Wefurthersplitthissample
the reddest and bluest quartiles at any redshift. Now,
into spheriods or disks based on the Galaxy Zoo classifi-
unlike with the elliptical probability, we see a residual
cations. The relationship is essentially identical regard-
clustering dependence on color at both fixed mass and
lessofmorphologyoverpracticallytheentiremassrange.
σ suchthatredgalaxiesclustermorestronglythanblue.
Thereisasmalldeviationatlowermassesbutthisisbe-
This is most likely the result of the truncation of star
low the mass range considered in this work. This gives
formation in satellite galaxies in massive dark halos, re-
us confidence that the velocity dispersion is measuring
sulting in satellite galaxies at fixed mass or σ having
the same physical quantity for both disks and ellipticals
redder colors than central galaxies with the same mass
within the SDSS fiber radius.
orσ(e.g.Yang et al.2009). Thefactthatweseeasignif-
The second approach is a more direct one: we can see
icantresidualcolordependence andnotmuch ofa resid-
if morphology itself has any effect on the clustering am-
ual morphology dependence suggests that the environ-
plitudeofourgalaxiesatfixedmassorσ. Wethusdefine
mentalmechanismfortransformingthecolorofagalaxy
samples with highand low elliptical probabilitiesin nar-
fromredtobluedoesnotsimultaneouslychangethemor-
row ranges of M and σ. We note that we are unable
dyn phology. A similar result was observed by Skibba et al.
to just take the quartiles of the distribution in elliptical
(2009) using the marked correlation function applied to
probability as we have done for parameters previously
SDSS galaxies.
as there is a redshift dependence to the probabilities as-
Since we do see some residual color dependence and
signed. Thisresultsfromgalaxiesbeinghardertoclassify
perhapsa hintofmorphologydependence inthe cluster-
at higher redshift due to their smaller angular size and
ingamplitudeatfixedmassitisworthaskingiftheσde-
surfacebrightnessdimming. Whilsttherehasbeensome
pendenceremainswhenmorphologyorcolorisfixed. We
correctionforthiseffectappliedtotheGalaxyZooprob-
show in Figure 9 the large scale w (r ) ratio ofhigh and
abilitieswefindthatitdoesnotentirelyremovethisbias p p
lowσsamplesatfixedM andhighandlowM sam-
resulting in the samples split into upper andlowerquar- dyn dyn
ples at fixed σ where we have restricted the galaxies to
tiles in P having different redshift distributions. We
el be either red (g−r > 0.9) or have high elliptical prob-
thus define our samples making use of the expectation
abilities (P > 0.6). The fit details given in Table 2
that the morphological mix does not vary significantly el
and the full w (r ) ratios shown in Figures 20 and 19
over our narrow redshift range. We find the best fit lin- p p
in Appendix A. Despite the sample sizes being reduced
ear relations between elliptical probability and redshift
andhencetheerrorsincreasing,theoriginaltrendofhigh
that produce the 25% most and least likely ellipticals at
σ galaxies having a higher clustering amplitude at fixed
each redshift and use these relations to define our sam-
massremainssignificantandthelargescaleratiosstayes-
ples. Details of these samples are given in Tables 9 and
sentiallythe samewithintheerrors. Thereisahintthat
10 and their distributions are plotted in Appendix A.
the w (r ) ratios are slightly reduced, but by removing
Figure 8 shows the large scale w (r ) ratios of these p p
p p the blue or spiral galaxies we have removed many lower
highandlowP samples atfixedmassandσ. Asbefore
el σ galaxies and reduced the difference between the high
we fit the w (r ) ratio (shown in Figure 17 in Appendix
p p and low σ samples. This is a clear demonstration that
A) between 1.6 < r < 25.1 h−1Mpc and determine the
p galaxies with higher σ cluster more strongly than galax-
probability that the ratio is consistent with one. Details
ieswithlowerσ regardlessoftheir mass,morphologyor
of these fits are give in Table 2. At fixed σ there is no
color.
10 Wake et al.
Table 2
Fitstocorrelationfuntionratiosforsamplesplitbymorphologyandcolor
Sample Bestfitratio Bestfitχ2 Ratio=1χ2 Ratio=1prob
Pel atσ 1 0.92±0.05 2.44 4.55 0.473
Pel atσ 2 1.06±0.09 4.29 4.71 0.452
Pel atσ 3 1.07±0.10 2.95 3.41 0.638
Pel atσ All - - 12.67 0.63
Pel atMdyn 1 1.12±0.05 2.60 6.16 0.291
Pel atMdyn 2 1.20±0.08 1.05 5.77 0.329
Pel atMdyn 3 1.10±0.12 1.43 2.02 0.847
Pel atMdyn All - - 13.95 0.53
g−r atσ 1 1.35±0.11 3.35 11.81 0.037
g−r atσ 2 1.36±0.14 4.55 11.34 0.045
g−r atσ 3 1.17±0.17 5.69 6.64 0.248
g−r atσ All - - 29.80 0.013
g−r atMdyn 1 1.34±0.10 6.66 16.34 0.006
g−r atMdyn 2 1.50±0.16 2.60 12.76 0.026
g−r atMdyn 3 1.13±0.11 3.83 5.19 0.394
g−r atMdyn All - - 34.29 0.0031
σ atMdyn El1 1.29±0.08 1.51 12.27 0.031
σ atMdyn El2 1.32±0.10 4.73 13.46 0.019
σ atMdyn El3 1.29±0.12 2.82 7.57 0.181
σ atMdyn ElAll - - 33.30 0.0043
σ atMdyn red1 1.21±0.06 2.32 12.08 0.034
σ atMdyn red2 1.15±0.08 1.08 4.49 0.482
σ atMdyn red3 1.48±0.15 1.34 10.71 0.058
σ atMdyn redAll - - 27.28 0.027
6. DISCUSSION σ and halo formation history (Section 6.3)
We have shown that at fixed mass (stellar or dynam- 6.1. Central galaxy halo mass correlations
ical) galaxies with high σ (or Σ) cluster more strongly
Since DM halo mass has the largest effect on cluster-
than galaxies with low σ. Conversely, at fixed σ (or Σ)
ingamplitude,the simplestexplanationforourresultsis
ourmeasurementsareconsistentwithhighandlowmass
that σ is better correlated with its host halo mass than
galaxies clustering the same. If the clustering of galax-
stellar or dynamical mass are. For galaxies living at the
ies is dominated by the clustering of the DM halos in
centersofhalos,centralgalaxies,thereisastrongcorrela-
which they reside, as expected in the CDM paradigm,
tionbetweenstellarmassandhalomass(e.g.Yang et al.
then this implies that even at fixed galaxy mass higher
2009; Wake et al. 2011), but with a significant scatter
σ galaxies live in more clustered halos than low σ galax-
(Yang et al. 2009, 2011). If the scatter between σ and
ies. The main determinant of a halo’s clustering ampli-
halomassforcentralsissubstantiallylowerthanthescat-
tudeisitsmass,butithasalsobeenshowninsimulations
terwithM thiscouldresultinclusteringobservations
that halos with different formation histories also cluster star
much as we have observed.
differently, with older more centrally concentrated halos
It is striking in Figures 3 and 4 that the clustering ra-
clusteringmorestrongly(Gao et al.2005;Wechsler et al.
tio betweenhighand lowσ samples becomes evenlarger
2006; Wetzel et al. 2007; Gao & White 2007; Jing et al.
asthe scaledecreases. The clustering amplitude onsuch
2007; Li et al. 2008). This means that galaxies with
scalesisdeterminedbytheseparationsofgalaxieswithin
higherσatfixedmasscouldbepreferentiallybelivingin
individual halos, so between central and satellite galax-
eithermoremassiveoroldermorecentrallyconcentrated
ies. Thus these smaller scale clustering measurements
halos.
can provide an additional constraint as to whether a
When considering how galaxies occupy halos it is also
tighter correlation between σ and halo mass (M ) for
important to make the distinction between the galaxy halo
central galaxies could produce the observed clustering
that has been formed at the center of a given DM halo,
results.
the centralgalaxy,andthosethat initially formedatthe
We show in Figure 10 an illustrative attempt at mod-
centers of other halos and have later been accreted into
eling the effect of a smaller scatter between σ and
this halo, satellite galaxies. The properties of a central
M thanbetweenM andM usingthehalomodel
galaxy will be largely determined by the formation his- halo star halo
(seeWake et al.2011,fordetailsofourmodel). Wehave
tory and properties of its host halo, whereas any effect
fitted halo occupation distributions (HOD), the mean
the current host halo has on satellite galaxies is limited
number of central and satellite galaxies as a function of
to the time after it has been accreted.
halo mass, to the clustering measurements for a series
In the following sections we address whether our re-
of samples of SDSS galaxies limited in stellar mass (i.e.
sult can be explained by a relationship between σ and
M >M ) (see Wake et al. 2012, for details of these
halo mass for central galaxies (Section 6.1) or for satel- star low
measurements). Subtracting two of the resulting HODs
lite galaxies (Section 6.2), or by a relationship between
withslightlydifferentM givestheHODforasampleof
low