Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups

0.98 MB·

by  Ran Li

Mon.Not.R.Astron.Soc.000,1–??(2008) Printed15January2009 (MNLATEXstylefilev2.2)
Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups
Ran Li1,2⋆, H.J. Mo2, Zuhui Fan1, Marcello Cacciato3, Frank C. van den Bosch3,
Xiaohu Yang4, Surhud More3
9 1Department of Astronomy, Peking University,Beijing 100871, China
0 2Department of Astronomy, Universityof Massachusetts, Amherst MA 01003, USA
0 3Max-Planck Institute for Astronomy, K¨onigstuhl 17, D-69117 Heidelberg, Germany
2 4Shanghai Astronomical Observatory, the Partner Group of MPA, Nandan Road 80, Shanghai 200030, China

n
a
J

5
1 ABSTRACT
We use galaxy groups selected from the Sloan Digital Sky Survey (SDSS) together

] with mass models for individual groups to study the galaxy-galaxy lensing signals
h
expected from galaxies of different luminosities and morphological types. We com-
p
- pare our model predictions with the observational results obtained from the SDSS
o by Mandelbaum et al. (2006) for the same samples of galaxies. The observational re-
r
sults are well reproduced in a ΛCDM model based on the WMAP 3-year data, but
t
s a ΛCDM model with higher σ8, such as the one based on the WMAP 1-year data,
a
significantly over-predicts the galaxy-galaxylensing signal. We model, separately,the
[
contributions to the galaxy-galaxylensing signals from different galaxies: central ver-

3 sussatellite,early-typeversuslate-type,andgalaxiesinhaloesofdifferentmasses.We
v also examine how the predicted galaxy-galaxy lensing signal depends on the shape,
4
density profile, and the location of the central galaxy with respect to its host halo.
3
9 Key words: dark matter - large-scale structure of the universe - galaxies: haloes -
4 methods: statistical
.
7
0
8
0
1 INTRODUCTION needs to be tested against observations. More recently, the
:
v halooccupation modelhasopenedanotheravenuetoprobe
According to the current paradigm of structure formation,
i the galaxy-dark matter halo connection (e.g. Jing, Mo &
X galaxies form and reside inside extended cold dark haloes.
B¨orner 1998; Peacock & Smith 2000; Berlind & Weinberg
r While the formation and evolution of dark matter haloes 2002; Cooray & Sheth 2002; Scranton 2003; Yang, Mo &
a in the cosmic density field is mainly determined by grav-
vandenBosch2003; vandenBosch,Yang&Mo2003; Yan,
itational processes, the formation and evolution of galax-
Madgwick & White 2003; Tinker et al. 2005; Zheng et al.
iesinvolvesmuchmorecomplicated,andpoorlyunderstood
2005; Cooray 2006; Vale & Ostriker 2006; van den Bosch
processes, such as radiative cooling, star formation, and
et al. 2007). This technique uses the observed galaxy lumi-
all kinds of feedback. One important step in understand-
nosity function and clustering properties to constrain the
ing how galaxies form and evolve in the cosmic density
average number of galaxies of given properties that occupy
fieldisthereforetounderstandhowthegalaxies ofdifferent
adarkmatterhaloofgivenmass.Althoughthemethodhas
physical properties occupy dark matter haloes of different
theadvantagethat it canyield muchbetterfitstothedata
masses. Theoretically, the connection between galaxies and
than the semi-analytical models or numerical simulations,
dark matter haloes can be studied using numerical simula-
onetypicallyneedstoassumeasomewhatad-hocfunctional
tions (e.g., Katz, Weinberg & Hernquist 1996; Pearce et al.
form to describe the halo occupation model.
2000; Springel2005; Springelet al.2005) orsemi-analytical
A more direct way of studyingthegalaxy-halo connec-
models (e.g. White & Frenk 1991; Kauffmann et al. 1993,
tionistousegalaxygroups1,providedthattheyaredefined
2004; Somerville & Primack 1999; Cole et al. 2000; vanden
assetsofgalaxies thatresideinthesamedarkmatterhalo.
Bosch 2002; Kang et al. 2005; Croton et al. 2006). These
Recently, Yang et al. (2005; 2007) have developed a halo-
approaches try to model the process of galaxy formation
based group finder that is optimized for grouping galaxies
from first principles. However, since our understanding of
thatresideinthesamedarkmatterhalo.Usingmockgalaxy
therelevantprocessesisstillpoor,thepredictedconnection
between the properties of galaxies and dark matter haloes
1 In this paper, we refer to a system of galaxies as a group re-
gardless of its richness, including isolated galaxies (i.e., groups
⋆ E-mail:[email protected] withasinglemember)andrichclustersofgalaxies.
2 Ran Li et. al
redshiftsurveysconstructedfromtheconditionalluminosity obtained by Mandelbaum et al. (2006) for the same galax-
functionmodel(e.g.Yangetal.2003) andasemi-analytical ies.Ourgoalisthreefold.First,wewanttotestwhetherthe
model (Kang et al. 2005), it is found that this group finder method of halo-mass assignment to groupsadopted by Y07
isverysuccessfulin associating galaxies with theircommon isreliable.Sincethemethodprovidesapotentiallypowerful
darkmatterhaloes(seeYangetal.2007;hereafterY07).The way to obtain the halo masses associated with the galaxy
groupfinderalsoperformsreliablyforpoorsystems,includ- groups, the test results have general implications for the
ingisolated galaxies in small mass haloes, makingit ideally study of the relationship between galaxies and dark matter
suitedforthestudyoftherelationshipbetweengalaxiesand haloes. Second, we want to examine in detail the contribu-
dark matter haloes over a wide range of halo masses. How- tions to the galaxy-galaxy lensing signal from different sys-
ever, in order to interpret the properties of the galaxy sys- tems,suchascentralversussatellitegalaxies,early-typever-
temsintermsofdarkmatterhaloes,oneneedstoknowthe sus late-type galaxies, and groups of different masses. Such
halomassassociatedwitheachofthegroups.Oneapproach analysis can help us interpreting the observational results.
commonlyadoptedistousesomehalomassindicator(such Finally, we would like to study how the predicted galaxy-
asthetotalstellarmassorluminositycontainedinmember galaxy lensing signal depends on model assumptions, such
galaxies) torank thegroups.With theassumption that the as the cosmological model and the density profiles of dark
corresponding halo masses have the same ranking and that matter haloes. In a companion paper (Cacciato et al. 2008,
themassfunctionofthehaloesassociatedwithgroupsisthe hereafterC08),weusetherelationshipbetweengalaxiesand
same as that given by a model of structure formation, one darkmatterhaloesobtainedfromtheconditionalluminosity
canassign ahalomasstoeachoftheobservedgroups.This function(CLF)modeling(Yangetal.2003;vandenBoschet
approach was adopted byY07for thegroup catalogue used al. 2007) to predict the galaxy-galaxy cross correlation and
in this paper. There are three potential problems with this tocalculatethelensingsignal,whileherewedirectlyusethe
approach. First, the approach is model-dependent, in the observed galaxy groups and theirgalaxy memberships.
sense that the assumption of a different model of structure This paper is organized as follows. In Section 2 we de-
formation will lead to a different halo mass function, and fine the statistical measure that characterizes the galaxy-
hence assign different masses to the groups. Second, even galaxy lensing effect expected from the mass distribution
if the assumed model of structure formation is correct, it associated with the galaxy groups. We provide a brief de-
is still not guaranteed that the mass assignment based on scription of the galaxy group catalogue and the models
the ranking of group stellar mass (or luminosity) is valid. of the mass distribution associated with galaxy groups in
Finally, even if all groups are assigned with accurate halo Section 3. We present our results in Section 4 and con-
masses, the question how dark matter is distributed within clude in Section 5. Unless specified otherwise, we adopt a
the galaxy groups remains open. Clearly, it is important to ΛCDM cosmology with parameters given by the WMAP
have independent mass measurements of the haloes asso- 3-year data (Spergel et al. 2007, hereafter WMAP3 cos-
ciated with galaxy groups to test the validity of the mass mology) in our analysis: Ω = 0.238, Ω = 0.762, and
m Λ
estimates based on the stellar mass (luminosity) ranking. h H /(100kms−1Mpc−1)=0.73, σ =0.75 .
0 8
≡
Gravitational lensing observations, which measure the
imagedistortionsofbackgroundgalaxiescausedbythegrav-
itational field of the matter distribution in the foreground, 2 GALAXY-GALAXY LENSING
provide a promising tool to probe the dark matter distri-
Galaxy-galaxy lensing provides a statistical measure of the
bution directly. In particular, galaxy-galaxy weak lensing,
profile of the tangential shear, γ (R), averaged over a thin
whichfocusesontheimagedistortionsaroundlensinggalax- t
annulus at the projected radius R around the lens galax-
ies, can be used to probe the distribution of dark mat-
ies. This quantity is related to the excess surface density
ter around galaxies, hence their dark matter haloes. The
(hereafter ESD) around thelens galaxy, ∆Σ, as
galaxy-galaxy lensing signal produced by individual galax-
ies is usually very weak, and so one has to stack the signal ∆Σ(R)=γ (R)Σ =Σ¯(<R) Σ(R), (1)
t crit
from many lens galaxies to have a statistical measurement. −
Thefirstattempttodetectsuchgalaxy-galaxylensingsignal whereΣ¯(<R)istheaveragesurfacemassdensitywithinR,
wasreportedbyTysonetal.(1984).Morerecently,withthe and Σ(R) is theazimuthally averaged surface density at R.
adventof wide and deep surveys,galaxy-galaxy lensing can Notethat,accordingtothisrelation,∆Σ(R)isindependent
be studied for lens galaxies of different luminosities, stellar of a uniform background. In theabove equation,
masses, colors and morphological types(e.g. Brainerd etal. c2 D
1996; Hudson et al. 1998; McKay et al. 2001; Hoekstra et Σcrit = 4πGDD (1s+z)2 (2)
l ls l
al. 2003; Hoekstra 2004; Sheldon et al. 2004; Mandelbaum
isthecritical surface densityin comovingcoordinates, with
etal.2005,2006;Sheldonetal.2007a;Johnstonetal.2007;
D andD theangulardistancesofthelensandsource,D
Sheldon et al. 2007b; Mandelbaum et al. 2008). Given that s l ls
the angular distance between the source and the lens, and
galaxies reside in dark matter haloes, these results provide
z theredshift of thelens.
important constraints on the mass distribution associated l
By definition, the surface density, Σ(R), is related to
with galaxies in a statistical way.
the projection of the galaxy-matter cross-correlation func-
Inthispaper,weusethegalaxygroupsofY07selected
tion,ξ (r),alongtheline-of-sight. Inthedistantobserver
from the Sloan Digital Sky Survey (SDSS), together with g,m
approximation
mass models for individual groups, to predict the galaxy-
galaxylensingsignalexpectedfromSDSSgalaxies.Wecom- ∞
Σ(R)=ρ¯ 1+ξ ( R2+χ2) dχ, (3)
pare our model predictions with the observational results g,m
Z−∞h p i
Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups 3
where ρ¯ is the mean density of the universe and χ is the
line-of-sight distance from thelens.
Thecross-correlationbetweengalaxiesanddarkmatter
can, in general, be divided into a 1-halo term and a 2-halo
term. The 1-halo term measures the cross-correlation be-
tween galaxies and dark matter particles in their own host
haloes,whilethe2-halotermmeasuresthecross-correlation
between galaxies and dark matter particles in other haloes.
Inthepresentwork, weare interested inthelensing signals
on scales R 6 2h−1Mpc where the observational measure-
ments are the most accurate. As we will show in 4, on
§
such scales the signal is mainly dominated by the 1-halo
term.Nevertheless,ourmodelalsotakesthecontributionof
the 2-halo term into account. More importantly, since cen-
tral galaxies (those residing at the center of a dark matter
halo) and satellite galaxies (those orbiting around a cen-
tral galaxy) contribute very different lensing signals (e.g.
Natarajan,Kneib&Smail2002;Yangetal.2006;Limousin
etal.2007),itisimportanttomodelthecontributionsfrom
central and satellite galaxies separately.
Asanillustration,inFig.1weshowtheESDsexpected
from a single galaxy in a host halo of mass 1014h−1M⊙.
The solid line representsthelensing signal expected for the
central galaxy of the halo. While the dotted and dashed
Figure 1. The ESD expected for a single galaxy. Here the host
lines show the lensing signal of a satellite galaxy residing
in a sub-halo of 1011h−1M⊙ with a projected halo-centric hthaelolemnsaisnsgissiagsnsaulmfoerdtthoebceen1t0r1a4lhg−al1aMxy⊙i.nTshuechsoalihdallion.eTrheeprdeostetnetds
distance r = 0.2h−1Mpc and r = 0.4h−1Mpc from the
p p linerepresents the lensing signal of a satellite galaxy residingin
center of the host halo, respectively. In the calculation, the a sub-halo of mass 1011h−1M⊙ which has a projected distance
darkmattermassdistributioninthehosthaloisassumedto rp=0.2h−1Mpcfromthecenterofthehosthalo.Thedashedline
follow the Navarro, Frank & White (1997) profile and that isthesameasthedottedline,exceptthatthesubhalo’sprojected
in the sub-haloes is assumed to follow the Hayashi et al. distanceis0.4h−1Mpcfromthecenter ofthehosthalo.
(2003) model.Thesemodelsaredescribedindetailin 3.3.
§
InordertoestimatetheESD,wesampletheseprofileswith
mass particles and project thepositions of all particles toa
plane perpendicular to the line of sight. The Σ(R) is then
estimated by counting the number of dark matter particles tivehalo-basedgroupfinderdevelopedbyYangetal.(2005),
inaannuluswithradiusRcentredontheselectedgalaxies. from the New York University Value Added Galaxy Cata-
Fig. 1 shows clearly that the lensing signals of the central log(NYU-VAGC;Blantonetal.2005)whichisbasedonthe
and the satellite are quite different. The ESD of the cen- SDSSDataRelease4(Adelman-McCarthyetal.2006).Only
tral galaxy follows the mass distribution of the host halo, galaxieswithredshiftsintherange0.016z 60.2,andwith
decreasing monotonically with R. The ESD of a satellite, redshift completeness >0.7, are used in thegroup identi-
on the other hand, consists of two parts: one from the sub- fication.ThemagnitudCesandcolorsofallgalaxiesarebased
halo associated the satellite, which contributes to the inner onthestandardSDSSPetrosian technique(Petrosian 1976;
part, and the other from the host halo, which dominates Strauss et al. 2002), and have been corrected for galactic
atlargerR.Thissimplemodeldemonstratesclearlythat,in extinction (Schlegel, Finkbeiner & Davis 1998). All mag-
ordertomodelthegalaxy-galaxylensingsignalproducedby nitudes have been K-corrected and evolution-corrected to
a population of lens galaxies, one needs to model carefully z = 0.1 following the method described in Blanton et al.
thedistributionofmatteraroundbothhosthaloesandsub- (2003). In Y07, three group samples were constructed us-
haloes.Todothis,weneednotonlytoidentifythehaloesin ing galaxy samples of different sources of galaxy redshifts.
which each lens galaxy resides, but also to model the mass Our analysis is based on Sample II, which includes 362,356
and density profile of each host halo and subhalo. In addi- galaxieswithredshiftsfromtheSDSSand7091galaxieswith
tion we also need to model the distribution of dark matter redshiftstakenfromalternativesurveys:2dFGRS(Collesset
relativetogalaxies.Inthefollowingsection,wedescribeour al. 2001), PSOz (Saunders et al.2000) or from the RC3 (de
modeling with theuse of observed galaxy groups. Vaucouleurset al.1991). Therearein total301,237 groups,
including those with only one member galaxy. The group
finder has been applied to mock catalogue to test the com-
3 MODELING THE MASS DISTRIBUTION pleteness and purity of the groups in Y07. About 90% of
ASSOCIATED WITH THE SDSS GROUPS the groups have a completeness f > 0.6 and 80% groups
c
with f > 0.8, where f is defined as the ratio between the
3.1 The SDSS Group Catalogue c c
number of true members that are selected as the members
Our analysis is based on the SDSS galaxy group catalogue of the group and the number of the total true members of
constructedbyY07.Thegroupsareselectedwiththeadap- thegroup.
4 Ran Li et. al
Figure2.ThehalomassMS (estimatedusingstellarmass),versusM∗(lowerpanels)andL(upperpanels)ofthegalaxiesinthehaloes.
Theleftpanels areforcentralgalaxiesandtherightpanelsareforsatellitegalaxies.
3.2 Halo Mass Assignment its absolute magnitude and color using the fitting formula
given by Bell et al. (2003).
An important aspect of the group catalog construction is
thedetermination of thehalo mass, M , of each group. In The basic assumption of the ranking method is that
vir
Y07,twoestimatorsareadopted.Thefirst,M ,isestimated thereisaone-to-onerelationbetweenM (orL )and
L stellar 19.5
usingtherankingofthecharacteristicluminosityofagroup, the group mass. Using the dark matter halo mass function
which is the total luminosity of all member galaxies in the predictedbyamodelofstructureformation, onecanassign
group with M 5logh 6 19.5 (hereafter referred to as ahalomasstoeachgroupaccordingtoitsM -ranking
r stellar
− −
L ). The second, M , is estimated using the ranking of (orL -ranking).Inthispaper,weusethemassfunction
19.5 S 19.5
thecharacteristicstellarmass,M whichisdefinedtobe obtainedbyWarrenetal.(2006). Notethatthisone-to-one
stellar
thetotalstellarmassofgroupmemberswithM 5logh6 mapping is applicable only when the group sample is com-
r
−
19.5. For each galaxy the stellar mass is estimated from plete. In Y07, three complete samples are constructed in
−
Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups 5
Figure3.Thedistributionofthehosthalomassesforthecentralandsatellitegalaxiesindifferentluminositybins,asindicatedbythe
r-bandabsolute-magnituderangeineachpanel.
three redshift ranges. Only groups in the complete samples thanthatbetweenL andhalomass,weadoptM asour
19.5 S
are used in the ranking. The mass of other groups are esti- fiducial halo mass throughout. As we demonstrate in 4.3,
§
mated by a linear interpolation based on the M -M using M instead yields results that are fairly similar.
stellar vir L
relation (or the L - M ) obtained from the complete
19.5 vir
sample. Detailed tests using mock galaxy redshift samples Fig. 2 shows the relation between the host halo mass,
have shown that the 1-σ error of the estimated halo mass MS, and thegalaxy stellar mass M∗ (thelower two panels)
is 0.3 dex (Y07). In addition, the two mass estimators, or the galaxy luminosity L (the upper two panels). Results
∼
M andM ,agree remarkablywell with each other,with a are shown separately for central galaxies (left panels) and
L S
scatter that decreases from about 0.1 dex at the low-mass satellite galaxies (right panels). As one can see, the stellar
end to about 0.05 dex at the high-mass end. Since the cor- mass (luminosity) of central galaxies is quite tightly cor-
relationbetweenM andhalomassissomewhattighter related with their host halo masses. However, for satellite
stellar
galaxies of a given stellar mass (or luminosity), their host
6 Ran Li et. al
halo mass covers a verylarge range, reflecting thefact that a set of masses for each group mass. We then set the mass
manylow-massgalaxiesaresatellitesinmassivehaloes.The originallyassociatedwithasatellitegalaxyaccordingtothe
distributions of host halo masses, M , for central or satel- stellar mass ranking of the satellites in the group. Here we
S
lite galaxies in different luminosity bins are shown in Fig. implicitly assume that the initial subhalo mass function is
3.Onaverage,brightercentralgalaxies resideinmoremas- the same as the mass function of the subhalos that host
sive haloes. For faint galaxies, the halo-mass distribution is satellites. This assumption is not proved by any observa-
broader, again because many faint galaxies are satellites in tions,andwehavetolivewithitsinceamorerealisticmodel
massive systems. is not currently available. Fortunately, subhalos only con-
In the group catalogue, the mass assignment described tribute a small fraction to the total lensing signal on small
above is used only for groups where the brightest galaxy scales. The uncertainty here will not have a significant im-
is brighter than M 5logh = 19.5. This is because the pact on any of our conclusions. To obtain the final mass in
r
− −
massrankingusedinthegroupcatalogisbasedonthetotal the subhalo at the present time, the evolution of the sub-
stellar mass (or total luminosity) of the member galaxies haloes needs to be taken into account. In other words, we
that are brighter than M 5logh = 19.5. The groups need to know the fraction of the mass that is stripped and
r
− −
with nogalaxies brighterthanthis magnitudethushaveno how the structure of a subhalo changes after the stripping.
assigned rank.Asdescribedin Y07,thereason forchoosing Here it is convenient to introduce a parameter f which is
m
thismaginitudeisacompromisebetweenhavingacomplete theretained mass fraction of thesubhalo. Gao et al. (2004)
sampleinarelatively largevolumeandhavingmoregroups studiedtheradial dependenceof theretained mass fraction
that are represented by a number of member galaxies. For f from alarge sampleofsubhaloesinalarge cosmological
m
groups in which all member galaxies have M 5logh > simulation. In their work, f is considered as a function of
r m
−
19.5,adifferentmethodhastobeadopted.Inmodelingthe r /r , where r is the distance of the subhalo from the
s vir,h s
−
luminosity function and stellar mass function of thecentral center of the host halo and r is the virial radius of the
vir,h
galaxies based on the same SDSS group catalogue as used host halo. The simulation of Gao et al. gives
here, Yanget al. (2008) obtain an average relation between f =0.65(r /r )2/3. (4)
m s vir,h
the luminosity (or stellar mass) of the central galaxy and
the halo mass down to M 5logh 17. We adopt this Wewilladoptthisinourmodelingofthemassesassociated
r
relation toassign halomasse−stoallg∼rou−ps(includingthose with subhaloes. However, in the group catalogue, only the
containingonlyoneisolated galaxy)representedbycentrals projected distance, rp, from the group center is available.
withMr 5logh> 19.5.Forconvenience,thehalomasses The 3D-distance, rs, is obtained by randomly sampling the
obtained−in this way−are also referred to as M (based on NFWprofileofthehosthalowiththegivenprojectedradius
S
the stellar mass of central galaxies) and ML (based on the rp.
r-bandluminosity of thecentral galaxies), respectively. Thus, the mass assigned to a subhalo is determined by
thefollowing threefactors: (1) thestellar mass of thesatel-
litegalaxy;(2)thehosthalomass;(3)thedistancebetween
3.3 Mass Distribution in Haloes and Subhaloes the satellite and the center of the host. Here the host halo
mass comes into our calculation in two ways. It not only
With the group catalogue described above, we can model
determines the subhalo mass function, but also affects the
the dark matter distribution by convolving the halo distri-
parameter f in Eqs.4. Note that the accretion history of
bution with thedensity profiles of individualhaloes. In our m
the host halo may also affect the value of f . We have to
modeling of the density profiles, thehost halo of a group is m
neglect such effect because it is unclear how to model the
assumedtobecenteredonthecentralgalaxy.Therearetwo
accretion histories for individualgroups.
waystodefineacentralgalaxy:oneistodefinethecentralin
For host haloes, we use the following NFW profile
a group to bethegalaxy with thehighest stellar mass, and
(Navarro,Frank&White1997)tomodelthemassdistribu-
theotheristodefinethecentraltobethebrightestmember.
tion:
Formostgroupsthesetwodefinitionsgivethesameresults,
butthereareveryfewcases(lessthan∼2%)wheredifferent ρ(r)= δ0ρ¯ . (5)
centralgalaxiesaredefined.Inourfiducialmodel,wedefine (r/rc)(1+r/rc)2
themostmassivegalaxies(intermofstellarmass)tobethe
where ρ¯ is the mean density of the universe, r is a scale
c
central galaxies.
radius,relatedtovirialradiusr bytheconcentration,c=
vir
In a hierarchical model, a dark matter halo forms
r /r ,andδ isacharacteristicover-densityrelatedtothe
vir c 0
through a series of merger events. During the assembly of
average over-density of a virialized halo, ∆ , by
vir
a halo, most of the mass in the merging progenitors is ex-
∆ c3
pected to be stripped. However, some of them may survive δ = vir . (6)
assubhaloes,althoughthetotalmasscontainedinsubhaloes 0 3 ln(1+c) c/(1+c)
−
issmall,typically 10%(vandenBosch,Tormen&Giocoli We adopt the value of ∆ given by the spherical collapse
vir
∼
2005). Some of the subhaloes are associated with ‘satellite model (see Nakamura & Suto 1997; Henry 2000). Numeri-
galaxies’ in a halo. In our modeling of the galaxy-galaxy calsimulationsshowthathaloconcentrationsarecorrelated
weak lensing, we only take into account subhaloes associ- with halo mass, and we use the relations given by Macci`o
atedwithsatellitegalaxies,treatingothersubhaloesaspart et al.(2007), convertedtoourdefinition ofhalo mass. Note
of the host halo. Giocoli, Tormen & van den Bosch (2008) thathereweuser ,insteadoftheconventionalnotationr ,
c s
provide a fitting function of the average mass function for todenotethescaleradiusoftheNFWprofile,asr hasbeen
s
subhaloesatthetimeoftheiraccretionintotheparenthalo used to denote the distance of a subhalo from the center of
of a given mass. Using this mass function, we first sample its host.
Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups 7
Forsub-haloes,wemodeltheirdensityprofilesusingthe
Table 1. The properties of galaxy samples. In each case, the
resultsobtainedbyHayashietal.(2003),whofoundthatthe absolute-magnituderange,themeanredshift,themeanluminos-
densityprofilesof strippedsub-haloescan beapproximated ity, and the fraction of late-types are listed. Note that L∗ =
as 1.2×1010h−2L⊙
f
ρs(r)= 1+(r/tr )3ρ(r), (7) Sample Mr hzi hL/L∗i flate
t,eff
where f is a dimensionless factor describing the reduction L1 −18<Mr <−17 0.031 0.075 0.81
in the cetntral density, and r is a cut-off radius imposed L2 −19<Mr <−18 0.048 0.191 0.70
t,eff L3 −20<Mr <−19 0.074 0.465 0.54
bythetidalforceofthehosthalo.Forf =1andr r ,
t t,eff ≫ c L4 −21<Mr <−20 0.111 1.13 0.35
ρs(r) reduces to the standard NFW profile ρ(r). Here ρ(r) L5f −21.5<Mr <−21 0.145 2.09 0.22
is calculated using the mass of the subhalo at the time of L5b −21.5<Mr <−22 0.150 3.22 0.12
itsaccretionintothehosthalo.Bothft andrt,eff dependon L6f −22<Mr <−22.5 0.152 5.01 0.04
the mass fraction of the sub-halo that remains bound, f .
m
Based on N-body simulations, Hayashi et al. obtained the
following fitting formulae relating f and r to f :
t t,eff m
galaxies issimilar tothegalaxy sample usedin Y07tocon-
log(r /r )=1.02+1.38logf +0.37(logf )2; (8)
t,eff c m m structthegroup catalogue usedhere. Theonly differenceis
that Mandelbaum et al’s sample includes all galaxies with
log(ft)= 0.007+0.35logfm+0.39(logfm)2+0.23(logfm)3. redshifts in the range 0.02 < z < 0.35, while the galaxies
−
(9) in Y07’s group catalogue are in 0.01 6 z 6 0.2. Since, as
Itshouldbepointedout,though,thattherearesubstantial to be described below, we are interested in the lensing sig-
uncertainties in modeling the mass distribution around in- nals around galaxies of given luminosity and morphological
dividualsatellitegalaxies.Inparticular,manyoftheresults type,thisdifferenceinredshiftrangeisnotexpectedtohave
aboutsubhaloesareobtainedfromN-bodysimulations,and a significant impact on our results. For thefaint luminosity
it is unclear how significant theeffect of including baryonic bins, our galaxy samples should be almost identical to that
matteris.Fortunately,thetotalmassassociated withsatel- of Mandelbaum et al. (2006), because all faint galaxies are
lite galaxies is small (see e.g. Weinberg et al. 2008). Fur- at z 6 0.2 in the SDSS catalog. For bright galaxies we also
thermore,thecontributionofthesubhaloesassociated with expectthestatisticpropertiesofthetwosamplestobesimi-
the satellite galaxies to the galaxy-galaxy lensing signal is lar.BothMandelbaumetal.(2006)andY07appliedsimilar
confinedtosmallscales. Wethereforeexpectthattheseun- evolutioncorrection andKcorrection,sothattheevolution
certainties will not change our results significantly. inthegalaxypopulationhasbeentakenintoaccount,albeit
With the mass distributions described above, we use inasimpleway.Aswewillsee,evenforthethetwobrightest
a Monte-Carlo method to sample each of the profiles with bins, the lensing signal is dominated by halos with masses
a random set of mass particles. Note that the halo mass 1014h−1M⊙, and the change in the halo mass function
∼
assigned to a group in the SDSS Group Catalog is M180, aroundthismassissmallbetweenz =0.2and0.35.Follow-
which is the mass enclosed in the radius, r180, defined such ing Mandelbaum et al. (2006), we split the galaxy sample
that M180 = 4πr1380(180ρ¯)/3. We therefore sample the par- into 7 subsamples according to galaxy luminosity. Table 1
ticle distribution within r180. After all the particles in each shows the properties of these subsamples: the luminosity
halo are sampled, we project the positions of all the par- range covered by each subsample, the mean redshift, the
ticles to a plane and calculate ∆Σ(R) by stacking galax- mean luminosity, and the fraction of late-type galaxies. As
ies in each of the luminosity bin. Since the mass distribu- expected the mean redshifts of brightest bins are different
tion is isotropic, an arbitrary direction can be chosen for from thecorresponding redshifts in Mandelbaum et al.
the stacking. Thus, the projection effect is naturally in- Also following Mandelbaum et al. (2006), we split each
cludedinourcalculation.Eachoftheparticleshasamassof galaxysubsampleintotwoaccordingtogalaxymorphology.
1010h−1M⊙. Ourtest using particles of lower masses shows Theseparationismadeaccordingtotheparameterfrac dev
that the mass resolution adopted here is sufficient for our generated by the PHOTO pipeline. The value of frac dev
purpose. Using 2 times more particles leads to a difference isobtained byfittingthegalaxy profile,in a given band,to
of about 5% at R 0.02h−1Mpc, and almost no difference amodelprofilegivenbyfrac dev F +(1 frac dev)
at R>0.1h−1Mpc∼. F , where F and F are th×e dedeVVaucou−leurs and ex×-
exp deV exp
ponentialprofiles,respectively.AsinMandelbaumetal.,we
usetheaverageoffrac devintheg,randibands.Galaxies
withfrac dev>0.5areclassified asearly-type,whilethose
4 RESULTS with frac dev<0.5 as late-type.
It should be pointed out that we did not carry a ray-
4.1 SDSS Lensing Data
tracing simulation to predict the galaxy-galaxy lensing re-
Before presenting our model predictions, we first describe sults. Instead, we directly calculate the excess surface den-
the observational results that we will use for comparison. sity around SDSS galaxies. Thus, our calculation doesn’t
TheobservationalresultstobeusedwereobtainedbyMan- dealwithsourcegalaxies.Ontheotherside,inMandelbaum
delbaumetal.(2006),whoanalyzedthegalaxy-galaxylens- et al. (2006) thesourcegalaxies arecarefully weighted, and
ing effects using galaxies in a sample constructed from the lensing signals are calibrated, to reduce any bias in the ob-
SDSS DR4 spectroscopic sample. Their sample of lensing servational measurements (see Mandelbaum et al. 2005 for
8 Ran Li et. al
Figure 4. Comparison of the lensing signal predicted by the fiducial model with the observational results. Here the ESD is plotted
as the function of the transverse distance R for lensing galaxies in different luminosity bins. Data points with error bars are the
observationalresultsofMandelbaumetal.,whilethelinesarethemodelpredictions.Thedotted,dashedanddot-dashedlinesrepresent
the contributions of the ‘1-halo term’ of central galaxies, the ‘1-halo term’ of satellite galaxies, and the ‘2-halo term’ (of both centrals
and satellites), respectively. The solid lines show the predicted total ESD. The r-band magnitude range for each case can be found in
Table1.
details).Thus,weassumetheobservationalresultsareunbi- parison of the lensing signal for galaxies divided according
ased,andcomparethemdirectlywithourmodelpredictions. to luminosity.
Theobservationdatausedherewaskindlyprovidedby Theerrorbarsontheobservationalpointsare1σ statis-
R.Mandelbaum.ThedatausedinFig.6hasbeenpublished ticalerror.Thesystematicerrorofthegalaxy-galaxylensing
in Mandelbaum et al. (2006) where lensing signal is calcu- has been discussed in detail in Mandelbaum et al. (2005).
latedforearlyandlatetypegalaxiesseparately.R.Mandel- The test has been carried out for three source samples:
baum also provided the lensing data combining early and r < 21, r > 21, and high redshift LRGs. The overall sys-
latetypegalaxiesforustomakethecomparisons presented tematicerrorisfoundtobecomparabletoorslightlylarger
in all other figures. Note that here we only show the com- than thestatistical error shown here.
Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups 9
Figure 5. The contribution to the ESD plotted separately for dark matter haloes of different masses. In each panel, the dotted line
shows the contribution from haloes with MS >1014h−1M⊙. The dashed line shows the contribution from haloes with 1013h−1M⊙ 6
MS <1014h−1M⊙,andthedot-dashedlineshowsthecontributionfromhaloeswithMS <1013h−1M⊙.Thesolidlineshowsthetotal
lensingsignalpredictedbythefiducialmodel.Forcomparison,theobservational dataareincludedasdatapointswitherror-bars.
4.2 The Fiducial Model vational results obtained by Mandelbaum et al. (2006) for
thesameluminositybins.Overall,ourfiducialmodelrepro-
InFig.4weshowthelensingsignalaroundgalaxiesindiffer-
duces the observational data reasonably well, especially for
entluminositybinsobtainedfromourfiducialmodel,which
bright galaxy bins where the observational results are the
has model parameters as described in the last section and
most reliable. The reduced χ2 is 3.2 combining all the lu-
assumes the WMAP3 cosmology. Here the ESD is plotted
minosity bins. The best match is for L5f, with a reduced
as a function of the projected distance R from galaxies.
χ2 of 0.9. Given that we do not adjust any model parame-
The solid line shows the averaged ESD of all galaxies in
ters, the χ2 indicates a good agreement. For the three low-
thecorrespondingluminositybin.Theamplitudeofthepre-
luminosity bins, the predicted ESD is lower than the corre-
dicted ESD increases with galaxy luminosity, reflecting the
sponding observational result. For L1 and L2, the observa-
fact that brighter galaxies on average reside in more mas-
tional data are very uncertain. For L3, if we take the ob-
sive haloes, as shown in Figs. 2 and 3. These results are to
servational data points at face-value, the discrepancy with
be compared with the data points which show the obser-
10 Ran Li et. al
Figure 6. The right panels show the ESD of early galaxies in different luminosity bins, while the left panels show the results for late
galaxies. The data points with error-bars show the observational results. The model predictions of the ESD using stellar mass as halo
massindicatorareshownasthesolidlines.Forcomparison,thedashedlinesshowthecorrespondingmodelpredictionsusingMLasthe
halomasses.
the model prediction is significant. As described in 3.2, and about 70% in L2 and L1. It is possible that this rela-
§
for groups which do not contain any member galaxies with tionunderestimatesthehalomass.Inordertoseetheeffect
M 5logh < 19.5, their halo masses are not obtained caused by such uncertainties, we have used a set of param-
r
− −
from the ranking of M , but from the average stellar eters from Yang et al. (2008) that are still allowed by the
stellar
mass-halo mass relation of central galaxies that is required observed stellar mass function but give larger halo masses
tomatchtheobservedstellarmassfunctionofcentralgalax- tothehostsoffaintcentralgalaxies. Thisincreasesthepre-
ies.Whileallthegalaxiesinthebrightluminositybinshave dicted ESD for L3 by 20%, not sufficient to explain the
∼
their host halo mass assigned by ranking method, the frac- discrepancy. Indeed, this discrepancy is not easy to fix. In
tion of the galaxies in halos that have masses assigned ac- the observational data, the amplitudes of the ESD for L3
cording to the mass-halo mass relation is about 30% in L3 at R 0.3 - 0.9h−1Mpc are actually slightly higher than
∼