The Galaxy-mass Correlation Function Measured from Weak Lensing in the SDSS

0.45 MB·English

by  E. S. Sheldon

Last revision 29-Nov-2003
PreprinttypesetusingLATEXstyleemulateapjv.7/8/03
THE GALAXY-MASS CORRELATION FUNCTION MEASURED FROM WEAK LENSING IN THE SDSS
Erin S. Sheldon,1,2 David E. Johnston,1,2 Joshua A. Frieman,1,2,3 Ryan Scranton,4 Timothy A. McKay,5 A. J.
Connolly,4 Tama´s Budava´ri,6,7 Idit Zehavi,8 Neta A. Bahcall,9 J. Brinkmann,10 and Masataka Fukugita11
Last revision 29-Nov-2003
ABSTRACT
We presentgalaxy-galaxylensing measurements overscales 0.025to 10 h−1 Mpc in the Sloan Digital
SkySurvey. Usingaflux-limitedsampleof127,001lensgalaxieswithspectroscopicredshiftsandmean
luminosity hLi ∼ L and 9,020,388 source galaxies with photometric redshifts, we invert the lensing
∗
signaltoobtainthegalaxy-masscorrelationfunctionξ . Wefindξ isconsistentwithapower-law,
4 gm gm
0 ξgm =(r/r0)−γ, with best-fit parameters γ =1.79±0.06 and r0 =(5.4±0.7)(0.27/Ωm)1/γh−1 Mpc.
0 At fixed separation, the ratio ξgg/ξgm = b/r where b is the bias and r is the correlation coefficient.
2 Comparing to the galaxy auto-correlation function for a similarly selected sample of SDSS galaxies,
we find that b/r is approximatelyscale independent overscales 0.2−6.7h−1 Mpc, with mean hb/ri=
n
(1.3±0.2)(Ω /0.27). Wealsofindnoscaledependenceinb/rforavolumelimitedsampleofluminous
a m
J galaxies (−23.0< Mr < −21.5). The mean b/r for this sample is hb/riVlim = (2.0±0.7)(Ωm/0.27).
We split the lens galaxy sample into subsets based on luminosity, color, spectral type, and velocity
8
dispersion, and see clear trends of the lensing signal with each of these parameters. The amplitude
2
and logarithmic slope of ξ increases with galaxy luminosity. For high luminosities (L∼5L ), ξ
gm ∗ gm
2 deviates significantly from a power law. These trends with luminosity also appear in the subsample
v of red galaxies,which are more strongly clustered than blue galaxies.
6 Subject headings: cosmology:observations—darkmatter—gravitationallensing—large-scalestruc-
3 ture of the universe
0
2
1 1. INTRODUCTION Inthecurrentparadigmofstructureformation,thefor-
3 mationofgalaxiesisheuristicallydividedintotwoparts.
The measurementofgalaxyclustering has long been a
0 On large scales, cosmological parameters and the prop-
primary tool in constraining structure formation models
/ erties of the dark matter determine the growth of den-
h and cosmology. Yet the power of galaxy surveys to dis-
sityperturbationsandtheeventualformationofmassive
p criminate between models is partially compromised by
dark halos. On smaller scales, hydrodynamic and other
- the fact that they provide only an indirect measure of
o processesshape how luminous galaxiesformwithin dark
the underlying mass distribution, subject to consider-
r matter halos and how they evolve as halos accrete and
t able uncertainties in the bias, that is, in how luminous
s merge. A natural consequence of this picture is that
galaxiestracethemass. Inthiscontext,thegalaxy-mass
a the galaxy distribution is related to but differs in de-
: cross-correlationfunctionξgm canprovideimportantad-
v tail from the mass distribution. This difference arises in
ditional information, since it is in some sense a ‘step
i part because halos are more strongly clustered than the
X closer’ to the clustering of mass. Moreover, comparing
darkmatterasawhole,andmoremassivehalosaremore
ξ with the galaxy auto-correlation function ξ yields
r gm gg strongly clustered than less massive ones (Kaiser 1984).
a a measure of the bias and therefore a constraint on the-
In addition, the efficiency of forming luminous galaxies
ories of galaxy formation. In this paper we measure the
ofdifferenttypesandluminositiesvaries(primarily)with
correlationbetween galaxies and mass using weak gravi-
halo mass.
tational lensing.
Thegalaxybiasparametercanbedefinedinanumber
1 Center for Cosmological Physics, The University of Chicago, ofways,butatraditionaloneisastheratioofthegalaxy
5640SouthEllisAvenueChicago,IL60637 and mass auto-correlation functions at fixed separation
2 Department of Astronomy and Astrophysics, The University (Kaiser 1984),
ofChicago,5640SouthEllisAvenue,Chicago, IL60637.
3FermiNationalAcceleratorLaboratory,P.O.Box500,Batavia, b2 = ξgg . (1)
IL60510.
4 Department of Physics and Astronomy, University of Pitts- ξmm
burgh,3941O’HaraStreet,Pittsburgh,PA15260.
5 DepartmentofPhysics,UniversityofMichigan,500EastUni- The amplitude of the galaxy-mass cross-correlation ξgm
versity,AnnArbor,MI48109-1120. relativeto ξ andξ canbe expressedin terms of the
6 Department of Physics of Complex Systems, Eo¨tvo¨s Univer- correlationcgogefficienmtm(Pen 1998),
sity,Budapest, Pf.32,H-1518Budapest, Hungary.
7 Department of Physics and Astronomy, The Johns Hopkins
ξ
Un8ivSetreswitayr,d34O0b0sNerovratthorCy,hUarnleivseSrtsriteyeto,fBAarltizimonoar,e,93M3DNo2r1t2h18C-h26er8r6y. r = (ξ gξm)1/2 , (2)
mm gg
Avenue,Tucson,AZ85721.
9 PrincetonUniversityObservatory,PeytonHall,Princeton,NJ
so that ξ = brξ . In general, b and r can be time-
08544. gm mm
10ApachePointObservatory,P.O.Box59,Sunspot,NM88349. dependent functions of the pair separation and depend
11InstituteforCosmicRayResearch,UniversityofTokyo,5-1-5 on galaxy properties. Since we will be comparing the
Kashiwa,KashiwaCity,Chiba277-8582, Japan. galaxy-galaxyandgalaxy-masscorrelationfunctions, we
2 Sheldon et al.
will constrain the ratio been used to measure the bias directly (Hoekstra et al.
ξ b 2001,2002a);theirresultsindicatethatbandrarescale-
gg = . (3) dependent over scales ∼ 0.1−5h−1 Mpc, but that the
ξ r
gm ratio b/r is nearly constant at b/r≃1.1 over this range.
Galaxy surveys have provided a wealth of informa- A significant step forward in galaxy-galaxy lensing
tion on the behavior of the galaxy bias b as a function came with the use of samples of lens galaxies with spec-
of galaxy luminosity and type. For example, on scales troscopicredshifts(Smithetal.2001;McKayetal.2002).
r < 20h−1 Mpc, the clustering amplitude ξ increases Lensingmeasurementscouldthen be made asafunction
gg
with luminosity (Norberg et al. 2001;Zehavi et al. 2002; of physical rather than angular separation, placing lens-
Norberg et al. 2002; Zehavi et al. 2004), while the am- ing correlation measurements on a par with the auto-
plitude and shape of ξ vary systematically from early correlation measurements from galaxy redshift surveys.
gg
tolategalaxytypes(Davis&Geller1976;Norbergetal. Incorporation of photometric redshifts for the source
2001;Zehavietal.2002). Moreover,thenearlypower-law galaxies (Hudson et al. 1998) also substantially reduces
behaviorofξ aswellasthesmalldeparturestherefrom errors in the lens mass calibration due to the breadth of
gg
(Zehaviet al.2003),combinedwith the assumptionthat the source galaxy redshift distribution.
the dark matter distribution is described by cold dark Inthispaper,westudygalaxy-masscorrelationsinthe
matter models, indicate that the bias is scale-dependent SDSS using weak gravitational lensing. Using a sam-
on these scales. ple of 127,001 galaxies with spectroscopic redshifts and
On larger scales, r > 20h−1 Mpc, there is evi- 9,020,388 galaxies with photometric redshifts, we mea-
dence from higher-order galaxy correlations (Frieman & sure the lensing signal with high S/N over scales from
Gaztan˜aga 1999;Szapudi et al. 2002;Verde et al. 2002), 0.025 − 10h−1 Mpc. This is the first galaxy lensing
from the cosmic shear weak lensing power spectrum study to incorporate both spectroscopic lens redshifts
(Hoekstraetal.2002b;Jarvisetal.2003),andfromcom- and photometric source redshifts, it is by far the largest
parisonofthe2dFgalaxypowerspectrum(Percivaletal. galaxy lens-source sample compiled, and it extends to
2001) with the WMAP cosmic microwave background scaleslargerthanpreviousgalaxy-galaxymeasurements.
(CMB) temperature angular power spectrum (Spergel A similar spectroscopic sample has been used for galaxy
et al. 2003) that the linear galaxy bias parameter b is auto-correlationmeasurementsintheSDSS(Zehavietal.
lin
of order unity for optically selected L galaxies. (Here, 2003). We compare the galaxy-mass and galaxy-galaxy
∗
b2 is the ratio of the galaxy correlation function to correlationsto constrain b/r over scales from 200 kpc to
lin
the mass correlation function computed in linear per- 10 Mpc. We also use the spectroscopic and photomet-
turbation theory; it is related but not identical to the ric data from the SDSS to divide the lens galaxysample
bias defined in Equation 1.) Measurement of the pa- by luminosity, color, spectral type, and velocity disper-
rameter β = Ω0m.6/blin from redshift space distortions sion. We see clear dependences of ξgm on each of these
in galaxy surveys, combined with independent evidence properties.
thatΩ ≃0.3,alsoindicatesb (L )≃1onlargescales The layout of the paper is as follows: In §2 we intro-
m lin ∗
(Peacock et al. 2001). duce lensing and the measurement methods. In §3 we
Inrecentyears,weakgravitationallensing hasbecome discussthe SDSSdata,reductions,andsampleselection.
a powerful tool for probing the distribution and cluster- The basic measurement of the lensing signal ∆Σ is pre-
ing of mass in the Universe. We focus on galaxy-galaxy sented in §4, and important checks and corrections for
lensing, the distortion induced in the images of back- systematic errors using random points are discussed in
ground(source)galaxiesbyforegroundlensgalaxies. Al- §4.1. In §5-7 we use the data to infer the galaxy-mass
thoughthe typicaldistortioninduced by a galaxylens is correlation function ξgm and compare it with indepen-
tiny(∼10−3)comparedtotheintrinsicellipticitiesofthe dent measurements of ξgg to constrain the bias. In §8-9
sourcegalaxies(∼0.3),thesignalsfromalargesampleof we explore the dependence of galaxy-mass correlations
lens galaxies can be stacked,providing a mean measure- on galaxy luminosity and type. We conclude in §10 and
ment with high signal to noise. The mean lensing signal discuss possible systematic errors in the Appendix.
can be used to infer the galaxy-mass cross-correlation Throughout this paper, where necessary we use a
function which, when compared with the galaxy auto- Friedman-Robertson-WalkercosmologywithΩM =0.27,
correlation function, constrains the amplitude and scale ΩΛ = 0.73, and H0 = 100 h km/s. All distances, densi-
dependence of the bias. ties, and luminosities are expressed in comoving coordi-
The first detection of galaxy-galaxy lensing was made nates.
by Brainerd et al. (1996), and the field has progressed
rapidly since then (dell’Antonio & Tyson 1996;Griffiths
etal.1996;Hudsonetal.1998;Fischeretal.2000;Wilson
2. LENSINGANDGALAXY-MASSCORRELATIONS
et al. 2001; Smith et al. 2001; McKay et al. 2002; Hoek-
2.1. Gravitational Shear and the Galaxy-mass
straetal.2003a). ThefirsthighS/Nmeasurementswere
Correlation Function
made in the Sloan Digital Sky Survey (SDSS) (Fischer
etal.2000). Recentstudieshavebenefitedfromimproved In this section we review the relation between the in-
dataanalysisandreductiontechniquesandfromsurveys duced shear,which can be estimated from source galaxy
whicharespecificallydesignedforlensing(Hoekstraetal. shape measurements, the galaxy-mass cross correlation
2001). Most work in galaxy lensing has concentrated on function ξ , and the projected cross-correlation func-
gm
its power to constrain galaxy halo parameters (Brain- tion w . The tangential shear, γ , azimuthally aver-
gm T
erd et al. 1996; Hudson et al. 1998; Fischer et al. 2000; aged over a thin annulus at projected radius R from a
Hoekstraet al. 2003a). However,galaxylensing has also lens galaxy, is directly related to the projected surface
Galaxy-mass Correlations in SDSS 3
mass density of the lens within the aperture, 2.2. Estimating ∆Σ
γT ×Σcrit =Σ(<R)−Σ(R)≡∆Σ , (4) We estimate the shear by measuring the tangential
where Σ(<R) is the mean surface density within radius componentofthe sourcegalaxyellipticity relativeto the
the lens center, e , also known as the E-mode. In gen-
R, andΣ(R) is the azimuthally averagedsurfacedensity +
eral,theshearisrelatedinacomplexwaytoe (Schnei-
at radius R (Miralda-Escude 1991; Kaiser et al. 1994; +
der & Seitz 1995), but in the weak lensing regime the
Wilson et al. 2001). The proportionality constant Σ
crit relationship is linear:
encodes the geometry of the lens-source system,
4πGD D e =2γ R+eint , (13)
Σ−1 = LS L , (5) + T +
crit c2DS where eint is the intrinsic ellipticity of the source, γ is
+ T
where D , D , and D are angular diameter distances
L S LS the shear,andR is the “responsivity”(see equation20).
to lens, source, and between lens and source.
The assumption behind weak lensing measurements is
Due to the subtraction in equation 4, uniform mass
thatthesourcegalaxiesarerandomlyorientedintheab-
sheets (such as the mean density of the universe ρ =
senceoflensing,inwhichcasetheir intrinsicshapes con-
Ω ρ )donotcontributeto∆Σ—itmeasuresthemean
m crit stitute a large but random source of error on the shear
excess projected mass density. The mean excess mass
measurement. This“shapenoise”isthedominantsource
density at radius r from a galaxy is ρ ξ (r). The mean
gm ofnoiseformostweaklensingmeasurements. Wediscuss
excess projected density Σ(R) is given by the radial in-
limits on intrinsic correlations between galaxy elliptici-
tegral:
ties in the appendix.
The othercomponentofthe ellipticity, e , alsoknown
hΣ(R)i= ρξ (x,y,z)dz ≡ρw (R) , (6) ×
gm gm as the B-mode, is measured at 45◦ with respect to the
Z
wherew istheprojectedgalaxy-masscorrelationfunc- tangent. The average B-mode should be zero if the in-
gm
tion and R = (x2 +y2)1/2 is the projected radius. The duced shear is due only to gravitational lensing (Kaiser
1995; Luppino & Kaiser 1997). This provides an impor-
observable ∆Σ is itself an integral over Σ(R) and hence
tant test for systematic errors,such as uncorrected PSF
w :
gm
smearing,since they generally contribute to both the E-
2 R and B-modes.
h∆Σ(R)i=ρ× R′dR′ w (R′)−w (R)
R2 gm gm Inordertoestimate∆Σfromtheshear,wemustknow
" Z0 # the angular diameter distances D , D , D for each
(7) L S LS
lens-source pair (see equations 4 and 5). In the SDSS,
If the cross-correlation function can be approximated
we have spectroscopic redshifts for all the lens galaxies,
bya power-lawinseparation,ξ =(r/r )−γ, thenw
gm 0 gm so that D is measured to high precision (assuming a
can be written as L
cosmological model). For the source galaxies, we have
w (R)=F(γ,r )R1−γ , (8)
gm 0 photometricredshiftestimates(photoz),withtypicalrel-
where F(γ,r0) = r0γΓ(0.5)Γ[0.5(γ −1)]/Γ(0.5γ) (Davis ativeerrorsof20-30%(see§3.1.5),sothereiscomparable
& Peebles 1983). In that case, the mean lensing signal uncertaintyinthevalueofΣ foreachlens-sourcepair.
crit
∆Σ is also a power law with index γ −1 and is simply Given the known redshifts of the lenses, the distribu-
proportional to ρwgm, tionoferrorsinthesourcegalaxyellipticity,andthedis-
γ−1 tribution of errors in the photometric redshift for each
h∆Σ(R)i= ρwgm(R) . (9) source,we can write the likelihood for ∆Σ from all lens-
3−γ
(cid:18) (cid:19) source pairs,
Moregenerally,thethree-dimensionalgalaxy-masscor-
relation function can be obtained by inverting ∆Σ di- NLensNSource
rectly. Differentiating equation 7, we find L(∆Σ)= dzsiP(zsi)P(γTi|zsi,zLj) , (14)
dw d∆Σ ∆Σ j=1 i=1 Z
gm Y Y
−ρ = +2 . (10)
dR dR R where γi = ei /2R is the shear estimator for the ith
T +
Thederivativedwgm/dRcanbeintegratedtoobtainξgm sourcegalaxy,P(zi)istheprobabilitydistributionforits
s
using an Abell formula (Saunders et al. 1992): redshift (the product of the Gaussian error distribution
1 ∞ −dw /dR returned by the photoz estimator and a prior based on
gm
ξ (r)= dR (11)
gm π (R2−r2)1/2 the redshift distribution for the source population; see
Zr §3.1.5), and P(γi|zi,zj) is the probability distribution
In practice, the data only cover a finite range of scales T s L
of the shear giventhe source and lens redshifts, which is
uptor =R . Theestimatedξ integratingtoR
max gm max
a function of the desired quantity ∆Σ:
is related to the true ξ by
gm
ξest(r)=ξ (r)− 1 ∞ dR −dwgm/dR (12) P(γTi|zsi,zLj)∝
gm gm π ZRmax (R2−r2)1/2 1 γi −∆Σ×Σ−1 (zi,zj) 2
where the last term reminds us of the (unknown) con- exp − T crit s L . (15)
tribution from scales beyond those for which we have  2" σ(γTi) # 
measurements. Provided the integrand falls sufficiently
 
fast with separation, this term is negligible for scales r Inequation15,4×σ2(γTi)=σ2(ei+)+σS2N: theshearun-
smallerthan a fractionofR . Furthermore,since ξ certaintyisthe sumofthemeasurementvarianceσ2(ei )
max gm +
is linear in ∆Σ, the covariance matrix of the latter can and the intrinsic variance in the shapes of the source
be straightforwardlypropagated to that of the former. galaxies σ2 = h(eint)2i. The shape noise measured
SN +
4 Sheldon et al.
from bright, well-resolved galaxies is σSN ≈ 0.32, and 3. DATA
the typical measurement error σ(e ) ranges from ∼0.05
+ The Sloan Digital Sky Survey (SDSS; York et al.
for r=18 to ∼0.4 for r=21.5. The intrinsic shape dis-
(2000)) is an ongoing project to map nearly 1/4 of the
tribution is not Gaussian as we have assumed in equa-
sky in the northern Galactic cap (centered at 12h20m,
tion 15, but it is symmetric. Monte Carlo simulations
+32.8◦). Using a dedicated 2.5 meter telescope lo-
indicate that this approximation does not bias the mea-
cated at Apache Point Observatory in New Mexico, the
surement of ∆Σ within our measurement uncertainties,
SDSS comprises a photometric survey in 5 bandpasses
provided we take σ as the standard deviation of the
SN (u,g,r,i,z; Fukugita et al. (1996)) to r ∼ 22 and a
non-Gaussian shape distribution.
spectroscopic survey of galaxies, luminous red galaxies,
Although the typical uncertainty in the source galaxy
quasars, stars, and other selected targets. In addition,
photometric redshifts is 20-30%, this is small compared
the survey covers 3 long, thin stripes in the southern
to the relative shear noise, which is typically σ /γ ∼
SN T Galactic hemisphere; the central southern stripe, cover-
300%. Assuming shape error is the dominant source of
ing∼200squaredegrees,willbeimagedmanytimes,al-
noise, we can approximate equation 14 as
lowingtime-domainstudies as wellas a deeper co-added
1 γi −∆Σ×hΣ−1 i 2 image.
logL(∆Σ)= − T crit j,i (16) WeselectourlensgalaxiesfromtheSDSSmaingalaxy
j,i 2(cid:20) σ(γTi) (cid:21) ! spectroscopicsample. Forourbackgroundsourcesweuse
X
only well-resolved galaxies drawn from the photometric
where we now use the critical surface density averaged survey with well-measured photometric redshifts. Each
over the photoz distribution for each source galaxy, sample is described in detail below.
hΣ−1 i = dziP(zi)Σ−1 (zi,zj) (17) 3.1. Imaging Data
crit j,i s s crit s L
Z Imaging data are acquiredin time-delay-and-integrate
MonteCarlosimulationsindicatethatthetruelikelihood (TDI) or drift scan mode. An object passes across the
approaches this Gaussian approximation after stacking camera (Gunn et al. 1998) at the same rate the CCDs
only a few hundred lenses. readout,whichoccurscontinuouslyduringtheexposure.
The maximum likelihood solution is the standard The object crosses each of the 5 SDSS filters in turn,
weighted average, resultinginnearlysimultaneousimagesineachbandpass.
In order for the object to pass directly down the CCD
Nj=L1ens Ni=S1ource∆Σj,iwj,i columns, the distortion across the field of view must be
∆Σ= , (18)
P Nj=L1ePns Ni=S1ourcewj,i edxiscteoerdtiinonglsyisnmtahlle. oTphtiiscsiscaaduvsaentaagbeioaussinforglaelnasxiyngs,hsainpcees
where P P (§3.1.3). The distortion in the optics of the SDSS 2.5m
telescopeisnegligible(Stoughtonetal.2002)andcanbe
∆Σj,i=γTi/hΣ−cr1itij,i ignored.
w =σ−2 The imaging data are reduced through various soft-
j,i j,i ware pipelines, including the photometric (PHOTO, Lup-
σj,i=σ(γTi)/hΣ−cr1itij,i. (19) tonetal.(2001)),astrometric(Pieretal.2003),andcal-
ibration (Hogg et al. 2001; Smith et al. 2002) pipelines,
Although this simple inverse variance weighting is not
leading to calibrated lists of detected objects. The cali-
optimal (Bernstein & Jarvis 2002), it does lead to unbi-
brated object lists are subsequently fed through various
ased results and only a slight increase in the variance of
targetselection pipelines (Eisenstein et al. 2001;Strauss
∆Σ.
etal.2002;Richardset al.2002)which selectobjectsfor
As indicated by equation13,the ellipticity inducedby
spectroscopic followup.
a shear depends on the object’s shear responsivity R.
The shape measurements discussed in §3.1.1 are im-
A measure of how an applied shear alters the shape of
plemented in PHOTO(v5 3), so we workdirectly with the
a source, R depends on the object’s intrinsic ellipticity
calibrated object lists. These lists contain, among many
and is similar to the shear polarizability of Kaiser et al.
otherthings,position(RA,DEC),severalmeasuresofthe
(1995). Following Bernstein & Jarvis (2002), we calcu-
flux, diagnostic flags for the processing,and moments of
late a mean responsivity as a weighted average over all
the light distribution for each object and for the local
tangential ellipticities.
Point Spread Function (PSF) (Stoughton et al. 2002).
w 1−k −k (ei )2 We augment the parameters measured by PHOTO with
j,i j,i 0 1 +
R= , (20) the probability that each object is a galaxy (§3.1.4) and
w
P (cid:2) i j,i (cid:3) with photometric redshifts.
where the weights are thePsame as in equation 19. We
3.1.1. Shape Measurements
have again assumed a Gaussian distribution of elliptici-
ties so that k and k are simple, Weak lensing measurements rely on the assumption
0 1
that source galaxy shapes are an unbiased, albeit rather
k =(1−f)σ2 , k =f2,
0 SN 1 noisy,measureoftheshearinducedbyforegroundlenses.
σ2 Therefore high S/N shape measurements and accurate
f= SN (21)
σ2 +σ2(ei ) correctionsforbiasarecrucialforweaklensingmeasure-
SN + ments. For both shape measurements and corrections
The mean R for our sources is 0.86. weusetechniquesdescribedinBernstein&Jarvis(2002)
Galaxy-mass Correlations in SDSS 5
(hereafter BJ02). 15−25 stars per frame. These stellar images P (u,v)
(i)
Wedeterminetheapparentshapesofobjectsfromtheir are used to form a set of KL basis functions or eigen-
flux-weighted second moments. images B (u,v), in terms of which the images can be
r
reconstructed by keeping the first n terms in the expan-
Qm,n = Im,nWm,nxmxn, (22) sion,
m,n n−1
X
P (u,v)= ar B (u,v) (25)
where Im,n is the intensity at pixel m,n and Wm,n is an (i) (i) r
elliptical Gaussian weight function, iteratively adapted r=0
X
totheshapeandsizeoftheobject. Initialguessesforthe where P denotes the ith star, and u,v are pixel posi-
(i)
size and position of the object are taken from the Pet-
tions relative to the object center.
rosianradiusandPHOTOcentroid. Objectsareremovedif The spatial dependence of the coefficients ar are de-
theiterationdoesnotconvergeoriftheiteratedcentroid (i)
termined via a polynomial fit,
wanders too far from the PHOTO centroid. The shape is
parametrized by the polarization, or ellipticity, compo- l+m≤N
nents, defined in terms of the second moments, ar ≈ br xl ym (26)
(i) lm (i) (i)
Q −Q l=m=0
11 22 X
e = (23)
1 Q +Q wherex,y arethecoordinatesofthecenteroftheith star
11 22
2Q relativetothecenteroftheframe,N isthehighestorder
12
e2= . (24) inx,y includedintheexpansion,andbr aredetermined
Q11+Q22 from minimizing lm
Thepolarizationisrelateddirectlytotheshearviaequa-
tion 13. n−1
χ2 = P (u,v)− ar B (u,v) (27)
Theerrorsinthemomentsarecalculatedfromthepho- (i) (i) r
!
ton noise under the assumption that each measurement i r=0
X X
is sky noise limited, which is a good approximation for Only the stars on ±1/2 a frame surrounding the given
the faint sources. Because the same pixels are used for frame are used to determine the spatial variation, while
measuring both components of the ellipticity, there is a starsfrom±2framesareusedtodeterminetheKLbasis
small covariance between the components of the ellip- functions.
ticity. This covariance is also calculated and properly InPHOTO,the numberoftermsusedfromthe KLbasis
transformed when rotating to the tangential frame for is usually n = 3; the order of the spatial fit is N = 2
shear measurements. unless there are too few stars, in which case the fit may
Because the natural coordinates for the SDSS are sur- beorder1oreven0(rare). Todeterminethecoefficients
vey coordinates (λ,η) (Stoughton et al. 2002), we ro- br ,atotalofn(N+1)(N+2)/2constraintsareneeded,
lm
tate the ellipticities into that coordinate system for the which may seem like too many for the typical number
lensing measurements. The full covariance matrix for of available stars. There are many pixels in each star,
(e1,e2) is used to transform the errors under rotations. however, so the number of spatial terms (N +1)(N +
Wecombinetheshapemeasurementsfromtheg,r,andi 2)/2 = 6 (for quadratic fits) should be compared with
bandpassesusingthecovariancematrices. Thisincreases the number of available stars.
the S/N, simplifies the analysis, and reduces bandpass- ThePSFisreconstructedatthepositionofeachobject
dependent systematic effects. The u and z bands are and its second moments are measured; these are used in
much less sensitive and would contribute little to the the analytic PSF correction scheme described in §3.1.3.
analysis.
3.1.3. Shape Corrections
3.1.2. PSF Reconstruction
TocorrectgalaxyshapesfortheeffectsofPSFdilution
AnanisotropicPSF,causedbyinstrumentalandatmo-
and anisotropy, we use the techniques of BJ02 with the
spheric effects, smears and alters the shapes of galaxies
modificationsspecifiedinHirata&Seljak(2002). Rather
inawaywhichcanmimiclensing. Inaddition,the finite
than a true deconvolution, this is an approximate ana-
sizeofthe PSF—the seeing—tendstocircularizethe im-
lytictechnique. ThePSFismodeledasatransformation
age,reducingthemeasuredellipticity. Inordertocorrect
of the preseeing shape; to removethe effects of the PSF,
for these effects, one should, in principle, determine the
the inverse transform must be calculated. This trans-
exact shape and size of the PSF at the position of every
formation is performed in shear space (or polarization
source galaxy. Imaging in drift scan mode produces im-
space):
ages that are long, thin stripes on the sky. Because the
PSF varies over time (along the direction of the scan) R∗[(−ePSF)⊕e]=(−ePSF)⊕e0 (28)
as well as across the camera, it must be tracked as a
where e is the preseeing polarization, e is the measured
function of position in the overallimage (see §3.1.3). 0
polarization, e is the polarization of the PSF, and
The photometric pipeline uses Karhunen-Lo`eve (KL) PSF
R is the resolution parameter, which is related to the
decomposition (Hotelling 1933; Karhunen 1947; Lo`eve
polarizability of Kaiser et al. (1995). The operator ⊕ is
1948) to model the PSF (Lupton et al. 2001). The PSF
theshearadditionoperatordefinedinBJ02. Iftheshear
is modeled on a frame-by-frame basis, where frames are
addition operator were a simple addition, then equation
defined as 2048×1490 pixel chunks composing the long
28 would reduce to:
SDSS image. A set of bright, isolated stars are chosen
from±2 frames aroundthe centralframe, with typically e=[e −(1−R)e ]/R (29)
0 PSF
6 Sheldon et al.
For unweighted moments (W =1) or if the objects givenits concentration,magnitude, andthe local seeing.
m,n
and PSF have Gaussian surface brightness profiles, the Foroursourcegalaxysample,wecanthen selectobjects
formulae in BJ02 are exact, and the resolution parame- which have high values of P . To map out the distri-
g
ter in this case is just R = 1−(s /s )2, where s is bution in concentration, we use regions from the SDSS
PSF obj
the linear size of the object or PSF. The light profiles of Southern Survey which have been imaged many times.
galaxies and of the PSF differ significantly from Gaus- Some of these regions have been imaged as many as 16
sian, however. Furthermore,as noted in §3.1.1,we use a times, with an average of about 8. By averaging the
Gaussian radial weight function to optimize the S/N of flux for each object from the multiple exposures to ob-
object shape measurements. As a consequence, the res- tainhigherS/N,acleanseparationbetweenthestarand
olution parameter R must be derived in an approximate galaxy concentration distributions is achieved to fainter
way that accounts for both the weight function and the magnitudes. We have maps of the concentration distri-
non-Gaussianity of the light profiles. For this work we bution for 16 < r < 22 and 0.9 < seeing < 1.8,
useaweightedfourthmomenttocorrectforhigherorder allowing us to accurately calibrate P and therefore de-
g
effects as discussed in BJ02 and Fischer et al. (2000). fine a clean galaxy sample. In the few regions of very
bad seeing, we extrapolate the concentration conserva-
3.1.4. Star-Galaxy Separation tively, erring on the side of including fewer galaxies in
the sample. In section 3.1.6 we discuss the cuts used for
To separate stars and galaxies cleanly at all magni-
our source sample.
tudes,weusetheBayesianmethoddiscussedinScranton
et al. (2002). The method makes use of the concentra-
tionparameter,whichcanbecalculatedfromparameters 3.1.5. Photometric Redshifts
outputby PHOTO.The concentrationis the difference be-
A photometric redshift (photoz) is estimated for each
tween the object’s PSF magnitude and exponential disk
object in the source catalog. The repaired template fit-
magnitude. The PSFmagnitudeisderivedbyfitting the
ting method is used, described in detail in Csabai et al.
local PSF shape to the object’s light profile. The only
(2000)andimplementedintheSDSSEDR(Csabaietal.
free parameter in this fit is the overall flux. An expo-
2003) as well as the DR1 (Abazajian et al. 2003). This
nential disk is also fit to the object, but in that case
techniqueuses the 5-bandphotometryforeachobjectas
the scale length is also a free parameter. Thus, large
a crude spectrum. The algorithm compares this spec-
objects have more flux in the exponential than the PSF
trum to templates for different galaxy types at different
fit and correspondingly large concentration, while stars
redshifts. The result is an estimate of the type and red-
have concentration around zero.
shift of each galaxy.
At bright magnitudes, galaxies and stars separate
There is a large covariance between the inferred type
cleanly in concentration space. At fainter magnitudes,
of the galaxy and its photometric redshift. The code
photometricerrorsincreaseandthedistributionsoverlap.
outputs a full covariance matrix for type and redshift.
This is demonstrated in figure 1, which showsthe distri-
Because we do not use the type information, we use the
bution of concentration for objects with 20 < r < 21
errormarginalizedovertype. Wefurtherassumethatthe
and21< r <22drawnfrom100framesofasingleSDSS
resultingerrorisGaussian,whichis onlya goodapprox-
imagingrun(3325),withmeanseeingof1.25′′(typicalof
imation for high S/N measurements. This introduces a
SDSSimagequality). Inbadseeingconditions,asmaller
bias in the estimate of hΣ−1 i, but Monte Carlo simula-
percentage of galaxies are larger than the PSF, again crit
tions indicate that this is a negligible effect.
making separation difficult.
From comparisons to galaxies with known redshifts,
If weknow the distributioninconcentrationfor galax-
the rms in the SDSS photoz estimates is found to be
ies and stars as a function of seeing and magnitude, we
∼0.035forr<18,increasingto∼0.1forr<21. About
canassigneachobjectaprobabilityP thatitisagalaxy,
g 30%ofoursourcegalaxysamplehasrbetween21and22.
Although the photozs are less reliable in this magnitude
range,these objects receiverelativelylittle weightin the
analysis, because they have large shape errors and large
500
photoz errors (and hence small hΣ−1 i).
20 < r < 21 crit
21 < r < 22 The photoz distribution for the sources used in this
400
study is shown in figure 2. The histogram shows the
photometric redshifts, and the smooth curve is the dis-
er 300 tribution calculated by summing the Gaussian distribu-
mb tionsforeachgalaxy. Weusethissmoothcurveasaprior
u
n 200 onthe photometricredshift whencalculatingthe inverse
critical density for each lens-source pair (see §2.2). The
large peaks in the distribution are most likely not real,
100
but rather the result of degeneracies in the photometric
redshiftestimation. Thisissueisaddressedfurtherinthe
0
appendix.
-0.2 0.0 0.2 0.4 0.6
concentration Although there is significant overlap between the dis-
tributionofphotozsandthedistributionoflensredshifts
Fig. 1.— Concentrationdistributionforobjectswith20< r <
(figure4),sourceswithphotozsinfrontofornearthelens
21 (dark curve) and 21 < r < 22 (light curve). Stars have
redshift are given appropriately small weight according
concentration near zero. At faint magnitudes, stars and galaxies
arenotaseasilyseparated. to equation 17.
Galaxy-mass Correlations in SDSS 7
4 thosewithphotozerrorsgreaterthan0.4,andwefurther
excludeobjectswithphotozlessthan0.02orgreaterthan
0.8sincefailedmeasurementstendtopileupataphotoz
3 of 0.0 or 1.0. This removes another 10% of the objects.
Thefinalsourcecatalogcontains9,020,388galaxies,cor-
dz responding to a density of about 1−2 source galaxies
N/
d 2 per square arcminute, depending on the local seeing.
N
1/
3.2. Spectroscopic Data
1
The lens galaxies are selected from the SDSS “main”
galaxy spectroscopic sample, which is magnitude-
(14.5 <r< 17.77) and surface brightness-limited (µ <
0 r
23.5), although these limits varied during the commis-
0.0 0.2 0.4 0.6 0.8
z sioning phase of the survey. See Strauss et al. (2002)for
Fig. 2.— Distributionofphotometricredshiftsforsourcesused a description of “main” galaxy target selection.
in this study. The histogram shows the photozs in bins of ∆z = SDSS spectroscopy is carried out using 640 optical
.01, and the smooth curve is derivedfrom summingthe Gaussian fibers positioned in pre-drilled holes in a large metal
distributionsassociatedwitheachobject.
plate in the focal plane. Targeted imaging regions are
assigned spectroscopic plates by an adaptive tiling algo-
rithm (Blanton et al. 2003), which also assigns each ob-
3.1.6. Defining the Source Sample ject a fiber. The spectroscopic data are reduced to 1-d
Source galaxies are drawn from SDSS imaging stripes spectra by SPECTRO2d,and the SPECTRO1dpipeline out-
9-15 and 27-37, covering a region of nearly 3800 square putsredshiftandanassociatedconfidencelevel,spectral
degrees. An Aitoff projection displaying the positions of classification (galaxy, quasar, star), line measurements,
these sources (as well as the lenses) is shown in figure 3. and spectral type for galaxies, among other parameters
Wemakeaseriesofcutsaimedatensuringthatthesam- (Stoughton et al. 2002). In addition, the velocity dis-
pleisofhighpurity(freefromstellarcontamination)and persionis measuredfor a large fractionof the early type
includes only well-resolvedobjects with usable shape in- galaxies.
formation. We firstrequirethatthe extinction-corrected Forthisanalysis,weuseasubsetofthe availablespec-
r-band Petrosian magnitude is less than 22. We next troscopic“main”galaxysampleknownasLSSsample12
make an object size cut, requiring that the resolution (M. Blanton 2003, private communication). Although
parameter R > 0.2. This removes most of the stars and we draw from a larger sample, the mask (see §3.2.3)
unresolved galaxies from the sample. However, at faint was produced for this subset. This sample is also be-
magnitudes (r> 21), many stars and galaxies have sim- ing usedfor analysisofthe galaxyauto-correlationfunc-
ilar values of R due to measurement error, so a further tion (Zehavi et al. 2004), while a slightly earlier sample
cut is needed. We employ the Bayesian galaxy proba- (sample11)has been used to estimate the galaxy power
bility (§3.1.4) and find that the combination R > 0.2, spectrum(Tegmarketal.2004). Usingthissampleallows
P > 0.8 guarantees that the source galaxy catalog is us to make meaningful comparisons between the auto-
g
greater than 99% pure for r < 21.5 and greater than correlation function and the galaxy-mass cross correla-
98% pure for 21.5<r<22. tionfunction. Thespectroscopicreductionsusedhereare
those of the SPECTRO1dpipeline, for which redshifts and
spectroscopic classifications differ negligibly from those
in the above references.
3.2.1. Redshifts
The SDSS spectra cover the wavelength range 3800-
9200˚A with a resolving power of 1800 (Stoughton et al.
2002). Repeated 15-minute exposures (totaling at least
45 minutes) are taken until the cumulative median
(S/N)2 per pixel in a fiber aperture is greater than 15
at g = 20.2 and i = 19.9 in all 4 spectrograph cameras.
Redshifts are extracted with a success rate greater than
99%,andredshiftconfidence levelsaregreaterthan98%
for 95% of the galaxies. Repeat exposures of a number
ofspectroscopicplatesindicatesthat“main”galaxyred-
shifts are reproducible to 30 km/s. We apply a cut on
the redshift confidence level at > 75%, which removes
0.5%ofthe galaxies. Many galaxiesare further removed
Fig. 3.— Aitoffprojectionshowingthepositionsofthesources during the lensing analysis, as discussed in §3.2.3 and
(black) and lenses (red) used in this study. The section covering
§3.2.4. A redshift histogram is shown in figure 4 for the
the equator is stripes 9-15. The higher latitude section is stripes
27-37. remaining 127,001galaxiesused in this work. The mean
redshift for our sample using the relative weights from
Additionally, we remove about ∼ 8% of the sources— the lensing analysis is hzi=0.1.
8 Sheldon et al.
10 3.2.4. Photometric Masks
AlthoughthecorrectionforPSFsmearing(§3.1.3)sub-
8
stantially reduces the galaxy shape bias, it does not
completely eliminate it—generally, a small, slowly vary-
z 6 ing residual remains. Fortunately, a residual PSF bias
d
N/ that is constant over the lens aperture cancels on aver-
d
N agefromtheazimuthallyaveragedtangentialshear,since
1/ 4 two aligned sources separatedby 90◦ relative to the lens
contribute equally but with opposite sign. To take ad-
2 vantageofthiscancellation,wedividethesourcegalaxies
aroundeachlensintoquadrantsanddemandthatatleast
two adjacent quadrants are free of edges and holes out
0
to the maximum search radius.
0.0 0.1 0.2 0.3 0.4
z We representthe geometryofthe sourcesusingthe hi-
Fig. 4.— Redshift distribution for galaxies used in this study. erarchicalpixelschemeSDSSPix1 (Scrantonetal.2003),
Thissamplecontainsonly“main”galaxytargets. modeledafterasimilarschemedevelopedforCMBanal-
ysis(Go`rskietal.1998). Thisschemerepresentswellthe
rectangulargeometryoftheSDSSstripes. Themaskisa
3.2.2. K-corrections collectionofpixels atvaryingresolutioncoveringregions
with holes or edges. We do not mask out bright stars
We apply K-correctionsusing the method discussedin
(which cover only a tiny fraction of the survey area) for
Blanton et al. (2002) (kcorrect v1 10). Linear combi-
the lensing analysis.
nationsoffourspectraltemplatesarefittothefiveSDSS
After checking the spherical polygon masks (§3.2.3),
magnitudesforeachgalaxygivenitsredshift. Rest-frame
each lens galaxy is checked against the pixel mask to
absolutemagnitudesandcolorsarethencalculated. Fig-
guarantee that it is within the allowed region. Each
ure 5 shows the distribution of absolute Petrosian mag-
quadrant around the lens is then checked to determine
nitude for each of the 5 SDSS bandpasses.
if it contains a hole or an edge. Lenses are excluded
from the sample if there are no adjacent quadrants that
3.2.3. Spectroscopic Masks
are completely unmasked. In addition to this cut, we
Because SDSS spectroscopy is taken through circular demand that the angular distribution of source galaxies
plateswithafinitenumberoffibersoffiniteangularsize, around the lens have ellipticity no greater than 20% in
the spectroscopic completeness varies across the survey order to ensure the availability of pairs with 90◦ separa-
area. Theresultingspectroscopicmaskisrepresentedby tion. Thesamecriteriaareappliedtotherandompoints
a combinationof disks and sphericalpolygons(Tegmark (see §4.1).
et al. 2004). Our spherical polygon mask contains 3844 We draw the final spectroscopic data set from SDSS
polygons covering an area of 2818 square degrees. Each stripes 9−12 (near the equator) and 28−37. We do
polygon also contains the completeness, a number be- not use galaxies from the 3 southern stripes (76,82,86),
tween 0 and 1 based on the fraction of targeted galaxies because they contain few lens-source pairs at large sep-
in that regionwhich were observed. We apply this mask aration. Figure 3 shows the distribution of the lens and
to the spectroscopy and include only galaxies from re- source galaxies as an Aitoff projection. After applying
gions where the completeness is greater than 90%. The the cuts described above, the final lens sample contains
same criteria are used to generate the random points as 127,001galaxies.
discussed in 4.1.
4. RESULTS:THEMEAN∆ΣFROM.02−10h−1 Mpc
104 Themeanlensingsignal∆ΣforthefullsampleofSDSS
u lens galaxies is shown in figure 6. The signal is unam-
g
r biguously detected from 25 h−1 kpc to 10 h−1Mpc. The
i
z correspondingmean shearγ is shownon the rightaxis.
T
103 Corrections have been made to this profile as described
N) in§4.1. Thesecorrectionsarerelativelysmallinallradial
g ( bins.
o
l Theerrorsinfigure6comefromjackknifere-sampling.
102
Although we expect statistical errors to dominate over
sample variance even on the largest scales shown here,
there are in addition large variations in the systematic
10 errors. Due to gaps between the 5 CCD columns, two
interleaving imaging runs make up a contiguous imag-
-14 -16 -18 -20 -22 -24 -26
ing stripe, and they are generally taken on different
M - 5 log(h)
nights under different photometric conditions. As a re-
Fig. 5.— Distributionofrest-frameabsolutemagnitudeineach
sult,the residualsfromthe PSFcorrectionvarybetween
ofthe 5SDSSbandpasses forlensesused inthis study. Thefilled
circles mark the magnitude range for the flux limited sample of the columns of interleaving runs, each of which is ∼ 0.2
Zehavietal.(2003). Ninetypercentofourlenssamplefallswithin
thisrange. 1 http://lahmu.phyast.pitt.edu/∼scranton/SDSSPix
Galaxy-mass Correlations in SDSS 9
102 10-2 χ2/ν =1.26. Thereis a20%chanceofχ2 exceeding this
value randomly.
An important check for systematic errors is the “B-
mode”, ∆Σ the average shear signal measured at 45◦
×
with respect to the tangential component. If the tan-
−2c] 10 10-3 gshenoutiladlbsheezaerrosi,gwnhaleriesadsuseysstoelmelyatticoelerrnosrinsgg,etnheeraBll-ymcoodne-
p
M O • γT tribute to both the E- and B-modes. The top panel of
h figure8shows∆Σ measuredusingthesamesource-lens
∆Σ [ sample used in fig×ure 6. This measurement is consistent
with zero.
1 10-4
4.1. Systematics Tests with Random Points
By replacing the lens galaxies with sets of random
points,wecangaugetwosystematiceffectson∆Σ: resid-
0.1 1 10 uals in the PSF correction and the radial bias due to
R [h−1 Mpc]
clustering of sources with the lenses. We generated a
Fig. 6.— Mean∆Σ=Σ(<R)−Σ(R)measuredforthefulllens random sample with ten times as many points as the
sample. The solidlineis the best-fitting power law ∆Σ∝R−0.76. lens sample, using the same masks and selection crite-
Therightaxisshowsthecorrespondingtangential shear γT. ria described in §3.2.3 and §3.2.4. The random points
aredrawnfromthesameredshiftdistributionasthelens
galaxies. Thesecriteriaguaranteethatthesameregions,
degrees wide. The residuals also vary over time along
and thus roughly the same systematics, are sampled by
the direction of the scan, with a typical scale of a few
the lenses and the random points. Any non-zero lensing
degrees. Thus the proper subsample size to account
signal for the random points we ascribe to residuals in
for this variation is about a square degree. We divide
the PSF correction.
the sample into 2,000 disjoint subsamples, each approx-
The bottom panel of figure 8 shows ∆Σ measured
imately a square degree in size (see §3.2.3) and remea-
around 1,270,010 random points, with errors from jack-
sure∆Σ2,000times,leavingouteachsubsampleinturn.
knife re-sampling. Note that the randomsample is large
Figure 7 shows an image of the resulting dimension-
enough that in this case sample variance dominates the
less correlation matrix Corri,j = Vi,j/ Vi,iVj,j, where error in the outer bins. There is a significant signal at
Vi,j = h(∆Σ(Ri) − h∆Σ(Ri)i)(∆Σ(Rj) − h∆Σ(Rj)i)i. large radius. The signal at smaller radii is less well de-
p
The off-diagonal terms are negligible in the inner bins termined, but it is in any case far below the signal due
but become important beyond R ∼ 1h−1 Mpc. The full to lenses. Interpreting the large-scale signal as system-
covariance matrix is used for all model fitting. atic error, we subtract it from the ∆Σ measured around
The mean ∆Σ for the full sample is well described by lenses and add the errors σ2 = σ2 + σ2 ; this
final lens rand
a power law correctionis incorporated in figure 6.
∆Σ(R)=A(R/1Mpc)−α (30)
with α=0.76±0.05 and A=(3.8±0.4)hM pc−2. The
⊙
outliers at intermediate radii make the reduced χ2 for 2
the power law fit somewhat poor but not unacceptable: −2c] 1
p
MO • 0
h
1.0 Σ [× -1
0.9 ∆ -2
0.5 0.8
2
0.7 −2c] 1
Mpc]) 0.0 0.6 pMO • 0
−1g(R [h −0.5 00..45 ∆Σrand [h --21
o 0.3
l
0.1 1 10
0.2
−1.0 R [h−1 Mpc]
0.1
Fig. 8.— Twotestsforsystematicsinthelensingmeasurement.
−1.5 0.0 The top panel shows the “B-mode” for lensing, ∆Σ×, measured
aroundthesamelensesusedinfigure6. Themeasurementiscon-
−1.5 −1.0 −0.5 0.0 0.5 sistentwithzero,asexpectedforlensing. Thebottompanelshows
log(R [h−1 Mpc]) ∆Σ measured around 1,270,010 random points. The detection at
large radius is indicative of systematic errors, most likely from
Fig. 7.— Correlationmatrixfor∆Σinfigure6calculatedusing residuals in the PSF correction. This has been subtracted from
jackknifere-sampling. thesignalaroundlensesforfigure6.
10 Sheldon et al.
The second systematic probed by the random sample some samples are large, the value of the correction is
involvesclusteringofthe sourcegalaxieswiththe lenses. well measured in each case.
The calculation of the mean inverse critical density in We have also calculated the average ∆Σ for the lumi-
§2.2 properly corrects for the fact that a fraction of the nositysubsamplesdefinedbelow (§8). BecauseC(R)de-
source galaxies are in front of the lenses, but only un- pendsonluminosity,theaverageofthesubsamplescould
der the assumptionthat the lens and sourcegalaxiesare inprinciple differ fromthe mean∆Σ estimatedfromthe
homogeneouslydistributed. Since galaxiesareclustered, full sample, as suggested by Guzik & Seljak (2002) re-
a small fraction of the sources are in fact physically as- garding the results of McKay et al. (2002). However,we
sociated with the lenses, causing a scale-dependent bias find no significant difference in the two methods for our
of the lensing signal. We correct for this by estimating weighting scheme, which results in smaller corrections
the excess of sources around lenses compared with the than those in McKay et al. (2002).
random points. The correction factor is the ratio of the Note our calculation of C(R) may be a slight over- or
sumsoftheweightsforsourcesaroundlensesandaround under-estimate,sincelensingitselfmayinducesomecor-
random points: relation between unassociated background sources and
the positions of the foreground lenses through magni-
C(R)= Nrand i,jwi,j (31) fication. Lensing magnification can bring galaxies into
Nlens Pk,lwk,l a magnitude limited sample that would otherwise have
been too faint to be included, and this effect will be a
where i,j indicates sources fouPnd around lenses, k,l in- function of scale. On the other hand, the geometric dis-
dicates sources found around random points, and w is
i,j tortion induced by the lens moves the apparent position
the weight for the lens-source pair (see §2.2). The value
of background galaxies radially outward, decreasing the
of C(R)−1 is shownfor the full sample in figure 9. The
number density. The net change in number density de-
useofphotometricredshiftsreducesthecorrectionsignif-
pends on magnification µ and the slope of the galaxy
icantly,sincesourcesassociatedwiththelensesaregiven
number counts s: the ratio of counts with and with out
little weight. We find that C(R) − 1 falls roughly as lensing is N/N ∝ µ(2.5s−1) (Broadhurst et al. 1995).
R−0.9. The correctionfor clustered sources is essentially 0
The magnification is of the same order as the shear,
negligiblebeyond50kpc. Thesignalinfigure6hasbeen which for our lens sample is . 10−3, and the slope s
multiplied by C(R).
is typically about 0.4, which results in negligible magni-
We measure this correction factor separately for each
fication bias.
of the lens subsamples presented in later sections. Since
galaxy clustering increases with luminosity, and higher 5. THEGALAXY-MASSCORRELATIONFUNCTION
luminosity lens galaxies are seen out to higher redshift
5.1. Power-law Fits to ∆Σ
where more of the faint sources are near the lenses, the
correction factor C(R) increases with the luminosity of Asnotedin§2.1,apower-law∆Σ isconsistent,within
the lenses. The correction for our highest luminosity the errors, with the galaxy-mass correlation function
samplesis2.0at25kpccomparedto1.1forthefullsam- ξgm also being a power-law, ξgm = (r/r0)−γ, with
ple. Similarly, the correction for early type lens galaxies slope γ = 1 + α. Fitting for γ and r0, we find best
is larger, while the correction for late types is smaller fit marginalized values of γ = 1.76 ± 0.05 and r =
0
than for the full sample. Although the corrections for (5.7±0.7)(0.27/Ω )1/γh−1 Mpc. Because∆Σispropor-
m
tionaltoρ,wehaveassumedafiducialvalueforthemean
densityρ=ρ Ω determinedbyWMAP+ACBAR+
crit m
CBI in combinationwith the power spectrum from2dF-
GRS and Lyman α data, which yields Ω =0.27±0.02
0.1 m
(Spergel et al. 2003). Marginalizing over this small un-
certainty in Ω does not change our error estimates.
m
5.2. Inversion to ξgm
10−2 Inadditiontofittingapowerlawto∆Σ,weperformed
1
− a direct inversion of ∆Σ to ξ as outlined in §2.1. ξ
R) for the main sample is showngmin Figure 10. Again, singcme
C( our measurements scale linearly with Ω , we have as-
m
sumed a fiducial value of Ω =0.27.
10−3 m
In principle, the inversion from ∆Σ to ξ at a given
gm
radius r requires data for ∆Σ to R = ∞. In practice,
the contribution from scales beyond our data is negligi-
ble in all radial bins but the last few. We estimate the
contributionfromlargescalesby extrapolatingthe best-
10−4
fit power law. Because the last point would be based
0.1 1 10
R [h−1 Mpc] entirely on the extrapolation, we exclude it from figure
10. Thecontributionoftheextrapolationissmallforthe
Fig. 9.— Correctionfactorfortheclusteringofsourcesaround remaining points except for the last bin at 6.7 Mpc, for
thelens galaxies. Thefunction C(R)−1isessentiallyaweighted
which it is a 40% effect.
cross-correlationfunctionbetween lenses andsources. The∆Σ in
figure6hasbeenmultipliedbyC(R),whichisanegligiblecorrec- Because we do not have any a prioriknowledge of ξgm
tionforradiilargerthan∼50kpc. at large scales, we must include this extrapolation in