The APM cluster-galaxy cross-correlation function : Constraints on Omega and galaxy bias

17 Pages·1.2 MB·English

by  Rupert Croft

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2008 (MNLATEXstylefilev1.4)
The APM cluster-galaxy cross-correlation function :
Ω
Constraints on and galaxy bias.
Rupert A. C. Croft1, Gavin B. Dalton2 & George Efstathiou3
1Department of Astronomy, The Ohio State University, Columbus, Ohio 43210, USA.
2Department of Physics, Universityof Oxford, Keble Road, Oxford, OX1 3RH, UK.
3Institute of Astronomy, Universityof Cambridge, Madingley Road, Cambridge, CB3 0HA, UK.
1February2008
8
9
9 ABSTRACT
1 We estimate the cluster-galaxycross-correlationfunction(ξ ), fromthe APMgalaxy
cg
andgalaxycluster surveys.We obtainestimates both in realspace fromthe inversion
n
a ofprojectedstatistics andinredshift spaceusing the galaxyandcluster redshiftsam-
J ples. The amplitude of ξ is found to be almost independent of cluster richness. At
cg
7 large separations, r ∼> 5 h−1Mpc ( h = H0/100kms−1, where H0 is the Hubble con-
2 stant),ξ hasasimilarshapetothegalaxy-galaxyandcluster-clusterautocorrelation
cg
functions.ξ inredshiftspacecanberelatedtotherealspaceξ byconvolutionwith
cg cg
1 an appropriate velocity field model. Here we apply a spherical collapse model, which
v
we have tested againstN-body simulations, finding that it provides a surprisingly ac-
4
curatedescriptionoftheaveragedinfallvelocityofmatterintogalaxyclusters.Weuse
5 thismodeltoestimateβ (β=Ω0.6/bwherebisthelinearbiasparameter)andfindthat
2
ittendstooverestimatethetrueresultinsimulationsbyonly∼10−30%.Application
1
0 to the APM results yields β =0.43 with β <0.87 at 95%confidence. This measure is
8 complementaryto the estimatesmade ofthe density parameterfromlargerscalebulk
9 flows and from the virialised regions of clusters on smaller scales. We also compare
/ theAPMξ andgalaxyautocorrelationsdirectlytothe masscorrelationandcluster-
h cg
mass correlations in COBE normalised simulations of popular cosmological models
p
and derive two independent estimates of the galaxy biasing expected as a function of
-
o scale.This revealsthatbothlowdensity andcriticaldensitycolddarkmatter(CDM)
r models require anti-biasing by a factor ∼ 2 on scales r ≤ 2 h−1Mpc and that the
t
s Mixed Dark Matter (MDM) model is consistent with a constant biasing factor on all
a
scales.The criticaldensity CDM modelalso suffers fromthe usual deficit ofpower on
v: large scales (r ∼> 20 h−1Mpc). We use the velocity fields predicted from the different
i modelstodistorttheAPMrealspacecross-correlationfunction.Comparisonwiththe
X
APM redshift space ξ yields an estimate of the value of Ω0.6 needed in each model.
cg
r
We find that only the low Ω model is fully consistent with observations, with MDM
a
marginally excluded at the ∼2σ level.
Key words: Galaxies : Clustering ; Large-scale structure of the Universe ; Clusters
of galaxies; Cosmology.
1 INTRODUCTION The first measurements of galaxy-cluster correlations were
made by Seldner & Peebles (1977) who measured the an-
The spatial cluster-galaxy cross-correlation function ξ (r)
cg gular cross-correlations of Lick galaxies around Abell clus-
is defined so that the probability dP of finding a galaxy in
ters. Their results suggested that ξ (r) was positive and
thevolume element dVat a distance r from a cluster is cg
significantly different from zero out to large spatial separa-
tions of r 100 h−1Mpc. A more recent analysis by Lilje
dP =n[1+ξ (r)]dV, (1)
cg ∼
& Efstathiou (1988) used cluster redshifts to determine the
where n is the mean space density of galaxies. ξ (r) is cross-correlation between Abell clusters and Lick galaxies
cg
therefore equivalent to the radially averaged overdensity as a function of metric separation (w(σ), see Section 2.2).
profile of galaxies centred on a typical cluster of galaxies. They found no convincing evidence for clustering on scales
(cid:13)c 0000RAS
2 R.A.C. Croft, G.B. Dalton & G. Efstathiou
r>20h−1Mpc and concluded that some of the signal seen dimensionalinformation,usingtheparentAPMgalaxysur-
by∼Selder and Peebles was due to artificial gradients in the vey and the cluster redshift survey (Section 2.2). We will
Lickcatalogue.However,LiljeandEfstathiou didfindsome also measure the 3 dimensional clustering directly from the
evidence for more large-scale power than predicted by the APM-Stromloredshiftsurveyandtheclusterredshiftsurvey
‘standard’(i.e.Ω=1,h=0.5,scale-invariant)CDMmodel. (Section 2.3).
Onscalessmallerthanr<10h−1Mpc,thereal-spaceξ (r)
cg
recovered by inverting th∼e observed form of w(σ) is well fit
by a power law:
2.2 The projected cross-correlation function
r γ
ξ (r)= 0 , (2)
cg (cid:16) r (cid:17) Thesimpleststatisticwhichcanbeusedtoconstrainξcg(r)
is the angular cross-correlation function, w (θ). However,
with γ =2.2 and r =8.8h−1Mpc. cg
0 the inversion of this quantity to find ξ (r) tends to be
Direct measurements of the spatial cross-correlation cg
ratherunstable.SinceweareusingtheAPMclusterredshift
functionfromredshiftsurveysofgalaxiesandclusters(Dal-
survey as our cluster sample we can make use of the clus-
ton, 1992, Efstathiou 1993, Mo, Peacock and Xia ,1993,
ter redshift information to determine the projected cross-
Moore et al. 1994) have confirmed that ξ has a similar
cg correlationfunction,w (σ),asdefinedbyLilje&Efstathiou
shape to the galaxy-galaxy and cluster-cluster correlation cg
(1988),whereσ=czθ/H isthemetricseparationofaclus-
functions, butwith an amplitude roughly equalto theirge- 0
ter with redshift z and a galaxy at angular distance θ from
ometricmean.TheanalysesofDalton(1992)andEfstathiou
the cluster centre. As our estimator for w (σ) we use the
(1993) wereperformedusingtheoriginal APMclustersam- cg
standard estimator,
pleofDaltonetal.(1992)andtheStromlo-APMgalaxiesof
Loveday et al. (1992). N N (σ)
w (σ)= ran CG 1, (3)
In this paper we use compute ξcg using the Stromlo- cg Ngal NCR(σ) −
APM galaxy redshift survey and thesample of 364 clusters
where CG and CR denote cluster–galaxy and cluster–
of Dalton et al. (1994b). We investigate its behaviour as a
randompairs,respectively.Weaccountforthewindowfunc-
function of cluster richness and compare it with the pre-
tionoftheAPMGalaxySurveyonaplate-by-platebasisby
dictions of popular comological models. We also calculate
generatingarandomcatalogueforasingleplatewith100000
ξ (σ,π), the cross-correlation as a function of separation
cg
randompointsandexcisingtheregionsmaskedfromthesur-
alongandperpendiculartothelineofsight.Thepeculiarve-
veyasthecatalogueisusedforeachplateinturn.Withthis
locity field around clusters influences the shape of ξ (σ,π)
cg
method, we can use individual galaxy positions from the
andisexpectedtodependonΩ,andthemasstolightratio
survey data rather than binned cell counts and so we can
around clusters, so that we can extract some information
measure w (σ) accurately at small scales. On larger scales
about the density parameter and galaxy biasing from our cg
wehavecheckedthatthismethoddoesnotintroducelarge-
measurements.Wealsocarryoutsomespecificcomparisons
scale power into our determination of w (σ) by comparing
withN-bodysimulations(soincludingnon-lineareffects)to cg
withtheresultsobtainedusingcellcountsforgalaxiesanda
investigate how the biasing of galaxies is expected to vary
singlerandomcatalogueforthewholesurvey(Dalton1992).
as a function of scale.
A similar method has been used more recently in the anal-
ysisoftheDurham–UKSTgalaxy redshift survey(Ratcliffe
et al. , 1997).
2 CLUSTER-GALAXY CORRELATIONS IN
Thedataforw (σ)areshowninFigure1fortheclus-
THE APM SURVEY cg
ter sample cross-correlated with all galaxies to three differ-
2.1 The APM data samples entmagnitudelimits.Theerrorbarsshownareobtainedby
dividingtheAPMsurveyregionintofourquadrantsandde-
In this paper we will use three different data samples, the
termining the error on the mean from the scatter between
APM angular galaxy catalogue, the APM-Stromlo galaxy
the four zones. Given the estimate of the depth of the Lick
redshift survey, and the APM cluster redshift survey. The
catalogue obtained byMaddox et al. (1990a), we would ex-
APM galaxy survey (Maddox et al. 1990a, Maddox et al.
pectthepointsform =18.5tocorrespondtotheresults
lim
1990b, Maddox, Efstathiou and Sutherland 1996) consists
for w (σ) obtained by Lilje & Efstathiou (1988). A com-
cg
of angular positions and other information, but not red-
parison of Figure 1 to Figure 9c of that paper reveals good
shifts, forover2million galaxies with ab magnitudelimit
J agreementovertherangeofσforwhichw canbemeasured
cg
of 20.5. The Stromlo-APM redshift survey (Loveday 1990,
reliably.
Loveday et al. 1992a, Loveday et al. 1992b) is a survey of
Theinversionofprojectedclusteringinformationinthe
1787galaxiesrandomlysampledatarateof1in20fromall
form of w (σ) to the three-dimensional statistic ξ (r) in-
cg cg
APMgalaxieswithb brighterthan17.15.Toconstructthe
J volves a weighted summation of the w (σ) points (Saun-
cg
APM cluster sample, an automated procedure was used to
ders, Rowan-Robinson & Lawrence, 1992):
select clusters from the angular APM survey (Dalton 1992,
Dalton et al. 1997). Cluster redshifts were then measured, ξ (r)= −1 w(σj+1)−w(σj)ln σj+1+ σj2+1−σi2 (4),
aclnudstuerssed(Dtoalctoonnsetrtuaclt.a1n99o2r)igainndalarnedesxhtieftnsciaotnaltoogu3e64ofcl1u9s0- cg πB Pj≥i σj+1−σj (cid:18) σj+pσj2−σi2 (cid:19)
p
ters (Dalton et al. 1994). It is the latter cluster catalogue where r = σ . The factor B in Equation 4 accounts for
i
which we will use in this paper. In our analyses we will use the difference in the selection functions of the clusters and
angular clustering and its inversion to obtain real space 3 galaxies and is definedas follows (Lilje & Efstathiou 1988):
(cid:13)c 0000RAS,MNRAS000,000–000
APM cluster-galaxy correlations 3
ψ(y )
B= i i . (5)
(1/yP) ∞ψ(x)x2dx
i i 0
P R
Hereψistheselectionfunctionofthegalaxysurveyandy is
i
theredshiftofclusteri.Theselectionfunctionψ wasevalu-
atedusingthetheluminosity function parametersobtained
from the Stromlo-APM survey by Loveday et al. (1992).
Again we show error bars obtained by inverting the w (σ)
cg
estimatesfromfourquadrantsofthesurveyforeachmagni-
tuderange.Thedatashowexcellentagreementbetweenthe
three different magnitude limited galaxy samples used, but
are not well represented bya single power law.
For the velocity field analysis in Section 4, we will use
a fit to the real space APM cross-correlation function. We
choose to fit an arbitrary function which is the sum of an
exponentiallytruncatedpowerlawandthelineartheorycor-
relationfunctionshapeofascale-invariantCDMmodelwith
Γ = Ωh = 0.2 (denoted as ξ (r) below) normalised so
Γ=0.2
⋆
that σ =1 :
8
r γ
ξ (r)= c eβreη/r+α ξ (r). (6)
cg r Γ=0.2
(cid:16) (cid:17)
Figure 1. The projected cluster-galaxy cross-correlation func-
tionforAPM galaxies and APMclusters. Results areshownus- Theparametercombinationrc =11.7,γ =2.6,η=0.6,
ing APM galaxies with different magnitude limits as indicated β = 1.7 and α = 1.8 gives a reasonable fit to the data
in the figure. The error bars are determined from the scatter in for all magnitude bins and is plotted as a solid line on
theresultsderivedfromfournearlyequalareazonesoftheAPM Figure 2. A fit is necessary because the noise level in the
survey. realspacecross-correlationfunctionbecomesratherlargefor
r > 10 h−1Mpc. The shape of the fit on these large scales
is∼motivated by theshape of theredshift space ξ (Section
cg
2.2).Wealsoplotonthesamefigurethepower-lawfit(given
by Equation 2) which Lilje and Efstathiou (1988) find is a
goodapproximationtotherealspacecross-correlationfunc-
tion derived from Abell clusters and Lick counts. We can
see that on large scales, thereis no evidencefor any signifi-
cantexcessofpoweroverthisfit.TheAPMresultstherefore
supporttheconclusionsofLiljeandEfstathiou (1988) sum-
marized in Section 1. Our APM results are in agreement
with the cross-correlation of APM clusters and Edinburgh-
Durham Sky Survey galaxies carried out by Merch´an et al.
(1997),althoughtheirestimatederrorsarelarge.Itisuseful
to note that on small scales, r < 2h−1Mpc, our error bars
on ξ (r) are very small, so th∼at we will be able to draw
cg
some interesting conclusions about galaxy biasing on small
scales from a comparison with theoretical models (Section
5).
2.3 The redshift space cross-correlation function
Figure 2. The real space cluster-galaxy cross-correlation func- We estimate ξcg(s), where s represents the separation of
tion for APM galaxies and APM clusters. Results areshown for cluster-galaxy pairs in redshift space, using the APM sam-
different magnitude limits, and the error bars have been calcu- ple of 364 clusters of Dalton et al. (1994b) and the 2000
∼
lated as described in the caption for Figure 1. For clarity, the galaxieswithredshiftsfromtheAPM-Stromlobrightgalaxy
mlim =18.5and mlim =20.5points have been slightlyoffset in redshift survey (Loveday et al. 1992a). The calculation of
the r direction. We also show a fit to the data with form given ξ (s)differsfromtheevaluationofthecluster-clustercorre-
cg
byEquation6andthepowerlaw(Equation2)whichLilje&Efs- lation function(Daltonetal.1994b) asthegalaxy selection
tathioufindisagoodfittothecross-correlationofAbellclusters
functionfallsverysteeplywithincreasingdistance.Weights
andLickcounts.
must be therefore be applied to the galaxy-cluster pairs to
⋆ Where σ8 denotes the rmsamplitudeofthe massfluctuations
inspheresofradius8h−1Mpc.
(cid:13)c 0000RAS,MNRAS000,000–000
4 R.A.C. Croft, G.B. Dalton & G. Efstathiou
recoverthemimimumvarianceestimateofξ (s).Thisop-
cg
timalweighting(atleast onscalesforwhichξ (s) 1)can
cg
≤
beshown to be (see Efstathiou 1988, Loveday 1990)
w 1/[1+4πn(r )Jcg(s )], (7)
ij ≃ i 3 ij
where
sij
Jcg(s )= ξ (x)x2dx, (8)
3 ij Z cg
0
r is the distance from the observer to galaxy i and s the
i ij
separation of galaxy i and cluster j. To use the formula
we must predict roughly what Jcg(s) will be - here we use
3
the weighting function resulting from a linear theory CDM
powerspectrumwithΓ=0.2andanamplitudetwicethatof
ξ (s) measured for APM-Stromlo galaxies (Loveday et al.
gg
1992). After cross-correlating the two catalogues to find all
galaxy-cluster pairs, we cross-correlate the clusters with a
catalogue of 100000 random points. This random catalogue
hasthesameboundariesandselectionfunctionasthegalaxy
Figure 3. The spatial cluster-galaxy cross-correlation function
sample. Wethen use thestandard estimator to find ξcg(s): for APM-Stromlo galaxies and samples of APM clusters with
different lower richness limits. All cluster subsamples are drawn
N N (s)
ξ (s)= ran CG 1, (9) from the sample of 364 clusters (APM R ≥ 50), so that 243,
cg Ngal NCR(s) − 113 and 60 clusters are used in the calculation of the curves for
where N (s) and N (s) are the galaxy-cluster and R≥60, R≥70 and R≥80 clusters respectively. The subsam-
CG CR pleshavemeaninterclusterseparations of36h−1Mpc(R≥60),
cluster-random pairs in the bin interval centred on s and
48 h−1Mpc ( R ≥ 70) and 59 h−1Mpc ( R ≥ 80). The solid
each are weighted using w from Equation 7.
ij lineisafit(withformgivenbyEquation6)totherealspaceξcg
Resultsforthe364 =50clustersare shownas trian-
R resultsfortheAPMR≥50clusters.
glesinFig.3.Itcanbeseenthatthethereissomecurvature
in the plot, with a definite break at s 30 50 h−1Mpc.
The curve crosses ξ (s) = 1 at roughl∼y 9 h−−1Mpc, which change in the shape or amplitude of ξ (s), given the er-
cg cg
is intermediate between the behaviour of the galaxy auto- rors. Our different samples have space densities n =3.5
c
correlationfunction,ξ (s)(s 5h−1Mpc,inthenotation 10−5 h3Mpc−3 ( = 50), 2.2 10−5 h3Mpc−3 ( = 60×),
gg 0
of Equation 2, Loveday et al.≃1992) and the cluster auto- 9.3 10−6 h3MpRc−3 ( = 70×) and 4.7 10−6 Rh3Mpc−3
correlation function ξ (s) (s 14 h−1Mpc, Dalton et al. ( =×80).WemightexpRectthecross-corre×lation functionof
cc 0
≃ R
1994). Theerrorbars,calculated from Poisson statisics and richerclusters tohaveahigheramplitude,at least on small
thenumberofcluster-galaxypairsineachbin,arerelatively scales,asclusterrichnessshouldberelatedtoξ withinthe
cg
small, indicating that ξ (s) will be an interesting statistic cluster selection radius. This was discussed by Seldner and
cg
to compare with theoretical models. We can compare the Peebles (1977), who found that the effect was smaller than
results for =50 clusters with the solid line in this figure, expected for Abell clusters. This is probably because the
R
which is the fit to the real space cross-correlation function distance indicator used by Abell depends on cluster rich-
(Equation 6). For the moment, we will note that ξ (s) is ness; for clusters at a given apparent distance, the richer
cg
marginally higher than ξ (r) on large scales, boosted by objects would actually befurtheraway,thusdepressing the
cg
streamingmotions(Kaiser1987)andsmallerforseparations amplitude of the angular cross-correlation function. These
less than 4h−1Mpcduetotheeffect oftheclusterveloc- problems should not affect the APM cluster sample as it
∼
itydispersion.Uncertaintiesintheclusterredshiftsprobably has been designed so that cluster richness does not affect
play a part in depressing the amplitude of ξ (s) on small apparent distance (Dalton et al. 1997). However, the situa-
cg
scales, as for many cluster we have redshifts for only 2 or tion here is complicated by the fact that we are measuring
3 galaxies (each with their own measurement errors). The redshift space clustering. Rich clusters will have their clus-
error in the cluster centre of mass velocity could, therefore, tering signal smeared out due to their high velocity disper-
be as much as a few hundredkms−1. Dalton et al. (1994a) sions,sothattheamplitudeofclusteringwillbemoreheav-
have compared results for APM clusters with many (>10) ily depressed on small scales than for poorer clusters. The
measured galaxy redshifts to the redshifts of the brightest underlyingsituationonsmallscalesisthereforenotentirely
galaxyineachcluster.Thermsscatterbetweenthetwoval- obvious. On larger scales (r > 1 h−1Mpc) though, our re-
ues is 512 kms−1, which should be higher than the error sults show that the amplitud∼e of ξ g(s) really does depend
c
on our cluster redshifts as we use 2 galaxy redshifts per only very weakly on cluster richness. We will show in the
≥
cluster. next section that this is consistent with model predictions.
Mooreetal.(1994)havefoundthatξ forIRASgalax- Wehavecalculated ξ fortheAPMsamplein redshift
cg cg
ies and Abell clusters is insensitive to cluster richness. We spaceasafunctionofpairseparation alongthelineofsight
haverepeated this typeof analysis using our APMsamples (π)andperpendiculartothelineofsight(σ).Theeffectsof
and the results are plotted in Figure 3. We have increased peculiarvelocities, which distort thepairseparations in the
the lower richness cutoff from = 50 (the full sample) up π direction areevidentintheresultsfor oursampleplotted
R
to = 80, but as the results show there is no detectable in Figure 4. We can see elongation present on small scales
R
(cid:13)c 0000RAS,MNRAS000,000–000
APM cluster-galaxy correlations 5
ous papers to study the cluster-cluster correlation function
(Croft & Efstathiou 1994, Dalton et al. 1994b). The sim-
ulations are of three different spatially flat universes, two
CDM models and one Mixed Dark Matter (MDM) model.
Onesetofsimulationsisof“standard”CDM(SCDM),with
Ω = 1,h = 0.5 and the other is of low density CDM
0
(LCDM) with Ω = 0.2,h=1 and a cosmological constant
0
Λ, where Λ=0.8 3H2. The power spectra for the SCDM
× 0
and LCDM models are as given Efstathiou, Bond & White
(1992). For the MDM model, we used the form given in
Klypin et al. (1993) with Ω = 1, h = 0.5, and a massive
0
neutrinocomponentcontributingΩ =0.3.Weassumescale
ν
invariant primordial fluctuationsfor all models.
Each simulation contains 106 particles in a box of co-
moving side-length 30000kms−1 and was run using a P3M
N-bodycode(Efstathiou et al.1985). Weuse5realisations
of each model with different random phases.
InthisSectionweusesimulatedclustercataloguescon-
structedfromtheN-bodysimulationstohavethesamemean
separationastheAPMsample(d =30h−1Mpc).Theclus-
Figure 4. The APM cluster-galaxy cross-correlation function c
tersareidentifiedfromsimulationsusingafriends-of-friends
ξcg(σ,π)(calculatedusingthefullsampleof364clusters)shown
algorithm to select candididate centres. The mass enclosed
as a function of pair separations perpendicular to the line of
within a fixed radius (in this case 0.5 h−1Mpc) of the cen-
sight (σ) and along the line of sight (π). Contour levels are at
ξcg =4,3,2,1,0.8,0.6,0.4,0.2,0.1,0.,−0.05.Thecontourlevelat tre of mass is computed and the clusters are ordered by
ξcg =1isshownbytheheavyline;negativecontoursareplotted mass. Finally, a mass limit is applied to generate a cluster
as dashed lines. ξcg(σ,π) was calculated using 16 bins in σ and catalogue of a specified mean space density. The procedure
π inthe interval 0−40h−1Mpc.For clarity,before plottingthe is described in more detail in Croft & Efstathiou (1994).
resultsinthisfiguretheyweresmoothedusingamovingwindow We note here that as long as the clusters are defined to be
average(3×3bins). collapsed objects, the set of objects identified in the simu-
lations isinsensitive totheselection criterion. Forexample,
Gaztan˜aga, Croft & Dalton (1995) identify clusters as high
which is caused by random galaxy velocities and redshift
peaks in the density field smoothed with small filters and
measurement errors. On larger scales, we can see a break
recover essentially the same catalogue of clusters (for a va-
in the contours around σ 6 h−1Mpc,π 10 h−1Mpc
≃ ≃ riety of filter sizes) as our percolation algorithm. We can
which could be caused by the infall region around theclus-
also reasonably expect the positions and mass rankings of
ter. This coherent infall should cause compression of the
galaxy clusters in our simulations to be insensitive to the
contoursalongtheσaxisonlargerscales(Kaiser1987,Lilje
details of the galaxy formation process, which in the real
& Efstathiou 1989), but there does not seem to be obvious
Universe would turn a large agglomeration of dark matter
evidenceof this in Figure 4.
into a galaxy cluster. These properties of galaxy clusters
The velocity field around clusters should be dependent
make them especially useful for elucidating the nature of
onthemassdistribution,andthereforeonthevalueofΩ.If
galaxybiasingandgalaxypeculiarvelocities.Inessencethey
wehaveasimplemodelforhowthetwomaineffectspresent
consist of a set of fixed reference points around which the
in velocity field arise (small scale dispersion and large scale
galaxy overdensity profiles and galaxy velocity profiles can
infall) we should be able to use distortions in ξ (σ,π) to
cg becompared directly with observations.
derive information on Ω and the amplitude of mass fluctu-
Wedonotattempttocarryoutanysortofdirectiden-
ations. To do this, we need to know ξ in real space and
cg
tification of galaxies in the simulations. Instead, we calcu-
to have a model which describes the behaviour of galaxy
latethecluster-particlecorrrelationfunction,ξ .TheAPM
velocities around clusters. We apply both of these to our cρ
ξ observations can be compared toξ to determine what
ξ (σ,π) data from the APM survey in Section 4.3 below. cg cρ
cg
sort of galaxy biasing is needed in each model. According
We will first examine the predictions of theoretical models
to linear perturbation theory, the amplitude of σ grows in
usingN-bodysimulations and usethem todevelopa model 8
proportion to the growth rate D(t) of linear density per-
of the velocity field around clusters.
turbations(seePeebles1980, Section10). Theamplitudeof
the two-point correlation function of the mass fluctuations
thus grows as σ2 D2(t) on scales on which linear the-
3 ξCG FROM SIMULATIONS OF ory is applicable8. O∝n the other hand, the rich cluster two-
COSMOLOGICAL MODELS pointcorrelationfunction,isalmostindependentoftime(see
Croft & Efstathiou 1994) because clusters are rare objects
3.1 Clusters and galaxies in simulations
and are strongly biased compared to the mass distribution.
Our predictions of the form of the cluster-galaxy corre- We therefore expect the linear theory growth rate of ξ
cρ
lation function in cosmological models come from study- to lie between these two cases, and to be proportional to
ing N-body simulations. We use catalogues of clusters con- σ , as in the high-peak model of Bardeen et al. 1986. This
8
structed from simulations which have been used in previ- means that the choice of output time (and hence normal-
(cid:13)c 0000RAS,MNRAS000,000–000
6 R.A.C. Croft, G.B. Dalton & G. Efstathiou
Figure 5. The cluster-mass cross-correlation function for simulations of (a) Standard CDM, (b) Low Density CDM and (c) Mixed
DarkMatter (see text forthe parameters of these models). Results areplotted inreal space (solidlines) andinredshift space (dashed
lines). All models are normalised to be consistent with the amplitude of fluctuations measured by COBE (ignoring tensor modes), so
that σ8 =1.0 for panels (a) and (b) and σ8 =0.67 in panel (c). The model curves inpanel (c) have been scaled upwards by a biasing
factor of 1.5inorder to makethe model consistent withthe amplitude of galaxy fluctuations. Alsoplotted ineach panel (as triangles)
isthecross-correlationfunctionofAPMStromlogalaxiesandtheAPMclustersampledescribedinSection2.3.Theerrorbarsonthese
pointshavebeencalculatedusingPoissonstatistics.Thedottedlinesshowafit(Equation6)totherealspacecluster-galaxycorrelation
function,consistentwithmeasurementsfromtheAPMsurveyandalsowiththecross-correlationofAbellclustersandLickcounts.
isation of the model) will make some difference to our re- real and redshift space. On smaller scales, MDM appears
sults.Thenormalisation wehaveusedisconsistentwiththe to give a reasonable fit to the shape of both the real space
amplitude of fluctuations inferred from COBE microwave results (Equation 6) and thosein redshift space. Weshould
background temperature anisotropies (Wright et al. 1994) becautiousaboutdrawingfirmconclusionsfromthis,asthe
so that σ = 1.0 for SCDM and LCDM and σ = 0.67 for MDMmodelhasbeensimulatedwithoutincludingthether-
8 8
MDM.Ifweassumethatσ forAPMgalaxiesiscloseto1.0 mal velocities of massive neutrinos and this may affect the
8
(Gaztan˜aga 1995) and density evolution on this scale has density profiles and internal structure of the clusters. The
been linear, then ξ for SCDM and LCDM should be very linearlybiasedMDMmodeldoesalsoseemtohavearather
cρ
nearly equaltoξ .Inthecase ofMDM, when plottingour high amplitude on intermediate scales, particularly in red-
cg
resultswemerelyscalethecurveupwardsbyaconstantlin- shift space(apoint that wewill returntoin Sections4 and
earbiasingfactor,1.0/σ .Adiscussionofmorecomplicated 5). In any case, it is possible that the efficiency of galaxy
8
biasing, including variations with spatial scale which might formation is different near clusters leading to scale depen-
berequired in some models, is deferred to Section 5. dent biasing. We will investigate this possibility in Section
5.
The richness dependence of ξ in the models is shown
cg
inFigure6,whereweplotξ (s)(inredshiftspace)forsim-
3.2 Results cg
ulatedclusterswithasimilarrangeofspacedensitiestothe
Our results for these three models are shown in Figure 5, 4differentAPMsamplesshowninFigure3.Thereisavery
both in real space (solid lines) and redshift space (dashed weak richness dependencein theamplitude of of ξ , which
cg
lines).Concentratingontherealspaceresultsfirst,itcanbe becomesevenweakeronlargescales(s 25h−1Mpc).This
≥
seen that ξcg(r) exhibits a sudden change of slope on small weak dependence of ξcg with cluster richness is compatible
scales in all cases. The SCDM and LCDM models have the with the observational results for the APM samples pre-
steepestslopesforthispartofthecurve,whichwemightex- sentedinFigure3,whichshownosignificant dependenceof
pect, given that they havethe most small scale power. The ξ with cluster richness.
cg
redshift space results show the usual depression on small
scales and amplification on large scales, both effects being
largest in thecase ofthetwoΩ=1models, which havethe
largest particle velocities. As far as a comparison with the
APMξ (r)isconcerned,wecanseethattheusualdiscrep-
cg
ancy on large scales with SCDM is evident, but that the 4 REDSHIFT SPACE DISTORTIONS AND
shape on these scales is consistent with LCDM. Indeed for ξ (σ,π)
CG
r > 3 h−1Mpc LCDM appears to give a good fit in both
(cid:13)c 0000RAS,MNRAS000,000–000
APM cluster-galaxy correlations 7
Figure6.Thecluster-masscross-correlationfunctioninredshiftspaceforclusterswithdifferentlowermasslimitsinsimulationsof(a)
Standard CDM, (b) Low density CDM and (c) Mixed Dark Matter. The curves are labelled with the mean intercluster separation of
eachsample.
4.1 The spherical infall model expression for vnon−lin(r) with an exponential e−δ(r)/δcut,
infall
with δ = 50. (See the comparisons with N-body simula-
Thereexistsaregimebetweenstreamingofgalaxiesonlarge- cut
tions in the next subsection for a justification of this value
scales and the virialised region of clusters which is impor-
of δ .)
tant in modelling observations of ξ (σ,π). As the density cut
cg Onefurtheringredientinourvelocitymodelistheran-
enhancement within a few h−1Mpc of clusters is greater
domvelocitydispersion aboutthesmooth infalling flow.To
thanunity,lineartheoryisnot expectedtodescribetheve-
makethingsassimpleaspossible,weassumeavelocitydis-
locity field accurately. However, if we assume that clusters
persionindependentofdistancefromthegalaxytotheclus-
are spherically symmetric, a solution for thenon-linear col-
ter,andindependentofdirection(whethertransversetothe
lapseofthesystemcanbefoundwhichisexactbeforeorbit
line between galaxy and cluster, or parallel to it, for exam-
crossing takes place. The solution is obtained by treating a
ple). We also assume that the velocities are drawn from a
proto-cluster with a top-hat profile as if it were an isolated
Gaussiandistribution.Theone-dimensionalvelocitydisper-
FriedmannuniversewithitsownvalueofΩ (seee.g.Reg¨os
0 sion will therefore beparametrised byone number, σ .
& Geller 1989 for details). Here, we use a good approxima- v
tion to theexact solution dueto Yahil(1985) (also used by
Lilje&Efstathiou1989),whogivesthefollowingexpression:
4.2 Direct tests of spherical infall on the velocity
1 δ(r) field in simulations.
vnon−lin(r)= Ω0.6H r , (10)
infall −3 0 0 (1+δ(r))0.25
In our analysis we need to assume that the velocity field
where δ(r) is the overdensityinside radius r, predicted from the average cluster density profile is equiv-
alent to the average cluster velocity field. This is a non-
3Jcρ(r) r
δ(r)= 3 , Jcρ(r)= ξ (x)x2dx. (11) trivial assumption and must be tested in some way (in our
r3 3 Z0 cρ casewewillusesimulations)beforewecanmakeanyclaims
for the reliablity of our results. The spherical infall model
By way of comparison, the linear theory prediction of the
has previously been used in many studies of the infall re-
infall velocity is
gions around individual rich clusters, particularily Virgo
1
vlin (r)= Ω0.6H rδ(r). (12) (e.g. Yahil, Sandage & Tammann 1980, Yahil 1985). How-
infall −3 0 0 ever,theassumptionofsphericalsymmetryforanyindivid-
Galaxysurveysprovidewithinformationontheoverdensity ualcluster is difficult tojustify empirically. In contrast, the
ofgalaxiesinsideradiusrandnotofthemass.Someassump- averageclusterprofilemeasuredbyξ (r)possessesspherical
cg
tions about the relationship between galaxies and mass are symmetry by construction. Here we test the validity of our
thereforerequiredtoinferavalueofΩ fromameasurement dynamical approximations by comparing our predicted ve-
0
of v (r). Here we use the simple linear biasing picture, locityfieldsdirectlywiththevelocityfieldsmeasuredaround
infall
so that Jcρ(r) = Jcg(r)/b. We therefore parametrise our N-bodyclusters.Wepresentresultsbothforthemeaninfall
3 3
infallmodelusingβ =Ω0.6/b.Aseventhenon-linearveloc- velocityasafunctionofradiusandthedispersionaboutthe
0
ity model is not expected to hold in the cores of clusters, mean infall.
near and in the virialised region, we choose to truncate the The average infall velocities of particles, are plotted as
(cid:13)c 0000RAS,MNRAS000,000–000
8 R.A.C. Croft, G.B. Dalton & G. Efstathiou
Figure7.Themeaninfallvelocityofparticlesaroundsimulatedclustersasafunctionofradius.Thefilledcirclesineachpanelcorrespond
totheinfallvelocitiesfoundinN-bodysimulationsof(a)SCDM,(b)LCDMand(c)MDM.Theerrorbarsshowtheerroronthemean
calculated fromthe scatter inanensembleof5realisations.Ineachpanel wehavealsoplottedthe predictions oflineartheory (dashed
lines)andanon-linearsphericalinfallmodel(solidlines),asdescribedinthetext.Thetwopairsofdashedandsolidlinesshowtheeffect
ofincludingandnotincludinganexponential truncationathighoverdensities asdescribedinSection4.1.
Figure 8. The one-dimensional velocity dispersion of particles around simulated clusters as a function of radius. The filled circles in
eachpanelcorrespondtothedispersioninthecomponentofrelativevelocityalongthelinefromparticletocluster(ie.dispersionabout
the mean streaming motionplotted inFigure7). The open circles show the one-dimensional dispersioninthe transverse direction. We
plotthe samemodelsas inprevious diagrams:(a) SCDM,(b)LCDM and(c) MDM.Theerrorbarsagain show theerroronthe mean
calculatedfromthescatter inanensembleof5realisations.
a function of distance from the cluster centre in Figure 7 region r > 2.5 h−1Mpc. Evidently, the spherical non-linear
(filledcircles).Wealsoplotthepredictionsofsphericalinfall model provides a much better match to the N-body results
(solid lines) and linear theory (dashed lines). The rapid de- thanthelinear.Overmuchoftherangeofdensitycontrasts
creaseofinfallvelocityatsmallrinthesimulationsisdueto relevant to our study of the infall region, thenon-linear ve-
particles reaching theboundaryof thevirialised region. We locity model underpredicts the infall by roughly constant
have roughly approximated this effect in our velocity mod- factor, but this is only 30% for LCDM and 10 20%
els by the exponential truncation described in Section 4.1. for SCDM and MDM. A∼t larger radii, r 20∼h−1M−pc we
≥
Thepairsofdashedandsolidcurvesshowthevelocitymod- see that both linear and non-linear models begin to match
elswithandwithouttheexponentialtruncation.Becauseof the N-body results even more closely. We therefore expect
the somewhat arbitrary nature of this truncation, we will that when we come to estimate β from distortions in the
restrict our quantitative use of the velocity model to the
(cid:13)c 0000RAS,MNRAS000,000–000
APM cluster-galaxy correlations 9
cross-correlation function our results will be fairly close to ingthatthesemodelsdohavearelativelyhighamplitudeof
thetrue value. mass fluctuations, being normalized to COBE rather than
The underprediction of the spherical infall model is in togivethecorrectabundanceofgalaxyclusters(White,Ef-
agreement with results found by Van Haarlem (1992) and stathiou & Frenk 1993). It is possible that lower amplitude
Diaferio and Geller (1996). As the velocity field in the im- models would conform betterto a smooth infall picture.
mediate vicinity of galaxy clusters has the potential to be Obvious signatures of radial infall, such as overdensity
complicated, there have been many mechanisms suggested “caustics”attheradiusofturnaroundfromtheHubbleflow
that might disrupt simple spherical infall. Amongthese are (Reg¨os&Geller1989)havebeendifficulttofindinobserva-
thepresenceof shear which should speed upcollapse (Hoff- tions of real clusters and in N-body simulations (van Haar-
man 1986, Lilje & Lahav 1991), which does not appear to lem et al. 1993). This does not seem surprising given the
be happening here, and formation of substructure and/or complexityevidentintheseplots.Thisisconsistentwiththe
ellipticity of the proto-cluster which slows infall (e.g. van work of Colberg et al. (1997) who have shown that matter
Haarlem & van de Weygaert 1993). The study of clusters infallingontoindividualclustersinsimulationsdoessofrom
in the SCDM scenario carried out by Villumsen & Davis specific directions which are correlated with the surround-
(1986) differs from ours in that they analysed the velocity ing large scale structure. It is possible that modelling the
field around individual clusters. Here we are dealing with velocity and density fields as triaxial ellipsoids (van Haar-
theaveraged velocity around all clusters. lem&vandeVeygaert1993)mayhelpintheunderstanding
Figure 8 is a plot of the1-dimensional relative velocity oftheflow.Inanycasetheaccuracyofoursphericallyaver-
dispersion as a function of distance from particle to cluster agedapproximationisgoodenoughtopermitaquantitative
in our three cosmological models. The filled symbols show analysis of theredshift space distortions.
the dispersion about the mean infall motion along the line
from particle to cluster and the open circles show the one-
dimensionaldispersioninthecomponentofrelativevelocity 4.3 Estimates of β and σv
transverse to this line. It can be seen that our assumption
Wearenowinthepositiontomakesomepredictionsofhow
thatσ isindependentofdistanceanddirectionturnsoutto
v therealspaceξ willbedistortedasafunctionofβandσ .
cg v
besurprisinglygoodTheplotsalsoshowthatthetransverse
Todothis,weshallconvolveξ (r)withourvelocitymodel,
cg
σ isslightlyhigherthantheradialvalue,intheinfallregion
v so that ξ in redshift space, ξ (σ,π), is given by (Peebles
between r 2 h−1Mpc and r 10 h−1Mpc. This might cg cg
∼ ∼ 1993, Section 20)
be the signature of some sort of “previrialisation” occuring
(Davis& Peebles 1977, Peebles 1993). ξcg(σ,π)=(√2πσv)−1 −∞∞[exp(v−[(vinnofna−lllin(r)y)/σv])2
Toillustratethecomplexityofthevelocityfieldaround (1+ξ R(r)) exp( v2/σ )2]dv, (13)
cg v
thesimulated clusters,andtoseeifwecanunderstandwhy × − −
some scenarios fit thespherical infall pictureslightly better where r = (σ2+π2) and y = (r2 σ2). To model the
−
thanothers,wehaveplottedtheparticlevelocitiesarounda observed esptimates of ξcg(σ,π), pwe model ξcg(r) in Equa-
sampleofindividualclusters.InFigures9to11weshowthe tion13withthefittingfunctionofEquation6whichiscon-
x and y components of thesmoothed peculiar velocity field sistent with the real space cross-correlation function of the
(a2h−1MpcGaussianfilterwasused)around9clustersfor APM sample (Section 2.2). We also test Equation 13 using
each model, in the rest frame of the cluster (plotted at the the N-body simulations and empirical estimates of ξ (r)
cg
centreofeachpanel).Wehavealsoplottedthedensityfield calculated in real space (plotted in Fig. 5).
smoothed with the same filteras a grayscale. Wecan reach To place constraints on β and σ we calculate the ex-
v
several conclusions from studyingtheseplots: pected ξ (σ,π) for a closely spaced grid of values, over the
cg
ThemagnitudeofthevelocityfieldaroundMDMclusters rangeβ=0 1.5andσ =0 1500kms−1.ξ (σ,π)iscal-
v cg
• − −
(andtoalesserextentSCDM)isanisotropic-therearesev- culated at 16 16 points, spanning the range of values for
×
eral clusters with small arrows on one side and large ones π andσ plotted in Figure4.This isdonebyconvolvingthe
on theother. realspaceξ (r)withourvelocitymodelusingEquation13.
cg
The larger coherence in the velocity field of the MDM In Figure 12 we show contour plots for a few sample values
•
model compared to the two CDM models is evident. Also ofβandσ .Wethenfindwhichofourgridofmodelshasthe
v
obvious is the greater magnitude of the velocities in this smallestχ2bycomparingwiththeobservedξ (σ,π).Wees-
cg
model. timateconfidenceboundsonβ andσ fromthedistribution
v
Thereissubstuctureinthedensityfieldforallmodelsand of ∆χ2 over the [β,σ ] plane. As the infall model does not
v
•
the cluster we are interested in at the centre of the panel give an accurate prediction for thevelocity in the virialised
is often part of an elongated system. The MDM model also region, we have chosen to exclude from the fit the ξ (σ,π)
cg
has a noticeably smoother density field on small scales. point with the smallest value of σ and π. Thus we exclude
Allmodelscontainsomeclusterswherethemainflowpat- any cluster-galaxy pairs with both σ and π<2.5h−1Mpc.
•
ternisnotcenteredontheclusterbutcontinuespastit,and Theχ2 contoursareshowninFigure13,wherewehave
othershavemorecomplexandnon-radialflows.Thepictures tested the SCDM, LCDM and MDM simulations. For the
do not inspire confidence in the idea of simple spherically test on the N-bodymodels, we haveused all ( 1000) clus-
≈
symmetricinfallaroundeachcluster.Itseemsthatthispic- ters in each box (and all 106 particles), as well as averag-
ture is only likely to apply after some sort of averaging is ing over 5 realisations. We can see straight away that we
carried out, at least around each cluster (e.g. as in Villum- are predicting values, for β at least, that are consistent
sen&Davis1986)oraroundastatisticallydefinedsampleof with the results of our direct comparison of infall veloci-
clusters,asdescribedinthispaper.Itisalsoworthconsider- ties plotted in Figure 7. For the LCDM model, for which
(cid:13)c 0000RAS,MNRAS000,000–000
10 R.A.C. Croft, G.B. Dalton & G. Efstathiou
Figure 9. The distribution of particles (with their projected velocities) surrounding 9 clusters taken from a simulation of a standard
CDMuniverse(withσ8 =1.0).Theclustersplottedcorrespondtothoserankedbymassasnumbers2,4,8,16,32,64,128,256and512from
a box of size 300h−1Mpc. In each panel, the cluster inquestion is located in the centre, and the velocity field of particles (in the rest
frameofthecluster)aroundtheclusterhasbeenassignedtoagridusinga2h−1MpcGaussiankernel.TheXandYcomponentsofthis
velocity fieldinaslicethrough the centre of thecluster are shownas arrows.Theparticles have alsobeen assigned to agridusingthe
samekernelandaslicethroughtheresultingdensityfieldisalsoshownoneachpanel asagrayscale(linearindensity).
the true value is β = 0.39 (Lahav et al. 1991), we find reveals that these values are in good agreement with the
β=0.52+0.06(95%confidencelimits).ForSCDMandMDM velocitydispersionofparticlesintheouterpartsofclusters.
−0.09
we measure β = 1.20+0.09 and β = 1.14+0.12, repectively. We now turn to the determination of β and σ from
−0.12 −0.18 v
The error bars on β given here are computed by marginali- the APM sample. The results of the maximum likelihood
sationoverallvaluesofσ .Theconfidencelimitsonthepair fitting are shown in panel (d) of Figure 13. We can see im-
v
ofparametersσ andβ areplottedascontoursinFigure13. mediatelythatwehavemuchlargerstatisticalerrorsthanin
v
This diagram shows that given enough clusters we should our simulations, as there are only 364 clusters and 2000
∼
beabletodistinguish betweenhighandlowβ models. This galaxies in the APM sample. The value of β we obtain is
constraintonβ fromtheinfall regionaroundclusters(most 0.43, with β < 0.87 at 95% confidence. The tail of proba-
ofthesignal comesfromr<10h−1Mpc)iscomplementary blilties towards rather high values of σ and the best fit σ
v v
both to measurements of the central mass of clusters from inferred, 660kms−1 (σ < 1070kms−1 at 95% confidence)
v
thevirialtheoremandtolargerscalebulkflowobservations are probably a consequence of uncertainties in the cluster
(e.g. Loveday et al. 1996). As the infall method tends to redshifts. There is not therefore a good constraint on the
systematically overestimate β byasmall amount,anymea- velocity dispersion in the outer regions of clusters from the
surement which gives a particularly low value, for example APM sample.
one inconsistent with high Ω models, will be especially in-
teresting.
The values of σ we measure using the maxi-
v 5 MODEL-DEPENDENT COMPARISONS:
mum likelihood fits are 370+60kms−1, 690+60kms−1, and
−90 −60 BIASING AS A FUNCTION OF SCALE.
570+30kms−1 for LCDM, SCDM and MDM respectively.
−60
These are again 95% confidence limits obtained after Tocalculatehowbiasingofgalaxyfluctuationsisexpectedto
marginalising over all values of β. Comparison with Fig. 8 changewith scaleforeachcosmological model,wecalculate
(cid:13)c 0000RAS,MNRAS000,000–000