The ESO Slice Project (ESP) Galaxy Redshift Survey. VII. The Redshift and Real-Space Correlation Functions

17 Pages·0.55 MB·English

by  L. Guzzo

A&A manuscript no.
ASTRONOMY
(will be inserted by hand later)
AND
Your thesaurus codes are:
ASTROPHYSICS
06 (03.11.1; 16.06.1; 19.06.1; 19.37.1; 19.53.1; 19.63.1)
The ESO Slice Project (ESP) Galaxy Redshift Survey ⋆
VII. The Redshift and Real–Space Correlation Functions
L. Guzzo1, J.G. Bartlett2, A. Cappi3, S. Maurogordato4, E. Zucca3,5, G. Zamorani3,5, C. Balkowski6,
A. Blanchard2, V. Cayatte6, G. Chincarini1,7, C.A. Collins8, D. Maccagni9, H. MacGillivray10, R.
Merighi3, M. Mignoli3, D. Proust6, M. Ramella11, R. Scaramella12, G.M. Stirpe3, and G. Vettolani5
1 Osservatorio Astronomico diBrera, via Bianchi 46, 23807 Merate (LC), Italy
2 Observatoire Astronomique, Universit´e LuisPasteur, 11 rue del’Universit´e, 67000 Strasbourg, France,
(unit´e associ`e au CNRS – UMR7550)
3 Osservatorio Astronomico diBologna, via Zamboni 33, 40126 Bologna, Italy
9
9 4 CERGA, Observatoire de la Cˆote d’Azur, 06304 Nice Cedex 4, France
9 5 Istituto di Radioastronomia del CNR,via Gobetti 101, 40129 Bologna, Italy
1 6 Observatoire de Paris, DAEC, 5 Pl. J.Janssen, 92195 Meudon, France
7 Dipartimento di Fisica, Universit`a degli Studidi Milano, via Celoria 16, 20133 Milano, Italy
n
8 School of Engineering, Liverpool John Moores University,Byrom Street, Liverpool L3 3AF, UnitedKingdom
a
J 9 Istituto di Fisica Cosmica e Tecnologie Relative, via Bassini 15, 20133 Milano, Italy
10 Royal Observatory Edinburgh,Blackford Hill, Edinburgh EH9 3HJ, United Kingdom
7
2 11 Osservatorio Astronomico diTrieste, via Tiepolo 11, 34131 Trieste, Italy
12 Osservatorio Astronomico diRoma, via Osservatorio 2, 00040 Monteporzio Catone (RM),Italy

1
v Received 00 - 00 - 0000; accepted 00 - 00 - 0000
8
7
3
Abstract. We present analyses of the two-point corre- agreement with the results obtained from angular sur-
1
0 lation properties of the ESO Slice Project (ESP) galaxy veys like the APM and EDSGC. Also the shape of the
9 redshift survey, both in redshift and real space. From the two–point correlation function is remarkably unanimous
9 redshift–space correlation function ξ(s) we are able to among these data sets, in all cases requiring more power
h/ trace positive clustering out to separations as large as above 5h−1Mpc (a ‘shoulder’), than a simple extrapola-
p 50h−1Mpc,afterwhichξ(s)smoothlybreaksdown,cross- tion of the canonical ξ(r) =(r/5)−1.8.
- ingthezerovaluebetween60and80h−1Mpc.Thisisbest The analysis of ξ(s) for volume–limited subsamples
o
r seenfromthewholemagnitude–limitedredshiftcatalogue, withdifferentluminosityshowsevidenceofluminosityseg-
st using the J3 minimum–varianceweightingestimator.ξ(s) regation only for the most luminous sample with MbJ ≤
a is reasonably well described by a shallow power law with −20.5. For these galaxies, the amplitude of clustering is
v: γ ∼1.5between3and50h−1Mpc,whileonsmallerscales on all scales > 4h−1Mpc about a factor of 2 above that
i (0.2−2h−1Mpc) it has a shallower slope (γ ∼ 1). This of all other subsamples containing less luminous galax-
X
flattening isshowntobe mostly dueto the redshift–space ies. When redshift–spacedistortions are removedthrough
r
a dampingproducedbyvirializedstructures,andislessevi- projectionofξ(rp,π),however,aweakdependenceonlu-
dentwhenvolume–limitedsamplesofthesurveyareanal- minosityisseenatsmallseparationsalsoatfaintermagni-
ysed. tudes, resulting in a growth of r from 3.45+0.21h−1Mpc
o −0.30
Weexaminethefulleffectofredshift–spacedistortions to 5.15+0.39h−1Mpc, when the limiting absolute magni-
−0.44
by computing the two–dimensional correlation function tudeofthesamplechangesfromM =−18.5toM =−20.
ξ(rp,π) , from which we project out the real–space ξ(r) This effect is masked in redshift space, as the mean pair-
below 10h−1Mpc. This function is well described by a wisevelocitydispersionexperiencesaparallelincrease,ba-
power–law model (r/ro)−γ, with ro = 4.15+−00..2201h−1 Mpc sically erasing the effect of the clustering growth on ξ(s)
and γ =1.67+0.07. .
−0.09
Comparison to other redshift surveys shows a consis-
tent picture in which galaxy clustering remains positive
out to separations of 50h−1Mpc or larger, in substantial
keywords – Cosmology – Large–Scale Structure
Send offprint requests to: L. Guzzo, [email protected]
⋆ Based on observationscollected attheEuropean Southern
Observatory,La Silla, Chile.
2 L. Guzzo et al.: Clustering in theESP Galaxy Redshift Survey
1. Introduction shift Survey (LCRS, Shectman et al. 1996). Our contri-
bution along this directionhas been the realizationof the
The spatial two-point correlation function, ξ(r) , is prob- ESO Slice Project(ESP) galaxy redshift survey,that was
ably the most classical statistic used in cosmology for completed between 1993 and 1996 (Vettolani et al. 1997,
clustering analyses. Since its early applications (e.g. Pee- Vettolani et al. 1998, V98 hereafter). The technical aims
bles 1980), obtaining reliable estimates of ξ(r) on large of the ESP survey were to exploit on one side the avail-
(> 5−10h−1Mpc) scales became one of the main sta- ability ofnew deepphotometric galaxycatalogues(in our
tisticalmotivationsforenlargingtheavailable3Dsamples casetheEDSGC,Heydon-Dumbleton et al. 1989),andon
throughnew,wideranddeeperredshiftsurveys.Theprin- the other the multiplexing performances of fibre spectro-
cipal reason for this is that on scales sufficiently large for graphs,asinthespecificcaseoftheOptopusfibrecoupler
the fluctuations to be still in the linear clustering regime, available at ESO (Avila et al. 1989). The main scientific
the shape of ξ(r) is expected to be preservedduring grav- goalsweretoestimatethegalaxyluminosityfunctionover
itational clustering growth. Comparison of observations a large and homogeneously selected sample with a large
with models is therefore simpler, because in this case the dynamic range in magnitudes (see Zucca et al. 1997, pa-
theoretical description of clustering does not require the per II hereafter), and to measure the clustering of galax-
full non-linear gravitational modelling, which is on the ies over a hopefully fair sample of the Universe. The ESP
contrary necessary at small separations. If we are able to survey, during about 25 nights of observations, produced
measure accurately ξ(r) [or its Fourier dual, the power a ∼ 85% complete sample of 3342 galaxies with reliable
spectrum P(k)], on large enough scales, we shall have a redshift. Its combination of depth and angular extension
measureofthedistributionoftheinitialfluctuationampli- is paralleled at present only by the Las Campanas Red-
tudes.Inaddition,iftheinitialdensityfieldwasdescribed shiftSurvey(LCRS,Shectmanetal.1996),whichhasthe
by a Gaussian statistics, this will be all the statistical in- advantageofcontaininga largernumber ofredshifts.One
formation we need to completely characterise the initial main difference of the LCRS with respect to the ESP, is
fielditself.Onefurtherreasonforpushingmeasuresofξ(r) that it is selected in the red, which makes comparison
to larger and larger scales, is the possibility to correctly with the results presented here particularly interesting.
estimateitszero–point,i.e.thescaleonwhichcorrelations Oneimportantadvantageofthe ESPis thatitisapurely
become negative. This is a specific prediction of any vi- magnitude–limited sample, which makes modelling of the
able model, as it reflects the turnover and peak scale in selection function easier than in the case of the LCRS,
the power spectrum, fingerprint of the horizonsize at the where an additional surface–brightness selection was ap-
matter–radiation equivalence epoch. Clearly, as much as plied.
the weakness of clustering in this regime is a benefit for In this paper we shall discuss the two-point correla-
the theory, it represents a hard challenge for the obser- tion properties of the ESP survey, both in redshift and
vations: statistical fluctuations destroy any possibility to in real space. Previous or parallel papers, in addition to
detectsignificantfeatures,suchasthe zero–pointofξ(r) , that describing the above mentioned bJ–band luminosity
ifthesampleunderstudydoesnotcontainenoughobjects function (Zucca et al. 1997), deal with the scaling prop-
at comparable separations. erties (Scaramella et al. 1998), the properties of groups
Historically,followingthepioneeringworksofthemid– (Ramella et al. 1998), and potential biases in the esti-
seventies (see Rood 1988 for a review), the industry of mate of galaxy redshifts (Cappi et al. 1998). The survey
redshift measurements exploded in the eighties, with the ingeneralisdescribedinVettolanietal.(1997),whilethe
completionfirstoftheCfA1(Davisetal.1982),thenofthe redshift data catalogue is presented in V98. Here we also
Perseus-Pisces (Giovanelli et al. 1986) and CfA2/SSRS2 marginally discuss the small–scale galaxy dynamics, for
(Geller & Huchra 1989; da Costa et al. 1994) surveys. its effects on redshift–space clustering. However,a proper
Thesefirstlargesurveys(containingseveral103redshifts), discussion of the small–scale pairwise velocity dispersion
producedsignificantadvancesintheestimationofξ(r) on and related topics, will be presented in a separate paper
small and intermediate scales (e.g. Davis & Peebles 1983, (Guzzo et al. , in preparation).
De Lapparent et al. 1988). However, they also showed The paper is organized as follows. Section 2 briefly
the existence of structures with dimensions comparable summarises the properties of the redshift catalogue. Sec-
to their depth. This,while onone side giving riseto spec- tion3discussesthetechniquesusedtoestimatetwo–point
ulations about the very existence of a transition to ho- correlationsandtheir errors.Sections 4 and 5 present the
mogeneity on large scales (see Guzzo 1997 for a recent redshift– and real–space correlation functions, discussing
review of this problem), explicitly indicated the need for theirdependenceonluminosityandcomparingresultswith
larger redshift samples. In more recent years, there have other surveys. In section 6 possible biases deriving from
been a few significant attempts to fulfil this need, as for theobservationalsetupareinvestigated.Themainresults
examplethesurveysbasedontheIRASsourcecatalogues are summarised in section 7.
(see e.g. Strauss 1996), the Stromlo–APM survey (Love-
day et al. 1992b), and especially the Las Campanas Red-
3
2. The Survey tion for galactic extinction to the observed magnitudes.
Observed heliocentric radial velocities were converted to
TheESPredshiftsurveyconsistsofastripofsky1o thick the reference frame of the Local Group using the trans-
(declination)by22o long(rightascension)nearthe South formation of Yahil et al. (1977). (We also checked that
Galactic Pole, plus an additional strip of length 5o situ- correctingvelocitiesto the CMB referenceframe doesnot
ated five degrees to the west of the main strip. In total, produce any difference in the results). We then computed
this coversabout25 squaredegreesata meandeclination comoving, luminosity, and angular diameter distances for
of −40.25o between rightascensions 22h30m and 01h20m. eachgalaxy,adoptingacosmologicalmodelwithHo =100
The target galaxies were selected from the Edinburgh– h kms−1 Mpc−1, and qo =0.5.
Durham Southern Galaxy Catalogue (EDSGC, Heydon– The correlation analyses are first performed on the
Dumbleton et al. 1989), which is complete to bJ = 20.5. whole apparent–magnitude limited catalogue. To do this,
The angular correlation properties of this catalogue were we apply the minimum variance technique (see § 3.1),
studiedindetailinCollinsetal.(1992).Thelimitingmag- for which we use the survey selection function as com-
nitude of the ESP is bJ = 19.4, chosen in order to have puted in Zucca et al. (1997). We also exclude from this
the best match of the number of targetsto the number of analysis those galaxies lying outside of the range of co-
fibres in the field of the multi–object spectrograph Opto- moving distances 100 ≤ D ≤ 500h−1Mpc, to avoid
com
pus, mounted at the Cassegrain focus of the ESO 3.6 m using those regions of the sample where respectively ei-
telescope. More details on the observing strategy and the ther the survey volume or the selection function are dan-
propertiesofthespectroscopicdataaregiveninVettolani gerously small. The sample trimmed in this way contains
et al. (1997), and in V98. 2850 galaxies, with a minimum luminosity corresponding
The final galaxy redshift catalogue contains 3342 en- to M = −15.6+ 5logh. Using the whole magnitude-
bJ
tries,andis85%redshift–completewithinthesurveyarea. limitedsampleisanattempttoextractthemaximumsig-
At its effective depth (z ≃0.16), the transverse linear di- nalfromtheavailabledata,butcanhavesomedrawbacks,
mensionsofthesurveyare∼210h−1Mpcby6.5h−1Mpc, as the contribution of galaxies with different luminosities
while its volume is ∼1.9·105h−3Mpc3. is not homogeneous over the sampled scales.
As can be seen in V98, the survey geometry is fairly
Therefore, to evidence possible biases and study the
complex, being composed by two rows of circular Opto-
general behaviour of clustering with luminosity, we also
pus fields, partly overlapping each other. This results in
construct a set of volume–limited subsamples, defined in
the presence of interstices in which galaxy redshifts were
Table1.Eachcolumnlists,respectively,(1)absolutemag-
not measured,and which obviously haveto be considered
nitude limit, (2) luminosity distance computed from the
carefully in the analysis of clustering.Also, the complete-
magnitude limit taking into accountthe K-correction,(3)
ness of each Optopus field is not strictly constant (V98).
corresponding comoving distance limit, (4) redshift limit,
This will require a weight to be applied to each object,
(5)lowercutincomovingdistance,(6)effectivevolumeof
depending on its parent field (see §3.1). The median in-
thesamplebetweenthetwodistanceboundaries,(7)total
ternalerroronthe redshiftmeasurementsis 64kms−1 for
number of galaxies.Given the small thickness of the ESP
absorption–lineestimates,and31kms−1foremission–line
slice, the lower distance limits are a safeguard to exclude
estimates (V98).
that part of the samples where galaxy density is poten-
In Figure 1 we show a cone diagramof the galaxydis-
tially undersampled by bright galaxies, thus introducing
tribution inthe survey.One firstqualitative remarkto be
shot noise. These low cut–off values have been computed
made, in view of the use we shall do of the data in this
asthe distancewithinwhich,forthatabsolutemagnitude
paper, is that the visual impression from this figure sug-
limit, one expects less than 10 galaxies within the corre-
gests - contrary to more ”local” surveys as, e.g. CfA2, a
spondingESPvolume,onthebasisoftheESPluminosity
typical size of structures which is smaller than the survey
function.Themainanalysesweshalldiscussinthefollow-
dimensions. This gives us hope that, possibly, clustering
ing are basedonthe firstfour samples inthe table.These
measures extracted from this survey would be sufficiently
offerthe bestcompromisebetweenhavingenoughvolume
representative of the general properties of our Universe.
sampled and a good statistics. However, we also select a
Clearly,the volume coveredby the ESP is still small, due
more “extreme” sample with M ≤−20.5, which contains
toitslimitedareacoverageonthesky,yetthelinearsam-
only 292 galaxies,to study in more detail the existence of
pling of large-scale structures in two dimensions is un-
luminosity segregationat high luminosities.
precedented, and matched only by the even larger LCRS.
Preliminary to the analysis, we performed a series of
necessary corrections to the raw data catalogue. We first
appliedK-correctionsto the observedbJ apparentmagni- 3. Computing the Two-Point Correlation
tudesinthesamewayasdiscussedindetailinZuccaetal. Function
(1997). Given the location of the ESP area in the region
of the South Galactic Pole, we did not apply any correc-
4 L. Guzzo et al.: Clustering in theESP Galaxy Redshift Survey
Fig.1. Cone diagram showing the large-scale distribution of galaxies in the ESP survey.
Mlim Dlum(max) Dcom(max) zmax Dcom(min) Vol Ngal
−18.5 328.5 296.8 0.106874 65 6.109·104 823
−19.0 398.7 353.1 0.129082 80 1.027·105 924
−19.5 483.4 418.3 0.155607 100 1.705·105 819
−20.0 590.3 496.6 0.188737 135 2.834·105 521
−20.5 738.4 598.4 0.234030 200 4.871·105 292
Table 1.Propertiesofthe volume–limitedsubsamples analysed.Distances (inMpc), volumes(inMpc3)andabsolute
magnitudes (bJ system), are computed for h=1, q◦ =0.5
5
3.1. Estimators We model ξ(r) as a power law, ξ(r) = (5/r)1.8 for r <
30h−1Mpc, and assume negligible correlations at larger
There is a long history and literature concerning the the- separations. This expression results in J (r) = 15.1r1.2,
3
ory of statistical estimators of the two–point correlation for r ≤30h−1Mpc, and J =894 at larger separations.
3
functionξ(r).Themostwidelyusedone(Davis&Peebles To test the sensitivity of the results to the adopted
1983), is given by model, we also compute J in a different way, i.e. using
3
2n GG(r) its definition in terms of the power spectrum P(k). In
ξ(r)= nGRGR(r) −1 , (1) thiscase,wehaveJ3(r)≡(4πr3/3) 0∞dkk2P(k)Wth(kr),
wheren andn arethemeanspacedensities(orequiva- where Wth(kr) is the Fourier transRform of the spherical
G R top-hat window function. We compute J for the CDM
lentlythetotalnumbers),ofobjectsinthetwocatalogues, 3
power spectrum normalized to unity in spheres of radius
GG(r)representsthenumberofindependentGalaxy-Galaxy
8h−1Mpc.[WenotethatvariantsofstandardCDM,such
pairswithseparationbetweenrandr+dr,whileGR(r)is
as an open model or a low-density flat model with non–
the number of Galaxy-Random pairs, computed with re-
zeroΛ,wouldprovideamorerealisticpowerspectrum,but
specttoarandomcatalogueofpointsdistributedwiththe
as it is clear from the results of the test, this would make
sameredshiftselectionfunctionandthesamegeometryof
norealdifference forourpurposes].We findno significant
the real sample (in other words, a Poisson realization of
deviations between the estimates of ξ(s) computed using
the catalogue). In our case, this implies reproducing the
thetwodifferentmodelsforJ ,andthereforeusedtheone
exactdistributionofOptopusfieldsonthesky,andapply- 3
based on ξ(r) in the rest of the computations.
ing the same selection process as done on the real data.
The weighting function has to be taken into account
Oneexampleisthe”drilling”ofsomeareasasitwasorig-
alsowhencalculatingthemeandensitiesn andn tobe
inally done on the EDSGC around the position of very G R
used in eq. 1. A full discussion of the merits of different
bright stars (Heydon-Dumbleton et al. 1989)1.
estimators for the mean density in a magnitude-limited
The number of GG(r) or GR(r) pairs in eq. 1 can be
sample is presentedin paperII. Ingeneral,anestimate of
formedingeneralincludinganarbitrarystatisticalweight,
the mean density, given a weight for each object and the
as GG(r) = w (r), that can depend both on the
(i,j) ij survey selection function, would be of the form
distances of tPhe two objects from the observer (d and
i
w
dj), and their separation (r = |di−dj|). Here the sum nG = i i , (3)
(i,j) extends over all independent pairs with separations VsdVPw(d)φ(d)
between r and r +dr. Any weighting which is indepen- whereRthesumextendsoverallgalaxies(orrandompoints),
dent of the local galaxy density will lead to an unbiased while the integralis extended to the whole survey volume
estimate of the correlationfunction, in the sense that the V .Inpractice,weareinterestedintheratioofn andn ,
s R G
mean value of the estimator, if averaged over an ensem- sothatweonlyneedtoestimateforthe twosetsofpoints
ble of similar surveys, would approach the true, under- the value of the quantity in the numerator (i.e. the effec-
lying galaxy correlation function. However, the variance tivetotalnumberofobjects).Thesameargumentsleading
of individual determinations of ξ around this mean will to eq. (2) result in a similar expression for the minimum
depend on the particular choice of wij(r); some choices variance,single–pointweights:wi =(1+nGφ(di)J3max)−1
leadtomoreaccurateestimatesthanothers.Byassuming (Davis & Huchra 1982);hereJmax is the maximum value
3
that two–point correlations dominate over higher orders ofJ .Note thatagainwehaveonequantity,n ,thatide-
3 G
(essentially the linear regime), one finds a useful approxi- allyshouldbecalculatediteratively.However,sinceinany
mationtothe minimum variance weights(Saunders etal. case our estimator will be unbiased, no matter the choice
1992; Hamilton 1993; Fisher et al. 1994a, F94 hereafter), oftheweight,weusehere-andcorrespondinglyinthepair
as weighting-themeandensityasestimatedinpaperIIfrom
1 1 this same sample, with the same M ≤ −12.4+5logh
w (r)= · , (2) bJ
ij
1+4πnGφ(di)J3(r) 1+4πnGφ(dj)J3(r) cut.
whereφ(d )isthesurveyselectionfunctionattheposition In our specific case, there is an additional complica-
i
of galaxy i, and J (r)≡ rdrr2ξ(r). The minimum vari- tion arising from the varying sampling of the individual
3 o
anceweightsthemselves,Rtherefore,requireknowledgeofξ, Optopusfields,asdiscussedin§2.Thisrequirestheintro-
i.e. we have a circular argument for which one must hope duction of another weight, Wi, to correct for the missed
thataniterativeprocedurewouldbeconvergent.Weshall galaxiesineachfield.Followingthe samereasoningasde-
instead adopt an a` priori model for ξ and use it to calcu- tailed in V98, we define the completeness in each field Ci
late the weights; given the host of assumptions employed as
toarriveat(2),thisapproachshouldnotbeconsideredas Ni
Ci = Z , (4)
too gross an approximation, as also shown, e.g., by F94. (1−f∗)Ni
T
1 Within the ESP area there are about 10 drill holes with a where Ni and Ni are the number of secured galaxy red-
Z T
typical diameter of 0.2 deg. shifts and the number of objects classified as galaxies in
6 L. Guzzo et al.: Clustering in theESP Galaxy Redshift Survey
the parent photometric catalogue, respectively, in each tions: ξ (r), with k =1,N .2 We refer to the set of sepa-
k B
fieldi,whilef∗ =0.122istheobservedfractionofmisclas- rationsri,oneforeachbinofsize∆r,asseparation space.
sifiedstars.Wethereforemultiplythew correspondingto The correlationfunction ξ(r) is a vector, ξS in this space
i
eachgalaxyby the weightpertaining to its field, that will withcomponentsξ(r ).Acovariancematrixinseparation
i
be given by Wi = 1/Ci . This is not necessary for the space may be constructed as
randomsample,that,byconstruction,isuniformoverthe C(r ,r )=<(ξ(r )−ξ¯(r ))(ξ(r )−ξ¯(r ))> , (5)
i j i i j j realiz
Optopus fields.
where’<> ’indicatesanaverageoverbootstrapreal-
Eq. (1), is based on the definition of ξ (e.g. Peebles realiz
izations and ξ¯(r ) is the ensemble average of the correla-
1980), in which it is understood that n represents the k
G
tionfunctionatseparationr .Sincethevaluesofξ attwo
universal mean galaxy density. Any sample mean density k
different separations are correlated, C is non–diagonal.
will vary about this value with a dispersion at least as
For this reason one cannot do a straightforward χ2 fit of
large as expected from simple Poisson statistics and, in
a model to the observedpoints. However,C is symmetric
fact, clustering on the scale of the survey volume will in-
(C=C˜) and real, and therefore hermitian, i.e. such that
crease this dispersion. The precision of the estimate of
it can be diagonalized by a unitary transformation if its
ξ at small values (large scales) is limited by this uncer-
determinant is non–vanishing. What happens in practice,
tainty, δn (the number of points in the random catalog
G
is that the estimated functions are oversampled, so that
N =V ·n may always be sufficiently increasedto elim-
r R
the effective number of degrees of freedom in the data is
inate its density as a source of uncertainty, and this is in
smallerthanthenumberofcomponentsofξS.Thismakes
fact what we do here by using typically N = 100,000).
r
C singular,andonehas toapply singularvalue decompo-
Giventhatn appearslinearlyinestimator(1),onewould
G
sition(asinF94),torecoveritseigenvectors.Thesedefine
conclude that the precision on ξ is also linear in δn . In
G
the required transformation T, to the orthogonal space,
fact,thisisnotcorrect,asemphasizedbyHamilton(1993)
i.e.C→D =T˜·C·TandξS →ξD =T·ξS,wherenow
andLandy&Szalay(1993),whosuggestedalternativees-
DisdiagonalandthecomponentsofξD areindependent.
timators in which the precision goes as δn2, as would
G In this space we can therefore define in the usual way the
seemmore appropriatefor the secondmomentofa distri-
quantity χ2 with respect to our model parameters, and
bution.Wehavecomputedξ(s)using(1),andwithHamil-
minimize it to find their best–fit values.
ton(1993)estimator,andfound for our surveyequivalent
resultswithintheerrors.Thisispartlyincontrasttowhat
was found by Lovedayet al. on the Stromlo–APMsurvey
4. The Redshift–Space Correlation Function ξ(s)
(1995),butagreestotheresultsofTuckeretal.(1997)on
the larger LCRS (cf. their Figure 1). The results we shall
4.1. Optimal weighting
show in the following are therefore all basedon the Davis
& Peebles (1983) estimator.
To maximise the large–scale signal from the data,we first
estimateξ(s)fromthewholeESPredshiftcatalogue,preparred
and trimmed as described in § 2. Figure 2 shows the re-
sultofapplyingtheJ optimal–weightingestimatortothis
3.2. Error estimation and model fitting 3
sample,containing2850galaxies.Between3and50h−1Mpc
ξ(s) is well described by a shallow (γ ≃ 1.5) power law,
with a redshift–space correlation length s ≃ 5h−1Mpc.
Statistical errorbars for our estimates of two-pointcorre- o
On larger scales, it smoothly breaks down, crossing the
lationfunctionsarecomputedusingbootstrapresampling
zerovaluebetween60and80h−1Mpc.Thebincenteredat
as discussed by Ling et al. (1986). F94 have discussed in
80h−1Mpchasindeedanegativevalue,althoughitisless
detailthereliabilityofthebootstraptechniqueinestimat-
than 1σ below zero. On larger scales (r > 100h−1Mpc),
ing errors for correlation functions; they show that, typi-
thereismarginalevidencethattheamplitudeofξ(s)rises
cally, bootstrap errors are a good description of true rms
errorsforseparationssmallerthan∼10h−1Mpc,whileon up again, a behaviour that seems to be shared also by
otherclusteringdata,asweshowbelow.Onscalessmaller
larger scales they tend to overestimate them by a factor
than 3h−1Mpc, ξ(s) flattens significantly, so that a sin-
of about 1.5. In the same work,they carefully discuss the
gle power–law is certainly not a fair description over the
technique for fitting of models to correlation functions,
whole explored range.
where the data points are not statistically independent.
A comprehensive comparison of the best-fit results of
We use the same procedure here, which we briefly sum-
a power–law ξ(s) to redshift–space correlation functions
marize in the following. Essentially, we create N = 100
B
from several surveys is given in Willmer et al. (1998). As
”perturbed”realizationsofthesample,byrandomlysam-
one could easily guess from our Figure 2, these results
pling the original catalog with replacement, (the ”boot-
strapping”). By subjecting each of these realizations to 2 Notethathereweshallfitmodelstoone–dimensionalquan-
the sameanalysis,weobtaina setofNB correlationfunc- tities only – i.e. ξ(s) or projections of ξ(rp,π)
7
Fig.2. Redshift–space correlation function from the whole ESP magnitude–limited catalogue, computed using the
minimum–variance, J –weighting scheme. For reference, the dashed line gives an arbitrary power-law function with
3
r =5h−1Mpc and γ =1.5.
o
are strongly dependent on the range of scales over which techniquefromtheLCRSandtheStromlo–APMsurveys3.
the fit is performed, that often is not the same for the The agreement between the three estimates is very good
differentdatasets.Thismakesthecomparisonofthecrude between2 and30h−1Mpc, givenalsothe different galaxy
best-fit values rather inconclusive. Equivalently difficult selectionfunctionsofthecorrespondingsurveys.Onlarger
is anyphysicalinterpretationof the measuredamplitudes scales we can note that the ESP and Stromlo–APM ξ(s)
and slopes, given that the fit sometimes includes scales seem to show slightly more power than the LCRS. This
wherepowersuppressionbyvirialisedstructures(“Fingers evidence for low–amplitude power on large scales, with a
of God”) is very effective (s<2h−1Mpc), while in other smooth decay from the power–law shape, resembles that
cases it starts above these, putting more weight on less observed in the IRAS 1.2Jy redshift survey (F94). One
nonlinear scales. Given these ambiguities, here we do not couldprobablyexplainthedifferencesinFigure3withthe
perform a formal fit to the ESP ξ(s) . We shall compute
3 We limit our explicit comparison to these two surveys,
a proper fit only to the real-space correlation function,
for which between 0.4 and 10h−1Mpc a power-law shape as the LCRS is the only other redshift survey with depth
and angular coverage comparable or superiorto theESP. The
is a rather good description of the data, and where the
Stromlo–APM survey, on the other hand, is less deep by ∼ 2
observed shape is at least freed of one major distorting
magnitudes and is sparsely sampled, but represents an inter-
effect.
esting comparison becauseof itslarge solid angle and thefact
In Figure 3, we compare the J3 estimate of ξ(s) from of being selected from the same photographic material (IIIaJ
the ESP on largescales to those computed with the same plates).
8 L. Guzzo et al.: Clustering in theESP Galaxy Redshift Survey
Fig.3.ξ(s)onlargescalesfromtheESP,LCRS(Tuckeretal.1997),andStromlo–APM(Lovedayetal.1992b),redshift
surveys.Thedashedlinesshowthereal–spaceξ(r)asobtainedthroughdeprojectionoftheangularcorrelationfunction
for the APM galaxy catalogue, for two clustering evolution models (Baugh 1996).
differentselectioncriteriainthethreeredshiftcatalogues: lowing section directly from the ESP itself, here we have
the LCRS is selected in r (with the additional complica- the chance to make already a few interesting remarks.
tion of a surface–brightness cut, whose effect is not fully Firstofall,thescaleofthebreakinbothξ(s)andξ(r)
clear), and thus should tend to favour earlier types on is consistentlyindicatedby the differentsurveysto be be-
the average, that we know reside preferentially in high– tween50and90h−1Mpc,withtheESPclearlypointingto
density regions. On the other hand, the b band used for a value >50h−1Mpc. The agreement between these new
J
the ESP (so as the IRAS infrared band), is expected to redshiftdataandtheAPMξ(r),directlyimpliesthatthe
better select star–forming objects in low–density regions. large–scale power originally seen in the angular correla-
Itshouldprobablybe expectedthatESPgalaxies,aswell tion function w(θ) from the APM (Maddox et al. 1990)
as IRAS galaxies, are better tracers of very large–scale, andEDSGC(Collinsetal.1992),galaxycataloguesisnot
low–amplitude fluctuations. significantlyenhancedbyerrorsinthemagnitudescale,as
InFigure3,wealsoplot(dashedlines),the real–space claimed by some authors (e.g. Bertin & Dennefeld 1997).
correlationfunction obtainedby deprojectingthe angular In fact, Figure 3 demonstrates that similar power is seen
correlationfunctionw(θ) fromthe APMgalaxycatalogue inredshiftsurveydata,eitherbasedonthesameb plates
J
(Baugh 1996). This is computed in the case of clustering (ESP) or CCD photometry (LCRS). Also concerning the
fixedin comovingcoordinates(bottom curve),or growing detailedshapeofgalaxycorrelationsabove3h−1Mpc,in-
according to linear theory (top). While the details of the dependent data sets show a high degree of unanimity:
real–spacecorrelationfunctionwillbe analysedinthe fol- if one ideally extrapolates to larger scales the ‘classic’
9
Fig.4. Redshift–space correlation function from four representative volume–limited samples extracted from the ESP
as described in Table 1. The dashed line with error bars reproduces the J -weighted estimate from the whole survey.
3
For clarity, points for M ≤−19 and M ≤−20 are displayed with a shift in log(s).
∼−1.8slopeobservedinrealspacebelow3h−1Mpcfrom verycloseto1.Forexample,avalueof1.2,consistentwith
theAPMξ(r)(butalsofrompreviousangularprojections, the observed data, would yield β =0.35.
see e.g. Davis & Peebles 1983), all surveys are systemat- It is interesting to note that the Stromlo–APMξ(s) is
ically above this extrapolation. To reproduce this feature theonlydatasetthataround∼10h−1Mpcseemstoshow
(described in the literature as a ‘shoulder’ or a ‘bump’, asignificantamplificationwithrespecttoξ(r).(Notethat
see Guzzo et al. 1991, and Peacock 1997), a rather steep thiscomparisonisactuallythesafest,beingtheAPMcat-
(∼k−2)powerspectrumP(k)isrequired(Branchinietal. alogue the parent photometric list of the Stromlo–APM
1994, Peacock 1997). redshift survey). The interpretation of this effect is how-
ever complicated by what we have pointed out in Paper
AsecondimportantpointaboutFigure3concernsthe II,whereweshowedhowthemeandensityinthisredshift
effectofredshift–spacedistortions.Onecanseehowsmall survey is about a factor of 2 lower with respect to that
istheamplitudedifferencebetweentheredshift–spacecor- deduced from deeper samples. The cause for this seems
relation functions and the APM real–space ξ(r) on large to be anegativedensityfluctuationthatweclearlydetect
scales. A linear amplification is indeed expected as a re- in the ESP data below z < 0.05, and that is also visible
sultofcoherentflowstowardslarge–scalestructures.This in other surveys. This ”local” underdensity, is shown to
canbeapproximatelyexpressedas∼1+1β+1β2,where be the explanation for both the low normalization of the
2 5
β =Ω0.6/bandbisthelinearbiasingfactor(Kaiser1987). Stromlo–APMluminosityfunction(Lovedayetal.1992a),
o
Figure 3 implies, within the errors,a value for this factor and the steep number counts observed at bright magni-
10 L. Guzzo et al.: Clustering in theESP Galaxy Redshift Survey
tudes in severalbands (e.g. Maddox et al. 1990).It is not
trivial to understand how this systematic 50% underesti-
mate of the mean density would affect the two–pointcor-
relation function, although it clearly reduces by the same
amount the number of pairs expected at a given separa-
tion. If this deficit is evenly spread on all scales, then the
measured ξ(s) will be practically unaffected, however if
it corresponds, for example, to a region with larger-than-
average voids, the net effect will be to boost up ξ(s) by
someamount.This wouldbe sufficienttoproduce the ob-
served discrepancy between ξ(s) and ξ(r) , and further
demonstrates how extended and accurate the data have
to be in order to extract dynamical information as the
value of β.
4.2. Volume–limited estimates
Fig.5.Afurthertestforluminositydependenceofcluster-
WhiletheJ –weightedestimatehastheadvantageofmax- ing in redshift space. ξ(s) for the brightest (M ≤ −20.5)
3
imisingtheinformationextractedfromtheavailabledata, volume-limited subsample of the ESP is compared to one
it has some important drawbacks that have been not al- of the main samples from the previous figure. Note that
waysappreciatedinpreviousapplications.Themainprob- the scale has been changed to better evidence the differ-
lem is that by its own definition it inevitably mixes the ence in amplitude on intermediate scales.
contribution of galaxies with different luminosities. This
would not be a problem if galaxy clustering were com-
pletelyindependentofluminosity,i.e.ifeachgalaxytraced displayedinFig.4.Inthefigure,wealsoreproducetheJ3–
fluctuations independently from its own absolute magni- weightedestimate(dashedline).Toeasevisualization,the
tude. As we discussed in §3.1, this technique weighs pairs data points for the M ≤−20and M ≤−19samples have
basedbothontheirseparationandontheirdistancefrom beenshiftedbyaconstantvalue(negativeandpositivere-
the observer. This implies that the main contribution to spectively) in log(s). It is clear how for the ESP data the
small–scalecorrelationscomes from low–luminositypairs, J3–based method produces a ξ(s) which is shallower be-
that are numerous (and dense) in the nearby part of the low3h−1Mpc.Onlargerscalestheshapeisconsistentbe-
sample and thus better trace clustering at small separa- tween the two methods, and for s>20h−1Mpc the opti-
tions.Onthecontrary,theestimateofξ(s)onlargescales malweightingperformsbetterintermsofsignal–to–noise,
is dominated by the contribution from luminous objects, than the single volume–limited estimates. The observed
that are in fact the only ones detectable in the distant ξ(s) is in practice the result of the convolution of the
part of the survey. Clearly, if there is any luminosity de- real–space correlation function ξ(r) with the distribution
pendence of clustering,this technique will tend to modify functionofpairwisevelocities.Thismeansthatthesmall–
in some way the shape of ξ(s) . In particular, if luminous scale flatter shape of ξ(s) in the case of the magnitude–
galaxiesaremoreclusteredthanfaintones,theJ weight- limited sample could be in principle produced either by a
3
ing will tend to give a shallower slope for a power–law smaller ξ(r) or by a higher pairwise velocity dispersions.
shapedcorrelationfunction. Therefore,while this method We find (Guzzo et al., in preparation), that most of the
is certainly optimal for maximising the clustering signal effect is produced by this second factor: the small–scale
on large scales (and so our conclusions on the large–scale pairwise velocity dispersion between 0 and 1h−1Mpc, is
shape of ξ(s) from the previous section should not be af- higher when measured on the whole survey using the J3–
fected), it might be dangerous to draw from its results weighted method. We measure σ12(1)≃600kms−1 using
far–reaching conclusions on the global shape of ξ(s) , es- theJ3 estimatefromthewholesample,whilethevolume–
pecially on small scales. A wise way to counter–check the limited samples give values between 300 and 400kms−1,
results obtained from the optimal weighting technique, is depending on the infall model adopted.
that of estimating ξ(s) also fromvolume–limited subsam- From this plot alone there is no evident sign of a lu-
ples extracted from the survey. In this case, each sample minosity dependence of ξ(s) , since the four estimates are
includesanarrowerrangeofluminositiesandnoweighting virtually the same within the errors. From their analy-
is required, the density of objects being the same every- sis of the SSRS2 sample, Benoist et al. (1996), also find
where. This means that in our case w (r) ≡ 1, and only that there are negligible signs of luminosity segregation
i
the Wi’s have to be taken into account. for galaxies fainter than M∗, which in our case corre-
We have performed this exercise on the ESP, and the sponds to −19.6. They also show, however, that a clear
results for the four main samples defined in Table 1 are effect becomes visible for absolute magnitudes brighter