E Variance catalogues Dalton et al Nichol et al and on ux Cluster Mon Not R Astron So c Peebles rst used to estimate theare correlation strongly functions correlatedEfstathiou so a that An even additional ato the small ep o ch sample at canmined which the b e from simulations numerical are simulations very analysed slowly Croft with time and so the clustering statistics deter use them to map out the distant and cation p eaks of Rich clusters of galaxies can b e used to trace the structure high in observations clusters have several imp ortant advantagesastro-ph/9501115 as tracers b oth 31 Jan 1995 CSIC Centre d Estudis Avan catsdeDepartment Blanes of Astrophysics Physics Astrophysics Blanes University Girona of Spain Oxford Nuclear Astrophysics Laboratory Keble Road Oxford OX RH Throughout the pap er we use the INTRODUCTION The p oint correlation function for galaxy clusters was Gazta naga economical analysed densities in Universe Recent of the clusters mass Bahcall in determinations in and in simulations Since clusters represent on way Redshift Ab ell s the distribution is very Skewness As galaxy more large clusters Soneira R A C di cult distributio n H based scales statistics in the cluster distribution follow the hierarchical pattern with p eculiar motions The standard Cold Dark Matter CDM mo del is inconsistent clusters and a meandensity depth uctuations in a complete sample from the APM Cluster Redshift Survey with of uctuations in spheres of radius We estimate the variance ABSTRACT hybrid Mixed Dark Matter relations quantitative predictions of approximation We argue that thiswith alternative metho d an is alternative not metho d reliablefrom of enough N b o dy for simulating making clusters which uses the truncated Zel dovich S Key observations numerical statistical J cluster catalogue advantage is that clusters are Universe in roughly constant are relatively insensitive either Survey than words on asso ciated Although Croft h Klypin km correlations automated to that it e g the Printed January s the is simulations of Mp c a systematic of Large scale p ossible Kurtosis second the observations Hauser with galaxies Kopylov cluster identi evolve G and S high 3 to third B 2 and structure the skewness The N b o dy simulation results follow similar hierarchical mo del and a low density di erent mo del Dalton or D with mutually APM limited mo dels Dalton et al shift b Survey has already b een used to constrain dark matter erarchical the estimates of it in the selection ofsured clusters from Ab e g ell s Efstathiou catalogueclusters et However is there al a ected is by evidenceinto and the inhomogeneities that underlying the matter clustering distribution mea and These the formation observations of couldpudi provide imp ortant insights et al tion e g Groth Peebles Fry Peebles Sharp and alogue rogordato Valdarnini Plionis e g Valdarnini Jing Cappi Mau Zhang T oth Holl osi Szalay Jing fourth S is R 4 kurtosis Previous Here we estimate the variance results S not Szalay Bouchet et al Gazta naga Szapudi Szalay Boschan Meiksin Sza MN L J Automatic of cluster samples have b een based on Ab ell s of and dark samples We analyse the distribution of clusters taken h almost una ected clear order consistent the relation These samples 1 a extensions T predictions Mp c matter 3 E analysis universe X style le v how and the kurtosis and h in statistics of X ray clusters 1 Plate the J Mp c with reasonable errorbars The We are able to measure the statistics from this results mo dels of by distribution CDM variant agree Measuring could J p oint Ab ell galaxies at The by found in the galaxy distribu all scales The a ect the redshift seem to repro duce p oint Corwin and Romer et al give 4 machine results of in the distribution of clustering the skewness the p oint density J APM correlation However distortions ab ove are S Olowin J Cluster with the statistics uctuations compared mentioned metho ds 2 J b oth 1 the hi of the from with Red cat and J a
2 Gazta naga Croft Dalton
sen to b e h Mp c and the mean intercluster separation in a sample from the APM Cluster Redshift Survey The
h Mp c Croft Efstathiou a selection of clusters in the APM Cluster Catalogue is based
on a computer algorithm applied to digitised photographic
We have also tested an alternative scheme for mak
ing theoretical predictions ab out cluster clustering This plates and so provides a more uniform sample of clusters
than Ab ell s We compare the results with previous analysis
scheme advocated by Plionis et al makes use of the
and with di erent cluster simulations
Zel dovich approximation to dynamicall y evolve the
density eld b efore clusters are selected In order to assess In next section we review the APM cluster data and
the validity of results derived using this metho d we have N b o dy mo dels We present the metho d of estimation and
the results in Section Sections and are devoted to
examined clustering in the Zel dovich approximation and
carried out a comparison with N b o dy clusters These tests the discussion and conclusions In an app endix we consider
are detailed in App endix A the prop erties of simulations generated using the truncated
Zel dovich approximation
CLUSTER CORRELATIONS
CLUSTER SAMPLES
Metho d of estimation
The APM Cluster Survey
We apply a variant of the metho d used by Efstathiou et al
We use sample B from Dalton et al b which contains
which accounts for selection e ects automatically by
clusters with mean redshift and space density
estimating the mean density and its uctuations in shells
h Mp c The clusters were selected from the
of mean constant redshift We divide space in the redshift
APM Galaxy Survey Maddox et al b c using an au
catalogue into a series of concentric shells of width R cen
tomatic selection algorithm describ ed in detail by Dalton
tred on the observer which are further sub divided into M
s
et al in preparation We have cut the sample to redshifts
spherical cells of radius R For each shell s we compute the
larger than km s to avoid uncertainties in the selec
mean number of particles N R in a cell and the moments
s
tion function at low redshift and imp osed an upp er limit of
m R
J
km s There are only clusters outside this range so
that the total number in the selected region is N As
M
s
X
J
the redshifts are obtained from only a few galaxies p er clus
m N N
J c s
ter the individu al cluster redshifts are uncertain to within
c
km s Dalton et al a
th
where N is the number of galaxies in the c cell The cor
c
J
N corrected for resp onding connected moments k
J
s
J
Simulations Poisson shot noise see section are given by
The cluster simulations are those generated by Croft Efs
k m N
s
tathiou a b using a particle particl e parti cl e mesh N
N k m m
s
b o dy co de Efstathiou Eastwoo d Efstathiou et al
k m m m m N
s
and are the same as those used by Dalton et al
b We consider three di erent mo dels First the stan
see e g Gazta naga Thus k N is the variance
s
dard CDM mo del which is scale invariant and has
k N is the skewness and k N also called
s s
hereafter h CDM Second a scale invariant
the kurtosis
low density CDM universe with h and a cos
To estimate the mean value of h i for the whole sample
mological constant such that H
we introduce the standard Gaussian likeliho o d function so
CDM The p ower sp ectrum for these two mo dels is taken
that the value of k in each shell is weighted by the inverse
from Efstathiou Bond White Third a scale
of its variance Var k
s
invariant mo del containing b oth a massive neutrino com
p onent eV neutrinos and CDM such that
total
k k k k
with h Mixed Dark Matter Var k
CDM s
M
s
MDM from Klypin et al The mo dels have b een nor
malised to b e consistent with the amplitude of uctuations
where we have made the assumption that the cells are in
in the microwave background Wright et al so that
dep endent Thus the weighting comp ensates for the e ects
the value of the linear theory rms mass uctuations in h
of uctuations from b oth shot noise imp ortant for distant
Mp c spheres for the two CDM mo dels and for
diluted shells or from having a small number of cells imp or
MDM realisations of each mo del were run using di erent
tant in the nearby shells We have tried two prescriptions
random phases in b oxes of side length h Mp c with
to estimate this variance The rst one uses the higher order
6
particles We have also run realisations of the moments m in the shell to estimate k in equation In
J J
the second prescription we assume a Gaussian distribution
CDM with the slightly higher normalisation Clus
ters of particles are lo cated using a p ercolation algorithm to estimate k from m as in Efstathiou et al Both
J
the cluster mass is de ned within a metric radius and the give similar results
richness limit of the cluster sample is chosen to match the For higher order moments we use the same weights for
observed cluster space density The metric radius is cho each shell as the ones used to estimate so that in general
Variance Skewness Kurtosis from galaxy clusters 3
Figure The variance R estimated from counts in cells in Figure The skewness R estimated from counts in cells in
a simulated mo ck catalogue of clusters symbols with error bars a simulated mo ck catalogue of clusters compared with the full
compared with the values from clusters selected from the same simulation as in Figure
realization of the simulation in a b ox lines The dashed line
and lled squares corresp ond to the standard CDM mo del
mask and the selection function obtained by Dalton et al
whereas solid lines and op en squares corresp ond to low density
CDM
b The ab ove counts in cells metho d is used to esti
mate h i in the mo ck survey and compare them with results
J
from direct estimation
In Figure we show the comparison for mo ck
M
s
X
and CDM catalogues against the results from the
N
s
i K h
J J
original simulated b ox In b oth cases we have included red
Var k
s
s
shift distortions Figure shows that there is go o d agree
M
s
X
ment although it should b e noted that a p erfect match is
N
s
K
not exp ected at large scales b ecause the mo ck catalogue only
Var k
s
s
samples a fraction of the total volume in the b ox These re
J
sults are for just one simulation not the average of real
N k with k evaluated for the s shell where
J J
s
J
izations showing that it is p ossible to recover the intrinsic
One of the most valuable merits of this approach is that
clustering from the catalogues using the ab ove metho d
it also provides an estimate of the variance on this mean
from the Figure shows the recovered skewness
Varh i as we have all the values in each shell and their
J
and CDM catalogues compared with the
relative weights
results from the full cubical b ox As exp ected the errors are
much larger here The errors in the mo del are even
h i i h i Varh
J J J
M
larger and it is di cult to obtain signi cant results from
just one catalogue given that the clustering amplitudes are
where M is the total number of shells and the averages use
smaller
the same weightings as equation
This is rep eated for di erent values of R the end pro d
R and its variance The cells in each uct b eing the mean
J
Results for APM clusters and dark matter
shell must not overlap to o much this is imp ortant b ecause
mo dels
otherwise uctuations in M are not taken into account in
s
the weighting and nearby shells tend to dominate the statis
Using the ab ove metho d we are able to measure correla
tics
tions with a very large cell radius up to R h Mp c
The smallest scales are limited by the shot noise in the cell
o cupancy to R h Mp c The error bars come from equa
Check of the metho d using simulated clusters
tion which provides a consistent estimate of the variance
of We have used the simulations describ ed ab ove to check our within the sample
J
metho d We rst estimate the true values of To make an accurate estimate of R using in the simulations
J J
counts in cells in redshift space in the simulation b ox in we use all clusters in each cubical b ox of side h Mp c
the standard way see e g Baugh Gazta naga Efstathiou There are clusters in each simulation In this way the
Next we transform the simulations to mimic the ge uncertainties are typically smaller than the symbols we use
ometry of the observational data applying the APM survey to compare with the data As mentioned ab ove we have also
4 Gazta naga Croft Dalton
Figure The variance R estimated from counts in cells in Figure The skewness R symbols with error bars esti
the APM clusters symbols with with error bars compared to the mated from counts in cells in the APM clusters compared to the
values in di erent simulations The continuous short dashed and values in di erent simulated mo dels as in Figure
long dashed lines corresp ond rep ectively to the estimation in real
space redshift space and redshift space with Gaussian velocity
errors The lines show the standard CDM mo del lower
lines the low density CDM variant middle lines and the MDM
mo del upp er lines
made mo ck catalogues of the simulations with the APM
mask and selection function and nd a go o d agreement
within the errors b etween the results from the mo ck cata
logue using the shell metho d and the direct results from
the cub e see x ab ove
R R and In Figures we show the values of
R obtained from the APM clusters symbols with error
bars The estimates for J b ecome very uncertain esp e
cially at large scales The real space results for in each
J
mo del are shown as continuous lines in Figures We es
for each mo del after applying redshift distortions timate
J
using the p eculiar velocities for the clusters in the simula
tions short dashed lines in gures Finally as the observa
tions are a ected by errors in the redshift we also show as
long dashed lines the estimates of in redshift space after
J
adding errors with a Gaussian distribution of disp ersion
Figure The kurtosis R estimated from counts in cells in
km s to the simulated cluster redshifts
the APM clusters symbols with error bars compared with the
Redshift distortions pro duce slightly larger clustering
values in di erent simulated mo dels as in Figure
amplitudes at large scales b ecause of coherent infall and
smaller amplitudes at small scales b ecause of random veloc
ities The crossing of the real space and the redshift space
scales to illustrate that even when R h Mp c there is
curves indicates the scale at which b oth e ects are compa
still a signi cant detection of S
rable We exp ect Gaussian errors of km s to reduce the
Note that the simulations show that redshift space dis
redshift space results at small scales H in agree
tortions pro duce only a small e ect in S as the distortions
ment with the gures J
J
In Figures and we show the hierarchical ratios S
in Fry tend to comp ensate for the distortions in
J
J
J
Gazta naga
with errors generated from the errors in The
J J
From Figure it is apparent that in the
mean values for the APM cluster are around S and
mo del has to o little p ower on scales R h Mp c This is
S At the largest scales b oth the data and the mo dels
have large errors nevertheless we show the results at these shown in Figure Although also evident in the skewness
Variance Skewness Kurtosis from galaxy clusters 5
Figure The hierarchical skewness S for the APM
clusters symbols with error bars compared with values from dif
Figure The variance R in the mo del normalized
ferent cosmological mo dels The lower panel shows the low density
to short dashed line and long dashed line
CDM mo del while the upp er panel shows the MDM mo del The
compared to R in the mo del
continuous short dashed and long dashed lines corresp ond rep ec
tively to the mo del estimation in real space redshift space and
redshift space with Gaussian velocity errors
Figure The skewness R in the mo del normalized
to short dashed line and to long dashed
Figure The hierarchical kurtosis S for the APM
line compared to R in the mo del normalized so
clusters symbols with error bars compared with the values from
that
di erent mo dels as in Figure
mo del these estimates corresp ond to just one set of random we b elieve this e ect is real it is not very signi cant if we
phases identical for each mo del With just one realization use errorbars
is not very sensitive to the it is di cult to distinguish b etween these three mo dels on While the amplitude of
normalizatio n of the mo del it do es dep end strongly on scales larger than h Mp c The results at intermediate
the value of To show this we have run one realization of scales have su ciently small errors to constrain the shap e
a CDM mo del with and h This parameter to have a value of This is in go o d
is shown in Figures and where we compare the results agreement with results from the APM galaxy angular cor
of the and mo dels with two normaliza relation function Maddox et al a Efstathiou Maddox
tions Because we only have one realization of the Sutherland
6 Gazta naga Croft Dalton
Discreteness
In b oth the observed cluster catalogues and the cluster sim
ulations the distribution of clusters is represented by a dis
crete number of p oints The main e ect of this discreteness is
to introduce additional correlations sometimes called shot
noise which are imp ortant at scales R d where d is
n n
the mean interparticle separation Typically if we have a
correlation length R which is de ned as R with
R d as in the case of galaxies then the intrinsic clus
n
d tering dominates over the shot noise For clusters R
n
so the shot noise dominates the measured clustering
This problem b ecomes evident when estimating from
J
moments of counts in cells b ecause one has to decide explic
itly what if any discreteness correction should b e applied
But the problem is equally imp ortant when estimating the
p oint correlation function directly In the latter case one
counts distances b etween pairs in the sample and compares
them with similar counts in a random Poisson distribu
tion of p oints Thus the Poisson shot noise correction has
b een widely used for clusters even when this is not always
explicitl y stated
Figure The uncorrected values of for the mo del
One could choose not to correct the measured corre
lled circles and the mo del lled triangles compared
lations for this discreteness as long as the same density of
with the values of after the discreteness Poisson shot noise
p oints is used in the comparison with mo dels and other cat
correction with op en circles triangles for the
alogues This is also the case if one regards the cluster distri
CDM mo del The Poisson shot noise correction N R
bution as intrinsically discrete and richness dep endent i e
is shown as a dashed line
characterised by its mean density and therefore by d
n
On the other hand one could try to correct for discrete
ness and estimate the intrinsic correlations of an underlying
correction on small scales where discreteness is imp ortant
density eld This estimate would b e indep endent of the
Given that we have not used the Poisson mo del in deciding
interparticle separation and would b e more useful in com
how to select clusters in the simulations this result indicates
y
parisons with mo dels or other catalogues The cluster dis
that Poisson shot noise mo del is indeed a go o d approxima
tribution can then b e imagined as a continuous eld which is
tion to the discreteness e ects
traced by our particular p oint distribution This continuous
eld is not the matter distribution but a biased transforma
tion of it As in the case of the galaxy distribution one can
further assume that the lo cation of the p oint drawn from the
DISCUSSION AND CONCLUSIONS
underlying uctuations is the result of a Poisson random
pro cess with probabili ty prop ortional to lo cal density In
We nd two common features in the cluster simulations of
this sense the Poisson shot noise mo del can b e applied to
di erent dark matter mo dels First redshift distortions are
correct for the discreteness
not very imp ortant This is in spite of the fact that di erent
In our case the Poisson correction is larger than unity
mo dels have quite di erent velocity elds e g Croft Ef
at scales R h Mp c as the mean cluster density N
stathiou b Second the values of S and S are quite
N This can b e in spheres with R h Mp c is
similar in all mo dels i e in Figures and This is in spite
seen in Figure where the uncorrected values of for
of the fact that di erent mo dels have quite di erent values
the mo del lled squares and the mo del
of the variance skewness or kurtosis Figures
after the lled triangles are compared with the values of
Both of these features can b e understo o d in terms of
discreteness correction with op en squares triangles for the
biasing Clusters are high p eaks in the galaxy mass dis
CDM mo del
tribution and they are therefore biased tracers of the mass
In the same Figure we have also plotted the Poisson
Kaiser On large scales one exp ects coherent stream
shot noise correction N R as a dashed line
ing motions to enhance redshift space uctuations Kaiser
but the relative e ect is suppressed by a factor b Note that for the mo del the shot noise contribu
tion is larger than the intrinsic clustering contribution at al l
the ratio of the mass value to the cluster value of for
scales Although we know that the intrinsic clustering prop
a biased distributio n On the other hand the amplitudes of
erties are di erent in these mo dels we can see in Figure S do not seem to b e much a ected by redshift distortions
J
how the uncorrected values of approach the Poisson
for either the mass distribution according to p erturbation
theory Bouchet et al or for galaxies according to
observations Fry Gazta naga This might explain
J
y
why for with J redshift distortions are S
J
One would then b e able to disentangle for example the ef
J
also not very imp ortant
fects of incompleteness such as random sampling and intrinsic
Why should S b e similar in all mo dels Following Fry richness dep endence in a catalogue of clusters J
Variance Skewness Kurtosis from galaxy clusters 7
with a Gaussian kernel radius R We nd that the vari Gazta naga a lo cal bias relating galaxy or matter
ance and skewness are largely overestimated in comparison uctuations to cluster uctuations
c
with the N b o dy results In our test the comparison is in
1
X
b
k
k
real space and the discrepancy is for R and R
x x f x
c
k
h Mp c This suggests that although Plionis et al use re
k
sults from a di erent observational catalogue of clusters the
where b b pro duces amplitudes for the cluster distribu
combined Ab ell ACO catalogue to make their quantitative
tion S and S of
comparison their conclusions are di erent and in our view
less trustworthy than those which would b e reached using
S b S c
m
full N b o dy simulations
S b S c S c c
m m
In summary we have shown that in the case of clus
ter samples discreteness has an imp ortant e ect on clus
where c b b and c b b are the resp ective quadratic
tering statistics at all scales The Poisson shot noise mo del
and cubic contributions from the biasing transformation in
has b een widely used implicitl y when estimating the p oint
equation and S are the hierarchical amplitudes in
J m
correlation function for clusters Here we have argued that
the underlying matter distribution Here we have used the
the Poisson shot noise correction should indeed b e applied
fact that b oth the galaxy distribution in the APM and the
when estimating correlations in the cluster distributio n
matter distribution in the simulations follow the hierarchi
J
J
We repro duce previous conclusions by Dalton et al b
cal relation see e g Gazta naga Baugh S
J
J
based on the p oint correlation function mainly that the
Gazta naga Efstathiou From the ab ove expression
standard CDM mo del is inconsistent with the second
we see that the underlying amplitudes S are typically
J m
order statistics at all scales As a new result we nd that this
suppressed by b when selecting high p eaks That the val
mo del is inconsistent with either the third or fourth order
ues of S and S we nd are quite similar in all mo dels
statistics at all scales A Mixed Dark Matter mo del gives
even when the matter amplitudes S and S are signi
m m
go o d agreement as do es a low density CDM variant b oth
cantly di erent see Juszkiewicz Bouchet Colombi
of these mo dels b eing consistent with the observations
Gazta naga Baugh indicates that the biasing contri
J
for J at the level We have also shown that it is
bution in equation is signi cant For example the typical
p ossible to measure S in the cluster distributio n at scales
variation b etween mo dels of S for the mass distribution
m
as large as R h Mp c The hierarchical correlations
at large scales is S Everything b eing equal the cor
m
found in the cluster distribution are consequence of similar
resp onding variation in the cluster distributio n from equa
relations in the underlying matter eld In a forthcoming
for the typical value b Given the tion is S
pap er we use the relations b etween the cluster values of
uncertainties in the measured values of S it is not surprising
S and the correp onding values for the matter distribution
J
that this small variation is undetected
e g equation to investigate the biasing pro cess in more
Recently Plionis Valdarnini and Cappi
detail in preparation
Maurogordato have estimated S and S from an
gular and redshift p ositions of Ab ell and ACO clusters The
Acknowledgements
hierarchial ratios shown in Figures and S and
S are similar to their redshift space results but are
We would like to thank George Efstathiou for helpful
slightly smaller than the ones found by Cappi Mauro
comments on a previous version of this manuscript EG ac
gordato in the angular distribution S and S
knowledges supp ort of a Fellowship from the Commission
It is not clear to us how signi cant this discrepancy is as
of Europ ean Communities and from CSIC Consejo Sup e
Cappi Maurogordato use b o otstrap errors which
rior de Investigaciones Cienti cas in Spain RACC acknowl
are unreliable see Baugh Gazta naga Efstathiou
edges receipt of a PPARC studentship
On the other hand uncertainties from the depro jection of an
gular into D statistics are not taken into account by Cappi
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Nichol R C Collins C A Guzzo L Lumsden S L MNRAS p
Variance Skewness Kurtosis from galaxy clusters 9
know whether clusters do indeed form at sites that can b e APPENDIX A CLUSTERS IN THE
predicted by weakly non linear evolution of the density eld ZEL DOVICH APPROXIMATION
If this do es o ccur then it could give supp ort to the idea that
In this app endix we are concerned with testing some asp ects
statistical clustering or high p eak biasing is mainly re
of the reliabili ty of theoretical predictions of cluster cluster
sp onsible for the distribution of clusters On the other hand
ing In particular we compare our N b o dy simulations with
if we are not able to nd a reliable corresp ondence b etween
simulations that use the Zel dovich approximation to
TZA and N b o dy clusters the matter distribution may still
evolve the density eld Such simulations have b een used
b e similar for b oth metho ds when smo othed on large scales
as a computationally inexp ensive way of making predictions
but in the TZA we will have b een unable to predict cor
ab out clustering and higher order moments of clusters for
rectly the lo cations of clusters that we are using to trace
a wide range of mo dels Plionis et al and references
that matter distribution
therein
A Comparison of Truncated Zel dovich
Approximation and N b o dy clusters A Alternative metho ds of simulating clusters
We test which of these is in fact the case by evolving the mat The two point correlation function of simulated rich clus
ters evolves weakly with time Croft Efstathiou a
ter distribtion for the same initial random phases using the
TZA and the P M N b o dy co de In this example case we will This is b ecause much of the clustering measured is statis
use the CDM mo dels normalised so that tical in the sense that the clusters were b orn clustered as
would happ en if they formed at the lo cations of high p eaks
The input p ower sp ectrum of the TZA run was truncated
Mp c which by use of a Gaussian lter of radius r h in the initial density eld Kaiser Less time consum
g
Plionis et al nd is optimal for this particular mo del ing metho ds than the use of full N b o dy simulations might
therefore b e adequate to mo del the statistical clustering
The nal particle distributio n for a h Mp c thick region
is shown in Figure A b ii and can b e compared directly prop erties of clusters Analytical approaches based on high
p eak theory eg Mann Heavens Peacock predict
with the N b o dy particle distribution Figure A a ii It
the same trends as N b o dy simulations the cluster correla
can b e seen that there are broad similarities b etween the
two density elds There are however fewer obvious clumps tion amplitude increases weakly with cluster richness for ex
ample To make accurate predictions which can b e compared
in the TZA density eld We are exp ecting this as small scale
quantitatively with observational samples though it is cer
p ower has b een removed from the p ower sp ectrum and also
no mechanism exists within the dynamical scheme to form tainly b etter to identify clusters with p eaks in the evolved
density eld
structure into virialised ob jects We must therefore decide
This is the view that has b een taken by Plionis et al
where such ob jects would have formed Plionis et al assume
that this would o ccur at p eaks in the TZA density eld and in their study of cluster correlations The dynamical scheme
we will follow them in de ning clusters to b e at the p ositions which they use is the Truncated Zel dovich Approximation
TZA which has b een tested by several authors e g Coles
of p eaks However examination of the TZA particle distri
bution shows that there are many smo oth lo oking regions Melott Shandarin Sathyaprakash et al and
which we would like to break up in order to repro duce the found to give a surprisingl y go o d indication of the b ehaviour
of gravitating matter even into the non linear regime It
N b o dy results
To nd clusters we grid the nal particle density elds us should b e noted that agreement found with N b o dy sim
ing a TSC scheme Ho ckney Eastwoo d onto a ulations was only go o d on scales much larger than the typ
ical radius of galaxy clusters We also note that it fails to
mesh and then apply Gaussian smo othing The p ositions of
the highest p eaks in the smo othed eld are taken as repro duce in detail the amplitudes of the higher order cor
the lo cations of the richest clusters This technique do es relations S of the mass at all scales e g Juszkiewicz et al
J
When the original Zel dovich approximation
have a number of disadvantages such as the fact that reso
lution is lost b elow the scale of the grid and also due to the is applied to a particle density eld all particles move along
smo othing pro cedure There is an additional anti correlation their initial tra jectories with constant velocity When par
ticles tra jectories cross one another structure which would
e ect arising from the fact that p eaks cannot lie in adjacent
grid cells This is discussed further b elow when we calculate have formed under the action of gravity is erased For this
clustering statistics To check the sensitivity to di erences reason in the improved version the truncated ZA the initial
in cluster nding techniques of the N b o dy results we ap p ower sp ectrum is truncated by smo othing with a gaussian
ply the ab ove pro cedure rst to the N b o dy outputs We try lter to remove the p ower at small wavelengths which is
lters of radius r and h Mp c As the inter resp onsible for the bulk of orbit crossing Using the TZA
f
particle separation in cluster cores is small we should not to evolve the density eld rather than a full N b o dy co de is
have to smo oth much in order to remove any discreteness computational ly much cheaper meaning that large numbers
e ects in fact using the unsmo othed grid do es give go o d of mo dels can b e run and tested This is p otentially of b ene
results In gure A a ii we compare the p ositions of these t as for example Plionis et al have run simulations
N b o dy clusters for r h Mp c with those identi ed of di erent universes with which it would b e interesting
f
using p ercolation It can b e seen that there is an extremely to compare our APM cluster statistics
go o d corresp ondence b etween the two cluster catalogues as It is therefore imp ortant to test the reliabili ty of this sort
we would exp ect The panel A a ii shows the distribution of scheme particularily as we have reached the stage where
of particles taken from the h Mp c square region indi cluster samples are large enough that we can measure higher
cated in the left hand picture order statistics with relatively small errors We would like to
10 Gazta naga Croft Dalton
Figure A h Mp c thickness slice through simulations of CDM universe with run using a a P M N b o dy
co de and b the truncated Zel dovich approximation The solid dots in panels a i and b i show the p ostions of clusters identi ed in
these simulations as high p eaks after smo othing the density with a Gaussian lter of radius h Mp c The op en circles show clusters
identi ed as p eaks with Gaussian smo othing of with lter radius h Mp c in panel b i and p ercolation groups in panel a i In panels
a ii and b ii we zo om in on the h Mp c square region enclosed by the solid lines in a i and b i showing the distribution of
particles in the N b o dy and Zel dovich simulations resp ectively
discreteness problems as the p eak regions do not have such We apply this cluster nding scheme to the TZA simula
a high particle density We have run TZA simulations with tion Because the nature of ob jects to b e identi ed with
particles in the same sized b ox in order to check on clusters is not obvious from the particle plot we try sev
the e ects of varying mass and spatial resolution Our re eral di erent lter sizes r and h Mp c We
f
covery of similar results to the tests with particles sug then lo ok for p eaks in the smo othed eld The results for
gests that the mismatch b etween TZA and N b o dy clusters r h Mp c and h Mp c are shown in Fig A b i
f
is caused mostly by the TZA dynamical scheme itself b eing where we plot those of the highest p eaks which fall in a
inadequate slice of thickness h Mp c Again it is instructive to com
pare with the N b o dy results We can see that neither value
of r pro duces a go o d corresp ondence b etween TZA clus
f
ters and the N b o dy clusters It also seems that the obvious
A Clustering statistics and Zel dovich clusters
caustics which the TZA pro duces are places where maxima
It is p ossible though that the two sets of ob jects can give
are preferentially selected particularil y when the smo othing
statisticall y similar results on the large h Mp c
radius is small A larger lter scale r than with the N b o dy
f
scales used in comparison with observations and on which
results should b e necessary in order to avoid any particle the TZA should b e most useful b ecause the density eld is
Variance Skewness Kurtosis from galaxy clusters 11
Figure A The variance r of clusters identi ed from simulations of a CDM universe with run using a P M N b o dy
co de and the truncated Zel dovich approximation The error bars represent the rms error on the mean calculated from the scatter b etween
realisations In the N b o dy results we show r for clusters selected from p ercolation groups solid line and from p eaks in the density
eld op en symbols smo othed with a Gaussian lter of radius and h Mp c The lled symbols corresp ond to the di erent
results obtained after using Gaussian lters of radius and h Mp c to smo oth the TZA evolved density eld Clusters were
identi ed as the highest p eaks in this eld giving a mean intercluster separation of h Mp c
strongly on the smo othing radius used to select clusters It more linear We have calculated the variance r for the
would app ear that we are selecting a di erent p opulation of TZA and N b o dy clusters the results b eing shown in Fig
ob jects as the smo othing radius is changed With a small ra A At this p oint it is worth mentioning that the metho d
dius there is marked excess clustering as those p eaks in the of selecting clusters from a grid which is necessary in the
initial density eld which would collapse to form large single TZA case a ects the correlations on small scales The anti
clusters are instead broken up into many smaller ob jects Us correlation introduced by the fact that p eaks must b e a min
ing r gives the closest results to the N b o dy clusters imum of grid cells apart will make r arti cially low on
f
although r is still marginally to o low It is imp ortant to small scales The Gaussian smo othing radius r used on the
f
r realise that when comparing a range of Zel dovich mo dels evolved eld b efore p eak selection will also depress
This e ect is visible in the plot for the N b o dy results us realistical ly with observations we should have an a priori
ing larger r results in deviations from the p ercolation curves reason for choosing the smo othing scale The b est scale may
f
out to larger r The TZA clusters should also b e a ected by dep end on the shap e of P k or the amplitude of uctua
b oth problems but it di cult to decide from the plot b e tions We have seen here that varying r b etween and
f
cause we do not know what prop erties a TZA sample with h Mp c gives enormously di erent results for r Also it
no gridding or smo othing would have We will therefore con apparent from the plots of cluster p ositions that even with
centrate on scales h Mp c in the rest of our discussion the b est radius we are not selecting the same ob jects as in
the N b o dy output This may imply that other prop erties
The N b o dy values of r calculated after using p ercola
of the cluster density eld are not reliably predicted by this
tion and p eaks to nd clusters are almost identical The
metho d
value of r which gives the worst agreement h Mp c
f
is probably to o large for the compact ob jects present in the
N b o dy output and in any case using a grid and lter is
not our metho d of choice for selecting N b o dy clusters For the TZA clusters however the correlation strengths dep end
12 Gazta naga Croft Dalton
A Discussion
As there is no clear visual corresp ondence b etween the N
b o dy and TZA clusters the exact nature of the ob jects we
are selecting in the TZA density eld is unclear When we
use a small r or the unsmo othed eld then it is p ossible
f
that we are a ected by particle discreteness although our
tests with higher resolution TZA simulations show that this
is not a ma jor problem here This suggests that some of the
ob jects found in the TZA output are artifacts of the limited
dynamics as are for example the shells of matter that have
undergone obvious orbit crossing By smo othing the initial
p ower sp ectrum as we are forced to do and b ecause the
TZA do es not work well on small scales we are losing infor
mation which we need to lo cate where virialis ed ob jects will
form This is related to the problem that the e cacy of the
TZA will dep end on the overall amplitude of uctuations
For higher normalisations we will need a larger truncation
scale on the p ower sp ectrum to suppress the increased orbit
crossing The ob jects that form will corresp ond even less to
those in the fully non linear calculation This implies that
the TZA could p erform b etter in comparison with N b o dy
simulations of the other mo dels with di erent p ower sp ectra
and lower which we have not tested here On the other
hand as the galaxy clusters which we are using to trace the
density eld are by de nition compact ob jects and proba
bly virialise d our theoretical predictions should b e able to
predict the lo cation of these ob jects The Zel dovich Approx
imation whilst repro ducing some other asp ects of the large
scale evolution of the density eld is unable to do this
In summary we nd that we can reliably select clumps of
matter which we identify as galaxy clusters from N b o dy
simulations with di erent selection metho ds pro ducing al
most identical results This gives us con dence that our the
oretical calculations of will b e robust We have also tested
J
a metho d for generating simulated cluster catalalogues us
ing the Zel dovich approximation and compared the clusters
selected using this metho d with those in our own fully non
linear calculation This leads us to conclude that the use of
clusters from the Zel dovich approximation is not really suit
able when engaging in a quantitative comparison of theories with observations of cluster clustering