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Variance, Skewness & Kurtosis: Results from the APM Cluster

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Variance, Skewness & Kurtosis: Results from the APM Cluster a Mon Not R Astron So c Printed January MN LT X style le v E Variance Skewness Kurtosis results from the APM Cluster Redshift Survey and mo del predictions E Gazta naga RAC Croft G B Dalton Department of Physics Astrophysics University of Oxford Nuclear Astrophysics Laboratory Keble Road Oxford OX RH CSIC Centre dEstudis Avancatsde Blanes Astrophysics Blanes Girona Spain ABSTRACT We estimate the variance the skewness and the kurtosis in the distribution of 2 3 4 density uctuations in a complete sample from the APM Cluster Redshift Survey with 1 clusters and a mean depth D h Mp c We are able to measure the statistics 1 of uctuations in spheres of radius R h Mp c with reasonable errorbars The J 1 statistics in the cluster distribution follow the hierarchical pattern S with J J 2 S roughly constant S and S We analyse the distribution of clusters taken J 3 4 from Nb o dy simulations of dierent dark matter mo dels The results are compared with an alternative metho d of simulating clusters which uses the truncated Zeldovich approximation We argue that this alternative metho d is not reliable enough for making The Nb o dy simulation results follow similar hierarchical quantitative predictions of relations to the observations with S almost unaected by redshift distortions from J p eculiar motions The standard Cold Dark Matter CDM mo del is inconsistent with either the second third or fourth order statistics at all scales However b oth a hybrid Mixed Dark Matter mo del and a low density CDM variant agree with the J observations Key words Largescale structure of the universe galaxies clustering metho ds numerical statistical limited samples of Xray clusters Romer et al give INTRODUCTION mutually consistent results The p oint correlation of the Rich clusters of galaxies can b e used to trace the structure APM Automatic Plate Measuring machine Cluster Red of the Universe on very large scales Although the identi shift Survey has already b een used to constrain dark matter astro-ph/9501115 31 Jan 1995 cation of clusters is more dicult than that of galaxies mo dels Dalton et al b clusters have several imp ortant advantages as tracers b oth Previous analysis of p oint and p oint statistics in observations and in simulations Since clusters represent and of cluster samples have b een based on Ab ells cat high densities in the galaxy distributio n it is p ossible to alogue and extensions by Ab ell Corwin Olowin use them to map out the distant Universe in a systematic eg Jing Zhang TothHollosi Szalay Jing and economical way As clusters are asso ciated with high Valdarnini Plionis Valdarnini Cappi Mau p eaks in the mass distribution cluster correlations evolve rogordato These samples seem to repro duce the hi very slowly with time and so the clustering statistics deter J found in the galaxy distribu erarchical relation J mined from numerical simulations are relatively insensitive tion eg Groth Peebles Fry Peebles Sharp to the ep o ch at which the simulations are analysed Croft et al Szapudi Szalay Boschan Meiksin Sza Efstathiou a An additional advantage is that clusters pudi Szalay Bouchet et al Gazta naga are strongly correlated so that even a small sample can b e These observations could provide imp ortant insights used to estimate the correlation functions into the underlying matter distribution and the formation of The p oint correlation function for galaxy clusters was clusters However there is evidence that the clustering mea rst analysed in Ab ells catalogue eg Hauser sured from Ab ells catalogue is aected by inhomogeneities Peebles Bahcall Soneira Klypin Kopylov in the selection of clusters eg Efstathiou et al and Recent determinations based on automated cluster it is not clear how this could aect the ab ove mentioned catalogues Dalton et al Nichol et al and on ux estimates of and Here we estimate the variance the skewness and Throughout the pap er we use H h kmsMp c the kurtosis in the distribution of density uctuations 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 ells We compare the results with previous analysis scheme advocated by Plionis et al makes use of the and with dierent cluster simulations Zeldovich 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 Nb 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 Zeldovich approximation and carried out a comparison with Nb 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 Zeldovich 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 eects automatically by clusters with mean redshift and spacedensity 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 bc 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 kms 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 kms 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 kms 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 shotnoise see section are given by The cluster simulations are those generated by Croft Efs k m N s tathiou ab using a particleparticl eparti cl emesh 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 dierent mo dels First the stan see eg Gazta naga Thus k N is the variance s dard CDM mo del which is scaleinvariant and has k N is the skewness and k N also called s s hereafter h CDM Second a scaleinvariant the kurtosis lowdensity 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 Vark 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 Vark 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 eects the value of the linear theory rms mass uctuations in h of uctuations from b oth shotnoise 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 dierent 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 dened 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 countsincells in Figure The skewness R estimated from countsincells in a simulated mo ck catalogue of clusters symbols with errorbars 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
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