aa r X i v : . [ a s t r o - ph ] O c t Draft version October 30, 2018
Preprint typeset using L A TEX style emulateapj v. 08/22/09
THE SHAPES OF GALAXY GROUPS - FOOTBALLS OR FRISBEES ?
Aaron Robotham, Steven Phillipps
H. H. Wills Physics Laboratory, University of Bristol,Tyndall Avenue, Bristol, BS8 1TL, United Kingdom andRoberto De Propris
Cerro Tololo Inter-American Observatory, Casilla 603, La Serena, Chile
Draft version October 30, 2018
ABSTRACTWe derive probability density functions for the projected axial ratios of the real and mock 2PIGGgalaxy groups, and use this data to investigate the intrinsic three dimensional shape of the dark matterellipsoids that they trace. As well as analysing the raw data for groups of varying multiplicities, aconvolution corrected form of the data is also considered which weights the probability density functionaccording to the results of multiple Monte-Carlo realizations of discrete samples from the input spatialdistributions. The important effect observed is that the best fit distribution for all the raw data isa prolate ellipsoid with a Gaussian distribution of axial ratios with ¯ β = 0 .
36 and σ = 0 .
14, whilstfor the convolved data the best fit solution is that of an oblate ellipsoid ¯ β = 0 .
22 and σ = 0 . σ confidence limits. Subject headings: galaxies: clusters: general; galaxies: halos INTRODUCTION
The spatial distribution of galaxies traces the shape ofthe dark matter potential in which they are embedded.Simulations show that dark matter halos are not spheri-cal, as one would naively expect because of dark matter’snon-dissipational nature, but are strongly flattened tri-axial ellipsoids (Dubinski & Carlberg 1991). Althoughprolate and oblate shapes are equally likely in the simu-lations, dissipative infall of baryonic gas eventually forcesthe halo shapes toward pronounced oblateness with ax-ial ratios b/a > . Electronic address: [email protected]
However, star counts (Lemon et al. 2004) and analysis ofthe stellar stream of the Sagittarius dwarf (Ibata et al.2001) support a spherical halo. It is also uncertainwhether the dwarf satellites can be used as test par-ticles, as they may not originate from cosmologicalsub-structures (Kroupa, Theis & Boily 2005).Groups of galaxies are likely to be the best testbedsto study the shapes of dark matter halos. Largegroup catalogs are now available from redshift sur-veys (e.g., the 2dF Percolation Inferred Galaxy Groups[2PIGG] of Eke et al. 2004 and the group catalogof Merch´an & Zandivarez 2005 from Data Release 3of the Sloan Digital Sky Survey) and these allowa statistical approach to the study of the shapesof groups and the shape dependence on richness,multiplicity and dynamical evolution. Early studiesmarginally favoured prolate shapes (Fasano et al. 1993;Orlov, Petrova & Tarantaev 2001), but not at the ex-clusion of oblate solutions. The most recent study of2PIGG groups by Plionis, Basilakos & Ragone-Figueroa(2006) favours prolate groups, but is obtained by meansof a multiplicity cut and as such represents a differentmethod to the one presented here. Prolate results indi-cate that the original (oblate ?) shapes may have beenstrongly modified during gravitational collapse, or thatfilamentary collapse is in fact being witnessed.In this paper we carry out a detailed analysis of theshape of 2PIGG groups, coupled with extensive MonteCarlo simulations and comparison with groups extractedfrom the 2dF mock catalogs used by Eke et al. (2004)to optimize group selection. In section 2 we present ourmethod for analysing the shape of observed groups andin section 3 apply this to the real and mock 2PIGG cat- Robotham et al.alog (Eke et al. 2004). Comparisons are made betweenthe raw data and data that is corrected for the finite(indeed, often sparse) sampling by factors which we de-termine from Monte-Carlo simulations of suitably popu-lated groups. Furthermore, the mock and real data areconsidered separately and KS tests are used to confirmtheir distributional similarities. MODELLING SHAPE DISTRIBUTIONS
To extract useful information from our projected groupshapes we first make some simplifying assumptions.With some theoretical justification (Jing & Suto 2002)we assume that our groups have three orthogonal axiswith two of equal length, but allow our groups to beeither prolate or oblate. This greatly simplifies the mod-elling compared to the case of general triaxial shapes.The problem is generally one of inversion, for which thereis no unique solution in case of triaxial shapes. However,it is still possible to recover whether the distribution ismore prolate or oblate-like, as will be discussed later.For now, we shall also make the assumption that weare able to measure the apparent group axial ratios per-fectly. Obviously this is not the case in practice sincethe group projection is represented by discrete galaxies,but we consider the effect of this in detail later. Figure 1describes the simplified problem we are considering. Inthe diagram shown the object is oblate and symmetricabout the z axis. C is where the tangent to the observer(at the point x , z in the x,z plane) cuts the z-axis, Ais the distance between the origin and the tangent to theobserver along the line normal to the tangent, a is wherethe ellipsoid cuts the z-axis and θ is the angle betweenthe z-axis and the tangent to the observer. The axialratio for this projected image is found by considering thedistance between the tangent lines to the observer (2A)and the larger axis that is projected normal to this on thesky (twice the radius of the ellipsoid in the x,y plane).The ratio between these values will tell us the observedaxial ratio for each given θ and ellipsoid shape.Knowing this we can investigate what the probabilitydensity function (PDF) of axial ratios would be for anygiven ellipsoid evenly sampled in spherical coordinate pa-rameter space. For a prolate or oblate object we have thestandard form ( ux ) + ( uy ) + z = a (1)where u > u < a would be the radius of the sphere for thecase u = 1.We can then calculate the projected shape of the el-lipsoid on the plane of the sky when viewed at an ar-bitrary angle θ by generalizing the results on the pro-jected ellipticities of disc or elliptical galaxies from, e.g.,Hubble (1926), Sandage, Freeman & Stokes (1970) andMilhalas & Binney (1981); see appendix for details. Theprojected axis length A (Fig. 1) is given by A = a (cid:18) a z (cid:19) sin θ = a (cid:18) cos θu + sin θ (cid:19) (2)so the apparent axial ratio q (defined to be less than 1for both oblate and prolate shapes: i.e. q = uAa or auA respectively) is given by u sin θ + cos θ = q ( oblate )= 1 q ( prolate )If we have a distribution of intrinsic shapes given by n ( u ) and random orientations then we can show (againsee appendix for details) that the distribution of apparentaxial ratios will be f ( q ) = q Z q n ( u ) du [(1 − u )( q − u )] . ( oblate ) (3) f ( q ) = 1 q Z q u n ( u ) du [(1 − u )( q − u )] . ( prolate ) (4)since only values of u < q can contribute. If we writethe intrinsic axial ratio as β < β = u for oblateand β = 1 /u for prolate shapes), then for a given β , thenumber of systems with an observed q between, say, q and q for oblate and prolate ellipsoids respectively is N ( β, q , q ) = 1 p − β Z q q qdq p q − β = hp q − β − p q − β i q q p − β = 1 p − β Z q q β dqq p q − β = (cid:20) √ q − β q − √ q − β q (cid:21) q q p − β So for a distribution of true axial ratios ˜ n ( β ), the ap-parent axial ratios for oblate ellipsoids follow N ( q , q ) = Z q ˜ n ( β ) 1 p − β (cid:20)q ( q − β ) − q ( q − β ) (cid:21) dβ (5)and for prolate ellipsoids N ( q , q ) = Z q ˜ n ( β ) 1 p − β " p ( q − β ) q − p ( q − β ) q dβ (6)The simplest distribution to try would obviously bea delta function ˜ n ( β ) = δ ( β ) for some β , however amore sensible assumption for the underlying axial ratiofunction (or at least the first that should be tried) wouldbe a Gaussian distribution, where˜ n ( β ) = r πσ exp (cid:18) − ( ¯ β − β ) σ (cid:19) (7)We will use this model in what follows. METHODOLOGY
The largest publicly available group samples are the2dF Percolation-Inferred Galaxy Groups (2PIGG) cat-alog (Eke et al. 2004), selected from the 2dF GalaxyRedshift Survey (2dF-GRS – Colless et al. 2001), theYang 2dF group catalog (Yang et al. 2005), and the YangSDSS group catalog (Weinmann et al. 2005). For thehe Shapes of Galaxy Groups 3
Fig. 1.—
Geometric description of ellipsoid viewing problem. purposes of this paper only the 2PIGG catalog and itsassociated mock catalog will be used; the other catalogswill be used in a more comprehensive paper (in prepa-ration) comparing the different grouping methods. Asin our previous work (Robotham et al. 2006) we beginby selecting all groups from this catalog with 0 . 10. The z = 0 . 05 lower limit is motivated by thesmall volume sampled by 2PIGG at low redshift (so thatgroups may not be representative), while at z = 0 . 10 the2dF GRS apparent magnitude limit begins to excludeeven moderate luminosity galaxies ( M B < − 18) fromthe sample (making the members less representative).It is not trivial to decide how many discrete objects yourequire in an elliptically distributed sample to accuratelydetermine the underlying shape. To resolve this problem,Monte-Carlo tests were performed using a range of initialspatial distributions randomly populated by a varyingnumber of discrete points. Using the moments method ofCarter & Metcalfe (1980) to determine ellipticity (foundto be preferential to the flattening technique of (Rood1979) in (Plionis, Basilakos & Tovmassian 2004)), thesesimulated discrete points were analysed and comparedto the true elliptical distributions that produced them.This method of simulating and measuring was carriedout 1,000 times for each multiplicity of group distributedwith an ellipticity defined by ε = 1 − β (i.e. an ellipticityof 0 would be a circle). The multiplicity was varied insteps of 5 up to 300, and ε in steps of 0.05 from 0 upto 0.95. With this extensive sample available, compar-isons of standard deviations in determined ε o for differentsimulated multiplicities were made versus the intrinsic ε i R e l a t i v e e rr o r Multiplicity Fig. 2.— Relative error of measured axial ratio, overplot withthe best fit function. used to generate the samples. It was discovered thatthe observed ellipticity correlates almost linearly withthe true ellipticity, error distributions are very close toGaussian, and the standard error for a measured ellip-ticity is directly proportional to the axial ratio (whendefined to be < n ) we can correct the ellipticity. Theempirical correction function is ε i = 1+ ε o − . − (cid:0) . N (cid:1) − (cid:0) . N (cid:1) + 1 . × − N ! (8)However, this systematic correction is not required Robotham et al.usually since it is so small compared to the standarderror of the distribution of ε i for a given ε o . Using thesame data, a fit has been made of the relative axial ratioerror against multiplicity (see figure 2). The standarderror is given by σ N = β (cid:18) . (cid:18) . N (cid:19) + (cid:18) . N (cid:19) − . × − N (cid:19) (9)This function is plotted against the data in figure 2.From this we see that independent of the underlying ε i ,20 objects are required to have a relative axial ratio errorof 20%. If we have 20 objects in a group with measuredellipticity of 0.2 ( β = 0 . σ will be 0.16, whilst for amore elliptical group with ε = 0 . β = 0 . σ as stated.So far we have considered the correction required foreach individual observation, i.e. the most likely intrin-sic ellipticity for a particular observed ellipticity for acertain multiplicity. In practice the rather complex dis-tortions for axial ratios near unity are best accountedfor by considering the amount of distortion between theinput and measured axial ratio distributions. Using thisMonte-Carlo data it is trivial to find the correction fac-tor between a uniform PDF of input axial ratios andthe measured non uniform distribution for each multi-plicity (see Fig. 3). This technique has the advantage ofcorrecting for the lack of axial ratios ∼ χ of 6.25 compared to 27.5 for the uncorrecteddata). For the oblate group of 20 objects with an ax-ial ratio of 0.5 the best fit was with the corrected data,returning an oblate fit with mean axial ratio of 0.48 (re-duced χ of 2.04 compared to 63.37 for the uncorrecteddata). In the later case the result was particularly sig-nificant because the oblate fit was strongly rejected bythe raw data alone, despite us knowing that this shouldbe the returned best fit distribution. These results areencouraging since the convolution correction is best ap-plied to a large amount of input data, such as the catalogdata that we have available. The caveat when using thistechnique is that extremely sharp features will always be P r obab ili t y D en s i t y Axial ratio of projected ellipsoid Original uniform p.d.f.multiplicity 5multiplicity 10multiplicity 20multiplicity 50multiplicity 100multiplicity 200multiplicity 300 Fig. 3.— Observed PDF for different multiplicities plotted to-gether with the uniform input PDF used for the Monte-Carlo tests. smoothed out in low multiplicity data since the axial ra-tios themselves are not re-binned, only re-weighted; thiseffect being obvious in 4. The type of distribution andthe mean are reliably recovered however, and this is thethe matter of most importance.These two different techniques, simply correcting themeans of the distributions for sampling effects or usingthe whole PDF, can be used to produce two different dis-tributions of axial ratios. When used on test data theydisplay good agreement for large multiplicity samples; di-vergence at large axial ratios is always present however,since for any multiplicity the Monte-Carlo tests stronglysuggest axial ratios near 1 are observationally inhibited.This is significant since it is the axial ratios near 1 thathelp us to distinguish between prolate and oblate under-lying distributions: oblate distributions plateau at largeaxial ratios, whilst prolate distributions will drop to ex-tremely low values.Considering the first of these two methods for creatingour PDFs, a cut-off was applied to the sample, requiringa minimum of 20 galaxies in order to achieve an accept-able level of confidence in the ε values returned. In addi-tion, a lack of sufficient multiplicity will always have theeffect of increasing the ellipticity measured; the obviousextreme example would be a multiplicity of 2, where themeasured ellipticity will always be 1, no matter the distri-bution that created them. Obviously, the ideal situationwould be to increase the multiplicity limit further, andit is true that this would increase the confidence in ourmeasured ellipticities, but two factors militate againstthis: first we are interested in a range of group multi-plicities, so would like to consider as broad a sample aspossible, and second we need as many measured elliptici-ties as possible in order to produce a well defined binnedPDF ( N ( q )). The remaining groups were analysed in thesame fashion as the Monte Carlo models and sorted intoaxial ratio bins of size 0.05 from 0 to 1, creating a binnedPDF that can be compared to the distributions predictedfor the models in Section 2.For the second method all group multiplicities are used,the only limitation being those of the original data set;i.e. multiplicities of 5 or more. A convolving routinewas written in order to weight the significance of eachresult, which is a combination of the multiplicity and theaxial ratio measured. These results were then placed inthe same bins as used for the previous technique, allow-ing for direct comparisons. To aid analysis Monte-Carlohe Shapes of Galaxy Groups 5 p r obab ili t y den s i t y axial ratio corrected dataraw datainput ellipsoid ob 0.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p r obab ili t y den s i t y axial ratio corrected dataraw datainput ellipsoid pro 0.5 Fig. 4.— Raw and corrected PDF plotted against the input ellipsoid data for a given group using the HEALPix method to sample therotation parameter space evenly. On the left is a prolate group of 30 objects with an axial ratio of 0.5. On the right is an oblate group of10 objects with an axial ratio of 0.5. distributions were also created, these correspond to thePDF that would have been observed had the true PDFof axial ratios been uniform for each group multiplicity.As a point of comparison figure 5 shows how our LocalGroup appears projected on the sky at different mag-nitude cuts (data taken from Pritchet & van den Bergh1999), evenly sampled using the HEALPix isolatitudepixelization method to ensure a representative distribu-tion of convolution corrected and non corrected projec-tions of the Local Group in our rotation parameter space.The effect of the correction is most evident on the lowmagnitude cuts where the plateau at larger axial ratiosis suppressed in the raw data, but after correction theplateau continues more evenly. This suggests the LocalGroup is oblate at low magnitude cuts. The extremelysmall axial ratios at high magnitudes is understandablein the context of the Local Group having significant sub-structure, and at high magnitude cuts only the regionsaround the Milky Way and M31 contribute. This createsa dumbbell structure that is increasingly smoothed outas fainter members are added, hence the systematicallylarger axial ratios as the magnitude limit is increased. CONTAMINATION BY INTERLOPERS A problem for any attempt to collect galaxies into as-sociated groups and clusters is the unavoidable presenceof interlopers, as such it seems prudent to quantitativelydefine what the effect is. The situation is improved bythe use of spectroscopy since this requires interlopers tohave chance projections and chance recessional velocities,however it is clear this still won’t eliminate all interlop-ers, just reduce the percentage of the population that arefalse. The 2PIGG catalog is spectroscopically selected, soit is as good as reasonably possible. Figure 4 of Eke et al.(2004) shows three plots to describe the typical interloperrates in groups of different mass, redshift and multiplic-ity. As might be expected, interloper rates get worse withthe largest multiplicities (hence mass) and at the high-est red shifts, however the redshift cut we have appliedat 0.1 indicates a very steady interloper rate of 20% to30%. Over our regime of group masses ( ∼ M ⊙ to ∼ . M ⊙ ) and multiplicity (5 to 163) the 2PIGG cat-alog is also predicted to maintain a consistant interloperrate.Since interlopers are an inevitable feature of our cata-log it is of paramount importance to discern what effect they have. Assuming a given group has an intrinsic shapeseen in projection, the consequence of interlopers will beto add a circularly distributed population on top of anyprojection of the same maximum allowed radius. Thisis obvious since any grouping algorithm merely considersthe magnitude, radial separation and, possibly, redshiftof any potential new member. Intrinsically no restrictionis made as to how objects affect the shape of the currentlygrouped objects. With this in mind a Monte-Carlo sim-ulation was designed to find how mixing together inrinsi-cally shaped ellipsoids of oblate and prolate varieties withan interloper population can distort the observed results.Of particular interest is how the same rate might affectdifferent multiplicities.For the sake of interest a large variety of interloperrates were investigated, from 2% to 50%, although ourmain interest will be the interloper rate ∼ p r obab ili t y den s i t y axial ratio Mv<-17.1Mv<-16Mv<13.1Mv<-11.3Mv<-8.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p r obab ili t y den s i t y axial ratio Mv<-17.1Mv<-16Mv<13.1Mv<-11.3Mv<-8.5 Fig. 5.— Local group projected axial ratios at different V magnitude cuts using the HEALPix method to sample the rotation parameterspace evenly. On the left are the uncorrected distributions. On the right are the convolution corrected distributions. . . . . . . Normal Axial Ratio I n t e r l ope r A x i a l R a t i o Multiplicity=5Multiplicity=10Multiplicity=20Multiplicity=50Multiplicity=200 0.0 0.2 0.4 0.6 0.8 1.0 . . . . . . Normal Axial Ratio I n t e r l ope r A x i a l R a t i o Multiplicity=5Multiplicity=10Multiplicity=20Multiplicity=50Multiplicity=200 Fig. 6.— The effect a 20% interloper rate has on prolate (left) and oblate right) best fit distributions. . . . . . . . . Multiplicity A x i a l R a t i o O b s e r v a t i on L i m i t Fig. 7.— Absolute detectable limits for given multiplicities andinterloper rates. tiplicity might not be so obvious. These trends are ob-served strongly for both prolate and oblate shapes.A significant issue is the confusion effect: does thepresence of interlopers make a prolate distribution lookoblate and vica-versa, and how does multiplicity and theinterloper rate affect the possibility of confusion? Whenconfusion occurs it is generally at the largest axial ra-tios, an understandable result since as prolate and oblateshapes move closer to spherical it requires a smalleramount of random distortion to transform one distribu-tion into the other, and hence it doesn’t affect prolate oroblate groups worse.It becomes easier to distinguish populations when theaxial ratios are lower ( ∼ . Axial Ratio P r obab ili t y D en s i t y ProlateAxial Ratio= 0.1ProlateAxial Ratio= 0.1Interloper Rate= 5% 0.0 0.2 0.4 0.6 0.8 1.0 Axial Ratio P r obab ili t y D en s i t y ProlateAxial Ratio= 0.1ProlateAxial Ratio= 0.1Interloper Rate= 5%0.0 0.2 0.4 0.6 0.8 1.0 Axial Ratio P r obab ili t y D en s i t y ProlateAxial Ratio= 0.1ProlateAxial Ratio= 0.1Interloper Rate= 5% 0.0 0.2 0.4 0.6 0.8 1.0 Axial Ratio P r obab ili t y D en s i t y ProlateAxial Ratio= 0.1ProlateAxial Ratio= 0.1Interloper Rate= 5% Fig. 8.— Distortion casued by 5% interloper rate for prolate groups of multiplicity 5 (top-left), 20 (top-right), 50 (bottom-left) and 200(bottom-right). from the QQ-plots in Figure 9.The last finding of note is that interloper rates aloneaccount for some degree of distribution broadening, ontop of the aforementioned shift to more spherical pop-ulations. This is understandable simply because weare modeling a binomial distribution (every Monte-Carlogalaxy has a chance of being an interloper as describedby the interloper rate), and as such there will be an as-sociated spread in observed shapes since when there aremany interlopers a given group will appear more circu-lar and when there are fewer this effect is reduced. Infact the number of interlopers will follow a near gaus-sian distribution when np ∼ 5, so groups with multiplic-ity 20 and interloper rates of 20% almost meet these con-ditions. Otherwise we’ll still observe a spread, just of amore discrete variety. This effect is clearly seen in Figure8 where only the multiplicity 200 population undergoesa smooth distortion; this is because np = 10 and meetsthe requirements to approximate a gaussian form for the number of interlopers. The multiplicity 50 populationhas np = 2 . 5, so whilst the distribution is broadened it isnot as smooth. The multiplicity 20 population has a verystrong feature from 0.2-0.5. This is because np = 1 andas such the event of no interlopers is quite likely (36%),in fact it is almost exactly the same chance as one in-terloper (38%). Thus the two main features–the peakbetween 0-0.2 and the plateau from 0.2-0.5– are causedby interloper numbers of 0 and 1 respectively. The slighthints of further plateaus beyond 0.5 are caused by inter-loper number beyond 1, and these only occur ∼ 26% ofthe time.These results demonstrate that any observed gaussiandistribution of ellipsoid axial ratios are, in general, de-scribing the upper limit of the axial axial ratio and theupper limit for the standard deviation. As the axial ratiois beyond the observing limit it is possible to make anestimate of the underlying distribution, bearing in mindthe interloper rate and multiplicity. Robotham et al. . . . . . . Normal Ax−Rat I n t e r l ope r A x − R a t Mlt=5Orig Ax−Rat=0.7Int Ax−Rat=0.65 0.0 0.2 0.4 0.6 0.8 1.0 . . . . . . Normal Ax−Rat I n t e r l ope r A x − R a t Mlt=20Orig Ax−Rat=0.7Int Ax−Rat=0.6 0.0 0.2 0.4 0.6 0.8 1.0 . . . . . . Normal Ax−Rat I n t e r l ope r A x − R a t Mlt=50Orig Ax−Rat=0.7Int Ax−Rat=0.6 0.0 0.2 0.4 0.6 0.8 1.0 . . . . . . Normal Ax−Rat I n t e r l ope r A x − R a t Mlt=200Orig Ax−Rat=0.7Int Ax−Rat=0.6 Fig. 9.— QQ-plots for best fit interloper groups where interloper rates= 20% and axial ratio= 0.7. TABLE 1Best fit parameters for raw group data Multiplicity ¯ β σ χ Reduced χ All groups (true oblate) 0 . 16 0 . 06 1719 . 41 95 . . 36 0 . 14 74 . 62 4 . . 14 0 . 08 1685 . 50 93 . . 36 0 . 16 75 . 96 4 . . 12 0 . 06 1836 . 35 102 . . 34 0 . 14 66 . 71 3 . . . 06 3028 . 97 168 . . . 14 79 . 57 4 . . 24 0 . 04 276 . 33 17 . . 42 0 . 12 33 . 99 2 . . 26 0 . . 30 12 . . 42 0 . 14 23 . 60 1 . . . 08 103 . 66 6 . . 46 0 . 14 15 . 97 1 . . . . 27 15 . . 46 0 . 12 23 . 32 1 . TABLE 2Best fit parameters for convolution corrected groupdata Multiplicity ¯ β σ χ Reduced χ All groups (true oblate) 0 . 22 0 . . 47 1 . . 44 0 . 18 110 . 27 6 . . 22 0 . 14 50 . 71 2 . . 42 0 . 18 60 . 49 3 . . . . 47 0 . . 42 0 . . 07 6 . . 16 0 . 12 38 . 09 2 . . . . 54 3 . . . . 57 2 . . 44 0 . 14 42 . 79 2 . . 28 0 . . 00 1 . . 44 0 . 16 33 . 96 2 . . 32 0 . . 31 2 . . 46 0 . 14 12 . 74 0 . . 36 0 . . 03 1 . . 48 0 . 14 9 . 91 0 . RESULTS The main results can be found in tables 1 and 2, andfigures 10, 11, 12 and 13 for the raw and the correcteddata respectively. The tables present parameter valuesfor the best fit χ prolate and oblate distributions ob-tained, whilst the figures display the raw and correcteddata overplotted with these best fit distributions alongwith their associated error ellipses.Firstly, we will consider the raw data with a multi-plicity cut-off of 20 (see figure 13), since for the multi- plicity range described it will be the most reliable rawresult and immediately comparable to other research.Assuming Gaussian PDF for either oblate or prolate el-lipsoids, χ tests were undertaken using the numericalintegration capacity of Maple to an accuracy of 0.02for both the mean (0.1 to 0.7) and the standard devi-ation (0.06 to 0.3). The minimum χ solution found thegroups to be prolate to a high degree of confidence (theminimal prolate χ solution returned 15.97 compared to103.66 for oblate), with a Gaussian distribution of mean¯ β = 0 . 46 and standard deviation σ β = 0 . 14. Thesevalues agree extremely well with those recently pre-sented by Plionis, Basilakos & Ragone-Figueroa (2006)and Paz et al. (2006). For this data cut we have 15 de-grees of freedom giving a reduced χ of 1.06, which iswithin 1 σ expectations.The parameter gradient for a prolate ellipsoid distri-bution is very steep for the raw data of groups with mul-tiplicity between 5 and 9. The results strongly implythe distribution being narrow (0 . < σ β < . β values of 0.44 to 0.48. More signifi-cantly the distribution is extremely far removed from anypossible oblate distribution. This is intuitively obviousfrom the lack a plateau at the near circular extreme ofthe binned axial ratio PDF This is an effect that will al-ways be present in oblate ellipsoids due to the observedaxial ratio always having to be greater or equal to theintrinsic axial ratio (true also for prolate ellipsoids), andbecause an oblate ellipsoid will possess a greater observedaxial ratio ( q ) when compared to a prolate ellipsoid shar-ing the same intrinsic axial ratio ( β ) inclined from ourline-of-sight by the same degree. The fact the measuredbinned PDF falls to almost zero for circular value axialratios strongly precludes an oblate contribution except atinsignificant levels. It is due to this effect of the observedaxial ratio, at a minimum, being equal to the intrinsicvalue that ¯ β will always be smaller than the peak in anyobserved PDF, hence in this case ¯ β is smaller than theobvious peak in the binned PDF at 0.6.The other raw measurements for different multiplicitygroups are presented in the corresponding graphs and ta-bles, however such is the inherent biasing in these lowermultiplicity data that little can be gleaned from theiranalysis alone. Instead, our last use of the raw data isto determine the consistency between the real universeand the ΛCDM dependent mock universe. It is com-pletely justifiable to make a comparison between the rawdata and their multiplicity associated mock data, sincethe shape biasing (being a sampling artifact) will be thesame. By directly comparing the minimal χ fits andhe Shapes of Galaxy Groups 9 p r obab ili t y den s i t y axial ratioBest fit plots for raw and corrected data using oblate and prolate distributionsraw datacorrected dataExpected uniformraw ob m=0.16 sd=0.06raw pro m=0.36 sd=0.14cor ob m=0.22 sd=0.1cor pro m=0.44 sd=0.18 σ mean x xx x raw obraw procorrected obcorrected pro 0 0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 p r obab ili t y den s i t y axial ratioBest fit plots for raw and corrected data using oblate and prolate distributionsraw datacorrected dataExpected uniformraw ob m=0.14 sd=0.08raw pro m=0.36 sd=0.16cor ob m=0.22 sd=0.14cor pro m=0.42 sd=0.18 σ mean x xx x raw obraw procorrected obcorrected pro 0 0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fig. 10.— All group multiplicities. Top plot shows best fits for the real 2PIGG data on the left and the relevant parameter contour plotson the right, whilst the bottom plot is a comparison to the mock 2PIGG data. considering the cross catalog KS tests we will have aninsight into the reliability of both the grouping algorithmand the cosmological model used. When considering ta-ble 1 and table 3 we can see remarkably good agreementbetween the real and mock data (c.f. Paz et al. 2006).Considering the fits, we find near perfect agreement for¯ β between the real and mock data, the major disagree-ments being for multiplicities of 5-9 (for prolate distribu-tions the real data has ¯ β = 0 . 34 compared to ¯ β = 0 . σ = 0 . 04 compared to σ = 0 . χ between the two data sets, another indicationthat they are truly describing the same underlying dis-tributions. Furthermore, when looking at the KS testresults we find that all direct comparisons can be drawnfrom the same underlying distribution to within 1 σ ex-pectations. The self similarities between the data for dif-ferent multiplicities are not quite as good (tables 4 and5), however, the real and mock data both reject sharedself similar distributions to a high degree of significance.As would be expected from the fitting results, the mostsimilar distributions are those for multiplicities of 20+and 10-20 for both the real and mock data (2 . 9% and0 . 14% respectively). It can be seen from the χ contourdata that these multiplicities only agree upon parame-ters when considering the 2 σ error contours, which isconsistent with this KS test statistic.We now consider the convolution corrected data withweighted histogram bins. The results for the real andmock forms of the 2PIGG catalog can be seen in figures10, 11, 12 and 13, and in table 2. Significantly, we now find that the distributions are not necessarily prolate.The complete data and the multiplicity cut for 5-9 isbetter fit by an oblate distribution, the cut-off for 10-19is equally well fit by either, and only the 20+ cut is bet-ter fit by a prolate distribution and not with a definiterejection of an oblate distribution. If oblate distribu-tions are accepted for the corrected data then we see astrong divide between the low multiplicity cut (5-9) andthe other two larger multiplicity cuts. The former has adistribution mean ¯ β = 0 . β ∼ . 3: in all cases σ β = 0 . 1. If we still assumeprolate distributions, despite evidence of strong rejec-tion in our 5-9 multiplicity cut, then we still see a smallvariation in distribution mean: ¯ β = 0 . ± . 01 for 5-9,¯ β = 0 . ± . 01 for 10-19 and ¯ β = 0 . ± . 02 for 20+.The latter two have σ β = 0 . 14, whilst the 5-9 cut has σ β = 0 . 2. Within the 1 σ error contours the two highermultiplicity cuts share the same distribution, ¯ β ∼ . σ β = 0 . 14, whilst the 5-9 cut is strongly indicatedto be a truly different distribution, lying outside the 3 σ error contours. This finding is consistant with the previ-ous KS tests which showed the two high multiplicity cutsto be the most similar; once the distributions are cor-rected they become identical. Evidence of oblate shapeshas been presented before in literature: Fasano et al.(1993) and Orlov, Petrova & Tarantaev (2001) being no-table examples, though with a smaller sample size anddifferent multiplicity limits respectively. These au-thors also found a non-rejectable oblate distribution,although Fasano et al. (1993) favoured the prolate fit.Orlov, Petrova & Tarantaev (2001) found an axial ratiomean of 3:1 for their sample of small galaxy groups (mul-0 Robotham et al. p r obab ili t y den s i t y axial ratioBest fit plots for raw and corrected data using oblate and prolate distributionsraw datacorrected dataExpected uniformraw ob m=0.12 sd=0.06raw pro m=0.34 sd=0.14cor ob m=0.2 sd=0.1cor pro m=0.42 sd=0.2 σ mean x xx x raw obraw procorrected obcorrected pro 0 0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 p r obab ili t y den s i t y axial ratioBest fit plots for raw and corrected data using oblate and prolate distributionsraw datacorrected dataExpected uniformraw ob m=0.1 sd=0.06raw pro m=0.3 sd=0.14cor ob m=0.16 sd=0.12cor pro m=0.4 sd=0.2 σ mean x xx x raw obraw procorrected obcorrected pro 0 0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fig. 11.— Group multiplicities of 5 to 9. Top plot shows best fits for the real 2PIGG data on the left and the relevant parameter contourplots on the right, whilst the bottom plot is a comparison to the mock 2PIGG data. TABLE 3KS test comparisons between true and mock data for different multiplicities Real raw All Real raw 5-9 Real raw 10-19 Real raw 20+Mock raw All 68 . 63% 1 . x − % 6 . × − % 1 . × − %Mock raw 5-9 4 . x − % 27 . 11% 5 . × − % 5 . × − %Mock raw 10-19 1 . × − % 5 . x − % 22 . 94% 0 . . × − % 4 . × − % 0 . 07% 67 . TABLE 4KS test comparisons between self similar true data for differentmultiplicities Real raw All Real raw 5-9 Real raw 10-19 Real raw 20+Real raw All 1 N/A N/A N/AReal raw 5-9 0 . 04% 1 N/A N/AReal raw 10-19 1 . × − % 5 . × − . × − % 8 . × − % 2 . 90% 1 tiplicities of 3-8), a figure that can be considered consis-tent with our data ( ¯ β = 0 . β = 0 . 28, ¯ β = 0 . 36 respectively)or prolate ( ¯ β = 0 . 44, ¯ β = 0 . 48 respectively) solutions. Itshould be noted that a KS test cannot be done in this sit-uation since it is only applicable to data with unweightedhistogram frequency points.Perhaps the most interesting aspect of this analysis ishe Shapes of Galaxy Groups 11 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p r obab ili t y den s i t y axial ratioBest fit plots for raw and corrected data using oblate and prolate distributionsraw datacorrected dataExpected uniformraw ob m=0.24 sd=0.04raw pro m=0.42 sd=0.12cor ob m=0.3 sd=0.1cor pro m=0.44 sd=0.14 σ mean x xx x raw obraw procorrected obcorrected pro 0 0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 p r obab ili t y den s i t y axial ratioBest fit plots for raw and corrected data using oblate and prolate distributionsraw datacorrected dataExpected uniformraw ob m=0.26 sd=0.1raw pro m=0.42 sd=0.14cor ob m=0.28 sd=0.1cor pro m=0.44 sd=0.16 σ mean x xx x raw obraw procorrected obcorrected pro 0 0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fig. 12.— Group multiplicities of 10 to 19. Top plot shows best fits for the real 2PIGG data on the left and the relevant parametercontour plots on the right, whilst the bottom plot is a comparison to the mock 2PIGG data. TABLE 5KS test comparisons between self similar mock data for differentmultiplicities Mock raw All Mock raw 5-9 Mock raw 19-20 Mock raw 20+Mock raw All 1 N/A N/A N/AMock raw 5-9 2 . × − % 1 N/A N/AMock raw 10-19 1 . × − % 1 . × − . × − % 4 . × − % 0 . 14% 1 the tentative evidence it shows for generally more spheri-cal groups for larger multiplicities, not just as a numericalartifact. This would be consistant with large multiplic-ity groups being more virialised and dynamically moreevolved, consistant with hierarchical formation of smallgroups collapsing along filaments and larger groups form-ing at nodes. The features that account for the high el-lipticity sub sample in low multiplicity groups are a low¯ β for an oblate solution, and a large σ for a prolate so-lution. Either way there does appear to be a low axialratio population that must be accounted for, and that isnot present in the higher multiplicity data. It is worthremembering that at low magnitude cuts (correspondingto higher multiplicities) the Local Group demonstratesoblate characteristics, so this result is not necessarilysuprising. This result is even more reliable occurring asit does in the region of the PDF that is least correctedin convolution. In fact in this region highly ellipticalresults for low multiplicity groups are reduced in weight-ing, so the fact the signal is still so strong indicates a real population difference. This trend would appear tobe in the opposite sense to what is expected in simula-tions (see (Allgood et al. 2006) and references therein),but it should be noted that these results are much closerto agreement after correction. CONCLUSION We found good agreement between our results for rawdata with a simple 20+ multiplicity cut and findings byother authors, indicating a strongly prolate distributionwith a mean ¯ β = 0 . 46 and σ β = 0 . 14. However, when fitswere applied to convolution corrected distributions whichallow for the error introduced by finite sampling we foundthat for the higher multiplicity cuts both oblate and pro-late distributions could produce reasonable fits. Moresignificantly, evidence suggests that these large multiplic-ity populations share the same underlying distribution (ifoblate: ¯ β ∼ . σ β = 0 . 1, if prolate ¯ β ∼ . σ β = 0 . p r obab ili t y den s i t y axial ratioBest fit plots for raw and corrected data using oblate and prolate distributionsraw datacorrected dataExpected uniformraw ob m=0.3 sd=0.1raw pro m=0.46 sd=0.12cor ob m=0.36 sd=0.1cor pro m=0.48 sd=0.14 σ mean x xx x raw obraw procorrected obcorrected pro 0 0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 p r obab ili t y den s i t y axial ratioBest fit plots for raw and corrected data using oblate and prolate distributionsraw datacorrected dataExpected uniformraw ob m=0.3 sd=0.08raw pro m=0.46 sd=0.14cor ob m=0.32 sd=0.1cor pro m=0.46 sd=0.14 σ mean x xx x raw obraw procorrected obcorrected pro 0 0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fig. 13.— Group multiplicities of 20 upwards. Top plot shows best fits for the real 2PIGG data on the left and the relevant parametercontour plots on the right, whilst the bottom plot is a comparison to the mock 2PIGG data. distributions produce a better quality of fit ( ¯ β = 0 . σ β = 0 . σ expectations, and the most self similarpopulations are for multiplicities 10-19 and 20+, whichappear consistent with being part of the same overallpopulation once the corrections have been applied.Further exploration of projected group shapes will bepresented in a future paper which will consider the dif-ferences we find when colour cuts and different groupingalgorithms are used. ACKNOWLEDGEMENTS AR acknowledges funding through the UK ParticlePhysics and Astrophysics Research Council (PPARC).We would like to thank the referee for their helpful sug-gestions, particularly the suggestion to discuss the effectof interlopers in more detail. REFERENCESAllgood, B. et al , 2006, MNRAS, 367, 1781Bardeen, J. M., Bond, J. R., Kaiser, N., Szalay, A. 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P. 2005b,MNRAS, 356, 1293APPENDIX For a prolate or oblate ellipsoid we know that( ux ) + ( uy ) + z = a (10)where u > u < ux ) + z = a (11) z = a − ( ux ) (12) z . δz δx = − u x (13)¿From standard trigonometry tan (cid:16) π − θ (cid:17) = cot( θ ) (14)so the gradient with respect to the observer is defined by δz δx = − u x z = cot θ (15)and x z = a u z − u = 1 u (cid:18) a z − (cid:19) (16)giving us cot θ = u x z = u (cid:18) a z − (cid:19) (17)¿From figure 1 we can see AC = sin θ (18) C = A sin θ = z − x .m (19)where m is the gradient, and will necessarily be the opposite sign to x . This gives the z value for the z-axis/tangentintercept. C = z − x − u x z = z + u x z = a z (20)Using (17) we get A = a (cid:18) a z (cid:19) sin θ = a (cid:18) cos θu + sin θ (cid:19) (21)We define the apparent axial ratio ( q ) to be less than 1 for both prolate and oblate ellipsoids. Thus for oblate ellipsoids q = uAa and for prolate ellipsoids q = auA . So u sin θ + cos θ = q ( oblate ) = 1 q ( prolate ) (22)To have q between q and q + δq , the symmetry axis has to lie at an angle between θ and θ + δθ . δθ = δq | δq/δθ | (23)4 Robotham et al.Mathematically this means if q changes rapidly with respect to θ then δθ is small. A simple example is to imaginea needle (an almost completely prolate object): when we view this end on and alter the angle, δq/δθ will be huge;our observed axial ratio q has changed from 1 (a perfect circle) to near 0 (by this definition almost a line) over aninfinitesimally small angle. So the range of angles over which we might say the axial ratio is similar ( δθ ) is very small.However, when we consider the same needle side on, and rotate as before, δq/δθ will be very small; q is changing verygradually. So for a prolate object viewed with the symmetry axis at an orthogonal angle, the range of angles possessingsimilar axial ratios is much larger than when viewed with the symmetry axis parallel to our line-of-sight.For a given u , and assuming a random orientation of symmetry axes for all the ellipsoids, the relative probabilityof observing the ellipsoid between two axial ratios (say q and q ) is given by the relative magnitude of the cosinebetween angles θ and θ that correspond to these observed ellipsoids. This is exactly the same factor that has tobe considered when trying to determine what fraction of the angular area of the celestial sphere is contained withindifferent declinations. As an example, if q varies the same amount between 0 ◦ and 10 ◦ as it does between 80 ◦ and 90 ◦ ,the difference in the solid angle of sky subtended will be cos 80 ◦ − cos 10 ◦ =11.43; thus it is 11.43 times more likely we will seethe ellipsoid in the larger θ range. Taking this to infinitesimal differences we have | cos( θ + δθ ) − cos θ | = | cos θ cos δθ − sin θ sin δθ − cos θ | = sin θδθ (24)So of the ellipsoids under consideration, a fraction sin θδθ will have their symmetry axes directed at an angle θ to theline-of-sight. Now we can write sin θδθ = sin θδq | δq/δθ | (25)Galaxies with u in the range ( u, u + δu ) contribute to the observed f ( q ) δq galaxies with axial ratios in the range( q, q + δq ) accordingly f ( q ) δq = n ( u ) δu sin θδq | δq/δθ | (26)So the probability density function (PDF) of observed axial ratios ( q ) is given by f ( q ) = Z n ( u ) (cid:18) sin θ | δq/δθ | (cid:19) du (27)For an oblate ellipsoid (from Eq. 3) u (1 − cos θ ) + cos θ = q (28) ∴ cos θ = q − u − u (29)Similarly u sin θ + 1 − sin θ = q (30) sin θ = 1 − q − u (31)It follows that δqδθ = − ( q − u ) . (1 − u ) . (1 − q ) . (1 − u ) . − u q (32) ∴ (cid:12)(cid:12)(cid:12)(cid:12) δqδθ (cid:12)(cid:12)(cid:12)(cid:12) = 1 q [(1 − q )( q − u )] . (33)Using equations (27), (31) and (33) we have f ( q ) = q Z q n ( u ) du [(1 − u )( q − u )] . (34)We now define the intrinsic axial ratio to be a value β always less than 1, for an oblate ellipsoid β = u . f ( q ) alone giveus a PDF of the observed axial ratios, however our data will have to be binned for analysis. So for a given value of β we integrate over a range of observed axial ratios (say q to q ) to construct a number density function for observedaxial ratios N ( β, q , q ) where N ( β, q , q ) = 1 p − β Z q q qdq p q − β (35)he Shapes of Galaxy Groups 15Considering the integral alone we use the substitution q = β cos θ (where δq = − β sin θδθ ), giving us Z q q − β cos θ sin θdθ p β cos θ − β = Z q q − β cos θ sin θdθiβ sin θ = Z q q iβ cos θdθ = [ iβ sin θ + C ] q q (36)Rearranging the original substitution and replacing sin θ accordingly, the integral becomes hp q − β i q q (37)so for an oblate ellipsoid with a given βN ( β, q , q ) = 1 p − β (cid:20)q ( q − β ) − q ( q − β ) (cid:21) (38)For a prolate ellipsoid β = u so the integral in (35) is multiplied by a factor β q , thus 36 becomes Z q q − β sin θδθβ cos θ ( p β cos θ − β ) = Z q q − δθi cos θ = (cid:20) i sin θ cos θ + C (cid:21) q q (39)so for a prolate ellipsoid with a given βN ( β, q , q ) = 1 p − β " p q − β q − p q − β q (40)the only difference is a factor of the apparent axial ratio being considered, (this makes sense because you would expecta prolate object to have a larger number density of small observed axial ratios). Putting this all together, for a function˜ n ( β ) using oblate ellipsoids we find N ( q , q ) = Z q ˜ n ( β ) 1 p − β (cid:20)q ( q − β ) − q ( q − β ) (cid:21) dβ (41)and for prolate ellipsoids N ( q , q ) = Z q ˜ n ( β ) 1 p − β " p ( q − β ) q − p ( q − β ) q dβdβ