# Non-Gaussian velocity distributions - The effect on virial mass estimates of galaxy groups

Abstract

We present a study of nine galaxy groups with evidence for non-Gaussianity in their velocity distributions out to 4R200. This sample is taken from the 57 groups selected from the 2dF Percolation-Inferred Galaxy Groups (2PIGG) catalogue of galaxy
groups. Statistical analysis indicates that the non-Gaussian groups have masses significantly higher than that of the Gaussian
groups. We also have found that all non-Gaussian systems seem to be composed of multiple velocity modes. Besides, our results
indicate that multimodal groups should be considered as a set of individual units with their own properties. In particular,
we have found that the mass distributions of such units are similar to that of the Gaussian groups. Our results reinforce
the idea of non-Gaussian systems as complex structures in the phase space, likely corresponding to secondary infall aggregations
at a stage before virialization. The understanding of these objects is relevant for cosmological studies using groups and
clusters through the mass function evolution.

4 Figures

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Available from: Marina TrevisanMon. Not. R. Astron. Soc. 413, L81–L85 (2011) doi:10.1111/j.1745-3933.2011.01038.x

Non-Gaussian velocity distributions – the effect on virial mass estimates

of galaxy groups

A. L. B. Ribeiro,1P. A. A. Lopes2and M. Trevisan3

1Laborat´

orio de Astrof´

ısica Te´

orica e Observacional, Universidade Estadual de Santa Cruz, 45650-000 Ilh´

eus-BA, Brazil

2Observat´

orio do Valongo, Universidade Federal do Rio de Janeiro, 20080-090 Brazil

3Instituto Astronˆ

omico e Geof´

ısico – USP, S ˜

ao Paulo, SP 05508-090, Brazil

Accepted 2011 February 22. Received 2011 February 6; in original form 2010 December 31

ABSTRACT

We present a study of nine galaxy groups with evidence for non-Gaussianity in their velocity

distributions out to 4R200. This sample is taken from the 57 groups selected from the 2dF

Percolation-Inferred Galaxy Groups (2PIGG) catalogue of galaxy groups. Statistical analysis

indicates that the non-Gaussian groups have masses signiﬁcantly higher than that of the

Gaussian groups. We also have found that all non-Gaussian systems seem to be composed

of multiple velocity modes. Besides, our results indicate that multimodal groups should be

considered as a set of individual units with their own properties. In particular, we have found

that the mass distributions of such units are similar to that of the Gaussian groups. Our

results reinforce the idea of non-Gaussian systems as complex structures in the phase space,

likely corresponding to secondary infall aggregations at a stage before virialization. The

understanding of these objects is relevant for cosmological studies using groups and clusters

through the mass function evolution.

Key words: galaxies: groups: general – galaxies: statistics.

1 INTRODUCTION

Groups of galaxies contain most of the galaxies in the Universe

and are the link between individual galaxies and larger structures

(e.g. Huchra & Geller 1982; Geller & Huchra 1983; Nolthenius &

White 1987; Ramella, Geller & Huchra 1989). The dissipationless

evolution of these systems is dominated by gravity. Interactions

over a relaxation time tend to distribute the velocities of the galaxy

members in a Gaussian distribution (e.g. Bird & Beers 1993). Thus,

a way to access the dynamical stage of galaxy groups is to study

their velocity distributions. Evolved systems are supposed to have

Gaussian velocity distributions, while those with deviations from

normality are understood as less evolved systems. Hou et al. (2009)

have examined three goodness-of-ﬁt tests [Anderson–Darling (AD),

Kolmogorov and χ2tests] to ﬁnd which statistical tool is best able to

distinguish between relaxed and non-relaxed galaxy groups. Using

Monte Carlo simulations and a sample of groups selected from the

CNOC2, they found that the AD test is far more reliable at detect-

ing real departures from normality in small samples. Their results

show that Gaussian and non-Gaussian groups present distinct ve-

locity dispersion proﬁles, suggesting that discrimination of groups

according to their velocity distributions may be a promising way to

access the dynamics of galaxy systems.

Recently, Ribeiro, Lopes & Trevisan (2010) extended up this

kind of analysis to the outermost edge of groups to probe regions

E-mail: albr@uesc.br

where galaxy systems might not be in dynamical equilibrium. They

found signiﬁcant segregation effects after splitting up the sample

in Gaussian and non-Gaussian systems. In the present work, we try

to further understand the nature of non-Gaussian groups. In partic-

ular, we investigate the problem of mass estimation for this class

of objects and their behaviour in phase space. The Letter is orga-

nized as follows: in Section 2 we present data and methodology;

Section 3 contains a statistical analysis with sample tests and

multimodality diagnostics for non-Gaussian groups; ﬁnally, in

Section 4 we summarize and discuss our ﬁndings.

2 DATA AND METHODOLOGY

2.1 2PIGG sample

We use a subset of the 2PIGG catalogue corresponding to groups

located in areas of at least 80 per cent redshift coverage in 2dF

data out to 10 times the radius of the systems, roughly estimated

from the projected harmonic mean (Eke et al. 2004). The idea of

working with such large areas is to probe the effect of secondary

infall on to groups. Members and interlopers were redeﬁned after

the identiﬁcation of gaps in the redshift distribution according to the

technique described by Lopes et al. (2009a). Finally, a virial analysis

is performed to estimate the groups’ properties. See details in Lopes

et al. (2009a), Ribeiro et al. (2009) and Ribeiro et al. (2010). We

have classiﬁed the groups after applying the AD test to their galaxy

velocity distributions (see Hou et al. 2009 for a good description

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L82 A. L. B. Ribeiro, P. A. A. Lopes and M. Trevisan

of the test). This is done for different distances, producing the

following ratios of non-Gaussian groups: 6 per cent (R≤1R200),

9 per cent (R≤2R200) and 16 per cent (R≤3R200 and R≤4R200 ). We

assume this latter ratio (equivalent to nine systems) to be correct if

one desires to extend up the analysis to regions where galaxy groups

might not be in dynamical equilibrium. Approximately 90 per cent

of all galaxies in the sample have distances ≤4R200. This is the

natural cut-off in space we have made in this work. Some properties

of galaxy groups are presented in Table 1, where non-Gaussian

groups are identiﬁed with an asterisk. Cosmology is deﬁned by

m=0.3, λ=0.7 and H0=100 hkm s−1Mpc−1. Distance-

dependent quantities are calculated using h=0.7.

3 NON-GAUSSIANITY AND MASS

3.1 Mass bias

We consider the two-sample statistical problem for Gaussian (G)

and non-Gaussian (NG) subsamples. We choose the mass result-

ing from virial analysis as the property to illustrate the compari-

son between the subsamples. Both Kolmogorov–Smirnov (KS) and

Cramer–von Mises (CvM) tests reject the hypothesis that the NG

subsample is distributed as the G subsample, with p-values 0.000 01

and 0.000 29, respectively. For these tests, we have used 1000 boot-

strap replicas of each subsample to alleviate the small-sample effect.

The result indicates an inconsistency between the mass distributions

of the G and NG groups. This could represent a real physical dif-

ference or, more probably, an indication of a signiﬁcant bias to

higher masses in NG groups. The median mass for this subsample

is MNG

200 =2.57 ×1014 M, while it is MG

200=8.85 ×1013 M

for the G subsample; thus MNG

200 is larger by a factor of ∼2.9. In

the following, we investigate this mass bias looking for features in

the velocity distributions of galaxy groups.

3.2 Exploring non-Gaussianity

The shape of the velocity distributions may reveal a signature of the

dynamical stage of the galaxy groups. For instance, systems with

heavier tails than that predicted by a normal parent distribution may

be contaminated by interlopers. Otherwise, systems with lighter tails

than a normal may be multimodal, consisting of overlapping distinct

populations (e.g. Bird & Beers 1993). We now try to understand the

mass bias in the NG subsample by studying non-Gaussianity in the

velocity distributions. Since we have carefully removed interlopers

from each ﬁeld (see Section 2.1; and Lopes et al. 2009a), we con-

sider here that the most probable cause of normality deviations in

our sample is due to a superposition of modes in the phase space

(e.g. Diemand & Kuhlen 2008). Visual inspection of radial velocity

histograms of NG systems suggests that multipeaks really happen

in most cases (see Fig. 1).

We statistically check multimodality by assuming the velocity

distributions as Gaussian mixtures with unknown number of com-

ponents. We use the Dirichlet process mixture (DPM) model to

study the velocity distributions. The DPM model is a Bayesian

non-parametric methodology that relies on Markov Chain Monte

Carlo (MCMC) simulations for exploring mixture models with an

unknown number of components (Diebolt & Robert 1994). It was

ﬁrst formalized in Ferguson (1973) for the general Bayesian sta-

tistical modelling. The DPM is a distribution over k-dimensional

discrete distributions, so each draw from a Dirichlet process is itself

a distribution. Here, we assume that a galaxy group is a set of k

components, k

i=1πif(y|θi), with galaxy velocities distributed ac-

Tab l e 1. Main properties of groups.

Group R200 (Mpc) M200 (1014 M)σ(km s−1)Nmemb N200

55 0.689 0.400 173.203 16 9

60 0.664 0.359 120.395 41 12

84 1.025 1.326 224.715 54 10

91 0.848 0.754 164.293 34 4

102∗1.550 4.602 433.218 32 8

130 0.999 1.245 240.908 43 11

138∗1.271 2.578 348.885 74 14

139 0.997 1.250 235.740 39 12

169 0.765 0.573 225.212 9 4

177∗1.178 2.100 336.309 45 14

179∗1.663 5.897 471.352 23 10

181 1.247 2.488 393.379 29 12

188 0.514 0.174 153.714 8 2

191 0.707 0.456 268.517 14 10

197 0.897 0.930 297.903 13 7

204 1.493 4.333 473.813 33 12

209 0.680 0.407 132.095 15 4

222 1.243 2.494 315.355 22 8

236 1.258 2.575 331.308 19 7

271 0.968 1.093 240.691 41 15

326 0.648 0.333 167.764 21 7

352∗1.656 5.618 496.694 31 13

353 0.873 0.826 220.025 34 12

374 0.902 0.919 217.038 21 10

377 0.480 0.138 132.466 13 5

387 0.649 0.345 125.308 32 5

398 0.781 0.601 182.800 29 12

399 0.917 0.973 183.397 34 7

409 1.078 1.583 251.071 64 11

410 0.847 0.769 199.465 42 8

428 1.201 2.191 342.440 26 14

435 0.606 0.283 123.549 23 4

444 0.820 0.697 244.491 13 7

447∗1.169 2.032 296.919 37 6

453 0.827 0.720 271.840 17 10

455 1.588 5.088 454.227 65 15

456 0.319 0.041 58.138 16 3

458 0.642 0.336 115.674 18 4

466 1.507 4.381 554.511 23 13

471 1.128 1.832 399.078 22 14

475∗1.152 1.952 294.413 28 6

479 0.806 0.671 160.887 17 6

480 0.923 1.006 272.058 23 12

482 0.663 0.373 202.903 17 9

484 1.099 1.702 268.116 32 7

485 1.679 6.076 559.140 29 18

488 1.220 2.334 326.566 42 12

489 0.872 0.852 188.283 30 6

493 1.170 2.061 323.268 30 10

504 1.319 2.985 429.207 30 13

505∗1.176 2.111 336.804 20 7

507∗1.459 4.036 436.498 27 11

513 0.724 0.495 178.637 19 6

515 1.117 1.817 298.220 30 13

519 0.808 0.689 207.011 17 4

525 0.495 0.158 108.230 9 3

536 2.003 10.576 676.902 45 24

cording to Gaussian distributions with mean and variance unknown.

In this framework, the numbers πiare the mixture coefﬁcients that

are drawn from a Dirichlet distribution. In the DPM model, the

actual number of components kused to model the data is not ﬁxed,

and can be automatically inferred from the data employing the usual

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Non-Gaussian velocity distributions L83

Figure 1. Histograms of radial velocities for all non-Gaussian groups.

Bayesian posterior inference framework. See Neal (2000) for a sur-

vey of MCMC inference procedures for DPM models.

In this work, we ﬁnd kusing the Rlanguage and environment

(RDevelopment Core Team) under the dpmixsim library (da Silva

2009). The code implements mixture models with normal structure

(conjugate normal–normal DPM model). First, it ﬁnds the coefﬁ-

cients πi, and then separates the components of the mixture, accord-

ing to the most probable values of πi, in the distributional space,

leading to a partition of this space into regions (da Silva 2009). The

results can be visually analysed by plotting the estimated kernel

densities for the MCMC simulations. In Fig. 2, we show the DPM

diagnostics for each group, that is the deblended modes in the ve-

locity distributions. We have found the following number of modes

Figure 2. DPM density probability decomposition. Velocities are rescaled

to the [0, 1] interval.

Tab l e 2. Main properties of individual modes.

Mode R200 (Mpc) M200 (1014 M)σ(km s−1)N200

102a1.077 1.549 337.482 21

102b0.728 0.477 153.372 4

138a1.550 4.677 498.927 35

177a1.249 2.481 369.569 11

177b1.212 2.289 362.462 14

179a1.326 3.001 336.265 8

352a0.487 0.142 158.571 6

352b0.941 1.033 218.210 6

352c0.697 0.416 208.670 3

352d0.558 0.213 181.206 5

447a1.016 1.330 240.437 3

447b1.192 2.155 441.382 5

475a1.500 4.308 470.516 19

475b2.345 16.347 829.977 10

505a0.928 1.040 228.411 3

The letters a, b, c, d indicate the individual velocity modes in each NG

system.

per group: 4/102, 3/138, 3/177, 2/179, 5/352, 3/447, 2/475, 3/505

and 3/507. Therefore, all non-Gaussian groups in our sample are

multimodal (reaching a total of 28 modes) according to the DPM

analysis. Unfortunately, we cannot compute the physical properties

of 13 modes, due to intrinsic scattering in velocity data (and/or the

smallness of the modes – those with less than four members). The

properties of the other 15 modes are presented in Table 2.

3.3 Mass bias revisited

Now, we perform again the statistical tests, comparing the distribu-

tion of mass in NG and G with the new sample (see Fig. 3.) First,

we compare the NG subsample and the sample of modes (M). Both

KS and CvM tests reject the hypothesis that M is distributed as NG,

with p=0.0211 and 0.0189, respectively. Then, we compare G

and M. Now, the tests accept the hypothesis that M is distributed

as G, with p=0.4875 and 0.4695. Hence, the distributions of the

Figure 3. Histograms for Gaussian (dark grey) and non-Gaussian (white)

groups. Histogram of modes is plotted in light grey. Density probability

comparison among systems. Intersection areas are in white.

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L84 A. L. B. Ribeiro, P. A. A. Lopes and M. Trevisan

modes deblended from non-Gaussian groups are themselves mass

distributed as Gaussian groups. Also, the median mass of the M

sample is MM

200=1.33 ×1014 M, a value larger than MG

200

only by a factor of ∼1.5 (before deblending the groups the factor

was ∼2.9). This consistency between G and M objects indicates

that the non-Gaussian groups are a set of smaller systems, prob-

ably forming an aggregate out of equilibrium. Indeed, groups are

the intermediate level between galaxies and larger galaxy systems.

Thus, our results suggest that we are witnessing secondary infall

(the secondary modes) on to a previously formed (or still forming)

galaxy system (the principal mode). Naturally, we cannot discard

the possibility of the NG groups being unbound systems of smaller

groups seen in projection, although the properties of the indepen-

dent modes are quite similar to those found in physically bound

groups.

4 DISCUSSION

We have classiﬁed galaxy systems after applying the AD normality

test to their velocity distributions up to the outermost edge of the

groups. The purpose was to investigate regions where galaxy sys-

tems might not be in dynamical equilibrium. We have studied 57

galaxy groups selected from the 2PIGG catalogue (Eke et al. 2004)

using 2dF data out to 4R200. This means that we probe galaxy dis-

tributions near to the turnaround radius, thus probably taking into

account all members in the infall pattern around the groups (e.g.

Rines & Diaferio 2006; Cupani, Mezzetti & Mardirossian 2008).

The corresponding velocity ﬁelds depend on the local density of

matter. High-density regions should drive the formation of virial-

ized objects, whereas low-density environments are more likely to

present streaming motions, i.e. galaxies falling towards larger po-

tential wells constantly increasing the amplitude of their clustering

strength (e.g. Diaferio & Geller 1997).

We have found that 84 per cent of the sample is composed of sys-

tems with Gaussian velocity distributions. These systems could re-

sult from the collapse and virialization of high-density regions with

no signiﬁcant secondary infall. They could be groups surrounded

by well-organized infalling motions, possibly reaching virialization

at larger radii. Theoretically, in regions outside R200, we should

apply the non-stationary Jeans formalism, leading us to the virial

theorem with some correction terms. These are due to the infall

velocity gradient along the radial coordinate and to the acceleration

of the mass accretion process (see Cupani 2008). Their contribution

is likely to be generally negligible in the halo core where the matter

is set to virial equilibrium and it becomes signiﬁcant in the halo

outskirts where the matter is still accreting (see e.g. Cupani et al.

2008). However, a slow and well-organized infalling motion to the

centre of the groups could diminish the importance of the correction

terms, i.e. the systems could be in a quasi-stationary state outside

R200. The high fraction of groups immersed in such surroundings

suggests that ordered infalling motion around galaxy groups might

happen quite frequently in the Universe.

In any case, Gaussian groups can be considered as dynamically

more evolved systems (see Ribeiro et al. 2010). The remaining 16

per cent of the sample is composed of non-Gaussian groups. We

found that these systems have masses signiﬁcantly larger than that

of Gaussian groups. This biasing effect in virial masses is basi-

cally due to the higher velocity dispersions in NG groups. Ribeiro

et al. (2010) found that NG groups have rising velocity dispersion

proﬁles. A similar result was found by Hou et al. (2009). Rising ve-

locity dispersion proﬁles could be related to the higher fraction of

Figure 4. Phase-space diagrams for a typical Gaussian (upper box), non-

Gaussian (middle box) and mode (lower box) systems. Density contours

indicate galaxy concentration across the diagrams. Distances to the centre

of the systems are normalized by R200. Radial velocities are subtracted from

the median velocity and divided by the velocity dispersion of the groups.

blue galaxies in the outskirts of some galaxy systems (see Popesso

et al. 2007; Ribeiro et al. 2010).

At the same time, the NG subsample is composed of multipeak

objects, identiﬁed by the DPM model analysis applied to the ve-

locity space. These results indicate that secondary infall might be

biasing the mass estimates of these groups. Thus, the NG systems

could result from the collapse of less dense regions with signiﬁcant

secondary infall input. Contrary to Gaussian groups, the surround-

ings of the NG systems do not seem to be in a quasi-stationary

state. They are dynamically complex. Actually, we have found that

these groups can be modelled as assemblies of smaller units. After

deblending the groups into a number of individual modes, we have

veriﬁed that the mass distribution of these objects is consistent with

that of Gaussian groups, suggesting that each unit is probably a

galaxy group itself, a system formed during the streaming motion

towards the potential wells of the ﬁeld.

Our results reinforce the idea of NG systems as complex struc-

tures in the phase space out to 4R200. This scenario is illustrated

in Fig. 4, where we show the stacked G and NG groups, and the

stacked modes. Galaxies in these composite groups have distances

and line-of-sight velocities with respect to the centres normalized

by R200 and σ, respectively. In the upper box, we present the G

stacked group. Note that galaxies are extensively concentrated in

the phase-space diagram, with a single density peak near to 0.5R200,

revealing a well-organized system around this point. Also, note that

the density peak has Vlos/σ ≈0. Contour density lines suggest that

ordered shells of matter are moving towards the centre. A differ-

ent result is found for the NG stacked group. In the middle box

of Fig. 4, we see a less concentrated galaxy distribution, with less

tight density contour levels, presenting a density peak slightly larger

than R200, and two additional peaks, around 2R200 , possibly inter-

acting with the central mode. The additional peaks are not aligned

at Vlos/σ ≈0, suggesting a less symmetrical galaxy distribution in

the phase space. These features suggest that non-Gaussian systems

are distinct, and dynamically younger than Gaussian groups, which

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Non-Gaussian velocity distributions L85

agrees well with the results of Ribeiro et al. (2010). Also in Fig. 4,

we present the mode stacked system in the lower box. Similar to

the Gaussian case, we have a single peak in the phase space, near to

1.5R200. Galaxy distribution however is still less symmetrical than

in Gaussian groups, with the peak not aligned at Vlos/σ ≈0. This

suggests that, although more organized than NG systems, modes are

probably dynamically distinct and younger than Gaussian groups

as well.

Our work points out the importance of studying NG systems both

to possibly correct their mass estimates and multiplicity functions

and to better understand galaxy clustering at group scale. The un-

derstanding of these objects is also relevant for cosmological studies

using groups and clusters through the evolution of the mass function

(Voit 2005). Using systems with overestimated properties may lead

to a larger scatter in the mass calibration (Lopes et al. 2009b) and

could also affect the mass function estimate (Voit 2005).

ACKNOWLEDGMENTS

We thank the referee for raising interesting points. We also thank A.

C. Schilling and S. Rembold for helpful discussions. ALBR thanks

the support of CNPq, grants 306870/2010-0 and 478753/2010-1.

PAAL thanks the support of FAPERJ, process 110.237/2010. MT

thanks the support of FAPESP, process 2008/50198-3.

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- CitationsCitations9
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