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

Article (PDF Available)inMonthly Notices of the Royal Astronomical Society 413(1) · March 2011with46 Reads
DOI: 10.1111/j.1745-3933.2011.01038.x · Source: arXiv
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.
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Mon. 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 (R1R200),
9 per cent (R2R200) and 16 per cent (R3R200 and R4R200 ). 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 s1Mpc1. 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 s1)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
1021.550 4.602 433.218 32 8
130 0.999 1.245 240.908 43 11
1381.271 2.578 348.885 74 14
139 0.997 1.250 235.740 39 12
169 0.765 0.573 225.212 9 4
1771.178 2.100 336.309 45 14
1791.663 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
3521.656 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
4471.169 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
4751.152 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
5051.176 2.111 336.804 20 7
5071.459 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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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 s1)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 Vlos0. 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 Vlos0, 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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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 Vlos0. 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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