University of Groningen Formation and Evolution of Galaxy ...Cover Image: A simulated galaxy cluster in a Cold Dark Matter Universe and its connectings ﬁla-ments, which channel material

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Formation and evolution of galaxy clusters in cold dark matter cosmologiesAraya-Melo, Pablo Andres

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Formation and Evolution of Galaxy Clusters

in Cold Dark Matter Cosmologies

Proefschrift

ter verkrijging van het doctoraat in de

Wiskunde en Natuurwetenschappen

aan de Rijksuniversiteit Groningen

op gezag van de

Rector Magnificus, dr. F. Zwarts,

in het openbaar te verdedigen op

maandag 19 mei 2008

om 16.15 uur

door

Pablo Andres Araya Melo

geboren op 2 juli 1979

te Santiago, Chili

Promotor: Prof. dr. M.A.M. van de Weijgaert

Beoordelingscommissie: Prof. dr. B.J.T. Jones

Prof. dr. O. Lopez-Cruz

Prof. dr. A. Reisenegger

ISBN-nummer: 978-90-367-3436-3ISBN-nummer: 978-90-367-3437-0 (electronic version)

A mis padres

Cover Image: A simulated galaxy cluster in a Cold Dark Matter Universe and its connectings fila-

ments, which channel material into it.

Back-cover Image: N-body simulation of the Large Scale Structure of the Universe. Clearly visible

are the clusters, filaments and voids.

The illustrations were created by Miguel Angel Aragon Calvo.

Contents

1 Introduction 1

1.1 Friedmann-Robertson-Walker Universe . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Effects of a cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Growth of Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Gravitational instability theory . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 The Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.3 Linear Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.4 Nonlinear evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Cluster of Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.1 Spherical Collapse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.2 Cluster Catalogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Galaxy Cluster as Cosmological Probes . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 Superclusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.6 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Simulations and Global Properties 21

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Cosmological Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 FRW Universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.2 Cosmic Structure Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.3 Cosmological Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 N-Body Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 The Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.1 Halo Catalogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.3 Mass Functions: Press-Schechter formalism . . . . . . . . . . . . . . . . . . . 32

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.A Spherical Collapse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.A.1 From initial time to turn around . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.A.2 Virialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.A.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Galaxy Cluster Evolution: Mass Growth and Virialization 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 The Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Halo identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.2 Halo properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.3 Halo merger trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

vi CONTENTS

3.2.4 Timing the simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Cosmological Cluster Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Cluster formation: the role of Ωm . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.2 Cluster formation: the role of ΩΛ . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Formation time and mass accretion history . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.1 Formation time and mass accretion history . . . . . . . . . . . . . . . . . . . 53

3.4.2 Global formation epoch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.3 Single halo MAH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4.4 General MAH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.5 General MAH: formation times . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 Virialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5.1 Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5.2 Case studies: single halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5.3 General view: virialization at z = 0 . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5.4 Virialization of Galaxy Clusters . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Cosmological Influence on Physical Characteristics of Clusters 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 The Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67


4.2.2 Halo properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Radial mass distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.1 Concentration parameter: definition . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.2 Individual halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.3 Mass dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.4 Evolution of the concentration parameter . . . . . . . . . . . . . . . . . . . . 73

4.4 Morphology in different cosmologies: dependence on mass, redshift and formation . . 74

4.4.1 Mass dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4.2 Shape Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5.1 Angular momentum in single halos . . . . . . . . . . . . . . . . . . . . . . . 79

4.5.2 Generic angular momentum growth . . . . . . . . . . . . . . . . . . . . . . . 79

4.5.3 Angular momentum and cosmological constant . . . . . . . . . . . . . . . . . 81

4.5.4 Angular momentum, merging and accretion . . . . . . . . . . . . . . . . . . . 81

4.5.5 The spin distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Cluster Halo Scaling Relations 87

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2.1 Fundamental Plane and Virial Theorem . . . . . . . . . . . . . . . . . . . . . 90

5.2.2 Fundamental Plane Evolution:

Mass-to-Light ratio and Galaxy Homology . . . . . . . . . . . . . . . . . . . 90

5.2.3 Cluster Fundamental Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3 The Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


5.3.2 Halo properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.3 Determination of Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . 95

5.4 Scaling Relations in Different Cosmologies: z=0 . . . . . . . . . . . . . . . . . . . . 96

5.4.1 Kormendy Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.4.2 Faber-Jackson Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

CONTENTS vii

5.4.3 Fundamental Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4.4 Halo size: radius definition and scaling relation . . . . . . . . . . . . . . . . . 100

5.5 Evolution of Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.6 Merging and accretion dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Galaxy Clusters into the Future 111

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2.1 The simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112


6.2.3 Halo properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3 Power Spectrum Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.4 Mass Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.5 Mass Accretion Histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.6 Shapes in the far future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.7 Angular Momentum and Spin Parameter into the far future . . . . . . . . . . . . . . . 120

6.8 Virialization towards the future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.8.1 Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.8.2 Virialization of Galaxy Clusters in the Far Future . . . . . . . . . . . . . . . . 124

6.9 Scaling Relations in the Far Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.9.1 Kormendy Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.9.2 Faber-Jackson Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.9.3 Fundamental Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7 Future Evolution of Superclusters in an Accelerating Universe 131

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.2 Spherical Collapse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.2.1 Criterion for Bound Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.2.2 Determining the linear overdensity for a marginally bound object . . . . . . . 134

7.3 The Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.3.1 HOP halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.3.2 Virialized groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136


7.3.4 Sample completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.4 Supercluster Mass Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.4.1 Press-Schechter mass functions . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.4.2 Mass functions in Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.4.3 Comparison of simulated and theoretical mass functions . . . . . . . . . . . . 141

7.5 Shapes of Bound Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.5.2 Shape evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.5.3 Mass dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.6 Density Profiles of Bound Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.7 Supercluster Multiplicity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.A Press-Schechter formalism and its variants . . . . . . . . . . . . . . . . . . . . . . . . 154

Nederlandse Samenvatting 157

Resumen en Espanol 165

viii

English Summary 173

Bibliography 184

Acknowledgements 185

1Introduction

The only true wisdom is in knowing you know

nothing.

Socrates

Early civilizations have been wondering where everything came from and how everything began.

For centuries, humanity has been wondering what our place in the vast world surrounding us is.

Tied in with this was the fundamental question of how the world came into being. Early civilizations

had a mythical view of the cosmos. For example, the Mesopotamian civilizations of Sumer and Baby-

lon had a cosmology in which Earth was a disk surrounded by the underground waters of the Apsu and

the underworld of the dead, with the heavens of the stars surrounding it all. Despite the highly sophis-

ticated level of astronomy in the neo-Babylonian world and its inheritants, it is with the ancient Greeks

that astronomy and cosmology entered the level of scientific inquiry. While many common Greeks still

shared a mythical view of the world, impressive philosophical and scientific advances led to a world

view based on mathematical and geometrical models of reality that remained virtually unchallenged

until the 16th century. Building upon the observations carefully obtained and archived by the Babylo-

nians, the Hellenistic Greeks were the first to use their geometric models to predict the observational

reality, creating a quantitative as well as a qualitative model of the Universe. Eratosthenes measured

the circumference of Earth, while Aristarchus measured size and distance of Earth and Moon. He even

forwarded the suggestion that the Sun was at the center of our world, a view which only became the

accepted view with Copernicus and Galilei in the 15th and 16th century.

It is with the Scientific Revolution of the 16th and 17th century that a truly scientific model of the

Universe came into being. With Nicolai Copernicus in 1543 Earth finally lost its privileged central

position in the cosmos. This prods us to refer to the Copernican principle when we wish to refer

to the fact that we can not be at a special spatial or temporal position in space-time. Standing on

the shoulders of Copernicus, Brahe, Kepler and Galilei, it was Isaac Newton who managed to frame

the laws of gravity and mechanics. This formed the foundations of classical physics. His view of

a static space-time and a gravitational force resulting from action at distance made it impossible to

open the view on the dynamic cosmological world view which we presently hold. It was Einstein’s

General Theory of Relativity that formed the final breakthrough towards turning cosmology into a

scientific inquiry. His metric theory of gravity turned space-time into a dynamic medium in which

gravity is a manifestation of the curvature of space-time. Soon it was realized that this implies that the

Universe could not be static and instead should be expanding or contracting. Friedmann and Lemaıtre

were the first who worked out the expanding solutions for a homogeneous and isotropic Universe.

Their theoretical ideas were soon confirmed in the seminal discovery in 1929 by Edwin Hubble of the

expanding system of galaxies around us. His “Hubble Law”, describing the fact that distant galaxies

are receding with velocities proportional to their distance, is still the fundamental basis for present day

cosmology.

It was Lemaıtre who realized the tremendous implications of this finding. At earlier times, the

Universe would have been a lot smaller, a lot denser and much hotter than the present Universe. This

gave rise to the Big Bang Theory, stating the fact that the Universe came into being at some finite point

2 Chapter 1: Introduction

Figure 1.1 — The “Bullet Cluster”. It provides the best evidence to date for the existence of dark matter. It is also

a nice example of hierarchical structure formation in action. Image Credit: X-ray: NASA/CXC/M.Markevitch et

al. Optical: NASA/STScI; Magellan/U.Arizona/D.Clowe et al. Lensing Map: NASA/STScI; ESO WFI; Magel-

lan/U.Arizona/D.Clowe et al.

in the past. We now know that the cosmos came into being 13.7 billion years ago in a seething sea

of radiation and matter. In the meantime, this theory has gained widespread acceptance as our world

view because of the mounting and impressive amount of observational evidence. It would explain

the darkness of the night sky, the so-called Olber’s paradox and the abundance of the light chemical

elements hydrogen and helium. Its validity got its final confirmation with the discovery of the cosmic

microwave background (CMB) radiation in 1965 by Penzias and Wilson. By reading its signal to ever

increasing sensitivity and accuracy, we have been able to read the value of the fundamental cosmo-

logical parameters to incredible precision, turning the CMB into the most important pillar of modern

cosmology.

Even despite its tremendous successes, the standard Big Bang theory does involve several unsolved

issues and coincidences. Some of the most precarious issues are that of the near flatness of the Universe

and its almost perfect isotropy (< 10−5), a fine tuning which is hard to understand within the context

of standard Big Bang theory. Also, it does not explain the origin of structure in the Universe. All these

issues may be solved simultaneously if the Universe underwent an inflationary exponential expansion

phase. During this cosmic inflation, 10−34 seconds after the Big Bang, the Universe blew up by a factor

of 1060.

While inflation has become an almost inescapable ingredient of our cosmological world view, we

are still left with the puzzle of the identity of the energy content of the Universe. Only 4% of its

energy content is in the form of known baryonic matter and radiation. Structure would never have

been able to form if it had not been for the dominant gravitational influence of a mysterious dark

matter component. Only sensitive to the force of gravity and perhaps to the weak nuclear force, its

insensitivity to the electromagnetic force means it is invisible, hence its name of darkness. Fig. 1.1

shows the “Bullet Cluster”, an impressive collision of two cluster of galaxies. The presence of dark

matter was detected indirectly by the gravitational lensing of background objects. Even though it may

represent more than 85% of matter and its gravitational influence has been recognized over a range of

scales, we have as yet been unable to pin down its identity. The presence of such a large amount of

1.1. FRIEDMANN-ROBERTSON-WALKER UNIVERSE 3

dark matter forms one of the major challenges for present day cosmology.

Even more mysterious and intriguing is the presence of dark energy. Discovered only ten years

ago, it has transformed our view of the Universe. Representing some 73% of the Universe’s energy, it

is responsible for the accelerated expansion of the Universe and assures its flat geometry. Its identity

is a total mystery for astronomers and physicists, though its overriding influence may provide the

key towards unraveling the dichotomy between quantum theory and general relativity on the highest

energy scales. While the most probable situation is that of the presence of a cosmological constant,

i.e. a modifying curvature term, another reading is the possibility that it involves a strange dark energy

medium. The pressure of this dark energy would be negative, translating into a repulsive gravitational

impact. While we have recognized its dominant influence on the largest cosmological scales, it remains

to be seen whether its impact can also be recognized in smaller structures.

One of the best studied and understood physical objects on extragalactic scales are the clusters of

galaxies. They are the most massive and most recent fully developed objects in the Universe. One

may therefore improve our understanding of dark matter and dark energy by studying their influence

on the structure of galaxy clusters. It is this which we attempt to do within this thesis. One particular

approach is to do this on the basis of a few model situations. We have chosen to investigate structure,

and in particular clusters, formation in a variety of cosmological models. By comparing the outcome

of the evolution of clusters in scenarios with different cosmological mass density and dark energy

content, we look for significant differences between the equivalent clusters in each of these scenarios

1.1 Friedmann-Robertson-Walker Universe

The general relativity theory of gravity is a metric theory. To describe the gravitational evolution of the

Universe we therefore need to constrain the geometry of the Universe. It is within this context that the

Cosmological Principle plays a fundamental role. Its statement that the Universe is homogeneous and

isotropic is clearly valid on large scales of several hundreds of megaparsecs and beyond, homogeneity

means that every region in space is the same and isotropy means that it looks the same in each direction.

The direct implication is that the Universe can only have one of three geometries: negatively curved

hyperbolic space, positively curved spherical space or flat space. The metric of these geometries

translate into the Robertson-Walker (RW) space-time metric,

ds2 = c2dt2−a2(t)

(

dr2

1− k r2+ r2 dθ2 + r2 sin2 θ dφ2

)

. (1.1)

In this equation, r is the radial coordinate, t is cosmic time and θ and φ specify the angle towards

an object. The curvature of space is specified in terms of a renormalized integer constant k, which

can attain three values: −1 (hyperbolic), 0 (flat) and 1 (spherical). The expansion of the Universe is

encapsulated in the cosmic scale factor a(t). By convention, at the present time t0, a(t0) = 1.

The Einstein field equations for a spacetime obeying the RW metric, assuming that the Universe

can be described as an ideal fluid, leads to the Friedmann equations, which describes the expansion

and evolution of the Universe:a

a= −

4πG

3

(

ρ+3p

c2

)

+Λ

3(1.2)

and(

a

a

)2

=8πGρ

3− kc2

a2+Λ

3(1.3)

where G is Newton’s gravitational constant, p is the pressure, ρ is the mass density andΛ is the vacuum

energy or cosmological constant, which acts as an energy density ρΛc2 = c4Λ/8πG.

By extrapolating these equations backwards in time, it is possible to see that an expanding Uni-

verse should have had a beginning. Edwin Hubble confirmed the expansion of the Universe, when he

discovered that galaxies recede from us with a velocity which increases with increasing distance,

v = Hr ; H(t) =a

a(1.4)


Figure 1.2 — An all-sky image of the Universe, when it had around three hundred thousands years (300.000

yrs), as measured by the WMAP. The shades of grey correspond to tiny temperature fluctuations. Courtesy

NASA/WMAP Science Team.

H(t) is known as the Hubble parameter, and it is the rate of expansion of the Universe. This universal

relation is known as the Hubble’s law. The Hubble constant, H0 is defined as the expansion rate at

present time t0, and is often expressed as H0 = 100 h km s−1 Mpc, where h is a dimensionless factor.

Hubble’s discovery may be seen as the beginning of modern cosmology. To assess the fate of the

Universe it is useful to express energy and matter densities in terms of the critical density ρc,

ρc =3H2

0

8πG, (1.5)

the density at which the Universe would be flat, i.e. k = 0. A Universe with a density higher than ρc

will be spherically curved, i.e. it is spatially closed, while a density lower than ρc would correspond to

a hyperbolic geometry and a spatially open Universe. Depending on the precise value of the Hubble

constant, currently estimated to be H0 = 71±1 km/s Mpc,the value of the critical density is 9.31×10−27

kg m−3.

With the definition of the critical density, we can define other useful cosmological parameters:

Ωm =ρm

ρc

, Ωrad =ρrel

ρc

, ΩΛ =Λ

3H20

, Ωk = −kc2

a20H2

0

, (1.6)

to refer to the density of ordinary matter, relativistic matter (radiation), vacuum energy and curvature.

Ignoring Ωrad, we have

Ωm+Ωk +ΩΛ = 1 . (1.7)

Note that they concern at present time, with Ωm,0 the density parameter at the current epoch. The

density parameter at other epochs will be denoted by Ω(z). Accordingly, the current density of the

Universe may be expressed as

ρ0 = 1.8789×10−26Ωh2 kg m−3 (1.8)

= 2.7752×1011Ωh2 M⊙Mpc−3 (1.9)

Introducing the deceleration parameter

q = −aa

a2(1.10)

In a Universe with matter and a cosmological constant, Eqn. 1.2 takes the form

q0 =1

2Ωm−ΩΛ (1.11)


where q0 is evaluated at present time. The deceleration parameter describes the rate at which the

expansion of the universe is slowing down. A matter dominated Universe (Ωm = 1) correspond to

q0 =12, while a negative value of of q0 corresponds to a universe in which the expansion is accelerating.

In principle, it should be possible to determine the value of q0 observationally. For example, for a

population of uniformly luminous sources because of the luminosity distance dependence on q0 (e.g., a

set of identical supernovae within remote galaxies) the deceleration parameter tend to favor q0 values

of less than 12. Since 1998 (Riess et al. 1998; Perlmutter et al. 1999), we know that q0 is negative,

therefore, we are living in an accelerating Universe.

Having presented all these definitions, it is possible to completely specify a FRW Universe by

the cosmological parameters (H0, Ω0, ΩΛ). Once these parameters are known, the evolution of a

cosmological model is given by

H(a) =a

a= H0 E(a) , (1.12)

where E is the normalized Hubble function defined as

E2(a) = Ω0a−3+ (1−Ω0−ΩΛ)a−2+ΩΛ , (1.13)

The hot Big Bang model is supported by a large amount of observational evidence. A few observa-

tions have become major pillars of the Big Bang model. One of them is that it explains Olber’s paradox.

Only in a Universe with finite age and with finite velocity of light the sky at night is dark. Hubble’s

law confirms the reality of an expanding Universe. The most important one, the Cosmic Microwave

Background (CMB) radiation. After the first minutes of the Big Bang, the temperature of the Universe

had cooled down to a few billion degrees. Photons were continually emitted and absorbed, giving the

radiation a blackbody spectrum. As the Universe expanded, it cooled to a temperature at which photons

could no longer be created or destroyed. When the temperature fell to some 3000 degrees, electrons

and nuclei began to combine to form atoms, a process known as recombination. Almost coincidental is

the resulting decoupling of radiation and matter, 380 000 years after the Big Bang. No longer scattered

by freely floating electrons, photons assumed a long journey along the depths if a virtually transparent

Universe. The photons make up the CMB that is observed today. Since then, gradual expansion of the

Universe goes along with a proportional cooling down of the photon temperature, having reached a

present day value of T ≈ 2.725K. A map of the temperature distribution of the CMB radiation is seen

in Fig. 1.2.

Current observations, in particular those of the microwave background temperature fluctuations by

WMAP, indicate that our Universe is nearly, perhaps perfectly, flat. From light element abundances, in

combination with primordial nucleosynthesis considerations, and from the WMAP determination of

the second acoustic peak in the CMB power spectrum we have learned that normal baryonic matter can

account for no more than 4.4% of the critical density. Numerous other observations indicate that there

is a substantial amount of non-baryonic dark matter. This is already found on galactic scales from e.g.

the rotation curves of disk galaxies. The dynamics of galaxies clusters indicate that the dark matter

may account for ∼25-30% of the energy density of the Universe. The recent WMAP5 (Dunkley et al.

2008) results list a value of 23% hidden non-baryonic dark matter.

1.1.1 Effects of a cosmological constant

Dark energy, in the form of a cosmological constant, has several strong effects on the evolution of

the Universe. Universes with a negative cosmological constant will recollapse due to the attractive

gravity of matter. A positive cosmological constant will resist the attraction of matter due to its neg-

ative pressure. In most Universes, a positive cosmological constant will eventually dominate over the

attraction of matter and will drive the Universe to expand exponentially. For a limited range of values,

the cosmological constant will never dominate over the matter, and therefore the Universe will recol-

lapse after some finite time. In the extreme case of a Universe with large cosmological constant, the

Universe may not experience a Big Bang. These Universes collapse from an infinite size, they turn

around and then expand to infinity again. They are called bouncing Universes.


Figure 1.3 — Gravitational lensing produced by the massive and compact galaxy cluster Abell 2218. Because

of its mass, gravity bends and focuses the light from galaxies that lie behind it. As a result, multiple images of

these background galaxies appear as stretched out arcs. Credits: W.Couch (University of New South Wales), R.

Ellis (Cambridge University), and NASA.

1.1.1.1 Observational evidence

Various pieces of evidence show that a Universe with a cosmological constant best fits observations.

It was not until recently that observations started to confirm this issue. The most compelling pieces of

evidence are:

• Observations of Type Ia supernovae: one of the most direct impacts of having a cosmological

constant is its influence on the cosmic and dynamical timescales. In principle, given a set of ob-

jects with either a standard proper size or luminosity, one could determine the physical distance

to the object. These particular objects are the supernovas type Ia. This type of supernovae ex-

hibit a behavior that allows the absolute magnitude of the supernovae to be determined from the

shape of their light curve and their time varying spectra. Once we know the absolute magnitude,

it is possible to determine their actual distance from us. Riess et al. (1998) and Perlmutter et al.

(1999) measured the light curves of distant type Ia supernovas and found direct evidence that

the Universe is expanding.

• Cosmic microwave background: the discovery of temperature anisotropies in the cosmic mi-

crowave background by the COBE satellite started a new era in the determination of cosmolog-

ical parameters.

The cosmic microwave background (CMB) is made up of photons that are coming to us since

they decoupled from matter. These photons are shifted to microwave wavelengths due to the

expansion of the Universe. This is the oldest light in the Universe.

The temperature fluctuations of the sky are decompose into spherical harmonics, yielding the

angular spectrum of the CMB. The Wilkinson Microwave Anisotropy Probe (WMAP) has stud-

ied this fluctuations, investigating the physical processes that happened when the Universe was

young. WMAP has matched the patterns of these fluctuations and matched them to the physics

we know, providing convincing results on the contents of the Universe. It has determined the

age of the Universe, the epochs of key transitions of the Universe, the geometry of the Universe

and the value of the cosmological constant ΩΛ.

• Integrated Sachs-Wolfe effect: this is a property of the cosmic microwave background radi-

ation. Photons of the CMB gain energy by falling into gravitational potential wells, and lose

energy when they climb out again. In a Universe in which the full critical energy density comes


from atoms and dark matter, the overall loss and gain of energy cancel out. However, in the

presence of dark energy, the photons gain more energy as they fall into an overdense region and

lose less energy as they come out due to the stretching of the potentials caused by the expansion

of the Universe. This is the Integrated Sachs-Wolfe effect. Hotter CMB photons are due to over-

dense regions in the Universe, while cold CMB photons correspond to underdense regions. This

behavior is seen in the CMB spectrum.

• Gravitational lensing: gravitational lensing is the process in which light from a very distant,

bright source (e.g. a quasar) is “bent” around a massive object (such as a galaxy cluster) between

the source and the observer. This process is one of the predictions of the general theory of

relativity.

Figure 1.4 — A composite of images for the evidence of dark energy. Top left figure: Hubble diagram (distance

modulus vs. redshift) from the SNIa observations. Top right image: a distant galaxy cluster. Bottom plot: the

“angular spectrum” of the fluctuations in the WMAP full sky-map. These combined data puts constraints on

the cosmological parameters, shown in the central figure. (WMAP spectrum: Courtesy NASA/WMAP Science

Team.)


The volume of space back to a specified redshift is sensitively dependent on the cosmological

constant ΩΛ. Therefore, counting the apparent density of observed objects provides a potential

test for ΩΛ. Using the statistics of gravitational lensing of distant galaxies it is possible to

infer the volume of space and, therefore, the value of the cosmological constant. An example

of a gravitational lens can be seen in Fig. 1.3. The galaxy cluster Abell 2218 is so massive

that gravity bends and focuses the light from galaxies that lie behind it. Images of background

galaxies are observed, which are larger and brighter than normal, optical images. As a result,

it is possible to observe a large amount of galaxies that otherwise would be extremely hard to

observe. Abell 2218 has produces more than 120 images of galaxies that are members of a

remote cluster.

• Number counts of galaxy clusters: they provide a direct probe of cosmology, complementary

to supernova Ia and CMB measurements. Catalogs are built using clusters found by the Sunyaev-

Zeldovich effect or with clusters in X-ray. The idea is to measure abundances of these objects as

a function of redshift and compare this to theoretical predictions.

The mentioned observational evidence does not provide independent identification of the cosmo-

logical parameters. Each one of them gives pieces of evidence on the values of Ωm and ΩΛ. Fig. 1.4

gives a more clear picture. SNIa gives information on the acceleration of the Universe via the decel-

eration parameter (see Eqn. 1.11), while the abundance of galaxy cluster tightly constrains the matter

abundance Ωm. The CMB spectrum tell us that the Universe is flat. Combined, these data allows us to

infer a “confidence interval” (image on the center of the figure) in the Ωm−ΩΛ plane.

Using these combined data, WMAP5 inferred values of Ωm ∼ 0.258, ΩΛ ∼ 0.742 and H0 ∼ 71.9

km/s/Mpc, resulting in an age of the Universe of 13.69 Gyrs. They are the main parameters defining

the concordance model.

1.2 Growth of Structure

The observed Universe is far from being homogeneous: there is an enormous richness of structure

ranging from dwarf galaxies to groups and clusters of galaxies.

1.2.1 Gravitational instability theory

The standard, and very important, ingredient of the theory of the evolution of the Universe is inflation

(Guth 1981). Inflation basically states that shortly after the Big Bang, the Universe entered a phase of

very rapid expansion.

In our current view of structure formation, we assume that the primordial Universe is not com-

pletely uniform. Instead, small density fluctuations were imprinted in the very early Universe. As

the Universe evolved, the small quantum fluctuations that were present in the first instants of the Uni-

verse were blown up to cosmological scales. This implies that after this rapid expansion, matter in

the Universe is inhomogeneously distributed, resulting in the plethora of structures we see in the Uni-

verse today (see Fig. 1.5). Inflation also predicts that the fluctuations have the character of a Gaussian

random field.

For a description of the inhomogeneities in the primordial density field it is convenient to write

δ(x) =ρ(x)−ρb

ρb

, (1.14)

where ρb is the background density. In comoving perturbation quantities, the continuity equation,

1.2. GROWTH OF STRUCTURE 9

Figure 1.5 — Gravitational collapse of primordial fluctuations. Tiny primordial fluctuations in the primordial

density field (left frame) get amplified by gravity. They will end up forming the large scale structure seen in the

right panel.

Euler equation and Poisson equation describing the evolution of a density perturbation are

∂δ

∂t+

1

a∇ · [(1+δ)v] =0 , (1.15)

∂v

∂t+

1

a(v · ∇)v+

a

av =− 1

ρa∇p− 1

a∇φ, (1.16)

∇2φ = 4πGρa2δ , (1.17)

in which v is the peculiar velocity and φ is the peculiar potential. The peculiar gravitational potential

φ is related to the peculiar gravitational field g by:

g(x) = −∇φa=Gaρ

∫

δ(x′)(x′−x)

|x′−x|3dx′ (1.18)

1.2.2 The Power Spectrum

The initial density field is fully characterized by its power spectrum P(k), which specifies the amplitude

of the fluctuations as a function of their spatial scale. In general, it is assumed that the power spectrum

has a power-law behavior P(k) ∝ kn, where the relative amplitude between scales is dictated by the

index n, i.e., n determines the balance between small and large scale power.

The initial perturbation spectrum is commonly assumed to be a power law,

P(k) = kn . (1.19)

There is a special case, where n = 1, in which Eqn. 1.19 has the property that the density contrast has

the same amplitude on all scales when the perturbations enter the horizon. This special case if often

referred to as the Harrison-Zeldovich (Zeldovich (1972), Harrison (1970)).

The primordial power spectrum is believed to change during the evolution of the early Universe

until the end of the epoch of recombination by various processes, such as growth under self-gravitation,

effects of pressure and dissipative processes. The overall effect can be encapsulated in the transfer


function, T (k), which gives the ratio of the later-time amplitude of a mode to its initial value:

P(k,z) = P0(k)T 2(k)D2(z)

D2(z0), (1.20)

The evolution of linear perturbations back to the surface of last scattering obeys the simple growth

laws given in Eqn. 1.20.

Calculations of the transfer function are a challenge, mainly because there is a mixture of matter

and relativistic particles. For Cold Dark Matter spectrum, Bardeen et al. (1986) found the approxima-

tion

T (k) =ln (1+2.34q)

2.34q[1+3.89q+ (16.1q)2+ (5.46q)3+ (6.71q)4]−1/4 , (1.21)

with q = kh/ΓMpc, and Γ is the shape parameter, Γ = Ωmh. Sugiyama (1995) obtained a more general

form for the shape parameter, which is given by

Γ = Ωmhexp

−Ωb

1+

√2h

Ωm

, (1.22)

with Ωb the baryon fraction density parameter.

To completely specify the power spectrum, we need to fix the overall amplitude. For P(k) with a

given shape, the amplitude is fixed if we know the value of P(k) at any k. One prescription for normal-

izing a theoretical power spectrum involves the variance of the galaxy distribution when sampled with

randomly placed spheres of radius R:

σ2(R) =1

2π

∫ ∞

0

k3P(k)W2(k,R)dk

k, (1.23)

where W(k,r) is the Fourier representation of a real space top-hat filter enclosing a mass M at the mean

density of the Universe, which is given by

W(kR) = 3

(

sin(kR)

(kR)3−

cos(kR)

(kR)2

)

. (1.24)

The value of σ(R) derived from the distribution of normal galaxies is approximately unity in spheres of

radius R = 8h−1Mpc, a quantity known as σ8. An alternative is to calculate the present day abundance

of rich clusters of galaxies: σ8Ωαm, with α ≈ 0.5−0.6. This quantity put constraints on both σ8 and the

matter density Ωm.

1.2.3 Linear Perturbation Theory

Assuming that the fluctuation field is small, one can linearize the equations of motion. One then obtains

∂2δ

∂t2+2

a

a

∂δ

∂t= 4πGρδ . (1.25)

The general solution to this equation consists of two modes:

δ(x, t) = A(x)D1(t)+B(x)D2(t) , (1.26)

where D1(t) and D2(t) are two time independent solutions. They are often called growing and decay-

ing modes. Usually, analysis concentrates on the growing mode, evidently the decaying solution is

damped. For a generic FRW Universe, the general expression for a the growing mode is given by

D1(z) =5Ωm,0H2

0

2H(z)

∫ ∞

z

1+ z′

H3(z′)dz′ . (1.27)


Figure 1.6 — Evolution as a function of redshift of a single dark matter halo in a ΛCDM flat Universe. The halo

at present time formed from a tiny perturbation in the primordial density field, and then subsequently accreted and

merged with small mass clumps of its surrounding, forming the massive object at present epoch.

In a Universe with Ωm = 1, the growing mode is D1 ≈ a ∝ t2/3.

For matter dominated Universes, the growth of structure closely resembles that in a Universe with

Ωm = 1. As it evolves and becomes increasingly empty, it enters a near free expanding phase. This

happens around redshift

1+ zm f =1

Ωm,0−1 . (1.28)

As a result, in matter dominated Ωm < 1 Universes at early times we see structure growing with a rate

D(a) proportional to a(t), while it freezes out after zm f .

In the case of of Λ dominated Universes, structure formation comes to a halt when the Universe

sets in its accelerated expansion at redshift zΛ f ,

1+ zΛ f =

(

2ΩΛ,0

Ωm,0

)

. (1.29)

In a Universe with Ωm = 0.3 and ΩΛ = 0.7, this corresponds to z ∼ 0.7.


1.2.4 Nonlinear evolution

Once density fluctuations approach unity, linear theory is not longer valid. Since the full nonlinear

solutions are in general too complex to solve analytically, one must rely on other alternatives, such as

computer simulations.

Structure in the mildly nonlinear regime is marked by some important characteristics. Amongst

these, the most important are:

• Hierarchical Clustering

• Anisotropic Collapse

• Cellular morphology

1.2.4.1 Hierarchical Clustering

In the cold dark matter scenario, for a power spectrum with n(k) > 3, small scale fluctuations collapse

first to form bound objects before larger structures do, resulting in a gradual building up of successively

larger structures by the clumping and merging of smaller structures. This process is called hierarchical

structure formation. An example of this is seen in Figs. 1.6 and 1.7, where the evolution of the same

cluster is seen in two different cosmologies, SCDM and ΛCDM.

A powerful description of hierarchical structure formation is the Press-Schechter theory (Press &

Schechter 1974; Bond et al. 1991), which describes the sample average characteristics of an emerging

population of nonlinear objects evolving from a linear density field of fluctuations in the primordial

cosmos.

For a Gaussian density field δ(M), with 〈δ〉 = 0 and 〈δ2〉 = σ2(M), one has that

p(δ) =1

√2σ(M)

exp−δ2c

2σ2(M), (1.30)

where δc is the density contrast associated with perturbations of mass M. Then, at any given time t, the

fraction of an object of mass M enclosed in a sphere of radius R within which the mean overdensity

exceeds δc is given by

f (δ > δc) =1

2erfc

(

δc√2σ(R)

)

. (1.31)

As M → 0, σ(R)→∞ and thus f → 1/2. The pure PS formula therefore predicts that only half of

the Universe forms lumps of any mass. This was corrected by Bond et al. (1991) using the extended

PS formalism or excursion set theory, showed that the factor of 2 can be justified with a sharp k-space

filter. The PS formalist leads to the following comoving number density of halos of mass M at time Z:

dn

dM(M,z) =

√

2

π

ρ

M2

δc(z)

σ(M)

∣

∣

∣

∣

∣

dlnσ(M)

d ln M

∣

∣

∣

∣

∣

exp

(

−δ2c(z)

2σ2(M)

)

, (1.32)

where δc(z) = δc/D(z) is the critical overdensity linearly extrapolated to the present time.

1.2.4.2 Anisotropic collapse

Another important characteristic of the nonlinear evolution is its anisotropic collapse: structures which

at early stages are slightly non-spherical tend to become more and more anisotropic as time evolves.

This characteristic was predicted by Zel’Dovich (1970), who explored the nonlinear regime by simply

assuming that linear conditions remain valid in the early nonlinear regime. Icke (1973) investigated

the evolution of homogeneous ellipsoidal configurations in an expanding FRW Universe and found

that the predominant final morphologies are flatness and elongated.

We can easily observe this by considering the initial displacement of particles and assume that they

continue to move in this initial direction. The physical position of a particle can be written as r = a(t)x,


Figure 1.7 — Formation and evolution as a function of redshift of a single dark matter halo in a SCDM Universe.

The hierarchical process is clearly visible: the halo gains mass by merging with smaller, surrounding objects.

where x is called the comoving position. Thus, for a given particle, the proper coordinate will be given

by

x(t) = q+D(t)f(q) . (1.33)

This looks like Hubble expansion with some perturbation, which will become negligible as t → 0.

Therefore, the coordinates q are equal to the usual comoving coordinates at t = 0, and D(t) is the linear

growth factor.

We concentrate here on the anisotropic collapse of a patch of matter. By applying a simple mass

conservation of the form ρ(x, t)dq = ρ(q)dq = ρbdq, we get

1+δ =

∣

∣

∣

∣

∣

∂x

∂q

∣

∣

∣

∣

∣

−1

=1

(1−D(t)λ1)(1−D(t)λ2)(1−D(t)λ3), (1.34)

where λ1, λ2 and λ3 are the three eigenvalues of the deformation tensor ∂ fi/∂q j. The vertical bars de-

note the Jacobian determinant of the transformation between r and q. Eqn. 1.34 describes the evolution

of the density field in the Zeldovich approximation. It predicts the collapse of matter into planar sheets

or pancakes. The subsequent collapse is determined by the second largest eigenvalue, which produces

a filament. The final collapse along the axis defined by the third eigenvalue will result in the formation

of galaxy clusters.


Figure 1.8 — The 2dF galaxy redshift survey. Shown is a map of how galaxies are distributed in space as a

function of distance from us. The foamy geometry of the cosmic web is clearly visible: matter is clumped in

some specific regions, representing clusters and superclusters of galaxies, while there are also empty regions,

representing voids. Courtesy: the 2dF Galaxy Redshift Survey Team.

The Zeldovich approximation normally breaks down later than Eulerian linear theory, i.e., first

order Lagrangian perturbation theory can give results comparable in accuracy to Eulerian theory with

high order terms. It is therefore commonly used to set up quasi-linear initial conditions for N-body

simulations. Also, it has been very successful in describing a variety of properties of the cosmic web.

1.2.4.3 Cellular Morphology

The third important characteristic of nonlinear regime is the appearance of a complex cellular geom-

etry, consisting of a foam-like network of filamentary and wall-like structures surrounding extended

empty regions, the Cosmic Web (Bond et al. 1996). Fig. 1.8 shows the galaxy distribution in the 2dF

Galaxy Redshift Survey. The web-like structure is clearly visible. Bond et al. (1996) showed that the

cosmic web is largely defined by the position and primordial tidal fields of rare events in the medium,

with the strongest filaments between nearby clusters whose tidal tensors are nearly aligned.

The rare high peaks in the cosmic web corresponding to clusters play a fundamental role. They are

the nodes that define the Cosmic Web.

1.3 Cluster of Galaxies

Galaxy clusters are the largest stable structures in the Universe. Typical properties of galaxy clusters

include:

• They contain 50 to 1000 galaxies, hot gas and large amounts of dark matter.

• They have total masses of ∼ 1014−1015h−1M⊙1.

• Their radius are in the order or ∼2-6 h−1Mpc2

11M⊙=1.989×1030 kg, is the mass of the Sun21 Mpc=3.086×1022 mts.

1.3. CLUSTER OF GALAXIES 15

• Galaxy members have velocity dispersions in the order of ∼800-1000 km/s.

Galaxy clusters have been key astrophysical objects in the development of our current understand-

ing of the large scale Universe. It was in galaxy clusters that dark matter was first detected. Clusters

are also very luminous X-ray sources, emitted by a tenuous extremely hot intracluster gas with a tem-

perature of T ∼ 107−108 K. The fact that they contain an atypical mixture of galaxies makes them into

important probes of the study of galaxy evolution.

When observed visually, cluster of galaxies appear to be collections of galaxies held together by

mutual gravitational attraction (see Fig. 1.9) However, their velocities are too large for them to remain

gravitationally bound by their mutual attraction. This implies that there must be an additional invisible

mass component or an additional attractive force besides gravity. Most of the mass of galaxy clusters

is in the form of hot gas, which emits in X-ray. In a typical cluster perhaps only ∼5% of the total mass

in in the form of galaxies, ∼10% in the form of hot X-ray emitting gas and the rest is in the form of

dark matter.

1.3.1 Spherical Collapse Model

The spherical collapse model (Gunn & Gott 1972) is a simple but very useful approximation to study

the formation and evolution of structures. It describes the evolution of an isolated spherical overdense

region in a homogeneous cosmological background of mean density ρb. Although isolated spherical

systems do not exist in reality, the spherical collapse model provides an excellent basis for understand-

ing and interpreting considerably more complicated evolution of generic systems.

In this model, the isolated spherical region starts to expand at the same rate as the background.

However, if its density is high enough, its rate of expansion will slow down sufficiently that it will

eventually stop at some point, in which it reach a maximum radius, and then it will turn around,

collapse and virialize. We can identify three stages of evolution in the spherical collapse model:

• Turn around: the galaxy cluster starts to slow down and eventually will decouple from the

expansion of the Universe. It will reach a point in which it will stop expanding. In this point, its

velocity is zero and has reached its maximum radius.

• Collapse: after reaching maximum expansion, the cluster starts to collapse and shrinks to a

small size.

• Virialization: in reality, the cluster will not collapse to a point. Before that happens, the ki-

netic energy of the region is converted into random motions. The perturbation reaches a bound

equilibrium state, virialization.

1.3.2 Cluster Catalogues

Galaxy cluster were first identified as overdensities in the spatial distribution of galaxies. This method

has obvious systematic problems, in particular due to projection effects, and also because of the diffi-

culties of identifying poor clusters.

The most extensive and widely used catalog of rich clusters was created by Abell (1958). Abell

surveyed plates taken from the Palomar Observatory Sky Survey (POSS) by eye, identifying clusters

such they contain at least 50 galaxies within a radius of ∼ 3 Mpc (known as the Abell radius) and within

two magnitudes of the third brightness member. Abell’s sample contained ∼ 1700 objects which met

his criteria. He also included an additional ∼ 1000 clusters, but were not part of the statistical sample.

Later, Zwicky et al. (1968) created another catalog from POSS, the Catalog of Galaxies and Clus-

ter of Galaxies (CGCG). His criteria was less strict than Abell’s. He drew isopleths at the level were

the cluster density was twice that of the background density of galaxies. The number of cluster mem-

bers was determined by counting all galaxies within the isopleth and within three magnitudes of the

brightest member. His galaxy cluster had at least 50 members, making them rich galaxy cluster.


Figure 1.9 — The Coma Cluster in the constellation of Coma Berenices, from the Spitzer infra-red satellite.

The image shows thousands of faint objects corresponding to dwarf galaxies that belong to the cluster. Two large

elliptical galaxies dominate the cluster’s center. Courtesy NASA/JPL-Caltech.

Clusters of galaxies contain substantially more mass in the form of hot gas, which is observable in

X-ray band. Many catalogs of galaxy clusters are also selected through their X-ray emission. Catalogs

are based on the ROSAT All Sky Survey (RASS). Amongst these, we can find the ROSAT Brightest

Cluster Sample(BCS, Ebeling et al. (1998)), the Northern ROSAT All Sky Galaxy Cluster Survey (NO-

RAS, Bohringer et al. (2000)) and the ROSAT-ESO Flux Limited X-ray catalog (REFLEX, Bohringer

et al. (2001), which contains clusters over a large part of the southern sky.

1.4 Galaxy Cluster as Cosmological Probes

Galaxy cluster have been (and are still) used to put constraints on cosmological parameters using a

variety of methods:

• The mass function, the number density of objects of a given mass, provides constraints on the

amplitude of the power spectrum at the cluster scale (e.g., Rosati et al. (2002)). Its evolution

also provides constraints on the linear growth rate of density perturbations, which translates into

dynamical constraints on the matter density parameter and dark energy density parameter.

• The power spectrum and the correlation function (clustering properties) of the large scale distri-

bution of galaxy clusters provide direct information on the shape and amplitude of the underlying

dark matter distribution. The evolution of these clustering properties is sensitive to the value of

the density parameters through the linear growth rate of perturbations (e.g., Moscardini et al.

(2001)).

1.5. SUPERCLUSTERS 17

Figure 1.10 — Optical map of galaxies in the core of the Shapley Concentration: A3556-A3558-A3562 chain.

Credit: T. Venturi, S. Bardelli, R. Morganti, & R. W. Hunstead, Istituto di Radioastronomia.

• The mass-to-light ratio in the optical band can be used to estimate the matter density parameter

once the mean luminosity density of the Universe is known. This is under the assumption that

mass traces light with the same efficiency both inside and outside clusters.

• By measuring the baryon fraction in nearby clusters it is possible to constrain the matter density

parameter, once the cosmic baryon density parameter is known. The baryon fraction of distant

clusters provide a geometrical constraint on the dark energy content. Two assumptions are made:

1) clusters are fair containers of baryons and 2) the baryon fraction inside clusters does not

evolve.

1.5 Superclusters

We can extend further in the hierarchy of structures in the Universe and define superclusters of galaxies.

As its name reveals, these structures consists of several galaxy clusters. Their sizes are in the order

of ∼100 Mpc. The supercluster formation is now at an early stage.They may be at the critical point

of maximum expansion and starting to collapse under their own gravity into an increasingly dense

superstructure. Due to their early formation stage, superclusters contain information on the large scale

initial density field and their properties can be used as a cosmological probe to discriminate between

different cosmological models.

Superclusters appear to surround large under-dense regions, called voids. The voids are of com-

parable sizes. Together, they create the cellular-like morphology of the Universe on large scales (see

Fig. 1.8).

Due to their recently formation, identifying superclusters is a very difficult task. Their overdensity

is thought to be small in comparison to the overdensity of galaxies or cluster of galaxies. It is important

to define a method that is best suited to identify superclusters. So far, superclusters have been defined

as clusters of clusters, using catalogues of superclusters of galaxies. Recent redshift surveys, such

as the 2dF Redshift Survey, have help to overcome to problem of identification, pinpointing potential

superclusters.

One of the largest supercluster in our local Universe is the Shapley Supercluster, in the constellation

of Centaurus. It consists of several hundred galaxies, and several of the clusters in Shapley are also

strong sources of X-rays. Fig. 1.10 shows a radio map of galaxies in the core of Shapley.


Figure 1.11 — Evolution of a single galaxy cluster halo in comoving (upper panels) and physical (lower panels)

coordinates.

1.6 Outline of this thesis

The aim of the thesis is to achieve a more profound understanding of the role of the cosmological

constant Λ and dark matter in the formation and evolution of cluster of galaxies. To this end, we have

performed a variety of N-body simulations. All simulations involve variants of the Cold Dark Matter

scenario, embedded with a range of cosmological parameters.

In chapter 2 we extensively describe the simulations we use throughout this thesis. This simulations

include open, flat and closed Universes, with or without a cosmological constant.An important property

of these simulations is that they all start from primordial Gaussian conditions with the same Fourier

phases. This allows us to follow the same structures in each of the simulations. We investigate the

influence of the cosmological constant on global properties of dark matter halos, in particular the

shape and evolution of the mass functions.

In chapter 3 we investigate the mass assembly and formation history of cluster halos in a range of

cold dark matter cosmologies. We also investigate the virialization of these clusters halos. We look

into the assembly history of identical clusters and assessed differences in its formation as a function of

three different time scales: redshift, lookback time and cosmic time. By doing this, we expect to obtain

lights on the influence of the cosmological constant in the formation of structures. We also compare

the degree of virialization in the different simulated cosmologies.

Chapter 4 is devoted to the study of individual properties of galaxy cluster halos. These properties

include the evolution of the angular momentum, morphology and density profile. This properties are

studied as a function of the underlying cosmology.

In chapter 5 we explore the effects of dark matter and dark energy on the dynamical scaling prop-

erties of cluster halos. We investigate the Kormendy, Faber-Jackson and Fundamental Plane relation

between the mass, radius and velocity dispersion of galaxy cluster halos. The validity and behavior of

these relations in the different cosmological models should provide information on the general virial

status of the cluster halo population.

In chapter 6 we probe the effects of future evolution in several properties of cluster sized halos.

Towards the future, structures such as galaxy clusters will grow in complete isolation in physical co-

ordinates (see Fig. 1.11). The effect of a nonzero cosmological constant drives the Universe towards

unbounded exponential expansion, while a zero cosmological constant makes the expansion to decel-

erate. This expansion will have an effect on the internal evolution of galaxy cluster. In order to study

1.6. OUTLINE OF THIS THESIS 19

Figure 1.12 — A simulated supercluster of galaxies defined as an overdensity of 2.36 times the critical density

of the Universe at present time (left frame) and as an overdensity of 2 times the critical density in the far future

(right frame), in a cosmology with Ωm = 0.3 and ΩΛ = 0.7. At present time, the supercluster presents various

substructures. In the far future, the supercluster has collapse, becoming a massive bound structure.

the influence of the cosmological constant, we extract information of the future gravitational growth

of the large scale structure of the Universe and of physical quantities such as morphology, angular

momentum, virialization and scaling relations. Global properties, such as the mass function and mass

accretion history are also explored. These properties will tell us when and how clusters of galaxies

reach dynamical equilibrium and, more importantly, they will allow us to determine the importance of

the cosmological constant in the fate of the Universe.

Chapter 7 presents the study of the mass functions of gravitationally bound structures, in particular,

the largest and most massive of these. The identification of the bound structures is done by using

the criterion presented in Dunner et al. (2006). We compare the identification at present time and

in the far future. We use this criterion as a physical definition for superclusters. Fig. 1.12 shows

a massive bound structure (a supercluster) at present time (left frame) and in the far future (right

frame). The supercluster has collapsed, forming the compact structure seen in the right frame of the

figure. By investigating the mass functions, we can identify qualitatively and quantitatively the largest

superclusters in our local Universe.


2Simulations and Global Properties

W study the influence of the cosmological constant on global properties of dark matter halos. In

particular, we study the shape and evolution of the mass functions. To this end, we perform

thirteen high resolution N-body simulations, which include open, flat and closed Universes, with or

without a cosmological constant. We find that the mass function of models with the same value of

the matter density are indistinguishable at low redshift, independent of the value of the cosmological

constant. We compare our simulated mass functions with the Press & Schechter formalism, and found

that it shows a rough agreement at low redshift, but it differs substantially at higher redshifts.

22 CHAPTER 2: Simulations and Global Properties

2.1 Introduction

Cosmological observations strongly indicate that we are living in a flat, accelerating Universe with a

low matter density. Observations of distant supernovae (Riess et al. 1998; Perlmutter et al. 1999) and

the precise measurements of cosmic microwave background fluctuations (Spergel et al. 2003) have

established a new cosmological paradigm. Most of the matter in the Universe is in the form of an

unknown species of dark matter, probably cold dark matter (CDM). While this is responsible for the

formation of structure and the corresponding clustering of matter in the Universe, most of its energy

is in the form of a mysterious dark energy. This dark energy behaves like Einstein’s cosmological

constant, Λ and it is responsible for the acceleration of the cosmic expansion. The estimated amount

of dark energy appears to be precisely sufficient to yield a flat geometry of our Universe. In all, it also

solves the apparent conflict suggested by the old age of globular cluster stars.

Structure in the Universe arose out of the gravitational growth of tiny primordial density and veloc-

ity perturbations. In the current standard view this process is hierarchical, with small clumps being the

first objects to form and gradually merging and accreting while assembling into ever larger structures.

The history of this process is highly dependent on the amount of (dark) matter in the Universe: struc-

ture formation in low Ωm cosmologies comes to a halt at much earlier times than that in cosmologies

with high density values.

An issue that remains to be clarified is the role of the cosmological constant in structure formation.

In this we may identify various influences of the cosmological constant. Here we discuss three effects.

• Dark energy strongly influences the dynamical time scales involved with the structure formation

process. Possibly this is its main effect.

• Dark energy implies a modified spectrum of primordial density fluctuations. Its main effect

concerns the amplitude of the perturbations.

• The dynamical accelerating influence of dark energy may also play a role in the dynamics of the

emerging and evolving structures. In the linear regime this is a minor effect, e.g., Lahav et al.

(1991) showed that it only has around ∼ 1/70 of the influence of matter perturbations.

There has not yet been a lot of attention to situations of an open or closed Universe with a cos-

mological constant. Nor, for that matter, on structure formation in closed pure matter-dominated Uni-

verses. Of the few studies that addressed such cosmologies we may mention Bjornsson & Gudmunds-

son (1995), who discussed how a closed Universe would appear to astronomers living at different

cosmic epochs. White & Scott (1996) considered structure formation and CMB anisotropies in a

closed Universe, both with and without cosmological constant. They found that there are a range

of closed models models that are consistent with observational constraints while being older than

flat models with a cosmological constant. While perhaps less feasible than the currently popular flat

Lambda-dominated Universes, or low-density matter-dominated Universes, such cosmologies are pos-

sible products of an early inflationary phase. For example, Linde (1995) showed that it is possible to

produce inflationary models that result in generic closed Universes.

One of the most important representatives in the cosmic hierarchy of structures, and therefore

important probes for the study of cosmic structure and evolution, are clusters of galaxies. They are

the most massive and most recently collapsed objects in the Universe. Their density is in the order of

several hundred times the critical density of the Universe, with collapse times comparable to the age

of the Universe. The substructure observed in many galaxy clusters reaffirms this idea.

Observationally, it is almost impossible to study the evolution of galaxy clusters, especially if one

wants to investigate differences between different cosmological models. This makes N-body simula-

tions a necessary tool. They represent a realistic description of the formation and evolution of galaxy

clusters.

To assess the influence of a positive cosmological constant on the formation and evolution of dark

matter halos and galaxy clusters we study this in a set of dissipationless N-body simulations. All

2.2. COSMOLOGICAL BACKGROUND 23

simulations involve variants of the Cold Dark Matter scenarios, embedded with a range of cosmolog-

ical parameters. By investigating structure formation for models with different values of Ωm < 1 and

ΩΛ , 0, we do seek to learn more about the influence of ΩΛ.

One important aspect is the mass function of emerging objects. Several authors have used mass

functions as a diagnostic for Ωm (e.g., Eke et al. 1996; Lee & Shandarin 1999; Governato et al. 1999;

Gardini et al. 1999; Pierpaoli et al. 2001; Sanchez et al. 2002; Reed et al. 2003; Younger et al. 2005).

Our approach will be similar to that of these authors, but will include a wider spectrum of cosmologies.

By carefully choosing the range of our cosmologies we hope to find more information on the various

influences of the cosmological constant on the structure formation process.

This chapter is organized as follows: in section 2.2 we give a description of the Friedmann-

Robertson-Walker equation and the various cosmologies. In section 2.3 we describe the different

N-body simulations. In section 2.4 we describe the techniques to construct the halo catalogues that

we use in order to calculate the different mass functions. We use these catalogues to extract the mass

function of each cosmology. We study the evolution of the mass function in the various cosmologies

and compare with the Press-Schechter mass function. Conclusions are presented in section 2.5.

2.2 Cosmological Background

We are going to investigate structure formation in Friedmann-Robertson-Walker Universes containing

matter and a cosmological constant, with a negligible radiation contribution, and with a generic, not

necessarily flat, geometry. Structure forms as a result of gravitational instability and we assume that

the dark matter is some cold dark matter particle.

2.2.1 FRW Universes

The general Friedmann-Robertson-Walker equation for the expansion of the Universe is (neglecting

the contribution by radiation)a

a= H0

√

Ωma−3+ΩKa−2+ΩΛ , (2.1)

where a is the expansion factor is related to the redshift via 1+ z = a−1. At present time, a0 = 1, Ωm

is the matter density parameter, ΩΛ is the vacuum energy density parameter and ΩK is the curvature

density parameter,

Ωm =ρm,0

ρc,0, ΩΛ =

Λ

3H20

, Ωk = −kc2

a20H2

0

, (2.2)

where ρc = 3H20/8πG is the critical density (the energy density needed to get a flat k = 0 Universe). If

k > 0, then the Universe is closed, if k = 0, it is flat and if k < 0, it is open. Dividing Eqn. 2.1 by H0

and evaluating at present time we get

ΩK = 1−Ωm−ΩΛ . (2.3)

This equations tells us that the sum of the matter density parameter and the cosmological constant

density parameter describes the geometry of the Universe. It is convenient to define Ωtotal ≡Ωm+ΩΛ.

Then, Eqn. 2.3 becomes ΩK = 1−Ωtotal.

In our study we assess and compare all three possible geometries. Table 2.1 shows a 4×4 matrix

with a combination of Ωm and ΩΛ values. The sums in light gray refer to those Universes modelled

for the present work. We chose the values in such a way that we could investigate systematically

the influence of ΩΛ, and reproduce earlier works (open models with Ωm = 0.3 and no cosmological

constant) and the accepted flat model (Ωm = 0.3 and ΩΛ = 0.7).

We study thirteen cosmologies in total. Six of them are open models, four are flat models and the

remaining three are closed Universes. Of the six open models, three of them are pure matter dominated


ΩΛ+ 0.0 0.5 0.7 0.9

0.1 0.1 0.6 0.8 1.0

Ωm 0.3 0.3 0.8 1.0 1.2

0.5 0.5 1.0 1.2 1.4

1.0 1.0

Table 2.1 — The studied cosmological models. Each model is specified by Ωm and ΩΛ. Each column corre-

sponds to cosmologies with the same ΩΛ, each row to cosmologies with the same Ωm. The grey cells contained

the value of Ωtotal = Ωm +ΩΛ of the corresponding cosmology.

while the other three have a cosmological constant. Of the flat models, one is an EdS Universe (for

the CDM structure formation scenarios which we investigate here this is known as SCDM). The other

three flat models involve a cosmological constant. The remaining three are closed Universes with a

cosmological constant.

2.2.2 Cosmic Structure Formation

In the homogeneous and isotropic FRW Universes, initially the tiny density perturbations δ(x, t) grow

linearly with a universal rate independent of their comoving spatial scale, the linear density growth

factor D(a),

δ(x, t) = D(a) ·δ0(x) , (2.4)

in which δ0(x) is the initial density fluctuation at comoving position x (linearly extrapolated to the

present time). The density growth factor D(a) is sensitively dependent on the cosmology at hand. An

explicit expression for D(a) is (Heath 1977)

D(a) =5ΩmH2

0

2H(a)

∫ a

0

da′

a′3H(a′)3

= ag(a) , (2.5)

where g(a) is the linear growth factor. An accurate approximation for g(a) in the case of Universes

with a cosmological constant is (Carroll et al. 1992):

g(a) ≈5

2Ωm(a)

[

Ωm(a)4/7−ΩΛ(a)+

(

1+Ωm(a)

2

)(

1+ΩΛ(a)

70

)]−1

. (2.6)

The density growth factor in a Einstein-De Sitter Universe is simply proportional to the cosmic expan-

sion factor a(t),

D(a) = a(t) ∝ t2/3 , (2.7)

while in an freely expanding empty Universe (Ωm = 0),D(a) becomes a constant and therefore structure

formation comes to a halt. This is the asymptotic situation for a low density matter dominated universe

(Ωm < 1). Such a Universe starts of as a near EdS Universe and attains free expansion at

1+ zm f =1

Ωm

−1 . (2.8)

As a result in matter dominated Ωm < 1 Universes at early times we see structure growing with a rate

D(a) proportional to a(t), while it freezes out after zm f .

2.2. COSMOLOGICAL BACKGROUND 25

Figure 2.1 — Left panel: linear power spectrum of the thirteen cosmological models in table 2.3. The amplitude

is scaled to z = 49. Right panel: the linear density growth factor for the thirteen models. Note that the color of the

lines corresponds to the particular value of Ωm specified in the upper box, while the line style specifies the value

of ΩΛ.

In the case of Λ dominated Universes structure formation comes to a halt when the Universes sets

in its accelerated expansion at redshifts zmΛ,

1+ zmΛ =

(

2ΩΛ

Ωm

)1/3

. (2.9)

For models with Ωm = 0.5 and ΩΛ = 0.7 and 0.9, this happens at z ∼ 0.40 and z ∼ 0.53, respectively.

For the cosmological models treated in this study, we have illustrated the density growth function

D(a) in the right hand frame of Fig. 2.1. In the figure we distinguish the various cosmologies by means

of grey scale values and linestyle. The linestyle represents the different values of ΩΛ while the grey

scale represents the various values of Ωm so that their combination forms a complete representation

of the cosmology at hand. For example, solid lines correspond to models without a cosmological

constant, ΩΛ = 0. Black lines are Ωm = 1 models and as we go from dark to light grey the value of Ωm

decreases. Throughout this chapter we will use these color scheme.

2.2.3 Cosmological Scenarios

The cosmological models which we study are variants of the cold dark matter (CDM) scenario. The

primordial density field is fully characterized by the power spectrum, for which we use the functional

form of the matter power spectrum Bardeen et al. (1986),

P(k) = AT 2(q)kn = Akn

[1+3.89q+ (16.1q)2+ (5.46q)3+ (6.71q)4]1/2× [ln(1+2.34q)]2

(2.34q)2, (2.10)

where T (q) is the transfer function of fluctuations, A is the amplitude and q = k/Γ. Γ is the shape

parameter and k = 2π/λ is the wavenumber in units of h−1Mpc. The index n is the slope of the pri-


Model Ωm ΩΛ Ωk Age mdm Γ

SCDM 1.0 0 0 9.31 13.23 0.7

OCDM01 0.1 0 0.9 12.55 1.32 0.07

OCDM03 0.3 0 0.7 11.30 3.97 0.21

OCDM05 0.5 0 0.5 10.53 6.62 0.35

ΛCDMO1 0.1 0.5 0.4 14.65 1.32 0.07

ΛCDMO2 0.1 0.7 0.2 15.96 1.32 0.07

ΛCDMF1 0.1 0.9 0 17.85 1.32 0.07

ΛCDMO3 0.3 0.5 0.2 12.70 3.97 0.21

ΛCDMF2 0.3 0.7 0 13.47 3.97 0.21

ΛCDMC1 0.3 0.9 -0.2 14.44 3.97 0.21

ΛCDMF3 0.5 0.5 0 11.61 6.62 0.35

ΛCDMC2 0.5 0.7 -0.2 12.17 6.62 0.35

ΛCDMC3 0.5 0.9 -0.4 12.84 6.62 0.35

Table 2.2 — Cosmological parameters for the runs. The columns give the identifications of the runs, the present

matter density parameter, the density parameter associated with the cosmological constant, the age of the Universe

in Gyr since the Big Bang, the mass per particle in units of 1010h−1M⊙ and the shape parameter of the power

spectrum. sigma8 and the Hubble parameter is the same for every model, σ8 = 0.8 and h = 0.7.

mordial power spectrum, for which we assume a Harrison-Zeldovich spectrum: n = 1 (Harrison 1970;

Zeldovich 1972)). The shape parameter Γ of the power spectrum completely describe the scale de-

pendence of the density fluctuations. For the shape parameter we use the form given by Sugiyama

(1995)

Γ = Ωm,0hexp

−Ωb

1+

√2h

Ωm,0

, (2.11)

where Ωm is the matter density and Ωb is the baryonic density. For our models we assume a baryon

density parameter of Ωb = 0.047 and a Hubble parameter of h = 0.7. The shape of the power spectrum,

via Γ, will be mainly determined by the value of Ωm (see Table 2.2).

The amplitude of the power spectrum is determined on the basis of σ8, the rms fluctuation (in

linear theory) of the mass contained in spheres of 8h−1Mpc, and for all our scenarios we use

σ8 = 0.8 . (2.12)

Note that we did not normalize according to the number of clusters present in each simulation.

To compare the linear power spectra P(k,z) of the models at a particular redshift z, we simply

compute the expression

P(k,z) = AT 2(k)kD2(z) . (2.13)

In this way, we can find the linear power spectrum at z = 49 in Fig. 2.1. To distinguish the power

spectra of the different scenarios we use the same color and linestyle scheme as described above for

the linear density growth factor D(a) (2.2.2). We may observe the following facts:

• Models without a cosmological constant have more power on small scales than models withΩΛ,

independent of the values of Ωm.

• The higher the value of Ωm, the less power on small scales, independent of the values of ΩΛ.

• Models with Ωm = 0.5 with a cosmological constant are the ones that have less power on both

scales, followed by the models with Ωm = 0.3

• Given a value of Ωm, the amplitude of the linear power spectrum increases as ΩΛ decreases.

2.3. N-BODY SIMULATIONS 27

2.3 N-Body Simulations

In order to study formation and evolution of structures in the thirteen different cosmologies, we per-

form an N-body simulation in each of these cosmological backgrounds. Each simulations consists

of 2563 dark matter particles in a box of size 200h−1Mpc with periodic boundary conditions. Every

simulation has the same Hubble parameter, h = 0.7 (where the Hubble constant is given by H0 = 100h

km s−1Mpc−1) and the same normalization of the power spectrum, σ8 = 0.8.

The simulations started from initial conditions at z = 49. In order to facilitate comparison between

the outcome of the simulations in the various cosmologies, the primordial Gaussian density fields were

assumed to have the same phases φ(k) for each of its Fourier components,

δ(x) =

∫

d3k

(2π)3δ(k)eik·x , (2.14)

with

δ(k) = |δ(k)| eiφ(k) . (2.15)

While the spectral dependence figures in via |δ(k)|, the choice for equal phases φ(k) in each of the

simulations means that we recognize the same morphological pattern in the cosmic mass distribution.

Each difference between the structure that form in each of the cosmologies can therefore be related to

the difference in cosmology.

We followed the gravitational evolution of the structures from the initial density fields at z= 49 until

the present, z = 0, using the massive parallel tree N-Body/SPH code GADGET-2 (Springel 2005). We

restricted ourselves to the dark matter particles, the gaseous SPH component was turned off. The

Plummer-equivalent softening was set at ǫPl = 15 h−1kpc in physical units from z = 2 to z = 0. At

higher redshifts the softening length was fixed in comoving units. Of each simulation we saved 100

snapshots from z = 4 till the present time. These time steps were equally spaced in log(a) (where a is

the expansion factor).

2.3.1 Simulation results

Fig. 2.2 show slices of 2h−1Mpc thick through the center of the simulation box of the ΛCDMF2 (top

panel), ΛCDMO2 (middle) and ΛCDMC2 model at z = 0. Visual inspection of these figure directly

reveals a few outstanding observations:

• The effect of choosing identical random phases is clearly visible: the patterns of the large scale

structure are similar. The differences between the cosmologies manifests itself in the different

levels and behavior of clustering.

• The ΛCDMO2 model contains less structure than the other ones. To a large extent this is the re-

sult of the extremely low value ofΩm in this scenario, possible in combination with the presence

of a cosmological constant. As a result, structure growth came to a halt at a significantly earlier

epoch.

• The detailed view in the zoom-ins strengthen these conclusions: higher Ωm produces consider-

ably more pronounced and evolved patterns, characterized by higher level of clustering.

Fig. 2.3 shows the evolution of the zoomed region in five different redshifts. We also show the time

(in Gyr) since the Big Bang for comparison. In different cosmologies the same redshift corresponds to

a different cosmic time, even for the same Hubble parameter. This depends sensitively on the values

of Ωm and ΩΛ. The low-Ωm model shows a faster growth and evolution of structure. In fact, we see

that at z = 4, it already shows some defined features, which are loosely present in the other models.

The formation and evolution of structures in the closed model is slower. But if we compare by cosmic

time, we see that the structure looks more or less the same. If we take t = 4.76 Gyr (z = 1.29) in the


Figure 2.2 — Slices of 2h−1Mpc thick through the center of the box of three different cosmological models:

ΛCDMF2 (top), ΛCDMO2 (middle) and ΛCDMC2 (bottom). On the right, a zoom into the region selected in the

slice.

2.3. N-BODY SIMULATIONS 29

Figure 2.3 — Evolution of zoomed regions of Fig. 2.2 in five different redshift. In the lower region, the time (in

Gyr) since the Big Bang is depicted. Time is different although redshift is the same.


ΛCDMF2 model and compare with the ΛCDMO2 model at t = 4.01 Gyr (z = 2.38), we see that the

structures are similar.

Perhaps here we see one of the main consequences of having a cosmological constant. It strongly

affects the available dynamical time scales for the formation of structure.

2.4 The Mass Function

The mass function is the number density of objects of a given mass. The abundance of the most massive

halos is sensitive to the overall amplitude of mass fluctuations, σ8. The evolution of this abundance

can give us clue on the cosmological density parameter, Ωm.

In order to calculate the mass function, we first need to construct the halo catalogues of each

cosmology.

Model Number of Clusters Model Number of Clusters

SCDM 1835 ΛCDMO3 192

OCDM01 31 ΛCDMF2 189

OCDM03 197 ΛCDMC1 189


ΛCDMO1 28 ΛCDMC2 538

ΛCDMO2 27 ΛCDMC3 527

ΛCDMF3 28

Table 2.3 — Number of cluster detected in each of the simulations of this study. For the parameters of each of

the models we refer to Table 2.2.

2.4.1 Halo Catalogues

To extract the groups present in the simulation, we use HOP (Eisenstein & Hut 1998). HOP associates

a density to every particle by smoothing the density field with a spline cubic kernel using the n nearest

neighbors of a given particle. Particles are then linked by associating each particle to the densest

particle from the list of its closest neighbor. This process is repeated until it reaches a particle that is its

own highest density neighbor. All particles linked to a local density maximum are identified as a group.

So far, no distinction between a high density region and its surrounding has been made.To identify

halos above a density threshold, a regrouping merging procedure is performed. This procedure is

based on three parameters. The code first includes only particles that are above some density threshold

δouter. It then merges all groups for which the boundary density between them exceeds δsaddle. Finally,

all groups identified must have one particle that exceeds δpeak to be accepted as an independent group.

We associate the value of δpeak with the virial density, which is given by the solution of the spherical

collapse model. For the other two density parameters, we follow the suggestion of Eisenstein & Hut

(1998), who claim that the values are in the ratio δouter:δsaddle:δpeak=1:2.5:3.

We apply HOP to every output of every run giving the corresponding value of ∆vir. We only

consider groups with more than 100 particles in each run, this means that the minimum mass will vary

for each cosmological model (see Table 2.2 for the masses of each particle). From each output of each

simulation we construct halo catalogues which will allow us to study the mass function. At z = 0, the

number of groups in each run ranges from ∼ 5300 for ΛCDMF1 to ∼ 14300 for SCDM. This indicates

that low Ωm Universe have less structure. The number of groups in cosmologies with the same Ωm but

different ΩΛ is similar (with the exception of SCDM), showing that the dynamics on this scale does

not tell much about the cosmological constant.

Table 2.3 shows the number of galaxy clusters in each cosmology at present time. We define a

galaxy cluster as an object with M > 1014h−1M⊙. We see that low matter universes have few clusters at

2.4. THE MASS FUNCTION 31

z= 0, independently of the value ofΩΛ. As the value ofΩm increases, the number of clusters increases,

which is to be expected, since there is more matter in the Universe to form structures. In models where

Ωm = 0.1, the amount of matter is too low to suppress the action of ΩΛ.

Figure 2.4 — Evolution of the mass function for the cosmological models discussed in 2.3. The panels de-

picts the mass function in four different redshifts. Colors represent models with the same value of Ωm, different

linestyles represent models with the same value of ΩΛ. At early times, z ∼ 3, Ωm dominates the evolution.


2.4.2 Results

Fig. 2.4 shows the evolution of the cumulative number of dark halos with mass above M per comoving

volume, N(> M), for all the cosmologies described. Each panel shows the mass function at a specific

redshift, z = 2.98, z = 1.49, z = 1.01 and z = 0. Colors and linestyle are as described in section 2.2.

Hierarchical clustering predicts that the number of objects of a certain mass increases as a function

of time, while lower mass structures start to form and get incorporated into more massive ones. This

is confirmed in Fig. 2.4, where we see that, for every model, the mass and the number of objects

increases as we go from z = 2.98 to z = 0. In the figure we observe the following:

• For the open models, the redshift at which structure growth stops is in the range of z ∼ 8 (Ωm =

0.1) to z ∼ 0 (Ωm = 0.5). For the other cases, the range is from z ∼ 1.62 for the ΛCDMF1 model

to z ∼ 0.25 for the ΛCDMF3 model.

• At low redshifts, open models (OCDM01, OCDM03 and OCDM05) show a higher number of

objects than the other models.

• As time evolves, structure formation in low Ωm models is suppressed, while in high Ωm objects

become more massive and more numerous.

• From z ∼ 1 to the present, it is possible to distinguish models with the same Ωm. The higher the

value of Ωm, the more massive the objects.

• The role ofΩΛ is less clear. This is specially obvious at z = 0, where we see that it is not possible

to distinguish models with the same Ωm and different ΩΛ. Their mass functions tend to largely

overlap.

2.4.3 Mass Functions: Press-Schechter formalism

The previous analysis showed us that the influence of ΩΛ is neglegible, to the point where at present

time is almost impossible to distinguish between models with sameΩm but differentΩΛ. In this section

we will study this effect together with the Press-Schechter formalism.

The Press-Schechter formalism (Press & Schechter 1974, hereafter PS, see also Bond et al. (1991))

states that the comoving number density of objects of mass M is

dn

dM=

√

2

π

ρb

M2

δc

σ(M,z)

∣

∣

∣

∣

∣

dlnσ(M,z)

d ln M

∣

∣

∣

∣

∣

e−δ2c

2σ2(M,z) , (2.16)

where ρb is the mean density of the Universe and δc is the effective linear overdensity required for the

collapse.

2.4.3.1 Critical collapse density and Cosmology

The critical value δc is only weakly dependent on Ωm and virtually independent of ΩΛ.

A good numerical approximation of δc is given by Navarro et al. (1997):

δc(Ωm) =

0.15(12π)2/3Ω0.0185m if Ωm < 1 and ΩΛ = 0,

0.15(12π)2/3Ω0.0055m if Ωm+ ΩΛ = 1.

(2.17)

Note that these approximations are not valid for Ωk > 0 or Ωk < 0 in the generic case of a non-zero

cosmological constant. To further investigate possible influences of ΩΛ on mass functions, we have

explicitly computed δc for the general spherical collapse model in section 2.A. Fig. 2.5 shows the

evolution of the density contrast as a function of the expansion factor in two cases, each showing four

different cosmologies: one where Ωm has a constant value but ΩΛ is different (left panel) and the other

whereΩΛ = 0 andΩm is different. In the first case, we see that the density contrast is weakly dependent

2.4. THE MASS FUNCTION 33

Figure 2.5 — Evolution of the critical linear collapse density as a function of the expansion factor for two

different cases. Left panel: the evolution for cosmologies with the same Ωm but different ΩΛ. We see that the

influence of the cosmological constant is almost imperceptible. Right panel: the evolution for cosmologies with

different values of Ωm and ΩΛ = 0. The influence of Ωm is more evident, specially between a cosmology with

almost no matter (Ωm = 0.1) and a SCDM one.

on ΩΛ. On the other hand, when ΩΛ is kept fixed (a value of zero in this case, right panel), and Ωm

changes, the spread in the values of δc between cosmologies with low matter to those with high matter

is larger. As expected, the lower the value of Ωm, the lower the value of δc.

The underestimation (overestimation) of the PS formalism at different times are due because the

original implementation of PS was based in spherical objects. This is not the case in hierarchical clus-

tering. At early times, mergers of low mass objects dominate the formation and growth of structures,

resulting in objects that are triaxial in shape. At present time, objects are more dynamically relaxed,

resulting in more spherical objects, which translates into a better fit of the PS formalism.

2.4.3.2 Comparison PS and simulated mass functions

Fig. 2.6 and 2.7 shows the mass functions of four cosmologies together with their respective PS fitted

mass function chosen as example. These are: SCDM, ΛCDMO2, ΛCDMF2 and ΛCDMC2. The

fitted PS mass functions of the other models with same geometry and/or same Ωm are similar to the

ones shown here. As observed in Fig. 2.4, the evolution of the mass function is characterize as an

increase in the number of objects and in their mass in each simulation as a function of time. The shape

is distinctive of each cosmology. At every redshift the SCDM scenario produces a high number of

objects spanning a wide range of masses. By contrast, the low density ΛCDMO2 model only contains

a relatively modest population of objects spread over a small mass range. The other two models,

ΛCDMF2 and ΛCDMC2, are intermediate cases. This can be directly related to their intermediate

value of Ωm.

The fitted PS mass functions show the same trend in evolution and shape, but do not agree with the

theoretical mass function at every redshift for some cosmologies. For the SCDM model, we find that

the PS mass function is consistent only at z = 0. At lower redshift, it underestimates the number

density of objects, with the critical case at z ∼ 2.98. This is also the case for the ΛCDMF2 and

ΛCDMC2 cosmologies. For theΛCDMO2 model the agreement is roughly consistent at every redshift.

We checked the other cosmologies with Ωm, and found the same behavior, except for the flat model

(Ωm=0.1, ΩΛ=0.9), where at low redshift the PS mass function significantly underestimates the true

mass distribution.


Figure 2.6 — Evolution of the mass function for the SCDM (left panel) and ΛCDMO2 (right panel) models in

solid lines. The dotted lines represent the fitted Press-Schechter mass functions.

2.5 Conclusions

We have investigated the evolution of dark matter halos in thirteen cosmological models by means of

numerical simulations. The investigated cosmologies include a SCDM Universe, six open Universes

with and without a cosmological constant, three flat models and three closed with cosmological con-

stant. The initial conditions were generated with identical phases for the Gaussian random field in

order to ensure the presence of identical morphological patterns in each of the simulations. Our sim-

ulations are set up in periodic boxes of size 200h−1Mpc containing 2563 particles. Every simulation

has the same Hubble parameter, h = 0.7, and the same normalization of the power spectrum amplitude,

σ8 = 0.8.

We studied global properties of halos, focusing on their mass function. As expected, models with

Figure 2.7 — Same as the previous figure, but now for the ΛCDMF2 and ΛCDMC2 models.

2.5. CONCLUSIONS 35

higher Ωm result in more and more massive objects, with SCDM being the most representative case.

Both in low density matter dominated Universes as well in Universes dominated by a cosmological

constant the formation and development of structures comes to a halt at early epochs.

We find that mass functions do distinguish between the Ωm of the different cosmologies. However,

at z = 0 we do not find any significant influence on the value of ΩΛ. While this relates to some extent

on the normalization of the power spectrum on mass fluctuations at the present epoch, we do notice

some noticeable effects at other redshifts. This is a result of the different dynamical timescales related

to these redshifts as a consequence of the different ΩΛ (Fig. 2.4).

These conclusions are reaffirmed by comparing our simulated mass functions to the PS prediction.

The PS formalism only leads to a rough agreement at low redshift (with the exception of low-Ωm

models). However, it does differ substantially at higher redshifts. This might be understood if we take

into account that the small objects forming at high redshifts are much more susceptible to external

dynamical influences while we also should take into account their anisotropic collapse via oblate or

prolate shapes. For example, Sheth & Tormen (1999) did demonstrate a substantial change in predicted

mass spectrum when taking into account these effects. In this respect it is good to realize that the Press-

Schechter formalism makes the implicit – and unrealistic – assumption that proto objects are spherical.


2.A Spherical Collapse Model

The spherical collapse model (Gunn & Gott 1972) describes the evolution of an isolated spherical over-

dense region in a homogeneous cosmological background of mean density ρb. Because it is straight-

forward to see that its equation of motion simplify to that of a one-dimensional equation for the motion

of a radial shell, it is the one physical system whose evolution can be analytically followed in its en-

tirely. While in reality isolated spherical systems do not exist, the model provides a necessary basis

for understanding and interpreting the considerably more complicated evolution of generic systems.

Moreover, it appears to provide a surprisingly accurate description of many physical systems lacking

spherical symmetry.

2.A.1 From initial time to turn around

Here we restrict ourselves to a sphere of mass M with a uniform mass distribution within a radius R(t).

This region starts to expand at the same rate as the background but, if its density is high enough, its

expansion will slow down so much that it will stop at some point, reaching a maximum radius, turn

around into contraction and finally collapse and virialize. Generically, we can identify three stages for

the evolution of such an overdense region:

• Turn around: the spherical region has stopped expanding and begins to collapse.

• Collapse: the spherical region begins to contract and collapses to a point.

• Virialization: in practice, collapse to a singularity does not occur. Before that happens, shell

crossing will occur and it will virialize.

The equation of motion which describes the evolution of the spherical overdense region is given by:

(

dR

dt

)2

=2GM

R+ΛR2

3+K . (A-1)

At maximum expansion, the velocity of the perturbation is zero, i.e., R = 0. By setting the latter

equation to zero, we obtain a cubic equation for the radius at maximum expansion or turn around, Rta

which has to be solved numerically. The time from t = 0 to turn around is then

tta =

∫ Rta

0

dr

r. (A-2)

By symmetry, we note that the time of collapse, tcoll, is always twice the time of turn around, tta,

tcoll = 2tta. We can obtain the mass and the radius at turn around, so we can calculate the density at

turn around relative to the background density at any expansion factor a:

∆ta =ρta

ρb

= a3Ωta,p

Ωm,0, (A-3)

where Ωta,p is the mean density of the perturbation relative to the background at turn around, which is

usually much larger than Ωm,0.

2.A.2 Virialization

After turn around, the spherical region starts to contract and collapse. Slight departures from the spher-

ical symmetry will cause the kinetic energy of the collapse to be converted into random motions. The

perturbation reaches some form of thermalized, bound, equilibrium state. At this time, the spherical

region is relaxed and bound.

2.A. SPHERICAL COLLAPSE MODEL 37

We do not know how long virialization takes. The standard assumption it is that is roughly the

collapse time. We can find the radius after virialization in terms of the turn around radius by using the

virial theorem. We follow the derivation described in Lahav et al. (1991). At turn around, the velocity

of the shell is zero, so the total energy is

Eta = UG,ta+UΛ,ta = −3

5

GM2

rta

− 1

10ΛMr2

ta , (A-4)

where we have integrated over the sphere. Since the energy is conserved, the uniform sphere has the

same energy at turn around and at virialization, Eta = Evir. Using the virial theorem, the energy at

collapse can then be written as

Kvir = −1

2UG,vir +UΛ,vir . (A-5)

Conservation of energy tells us that Eta = Evir, i.e., Kvir +UG,vir +UΛ,vir = UG,ta +UΛ,ta. Using Eqn.

A-5, we get1

2UG,vir +2UΛ,vir = UG,ta+UΛ,ta . (A-6)

Assuming that the sphere remains uniform, we can define the effective virial radius of the system (in

analogy to the initial energies). This leads to the a cubic equation for the ratio rvir/rta:

2η

(

Rvir

Rta

)3

− (2+η)

(

Rvir

Rta

)

+1 = 0 , η ≡Λr3

ta

3GM. (A-7)

Note that if η = 0 (Λ = 0), Rvir/Rta = 1/2. The density after virialization is then given by

∆vir =ρvir

ρ=

a3Ωp,ta

Ωm,0

(

Rta

Rvir

)3

(A-8)

Following the same procedure, it is possible to obtain the linearly extrapolated overdensity at virial-

ization. This is given by (Gross 1997)

δ f =3

5D(a)

Ωk,0

Ωm,0−Ωk,ta

Ω2/3p,taΩ

1/3

m,0

, (A-9)

where D(a) is the growth factor and Ωk,ta = −(Ωp,ta+ΩΛ).

2.A.3 Results

We compare our numerical results with the approximations given by Bryan & Norman (1998) for a flat

universe and that of Pierpaoli et al. (2001), whose fit is for general cosmologies. The approximation

given by Bryan & Norman (1998) is

∆vir(z) =18π2+82x−39x2

1+ x, (A-10)

where x = Ωm(z)− 1. This relation is accurate in the range Ω(z) = 0.1− 1. The fit by Pierpaoli et al.

(2001) is given by

∆vir(z) = Ωm(z)

4∑

i, j=0

ci jxiy j , (A-11)

where x ≡Ωm(z)−0.2, y ≡ΩΛ(z), and the coefficients ci j are given in their Table 1. Their fit is accurate

within 2% in the range 0.2 ≤Ωm ≤ 1.1 and 0 ≤ΩΛ ≤ 1.

As an example, Fig. 2.8 shows ∆vir versus Ωm for flat models using both fits and our numerical

results. They agree quite well in the range 0.2 ≤ Ωm ≤ 1 (where the fit given by Pierpaoli et al. (2001)


Figure 2.8 — Left panel: values of ∆vir for a flat Universe. Solid line shows the numerical solution, dotted line

shows the fit given by Eqn. A-11 and dashed line shows the fit given by Eqn. A-10. Right panel: evolution of ∆vir

for a Ωm +ΩΛ = 0.6 Universe. Lines as before, although here we do not plot the fit given by Eqn. A-10.

is accurate). For lower values of Ωm the fits presents problems in comparison with the numerical

result. The restriction in the fit give by Pierpaoli et al. (2001) is more evident in the right panel of the

same figure. We plot the evolution of ∆vir for one of our models (Ωm +ΩΛ = 0.6) using the numerical

solution (solid line) and the fit given by Eqn. A-11 (dashed line). As expected, for lower values of the

expansion factor, both curves agree quite well but as we approach a = 1, the value of Ωm tends to 0.1,

and the fit is no longer applicable.

3Galaxy Cluster Evolution: Mass Growth

and Virialization

We investigate the formation history and virialization of cluster sized dark matter halos in a range

of cosmologies. With the help of a set of thirteen N-body cosmological simulations, we seek

to identify the influence of the cosmological density parameter Ωm and the cosmological constant ΩΛon the formation of clusters. We find that the accretion and merging history of clusters is sensitive to

the cosmological density parameter Ωm. The only identifiable influence of a cosmological constant

is that via the corresponding dynamical timescales. The formation redshift of clusters is substantially

higher in low Ωm Universes. We compare our simulations to a few analytical prescriptions of the

halo formation history and find that they manage to reasonably reproduce the complete mass assembly

history of clusters. We also address the virialization of clusters in the various cosmologies. In none of

the cosmologies we find perfectly virialized clusters. Low mass halos display a large range of virial

states, ranging from highly virialized to hardly bound. High mass cluster halos have a virial state

|U | ∼ 1.60− 1.65K which turns out to be the same for all simulated cosmologies. The one difference

concerns the spread around this relation: it is larger in high Ωm Universes.

40 Chapter 3: Galaxy Cluster Evolution: Mass Growth and Virialization

3.1 Introduction

Clusters of galaxies are the most massive and most recently collapsed and virialized objects in our Uni-

verse. As such, they represent ideal probes for understanding the complex formation history of objects

in the Universe. In the currently favored structure formation scenarios, mostly involving a dominant

cold dark matter component, structure arose through the gravitational growth of tiny primordial Gaus-

sian density and velocity perturbations.

This process involves a complex hierarchical evolution in which small scale density fluctuations

are the first to collapse and form objects. As larger and larger fluctuations start to mature, the small

clumps merge with each other into ever larger objects. Along with the more pronounced mergers, the

process also involves a gradual accretion of more modest mass concentrations. By means of the gradual

accretion and merging, massive structures finally emerge out of the almost pristine and featureless

primordial Universe.

Turning to individual objects like clusters, we may wonder to what extent their internal structure

and dynamics is affected by the global cosmological background. By identifying possible influences

– of which those of the cosmic density Ωm, the cosmological constant ΩΛ and the amplitude of mass

fluctuations quantified by σ8 are the most straightforward ones – potentially we would be able to infer

the cosmology from observations of clusters.

In this chapter we are particularly interested in three closely related aspects of cluster evolution.

These are the mass assembly history of clusters, distinguished in terms of the specific merging and

gradual accretion mode of mass growth, and the final dynamical state of the emerged cluster halos.

The latter is usually quantified in terms of the virialization of the cluster.

Merging and accretion, together with virialization, are tightly related to the formation time. It

is important to note that there is no proper definition of formation time. This makes it difficult to

compare studies between authors. A few definitions of formation time are based on idealized analytical

prescriptions of the mass accretion history of halos. None of these definitions are compelling and rather

arbitrary. As we will show in this study, they not only fail to describe the mass accretion history over

the complete formation time of cosmic objects but also appear to conflict with each other (see also,

Lacey & Cole 1993; Wechsler et al. 2002; van den Bosch 2002; Tasitsiomi et al. 2004; Cohn & White

2005)

Virialization is the ultimate dynamical state of any collapsing structure turning into an individual

and recognizable cosmic object. During the virialization process, the matter contained in the object ex-

changes energy to the extent of reaching an equilibrium state. Virialization would offer the possibility

to infer the mass of the object on the basis of the thermal velocity of its matter content. Nevertheless,

the presence of substructure tells us that this assumption is usually incorrect and not warranted. X-ray

observations of galaxy clusters show a large variety of substructure and morphology (e.g. Mohr et al.

1995). There is a variety of studies addressing the virialization of galaxy clusters, usually involving

simulated clusters (eg., Knebe & Muller 1999; Maccio et al. 2003; Shaw et al. 2006). Knebe & Muller

(1999) studied virialization of their simulated cluster halos and found that the virial theorem does not

hold in any of their simulated cosmologies, arguing that this is due to the influence of an outer pressure

of radially infalling particles into the halos. Maccio et al. (2003) also applied the virial theorem, but

they added a pressure term to take into account the external material. Shaw et al. (2006) also work

with the virial theorem plus a surface pressure term. They showed that by adding this term, the virial

theorem holds for their clusters.

In this chapter we investigate the formation history and virialization of galaxy cluster halos and

its dependence on the background cosmology. This formation history is tightly related to the internal

kinetic and potential energy they acquire during their evolution. The structure of the chapter is as

follows. In section 3.2 we briefly describe the simulations we use. Section 3.3 presents the study of

merger and accretion of cluster halos. Section 3.4 presents the formation time and their dependence on

the background cosmology. In section 3.5 we study the virial theorem of galaxy clusters. Conclusions

are presented in section 3.6.

3.2. THE SIMULATIONS 41

Model Ωm ΩΛ Ωk Age mdm mcut ∆vir,b ∆vir,c

SCDM 1.0 0 0 9.31 13.23 1323 177.65 177.65

OCDM01 0.1 0 0.9 12.55 1.32 132 978.83 97.88

OCDM03 0.3 0 0.7 11.30 3.97 397 402.34 120.70

OCDM05 0.5 0 0.5 10.53 6.62 662 278.10 139.05

ΛCDMO1 0.1 0.5 0.4 14.65 1.32 132 838.30 83.83

ΛCDMO2 0.1 0.7 0.2 15.96 1.32 132 778.30 77.83

ΛCDMF1 0.1 0.9 0 17.85 1.32 132 715.12 71.51

ΛCDMO3 0.3 0.5 0.2 12.70 3.97 397 358.21 107.46

ΛCDMF2 0.3 0.7 0 13.47 3.97 397 339.78 101.93

ΛCDMC1 0.3 0.9 -0.2 14.44 3.97 397 320.79 96.237

ΛCDMF3 0.5 0.5 0 11.61 6.62 662 252.38 126.19

ΛCDMC2 0.5 0.7 -0.2 12.17 6.62 662 241.74 120.87

ΛCDMC3 0.5 0.9 -0.4 12.84 6.62 6622 30.85 115.43

Table 3.1 — Cosmological parameters for the runs. The columns give the identification of the runs, the present


in Gyr since the Big Bang, the mass per particle in units of 1010h−1M⊙, the mass cut of the groups given by HOP

in units of 1010h−1M⊙, the value of the density needed to have virialized objects with respect to the background

density, and the same as before, but now with respect to the critical density.

3.2 The Simulations

The simulations and the method to identify halos are extensively described in chapter 2. Here, we

summarize this description.

We perform thirteen N-body simulations that follows the dynamics of N = 2563 particles in a

periodic box of size L = 200h−1Mpc. The initial conditions are generated with identical phases for

Fourier components of the Gaussian random field. In this way each cosmological model contains the

same morphological structures. For all models we chose the same Hubble parameter, h = 0.7, and the

same normalization of the power spectrum,σ8 = 0.8. The principal differences between the simulations

are the values of the matter density and vacuum energy density parameters, Ωm andΩΛ. By combining

these parameters, we get models describing the three possible geometries of the Universe: open, flat

and closed. The effect of having the same Hubble parameter and different cosmological constants

translates into having different cosmic times.

The initial conditions are evolved until the present time (z = 0) using the massive parallel tree N-

body code GADGET2 (Springel 2005). The Plummer-equivalent softening was set at ǫpl = 15h−1kpc

in physical units from z= 2 to z= 0, while it was taken to be fixed in comoving units at higher redshifts.

For each cosmological model we wrote the output of 100 snapshots, from a = 0.2 (z = 4) to the present

time, a = 1 (z = 0), equally spaced in log(a).

3.2.1 Halo identification

We use the HOP algorithm (Eisenstein & Hut 1998) to extract the groups present in the simulations.

HOP associates a density to every particle. In a first step, a group is defined as a collection of particles

linked to a local density maximum. To make a distinction between a high density region and its

surroundings, HOP uses a regrouping procedure. This procedure identifies a group as an individual

object on the basis of a specific density value. Important for our study is the fact that for this critical

value we chose the virial density value ∆c following from the spherical collapse model. In order to

have the proper ∆c we numerically compute its value for each of the cosmologies (see appendix 2.A).

Table 3.1 lists the values of the cosmological parameters and the values of the virial density for each


cosmology at z = 0. For the latter we list two values: the virial overdensity ∆vir,b with respect to the

background density ρb of the corresponding cosmology, and the related virial overdensity ∆vir,c with

respect to the critical density.

Note that we only consider groups containing more than 100 particles. Because the particle mass

depends on the cosmological scenario, this implies a different mass cut for the halos in each of our

simulations. As a result, SCDM does not have groups with masses lower than 1013h−1M⊙. We have

to keep in mind this artificial constraint when considering collapse and virialization in hierarchical

scenarios at high redshifts. When structure growth is still continuing vigorously at the current epoch,

the collapsed halos at high redshifts will have been small. Our simulations would not be able to resolve

this.

3.2.2 Halo properties

In this chapter we analyze the complete sample of halos, although in some situations we limit ourselves

to those dark matter halos that would correspond to rich galaxy clusters. A galaxy cluster is defined to

have a dark matter mass M of M> 1014M⊙.

The particle mass is a function of the cosmology. With the N particles contained within a sim-

ulation volume V = L3, the mass mpart of a particle in a cosmology with density parameter Ωm is

simply

mpart = Ωm ρc

(

V

N

)

= Ωm

(

3H2

8πG

)

(

V

N

)

(3.1)

= Ωmh2(

V

N

)

2.7755×1011 M⊙

To investigate the virial properties of the halos we determine of each halo the mass M, kinetic energy

K and gravitational potential energy U. Given a halo of N particles we compute these quantities as

follows:

• Mass: the number of halo particles multiplied by the mass per particle present in each halo:

M = Nhalompart , (3.2)

where Nhalo is the number of particles in the halo.

• Kinetic Energy: the total sum of the particle kinetic energies (with respect to the center of the

halo):

K =1

2

Nhalo∑

i=1

mi(vi−vcenter)2 , (3.3)

where vi is the physical velocity of particle i and vcenter the physical velocity of the halos’ center

of mass.

• Potential Energy: defined as

U = −N

∑

i=1

N∑

j=i+1

Gmim j

|ri− r j|. (3.4)

with ri and r j the locations of particles i and j.

• Virial ratio: defined as the ratio of the kinetic energy and the potential energy,

V ≡ 2K

U(3.5)


3.2.3 Halo merger trees

The hierarchical scenario of structure formation states that small clumps form first, and gradually

merge and accrete into larger structures.

Under this process, the formation history of a halo is a complex story of the formation and evolution

of many individual clumps that ultimately end up in a given halo at some cosmic epoch. To this, we

have to add all the field material that meanwhile accreted quiescently on to these proto-halos. One

practical approach to deal with this is to trace back in time all the particles that have ended up in

the halo. By identifying the halos to which they previously belonged we may infer the merging and

accretion history of the halo.

By continuing the proto-halo identification process until the epoch at which all contributing par-

ticles are field particles that do not belong to any proto-halo, we have produced what is commonly

known as merging tree (Kauffmann & White 1993; Lacey & Cole 1993). In Fig. 3.1 we depict the first

stage of this process: a halo and the four proto-halos that finally merged into this halo.

Folklore calls the most recent halo the child halo. Of course, this poses the question which of the

four proto-halos - or even all them - should be regarded as the real progenitor of the present halo. We

assign, arguably a somewhat artificial choice, this quality to the most massive progenitor of a given

dark matter halo. In Fig. 3.1 the progenitor of the present halo is proto-halo C.

Figure 3.1 — Simple description of the merging tree of a dark matter halo. In this case, a given halo has four

progenitors at a previous redshift, but only one, C, is the most massive one.

Given the groups found by HOP, it is straightforward to construct the merging tree of every single

dark matter halo in each cosmological model.

3.2.4 Timing the simulations

When evaluating the growth of cluster halos, we do so in terms of three time related quantities.

The first is the redshift z, the prime observational time related cosmological quantity. It directly

relates to what an observational cosmologist would infer from his/her observations. Cosmic redshift z

is directly related to the cosmic expansion factor,

z =1

a−1 . (3.6)

The time t(z) is related to redshift, but in a way which depends on the cosmology. The same

redshift z corresponds to different times for different cosmologies. Here we distinguish two different


time measures. The first, lookback time, relates directly to the redshift z: it is the time interval between

the present epoch and the time of “emission” by the observed object at redshift z. For a FRW Universe,

with a Hubble constant H0, with a matter density parameter Ωm,0, a cosmological constant ΩΛ,0, the

lookback time tl(z) is given by

tl(z) =1

H0

∫ 1

1/(1+z)

xdx√

Ωm,0x + ΩΛ,0x4 + (1−Ω0)x2, (3.7)

with Ω0 = Ωm,0+ΩΛ,0.

An equally interesting quantity for dynamically evolving systems is what we call cosmic time,

tc(z). It is nothing else than the age of the Universe at that redshift, i.e. the time that passed since the

Big Bang. It is a measure for the available dynamical time of the system. For a FRW Universe, with

the same parameters as specified above, it is simply given by

tc(z) =1

H0

∫ 1/(1+z)

0

xdx√

Ωm,0x + ΩΛ,0x4 + (1−Ω0)x2. (3.8)

To appreciate the relation between redshift, cosmic time and lookback time, we have plotted red-

shift as a function of these time definitions for four different cosmologies in Fig. 3.2. Objects at a

particular redshift z in the SCDM Universe are considerably younger than those in the Universes with

a cosmological constant. Also, we see that Universes with a lower Ωm are older than the ones with a

higher Ωm (left hand frame). Conversely, the lookback time to an object with redshift z in a SCDM

cosmology is substantially smaller than that in the Universes dominated by a positive ΩΛ. When com-

paring the ΛCDM cosmologies we see that the one with the higher Ωm has a shorter lookback time for

a given redshift.

Figure 3.2 — Redshift as a function of cosmic time (in units of Gyr, left panel) and lookback time (in units of

the Hubble time, right panel for the four cosmological models described in the text.

3.3 Cosmological Cluster Formation

The formation of structure in the Universe is the result of the gravitational growth of primordial tiny

density and velocity perturbations. Perhaps the most important aspect of the formation process is its

hierarchical character. The first objects to form are small, these small clumps subsequently merge into

3.3. COSMOLOGICAL CLUSTER FORMATION 45

ever larger objects. Hence, clusters of galaxies emerged as the result of some massive mergers with

peers and a more continuous process of gradual accretion of mass from their surroundings.

The history of this process is highly dependent on the cosmology. The first factor of importance

is the amount of dark matter in the Universe. The structure formation process is driven by the growth

of fluctuations in the density of dark matter. In low Ωm Universes, the rate of growth is therefore

slower than that in high Ωm Universes. In addition, the cosmic expansion history bears strongly on

the formation process. Negatively curved, matter dominated Universes tend to evolve towards to a

free cosmic expansion, while the Universe would even assume an exponential De Sitter expansion in

case a cosmological constant assumes dominance over the cosmic dynamics. In both cases, the fast

expansion rate of the Universe stifles the growth of matter perturbations so that the structure formation

process comes to a halt. As a result, low Ωm and/or high ΩΛ Universes will stop forming clusters at

some specific epoch.

The presence of a cosmological constant may also leave its imprint in a few alternative ways. One

effect is that of the lengthening of the age of the Universe. Given sufficient amounts of dark matter

this would lead to a more substantial level of structure formation at higher redshifts. On the other

hand, it might have a negative influence because of the repulsive nature of the cosmological constant.

However, this effect does seem to be of minor influence: e.g. Lahav et al. (1991) found it would have

no more than ∼ 1/70 of the influence of matter perturbations.

In this section we follow the evolution of one particular cluster in a range of cosmologies in order

to appreciate the evolutionary status of that cluster at similar redshifts, lookback time and cosmic time.

3.3.1 Cluster formation: the role of Ωm

To get a visual appreciation of the cosmological influences on the hierarchical buildup of a cluster

sized halo, we follow the evolution of one particular cluster in a range of simulated cosmologies. We

are able to do so as we can identify the same cluster halo in all our simulations as we crafted our initial

conditions such that they would contain the same morphological make-up, by taking the same Fourier

phases for the initial Gaussian density fields.

In the following three figures, we follow the evolution of this dark matter halo, with mass M>

1014h−1M⊙, in four different cosmologies. These are theΛCDMO2,ΛCDMF2,ΛCDMC2 and SCDM.

In each of these cosmologies we show the evolution of the halo at six different time steps. We show the

mass distribution in and around the cluster, and its progenitors, in a box of comoving size 5h−1Mpc.

Circles enclose halos identified by HOP, with the circle radius proportional to virial radius of the group

(i.e. the distance from the center of mass to the outermost particle of the group). Sometimes we see

circles within circles, this is just a projection effect. In Fig. 3.3 we show the state of the cluster at

the same z in all four cosmologies. The equivalent Fig. 3.4 does the same thing, but then at the same

lookback time tl(z). Finally, in Fig. 3.5 we try to do the same in terms of cosmic time by depicting the

cluster at a more or less comparable cosmic time.

Perhaps most illustrative for the evolutionary trends in the different cosmologies is the redshift

evolution in Fig. 3.3. The sequence ΛCDMO2, ΛCDMF2, ΛCDMC2 and SCDM clearly corresponds

to a sequence in which the formation of the halo shifts to later and later epochs. Taking into account

that all models were normalized to the present epoch, so that it comes as no surprise that the cluster

halo at z = 0 looks similar in all four cosmologies, we find that at all depicted redshifts the cluster in

the ΛCDMO2 is the most pronounced and evolved mass concentration.

In all four cosmologies, we clearly see that the buildup of the halo involves the merging of several

smaller mass clumps, some of which are identified as genuine proto-halos by means of circles. Cer-

tainly at the first two to three time steps, we see that particles and protohalos fall in into the cluster

via the filamentary structure running from the top to the bottom of the box. There is a substantial

difference between the coherence and prominence of the filament in the ΛCDMO2 cosmology and the

equivalent one in the higher Ωm Universes of the ΛCDMF2, ΛCDMC2 and SCDM. Over the whole

depicted redshift range in the ΛCDMO2 cosmology we see a strong massive filament connecting the

most massive clumps in the environment of our (proto) cluster. By contrast, the matter distribution


Figure 3.3 — Cluster evolution: Ωm influence. Evolution as a function of redshift of a single dark matter halo

in different cosmological models: ΛCDMO2, ΛCDMF2, ΛCDMC2 and SCDM.


Figure 3.4 — Cluster evolution: Ωm influence. Evolution as a function of lookback time of a single dark matter

halo in different cosmological models: ΛCDMO2, ΛCDMF2, ΛCDMC2, and SCDM.


Figure 3.5 — Cluster evolution: Ωm influence. Evolution as a function of cosmic time of a single dark matter

halo in different cosmological models: ΛCDMO2, ΛCDMF2, ΛCDMC2 and SCDM.


around the cluster in the SCDM cosmology seems to be much more clumpy and less concentrated in

the filament. Particularly at z ∼ 1.5 and z =∼ 1 the clump distribution around the central cluster is quite

isotropic, and we still see the imprint of this distribution at the more recent redshifts (z ∼ 0.5 and z = 0).

The prominence of the filamentary mass distribution is clearly related to the underlying cosmology,

and will depend on the slope of the power spectrum at cluster scales.

Model % Model %

SCDM 80 ΛCDMO3 83


OCDM03 84 ΛCDMC1 82


ΛCDM01 84 ΛCDMC2 82

ΛCDM02 84 ΛCDMC3 81

ΛCDMF1 82

Table 3.2 — Percentage of halos formed via accretion in every cosmological model.

At later times we see that most matter in the surroundings has accreted onto the massive central clus-

ter. Interesting in this respect is the question how much of the surrounding material has accreted

quiescently and how much came in with merging subclumps. We distinguish merging and accretion

on the basis of the relative amount of mass gain by an absorbing cluster halo. A merger is one where

the mass of the protocluster grows by more than 30%. From Table 3.2 we learn that most of its mass

has accreted gradually onto the forming cluster. In all cosmologies this concerns at least 80% of the

cluster mass.

It is also clear that the protohalo in the flat cosmology and the SCDM cosmology at the first

redshift bin z ∼ 3 is a rather underdeveloped mass clump. In these cosmologies the growth proceeds

almost linearly in redshift, in each frame we notice a strong development of the cluster with respect

to the previous frame. By contrast, the lower density ΛCDMO2 cosmology, and to some extent also

ΛCDMF2 after z ∼ 0.5, testify of a significant slow down in cluster growth. The cluster in ΛCDMO2

at z ∼ 0.5, and even that at z ∼ 1, looks pretty much like the cluster nowadays.

There is a clear trend of a more advanced cluster state as we go from the left hand column

(ΛCDMO2) to the right hand column (SCDM). Evolution in the SCDM cosmology is considerably

stronger in the last few Gyrs than in the lower density ΛCDM models. Comparison between the four

columns reveals the dominant influence of the matter density in the growth of structure: the larger Ωm,

the stronger the evolution has been in the same time interval.

Fig. 3.4 shows the same comparison, but as a function of the lookback time. It provides the same

impression, but at a larger contrast. Because cosmic time is stretched in the more open cosmologies a

time span in the ΛCDMO2 cosmology corresponds to a shorter redshift interval than that in the SCDM

(see Fig. 3.2). In the last 0.6 Gyr the cluster in ΛCDMO2 hardly underwent any significant develop-

ment. In the same time span, we see a substantial amount of evolution in the ΛCDMC2 cosmology, an

evolution which is mimicked to a weaker extent in the ΛCDMF2 cosmology. The situation is radically

different for the SCDM clustering: 0.6 Gyr ago there were hardly any massive clusters and the ones of

today were still minute clumps of matter embedded within a faint filamentary environment.

The differences between the cluster evolution as a function of redshift or lookback time to a sub-

stantial extent may be ascribed to the differences in the evolutionary state of the clusters at any one

of these particular epochs. Taking the reverse view and starting at the same cosmic time, we follow

the clusters at comparable dynamical stages of evolution and during the same time span in each of the

cosmologies. Fig. 3.5 nicely confirms our expectations: the evolution in the higher density SCDM cos-

mology is more rapid than in the other cosmologies. In fact, we see a nice sequence of more prominent

evolution as we go from the right hand column (SCDM) to the left hand column (ΛCDMO2). It is like

a mirror for our earlier assessment in terms of lookback time. The reason remains the same: driven by


the higher mass density the structure evolution in high density Universes proceeds more rapidly in the

same time interval.

While these three sets of images do illustrate the decisive role of Ωm in the buildup of clusters, the

role of ΩΛ remains more subtle.

3.3.2 Cluster formation: the role of ΩΛ

Figs. 3.6 and 3.7 show the evolution of the same cluster as in the previous subsection, but then for a

set of four cosmologies with the same Ωm = 0.3, yet systematically different ΩΛ.

Clearly, the differences between these four cosmologies are not very large. The one noticeable

trend is that of the more rapid evolution in the higher Λ cosmologies than in one with a lower Λ. At

z ∼ 3, the cluster in the OCDM03 model is clearly more substantial than the one in the ΛCDMC1

model. This remains true at nearly all redshift steps, except of course at the current epoch.

We also note that the surrounding mass distribution in the different cosmologies is comparable, be

it at different redshifts. For example, the configuration at z ∼ 2.3 in OCDM03 is almost the same as

the one at z ∼ 1.5 in the ΛCDMC1 cosmology. There does not seem to be a difference in local large

scale geometry between these models. Earlier, we had seen that there was a substantial difference in

filamentary character of the large scale structure between cosmologies with different Ωm. Apparently,

ΩΛ does not bear strongly on the coherence of the cosmic web.

The same impression is obtained from Fig. 3.7, where we compare the same cosmologies over a

similar cosmic time. Also here we see a more rapid evolution of the ΛCDMC1 cluster in comparison

to the OCDM03 cluster. The only conclusion we may draw here is that of ΩΛ having some impact

through its influence on the timescales in the Universe. We can not infer any significant dynamical

influence.

The factor that we need to assess in somewhat more detail is the history of the accretion of mass

towards the buildup of the cluster. The merging trees of the cluster halos may contain some more

information on the underlying cosmology. We investigate this in the next section.

3.4 Formation time and mass accretion history

In hierarchical structure formation scenarios the buildup of a galaxy or cluster halo is a complex pro-

cess of accretion and merging. It brings up the question what exactly the nature is of a protocluster.

Any cluster combines the matter content of many previous mass clumps. Which of these or how many

of these should be considered progenitors. And at which state should we consider such a halo as a

matured object. It is clear that in such a situation the concept of progenitor or formation time may not

be clearly and unequivocally defined. In other words, such a definition is imperfect and in many cases

arbitrary.

To some extent, formation time involves a measure of arbitrariness. There are a few possible defi-

nitions of formation time in use, certainly not always in agreement with each other. The most idealized

one is the one based on the spherical collapse model. Given a particular density threshold it would im-

mediately provide a theoretical expression for collapse or virialization time (Gunn & Gott 1972; Lahav

et al. 1991). However, spherical systems do not exist in reality. We know that the primordial density

peaks are anisotropic, close to triaxial (Bardeen et al. 1986). Better approximations for collapse time(s)

for such objects are provided by the ellipsoidal model (Icke 1973; Eisenstein & Loeb 1995; Bond &

Myers 1996; Sheth et al. 2001). Even though it leads to considerable improvement of, e.g. the mass

spectrum of halos, it does not fully take into account the complex, non local hierarchical buildup of

halos (with the exception of the Peak Patch description of Bond & Myers (1996)).

To take into account the more realistic circumstances of the hierarchical clustering evolution, we

will mostly follow Lacey & Cole (1993) in adopting the general definition of formation time being

the time (redshift) z f at which the parent protocluster contains half (or more) of its current mass. An

alternative definition of formation time is the time at which the potential well of the halo becomes deep

3.4. FORMATION TIME AND MASS ACCRETION HISTORY 51

Figure 3.6 — Cluster evolution: ΩΛ influence. Evolution as a function of redshift of a single cluster sized dark

matter halo in different cosmological models: OCDM03, ΛCDMO3, ΛCDMF2 and ΛCDMC1. Note that each of

these cosmologies have the same Ωm = 0.3.


Figure 3.7 — Cluster evolution: ΩΛ influence. Evolution as a function of cosmic time of a single cluster sized

dark matter halo in different cosmological models: OCDM03, ΛCDMO3, ΛCDMF2 and ΛCDMC1. Note that

each of these cosmologies have the same Ωm = 0.3.


enough to be considered a cluster, for which e.g. one could assume an X-ray emission criterion when

looking at rich clusters. Probably the most extensive discussion of the various cluster formation time

definitions is the one by Cohn & White (2005).

3.4.1 Formation time and mass accretion history

In the context of hierarchical scenarios, possibly the most objective path towards defining the formation

time of a particular object is to assess its complete history of merging and accretion. Pursuing the path

of the most massive progenitor of a halo yields the so called mass accretion history (MAH). Different

studies have found that MAHs may affect the final properties of halos, something which is to expect

given the hierarchical scenario of structure formation (e.g., Wechsler et al. 2002; van den Bosch 2002;

Zhao et al. 2003; Tasitsiomi et al. 2004).

On the basis of the MAH, through modelling by an idealized analytical expression, a few different

definitions of formation time were forwarded. We have assess three of these.

Wechsler et al. (2002) (hereafter W02) proposed a fit to the MAHs by an exponential function:

M(a) = M0e−α

(

1a−1

)

, (3.9)

where a is the expansion factor and M0 is the final mass. The single parameter α in Eqn. 3.9 is

related to a characteristic formation epoch a f . It is defined as the expansion scale factor a at which

the logarithmic slope of the accretion rate, dlogM/dloga falls below some specified value S (Wechsler

et al. 2002). The value of S is arbitrary, we follow W02 and take S = 2. The resulting expression for

the formation redshift z f is given by

z f =2

α−1 . (3.10)

In our fitting procedure we choose to keep the M0 fixed , and not use it as an extra free parameter as in

e.g. Allgood et al. (2006); Aragon-Calvo (2007).

van den Bosch (2002) (hereafter VDB02) defined the formation redshift z f from a theoretical fit to the

mass accretion history inferred from the extended Press-Schechter formalism (Bond et al. 1991). This

fitting formula is given by

log(Ψ(M0,z)) = −0.301

(

log(1+ z)

log(1+ z f )

)ν

, (3.11)

where z f and ν are parameters that follow from fitting the expression to the mass history of a halo.

The parameter z f would then correspond to the formation redshift, the epoch at which the protohalo

contains half or more of the mass of the current cluster.

Arguing that the MAH description by W02 (see Eqn. 3.9) provides a poor fit in a variety of situations,

Tasitsiomi et al. (2004) (hereafter T04) proposed a more general fitting equation,

M(a) = M0ape−α(1/a−1) , (3.12)

which will reduce to Eqn. 3.9 for p = 0.

3.4.2 Global formation epoch

The first step of our procedure is the construction of the merging tree of each halo. We do this by

tracking backwards in time every progenitor of the present day halo (see section 3.2.3).

Once we obtain the merger tree of a given halo, we track down its most massive progenitor. In

a first approximation of its formation time, we determined the redshift at which its most massive

progenitor contains half or more of its present mass. By averaging the formation times of all cluster

halos we obtain an average formation redshift and formation time for any particular cosmology. These

are listed in Table 3.3.


From Table 3.3 that the formation redshift in low matter Universes (Ωm = 0.1) is high compared

to models with high density values. This is in line with our expectations based on structure formation

in low density Universes. We also notice an effect of the cosmological constant. For cosmologies

with the same Ωm, the formation redshift is higher for models with decreasing ΩΛ. This conclusion

is slightly modified when we assess formation epoch in terms of formation time. Illustrative is the

fact that the formation time in the SCDM model is significantly shorter than in the ΛCDMF1 model.

This is in line with what we discussed in section 3.3, where we noticed the same trend of a more rapid

evolution driven by a higher matter density.

While these estimates of average formation epoch do provide some impression of general trends,

they do not take into account important factors like the accretion and merging history of the halos

and their mass. These will bear strongly on the spread around the mean formation epoch. In order

to investigate this issue, we turn towards fitting the full MAH on the basis of the model equations

discussed in the previous section 3.4.1.

3.4.3 Single halo MAH

The mass accretion histories of four individual halos in a set of four cosmologies are shown in Fig. 3.8.

In all situations we see a steady increase of the mass as a function of expansion factor a(t). It is clear

that all four halos formed first in the open cosmology. Later, after a ∼ 0.5, it is these halos that hardly

grow in mass anymore.

There are some interesting differences between these halos. The halos in the lefthand panels

evolved by steady accretion. The ones in the righthand panels do suffer major mass jumps as a re-

sult of a massive merger. In the case of the latter, the merger happened recently at a ∼ 0.9 for the one

in the upper panel, at least for the high density cosmologies. The cluster in the low density Universe

experienced a similar merger at a considerably earlier time. The merger in the other halo (bottom

panel) happened at different times for each of the cosmologies: the earliest in the ΛCDMO2, last in

the SCDM cosmology.

The validity of the approximation by the W02, VDB02 and T04 expression for the theoretical mass

accretion history may be appreciated from Fig. 3.9. As long as the halo formed via gentle accretion

(lefthand frame), these fits seem to be quite reasonable. The VBD02 formula produces the best results

of all three. The difference with reality is substantially larger for the halo that underwent a massive

Figure 3.8 — Mass accretion history (MAH) of four different dark matter halos in four cosmological models.


Figure 3.9 — Mass accretion history for two halos in the SCDM model normalized by the final mass (solid line).

The dotted line indicates the best fit from the W02 model, the dashed line the T04 model and the dash-dotted line,

the VDB02 model.

merger. The W02 expression fails over nearly the entire formation history. Although VDB02 and T04

represent better fits, they do not manage to accurately reproduce the entire formation history of this

individual halo.

3.4.4 General MAH

We determine the general mass accretion history by averaging all individual mass accretion history of

the halos in each cosmology sample. The average MAH of these galaxy clusters is shown in Fig. 3.10.

Careful inspection of the inferred MAHs reveals a few facts. Not surprisingly, galaxy clusters tend

to form earlier as Ωm is lower. This is particularly clear when comparing the sequence ΛCDMO2,

ΛCDMF2, ΛCDMC2. Also we find that the scatter in mass accretion histories is more substantial in

the higher Ωm ΛCDMC2, and even more so in the SCDM cosmology. This may be tied in with the

observation by Wechsler et al. (2002) that the MAH scatter is tightly correlated with the concentration

parameter of the halos. In SCDM the clusters are much less concentrated than the halos in the low Ωm

Universes. This we also found in chapter 4.

In each cosmology we also found that the general MAH is shifted to earlier times for samples of

lower mass halos. It reflects the general tendency of low mass halo to form earlier.

Model Average W02 T04 VDB02

z f Age z f z f z f

SCDM 0.61 ± 0.34 4.56 0.86 ± 0.08 1.00 ± 0.46 0.62 ± 0.04

OCDM01 1.08 ± 0.54 5.44 2.20 ± 0.12 2.92 ± 0.92 1.00 ± 0.07

OCDM03 0.77 ± 0.40 5.49 1.37 ± 0.09 2.21 ± 0.73 0.73 ± 0.05

OCDM05 0.69 ± 0.37 5.17 1.14 ± 0.09 1.82 ± 0.70 0.67 ± 0.05

ΛCDMO1 1.03 ± 0.42 6.82 1.70 ± 0.10 1.06 ± 0.29 0.99 ± 0.06

ΛCDMO2 0.98 ± 0.38 7.90 1.41 ± 0.08 0.65 ± 0.20 0.95 ± 0.05

ΛCDMF1 0.86 ± 0.35 9.87 0.93 ± 0.06 0.21 ± 0.12 0.88 ± 0.05

ΛCDMO3 0.69 ± 0.33 6.73 0.95 ± 0.07 0.86 ± 0.30 0.68 ± 0.04

ΛCDMF2 0.66 ± 0.30 7.41 0.75 ± 0.06 0.37 ± 0.19 0.67 ± 0.04

ΛCDMC1 0.61 ± 0.25 8.38 0.44 ± 0.04 -0.07 ± 0.10 0.67 ± 0.04

ΛCDMF3 0.62 ± 0.31 6.16 0.78 ± 0.07 0.63 ± 0.29 0.64 ± 0.04

ΛCDMC2 0.60 ± 0.29 6.66 0.60 ± 0.06 0.24 ± 0.19 0.63 ± 0.04

ΛCDMC3 0.56 ± 0.26 7.35 0.37 ± 0.05 -0.09 ± 0.12 0.62 ± 0.04

Table 3.3 — Formation redshifts for cluster halos in all cosmological simulations. The second column indicates

the average formation redshift and the third column indicates the cosmic time of this average formation redshift.

The last three columns indicates the formation redshift inferred from the W02, T04 and VDB02 models.


Figure 3.10 — Average mass accretion histories for galaxy clusters in four cosmological models. The shaded

area denotes the standard deviations.

While we do find these general trends as a function of underlying cosmology, perhaps as important

is the observation that the width of the MAHs is so large that there is a large overlap between the MAHs

of different cosmologies. The impression is that it is possible to discriminate between cosmologies of

differentΩm, but not between comparable cosmologies with differentΩΛ. This makes attempts to infer

cosmological parameters like ΩΛ on the basis of the growth history of cluster sized objects, in other

words the mass spectrum as a function of redshift, a nontrivial affair.

Figure 3.11 — Average mass accretion histories for galaxy clusters in four cosmological models. The fitting

functions of W02, T04 and VDB02 are also shown.

3.5. VIRIALIZATION 57

3.4.5 General MAH: formation times

In Fig. 3.11 we compare the three model fits – W02: Eqn. 3.9, VBD02: Eqn. 3.11 and T04: Eqn. 3.12

– to the average MAH in each cosmology shown in Fig. 3.10. In general, we find that the models fits

manage to reproduce the mass accretion history over a substantial timespan of our simulations. The

only significant deficiencies occur at the earliest simulated epochs in the higher Ωm Universes. This

may be ascribed for a substantial extent to the poor low mass resolution of these simulations.

However, when we infer the formation redshifts z f implied by each of these model fits we find

strong inconsistencies (see table 3.3, three last columns). Comparing each of the inferred model for-

mation redshifts to the directly inferred one (section 3.4.2 and second column of table 3.3) there is

some similarity with VDB02. This may be related to the fact that the latter also concerns the epoch at

which the halos contain half of their final mass, i.e. according to Press-Schechter theory.

3.5 Virialization

The end stage of any evolving halo is its final and complete collapse, followed by virialization. During

virialization the internal energy of the object is distributed such that it attains a perfect equilibrium

configuration.

Virialization is a complex dynamical process. The simple and idealized spherical model leads

to a definitive prediction of virial state and virial time. However, in reality the process will be less

straightforward. Factors such as the nonspherical shape of the halo as well as the substantial level of

substructure expected in hierarchical models will modify the virialization time and the virialization

process itself. This will be exacerbated by the fact that the halo will not be an isolated island in the

Universe but an organic part of the Cosmic Web, responsible for a constant influx of matter and matter

clumps from the immediate surroundings to the halo.

In this section, we investigate the cluster virialization process in our set of cosmological simula-

tions. This should provide information on the global cosmological as well as environmental influences

on the virialization.

3.5.1 Virial Theorem

The exact expression of the virial equation for a self gravitating system, not necessarily isolated, is

given by:

1

2

d2I

dt2= 2K +U −Ep , (3.13)

where K is the kinetic energy, defined as

K =1

2

N∑

i=1


where the sum is over all particles velocities within any given region. The potential energy U of the

system is given by

U = −N

∑

i=1

N∑

j=i+1

Gmim j

|ri− r j|. (3.15)

where the sum of the distances is over all particle pairs. The environmental influences are incorporated

via a surface pressure term Ep (see e.g. Chandrasekhar 1961)

Ep =

∫

Ps(r)r ·dS (3.16)


Limiting ourselves to an isolated object, i.e. Ep = 0, a system is virialized if it fulfills the condition

I→ 0

2K +U = 0 . (3.17)

This is known as the Virial Theorem for a perfect isolated self gravitating system.

To get a good measure for the virial state of an object may therefore obtained from the virial ratio

V =2K

|U |. (3.18)

This expression should approach unity for virialized systems, the virial ratio 1 <V < 2 for a bound

system, whileV > 2 implies the system is unbound.

Figure 3.12 — Evolution of the kinetic energy (left hand panel) and potential energy (right hand panel) as a

function of expansion factor for four different halos in four cosmological models. Units are arbitrary.


3.5.2 Case studies: single halos

A first impression of the virialization process should concern that of a set of individual halos in each of

the the cosmological simulations. On the basis of the SCDM simulation we identified four halos and

we extracted them, and their equivalents, from each of the simulations. We select cluster sized halos

with masses larger than 6×1014h−1M⊙.

Following our definition that a merger is a halo that at least at one particular epoch absorbed an

infalling clump of at least 30% of its mass, we selected two merging halos and two accreting halos.

One of these halos is a halo that formed by truly quiescent accretion (D), the other accreting halo did

suffer some major impact in one or more of the cosmologies (A).

Fig. 3.12 shows the evolution of the kinetic energy (left column) and the potential energy (right

column) of the four halos in four of the simulated cosmologies (SCDM, ΛCDMO2, ΛCDMF2 and

ΛCDMC2). The first impression is that of the kinetic energy and potential energy evolution following

each other quiet well.

It is interesting to see the influence of major mass infall on both the kinetic energy and potential

energy of a halo. Major mergers are always reflected in sudden jumps in kinetic and potential energy

(see halos B and C, and A in SCDM). In cases of merger or violent mass gain, the potential drops

substantially while accretion manifests itself by a gradual decrease of the potential. Also note that the

ΛCDMO2 halos stop evolving their potential and kinetic energy at an early epoch, and in the case of

halo C even starts to lose energy after a ∼ 0.6, following a substantial merger. It is a clear reflection of

the global cosmological influence on the development of individual clusters. In that spirit we do not

find any trace of a role of the cosmological constant.

Figure 3.13 — Evolution of the virial ratio as a function of expansion factor for four different halos in four

cosmological models. Units are arbitrary.

To evaluate the virial state, and its evolution, of each halo we turn to Fig. 3.13, which depicts the

virial ratio V for each of the analyzed halos. Most halos have a virial ratio consistently near unity

but never really converging towards unity. Instead, they seem to linger erratically around values V ∼1.1−1.4. Halo A did seem to have reached virial equilibrium at a very early stage, but subsequently it

started to drift away from V ∼ 1. If we wish to relate the behavior of V to the nature of the infall of

matter it is perhaps that the fluctuations inV seem to be somewhat milder for the accreting halo D.

The fact that unity is never reached might be an indication for environmental influences. One factor

that will play a role is that of the gradual infall of matter from the surroundings, which would account

for the pressure term Ep. On the other hand, we should not exclude the influence of numerical artifacts


such as that of force softening in the N-body simulations. The fact that we also do not observe a

convergence for the case of the ΛCDMO2 clusters might be indicative of such a factor. We are in the

process of evaluating this systematically.

3.5.3 General view: virialization at z = 0

Turning from the virial state of individual halos we wish to see what the state of affairs is for the entire

sample of halos and investigate whether we can detect any systematic trends with cosmology.

As a first step we investigate in how far the whole sample fits to the relation between potential and

kinetic energy predicted by the virial theorem. In order to do so, we followed a procedure that remains

close to the full form of the virial theorem for a self gravitating system:

1

2

d2I

dt2= 2K +U −Ep . (3.19)

This we achieved by modelling this virial relation in the form of the linear “virial line” |U | −K,

|U | = µK + λ. (3.20)

For a perfect, isolated virialized object, µ = 2 and the pressure term λ = 0. Also note that for a

marginally bound object, µ = 1 and λ = 0. If a halo would not be bound, i.e., K > |U | we would

have µ < 1.

Each cosmological simulation provides us with a set of cluster halos and their kinetic energy K and

potential energy U (Eqns. 3.14 and 3.15). On the basis of the resulting |U |−K plot (Fig. 3.14) we seek

to infer the corresponding virial line, i.e. its parameter µ and λ. In our analysis we assume λ = 0, as

we found that variation of this parameter did not lead to results that would imply significantly different

physical interpretations.

In order to prevent unphysical fits and sensitivity to halos that were evidently not bound or virial-

ized, we proceeded as follows. We choose to discard the 10% halos that would fall below our infer

virial line: they would be considered unbound. According to this choice, we simply determining the

value µ such that it is the value for which 10% of the sample points fall below the line |U | = µK. One

can think of this virial line as a lower virial relation (see also Knebe & Muller 1999).

We applied the described procedure to the complete set of dark matter halos. This also includes

halos with a mass less than M< 1014h−1M⊙, halos we do not necessarily brand as clusters. In Fig. 3.14

Model µ Unvirialized Halos

SCDM 1.54 19%

OCDM01 1.63 10%

OCDM03 1.56 16%

OCDM05 1.54 18%

ΛCDMO1 1.55 17%

ΛCDMO2 1.51 22%

ΛCDMF1 1.46 29%

ΛCDMO3 1.52 22%

ΛCDMF2 1.49 26%

ΛCDMC1 1.46 32%

ΛCDMF3 1.50 24%

ΛCDMC2 1.48 28%

ΛCDMC3 1.46 32%

Table 3.4 — Virial ratio µ of the halo sample in each cosmology, (see Eqn. 3.20) and percentage of unvirialized

halos with respect to the OCDM01 virial ratio µ = 1.63.


Figure 3.14 — The virial relation for simulated halo samples in four cosmologies. Plotted is the potential energy

|U | vs. the kinetic energy K of each halo in SCDM (top left frame),ΛCDMO2 (top right frame),ΛCDMF2 (bottom

left frame) and ΛCDMC2 (bottom right frame). The solid line is the fitted relation |U | = µK (see text), the dashed

lines are the relations |U | = 2K, the perfect virial relation, and |U | = K, the criterion for a gravitationally bound

configuration.

we have plotted the potential energy |U | vs. the kinetic energy K for all halos in four different cosmolo-

gies: SCDM, ΛCDMO2, ΛCDMF2 and ΛCDMC2, all at the present epoch. The solid line in each

frame is the fitted line |U | = µK, the dashed lines delineate the region between perfect virialization

(upper line) and mere boundness (lower line). We see that in all four cases the halo sample is quite

close to the ideal virial state. The exception are the low mass halos: we find many that lie so far away

from the virial line that obviously they are not even bound. Given that this even occurs in the case of a

few moderately massive halos means that this can not be ascribed only to resolution effects.

Fig. 3.15 is a complementary image of the virial state of our halo sample. It plots the virial ratio

|U |/K as function of the mass of the halos. The plots are scatter plots, with each point representing

a halo in the simulation. The density of the points in the scatter plot can be inferred from the super-

imposed density grey scale plot. By means of the horizontal bars the perfect virial state, |U |/K = 2,

and the criterion for a gravitationally bound configuration, |U |/K = 1. In each of the four cosmologies

we see that over the whole mass range the majority of halos linger around a similar virial ratio of

|U |/K ∼ 1.5−1.6.

Interesting to see is the wide spread in the case of the low mass halos: some of these are highly

virialized (|U |/K ∼ 2), while there is a group of low mass halos that may not even be considered as

gravitationally bound. In how far this maybe ascribed to an artifact of the HOP halo identification

procedure or to some real intrinsic physical effect cannot be judged within the context of this work.

Also interesting would be the question in how far it could be related to the formation history and time

of this halos, with the strongly bound ones presumably corresponding to early formed halos. Also e.g.


Figure 3.15 — The virial ratio |U |/K as function of the mass of the halos in our simulated halo samples. The

four represented cosmologies are: SCDM (top left frame), ΛCDMO2 (top right frame), ΛCDMF2 (bottom left

frame) and ΛCDMC2 (bottom right frame). The plots are scatter plots, with each point representing a halo in the

simulation. The density of the points in the scatter plot can be inferred from the superimposed density grey scale

plot. In each frame we indicate by means of the horizontal bars the perfect virial state, |U |/K = 2, and the criterion

for a gravitationally bound configuration, |U |/K = 1.

Knebe & Muller (1999) found similar trends, leading them to suggest that the loosely bound halos

are the result of recent soft mergers into halos that as yet are more anisotropic than the major share

of halos. They would virialize at a later time. Also, it remains a particularly interesting speculation

whether there are environmental factors at play: isolated halos in low density areas may evolve into

substantially stronger bound objects. These aspects are the subject of a presently ongoing study.

3.5.4 Virialization of Galaxy Clusters

Limiting ourselves to the subset of halos that may be branded as genuine cluster sized halos with

a mass larger than M > 1014h−1M⊙, we find an interesting contrast with respect to the virial state

of the complete sample of halos: all cluster halos center around the virial line |U | = µK, with µ ∼1.60− 1.65 (Fig. 3.16). There is hardly any noticeable difference between the different cosmologies.

The one important exception is the spread around the virial line: the high Ωm cosmologies, SCDM and

ΛCDMC2 have a larger width. Later on we will see that this is reflected in the related width of the

cluster Fundamental Plane (see chapter 5).

3.6. CONCLUSIONS 63

Figure 3.16 — The virial ratio |U |/K as function of the mass of the cluster halos in our simulations. The four

represented cosmologies are: SCDM (top left frame), ΛCDMO2 (top right frame), ΛCDMF2 (bottom left frame)

and ΛCDMC2 (bottom right frame). The plots are scatter plots, with each point representing a cluster halo in the

simulation. In each frame we indicate by means of the horizontal bars the perfect virial state, |U |/K = 2, and the

criterion for a gravitationally bound configuration, |U |/K = 1.

3.6 Conclusions

In this chapter we have studied the mass assembly and formation history of cluster halos in a range of

CDM dominated cosmologies. These cosmologies all concern hierarchical formation scenarios. The

simulations all start from primordial Gaussian conditions with the same Fourier phases. This allows

us to follow the same structures, and therefore clusters, in each of the simulations.

We first looked into the assembly history of a few identical clusters and assessed differences in its

formation as a function of redshift, lookback time and also cosmic time, i.e. the time since the Big

Bang. We found that nearly all the differences have to be ascribed to the difference in density of the

cosmological background, cq. Ωm. The formation redshift of clusters is substantially higher in low

Ωm Universes. The only noticeable influence of ΩΛ on the evolution of the clusters is its impact on the

cosmic time corresponding to a particular cosmology. It either stretches or compresses the available

dynamical timescales for cluster evolution.

We proceeded with a study of the mass accretion history (MAH) of the halos in our sample. We

evaluated the MAH of a few individual cluster halos, and that of the average MAH in each of the

cosmologies. As for the individual halos, we did find that the accretion and merging history of halos is

the one dominant noticeable influence. This appears to be regulated to some extent by the background


cosmology: in low Ωm Universes most evolution takes place at early times, often accompanied by

massive mergers at high redshifts. In high Ωm Universes such mergers seem to be more frequent at

recent epochs.

When comparing the average mass accretion history with the predictions of analytic model de-

scriptions such as Wechsler et al. (2002); van den Bosch (2002); Tasitsiomi et al. (2004), we find that

they manage to reproduce the MAH over a large range of the cosmic expansion history. The only

major differences occur at the earliest epochs, presumably the product of the poor mass resolution of

the simulations. The spread of the MAHs around the average appears to be significant. It is somewhat

larger in high Ωm Universes than in low Ωm ones, and most of them do overlap with the MAH in other

cosmologies. It will render any conclusions on cosmological parameters like ΩΛ on the basis of the

cluster population rather cumbersome.

Finally, we studied the virialization of the emerging halos. In general, the halo population appears

to be close to a virial state. However, almost independent of cosmology the halos attain a relation

|U | = µK with µ ∼ 1.5− 1.6 instead of the perfect virial ratio µ = 2. Low mass halos display a large

spread, with some halos being highly virialized, while others can hardly be characterized as a singly

gravitationally bound objects.

Cluster halos, i.e. halos with M > 1014h−1M⊙, are perhaps the most well behaved halos. Nearly

all of them obey the same virial relation, independent of cosmology. The only noticeable influence

of cosmological background is through the spread in the virial relation. In high Ωm Universes it is

somewhat larger than in low Ωm ones.

Neither in the mass accretion history of in the virial state of cluster halos have we been able to find

any influence of a cosmological constant.

4Cosmological Influence on Physical

Characteristics of Clusters†

W study diverse individual properties of galaxy clusters in cosmological N-Body simulations.

These properties include the angular momentum, morphology and density profile. We focus on

the possible effects of dark matter and dark energy on these properties. The simulations span a wide

range of cosmological parameters, representing open, flat and closed Universes. The first aspect we

address is the internal mass distribution of halos, specified in terms of the radial density profile. We

find that the form of the density profile is independent of cosmology. The one major difference between

the density profiles of clusters in different cosmologies is concerns the concentration of halos. Halos

in low Ωm are more concentrated that halos in higher Ωm Universes. We did not detect any influence

of the cosmological constant. A second aspect is that of the shape and morphology of the emerging

cluster halos. We find that clusters in low Ωm Universes are more spherical. In all cosmologies we find

comparable triaxial shape distributions, with a slight preference for a prolate shape. Finally, the third

intrinsic characteristic that we address is the angular momentum of the galaxy clusters. There is no

detectable cosmological influence on the angular momentum of halos, not even on the general angular

momentum distribution. There seems to be some differences in angular momentum of clusters that

underwent a massive merger and ones that merely evolved through quiscent accretion. The steadily

accreting halos have a lower spin parameter than those that underwent major mergers. The imprint of

the background cosmology on this process is that on the epoch at which we observe major mergers,

occurring much earlier in low Ωm.

†Pablo A. Araya-Melo, Rien van de Weygaert & Bernard J.T. Jones, 2008, in preparation.

66 CHAPTER 4: Cosmological Influence on Physical Characteristics of Clusters

4.1 Introduction

The hierarchical clustering process is the accepted mechanishm by which structures form in the Uni-

verse. Small clumps emerge from the primordial density field and subsequently merge to ever bigger

ones. While this process is highly sensitive to the amount of matter present in the Universe, the influ-

ence of the cosmological constant is less clear.

N-body simulations of structure formation are one of the most important tools to study the process

of hierarchical clustering. They can be used to study the physical characteristic of dark matter halos,

within their cosmological context. For our purpose, they serve as an excellent laboratory to study

the influence of various cosmological factors, such as the cosmological density parameter Ωm and

cosmological constant ΩΛ, on the properties of the cluster population. In this chapter we specifically

focus on the following intrinsic properties of clusters:

• Internal mass distribution, specified in terms of its concentration and radial density profile

• Shape and Morphology

• Angular Momentum

We investigate these properties using dissipationless N-body simulations of variants of the CDM model

with different cosmological parameters. We consider thirteen different cosmologies, combinations with

Ωm < 1 and/or ΩΛ , 0, representing a choice of open, flat and closed Universes.

One of the most fundamental intrinsic properties of halos is that of their internal mass distribution.

We are interested in two properties. The extent towards which the mass distribution is concentrated

is quantified by means of a concentration parameter. More information is contained in the radially

averaged density profile. A lot of effort has been devoted in trying to understand and model the

radial density profile of dark matter halos (e.g., Navarro et al. 1997; Moore et al. 1998; Wechsler

et al. 2002; Tasitsiomi et al. 2004). This interest got much emphasize by the finding that halo density

profiles do obey a “universal” form, the socalled NFW profile (Navarro et al. 1997). The fact that

it can be described by the same expression independent of mass and/or cosmological scenario may

provide a hint of a profound physical manifestation of gravitational clustering. Despite numerous

studies, this has not yet been satisfactorily explained. While this universal NFW profile does contain a

concentration parameter, we find it more objective to use a concentration parameter that is independent

of density profile. We therefore introduce an alternative parameter in terms of the ratio of the mean

harmonic radius and the mean separation radius.

A fundamental additional characteristic of any mass distribution is that of its shape and morphol-

ogy. Clusters arose out of a primordial Gaussian density field, from which they evolved around a

density peak. Density peaks in such Gaussian random fields are never spherical (Bardeen et al. 1986).

During the subsequent nonlinear evolution and collapse of such primordial density peaks, any initial

asphericity will be strongly enlarged (e.g. Icke 1973; Lynden-Bell 1964; Eisenstein & Loeb 1995). In

how far the emerging halo retains the memory of this history during its collapse and virialization is not

directly clear but may be inferred from cosmological N-body simulations. These shows that they retain

a strongly flattened triaxial shape (Dubinski & Carlberg 1991; van Haarlem & van de Weygaert 1993).

Part of it is a sensitive function of the virialization process. Dissipative processes involved with the

setttling of baryonic gas in the dark matter potential is perhaps an equally important factor. Also, the

environmental influences are a major factor determining the final shape of a halo. One factor that plays

a role since primoridal times is that of the tidal influences exerted by the surrounding inhomogenous

large scale matter distribution. This is augmented by the resulting weblike cosmic matter distribution

which ties the emerging cluster to its direct surroundings. As first described by van Haarlem & van de

Weygaert (1993), the resulting anisotropic infall of matter through the filamentary extensions have a

deep and immediate impact on the shape of the cluster halo.

The final intrinsic property of halos that we wish to address is that of the angular momentum.

According to the Tidal Torque Theory, the protohalo starts to acquire angular momentum as a con-

sequence of the tidal shear produced by the neighboring large scale matter distribution (Hoyle 1949;


Peebles 1969; Doroshkevich 1970; Efstathiou & Jones 1979; White 1984; Barnes & Efstathiou 1987;

Catelan & Theuns 1996; Porciani et al. 2002; Bullock et al. 2001; van den Bosch et al. 2002; Bett

et al. 2007). Most of the angular momemtun is gained during the linear phase of collapse. It increases

in proportion to the corresponding cosmic time interval (White 1984): it grows like a2D, where a is

the expansion factor and D is the linear growth factor. As soon as nonlinear collapse sets in and the

halo decouples itself from the expanding Universe, it becomes increasingly difficult for the tidal field

to impart a substantial increase of angular momentum on the steadily contracting halo (Peebles 1969).

The outline of the chapter is as follows. In section 4.2 we described the different cosmological

models used. The density profile and the concentration is investigated in section 4.3. Shape of dark

matter halos is discussed in section 4.4, while then angular momentum and its dependence on the

background cosmology is investigated in section 4.5. Conclusions are presented in section 4.6.

4.2 The Simulations



We carry out thirteen N-body simulations that follows the dynamics of 2563 particles in a box

of periodic size 200h−1Mpc. The initial conditions are generated with identical phases for Fourier

components of the Gaussian random field. In this way each cosmological model contains the same

morphological structures. For all models we chose the same Hubble parameter, h = 0.7, and the same

normalization of the power spectrum, σ8 = 0.8. The principal differences between the simulations are

the values of the matter density and vacuum energy density parameters, Ωm and ΩΛ. By combining



























this.



SCDM 1.0 0 0 9.31 13.23 1323 177.65 177.65

OCDM01 0.1 0 0.9 12.55 1.32 132 978.83 97.88

OCDM03 0.3 0 0.7 11.30 3.97 397 402.34 120.70

OCDM05 0.5 0 0.5 10.53 6.62 662 278.10 139.05

ΛCDMO1 0.1 0.5 0.4 14.65 1.32 132 838.30 83.83

ΛCDMO2 0.1 0.7 0.2 15.96 1.32 132 778.30 77.83

ΛCDMF1 0.1 0.9 0 17.85 1.32 132 715.12 71.51

ΛCDMO3 0.3 0.5 0.2 12.70 3.97 397 358.21 107.46

ΛCDMF2 0.3 0.7 0 13.47 3.97 397 339.78 101.93

ΛCDMC1 0.3 0.9 -0.2 14.44 3.97 397 320.79 96.237

ΛCDMF3 0.5 0.5 0 11.61 6.62 662 252.38 126.19

ΛCDMC2 0.5 0.7 -0.2 12.17 6.62 662 241.74 120.87

ΛCDMC3 0.5 0.9 -0.4 12.84 6.62 6622 30.85 115.43






In this study we define a galaxy cluster halo as a dark matter halo with a mass M> 1014h−1M⊙.


When assesing the angular momentum, shape and concentration for the simulated dark matter halos

we chose the following quantities:

• Mass: defined as the number of particles multiplied by the mass per particle present in each

group:

M = npartmpart , (4.1)

where npart is the number of particles in the halo and mpart is the mass of each particles. The

mass of the particle is different for each cosmology.

• Center of mass: computed as the mean position of all the particles in the halo

xcom,i =

∑npart

jxi, j

npart

, (4.2)

where the subscript i goes from 1 to 3, denoting the three dimensions (x,y,z).

• Angular momentum: defined as

J =

N∑

i=0

miri×vi , (4.3)

where ri and vi are the position and velocity of the ith particle with respect to the center of mass

of the group.

It is often useful to define the spin parameter, a dimensionless quantity which relates the angular

momentum and the energy of a group (Peebles 1971),

λ =J√|E|

GM5/2; (4.4)

4.3. RADIAL MASS DISTRIBUTION 69

where J is the angular momentum of the group (see eqn. 6.4), E is the total energy, M its

mass and G is the gravitational constant. Note that its dependence on the total energy of the

system is rather weak. The spin parameter is essentially the ratio of the angular momentum of

an object to that required for rotational support. A value of λ = 0.05, for example, implies very

little systematic rotation and negligible rotational support. Typical values for individual halos in

simulations are between 0.2 and 0.11 (see, e.g., Bullock et al. 2001; Vitvitska et al. 2002; Peirani

et al. 2004; Bett et al. 2007).

• Shape: To calculate the shape of a halo, we calculate the inertia tensor using all particles inside

the region of interest:

Ii j =∑

xix j , (4.5)

Our coordinate system will be chosen with respect to the center of mass of the halos. Diagonal-

izing the matrix, we obtain the eigenvalues a1 > a2 > a3. The eigenvalues of the inertia tensor

are a quantitative measure of the degree of symmetry of the distribution of particles. The axis

ratios follow from the ratio of the eigenvalues

b

a=

√

a2

a1,

c

a=

√

a3

a1, (4.6)

with a > b > c the axes of the object. The sphericity s of an object is defined as the ratio between

the smallest axis of the mass clump and its largest axis,

s =c

a. (4.7)

4.3 Radial mass distribution

Fig. 4.1 shows the evolution of a single cluster in the SCDM model. Every dark matter halo in in the

cold dark matter scenario grows in a similar way: small clumps emerge from the primordial density

field and accrete and merge with surrounding mass concentrations, gaining mass and growing in size.

The first measure of the mass distribution around a cluster peak is its radial density distribution. In this

section we investigate this aspect of the cluster halos.

Figure 4.1 — Evolution of a single cluster halo in the SCDM model. Plotted are the particles belonging to the

cluster halo at the specified redshift.


An important aspect of the halo’s mass distribution is its radial density profile. A practical problem

in inferring this from HOP halos is their often irregular shape. In that situation the center of mass can

can fall in an empty region. This results in irregularities of the calculated radial profile. To avoid this,

we do not determine the radial profile around the center of mass of the HOP halo. Instead, we shift the

halo’s central position towards the most massive concentration of particles. This is accomplished by

means of an iterative procedure.

Fig. 4.2 shows the density profile of the SCDM cluster depicted in Fig. 4.1. The evolution and

growth of the cluster can be clearly recognized in the density profile. The central density, with respect

to the critical density, increases continuously. Meanwhile, the cluster also grows in extent as the radial

profile reaches out to increasingly larger distances. The density profiles of the peers of the depicted

SCDM halo in the other cosmologies do in general agree in shape, be it with the relevant differences

in growth of the density and size of the halos.

Overall, we have the impression that the different density profiles are rather (self-) similar, the only

difference being the concentration parameter c (see also Huss et al. (1999)).

Figure 4.2 — Evolution of the radial density profile for the SCDM dark matter halo shown in Fig. 4.1. The den-

sity profile ρ/ρc, with ρc the critical density of the corresponding cosmology, is shown at 6 subsequent redshifts.

4.3.1 Concentration parameter: definition

The standard lore is to describe the density profile in terms of the fit to the NFW (Navarro et al.

1997) fitting formula. In assessing the concentration of the halos in our sample, we want to remain

independent of such profile prescription. By defining a model independent measure of concentration

we seek to base our conclusions on an objective measure. Moreover, the NFW profile determination

may be affected by several artifacts. Examples are the binning size, the merit function, the weights

assigned to the data points and the used range of radii (Tasitsiomi et al. 2004). The often irregular

distribution of the particles within the sample halos makes the results highly sensitive to these artifacts.

We define a concentration parameter c as the ratio between mean harmonic radius rh and the mean

separation radius rmsep of the halo,

c =rmsep

rh

, (4.8)


in which rh and rmsep are defined as

rmsep =1

N

∑

i, j

|ri j| ,1

rh

=1

N

∑

i, j

1

|ri j|, , (4.9)

where ri j is the separation vector between the ith and the jth particle and N is the number of particle

pairs within the halo containing n particles,

N =

(

n

2

)

=n(n−1)

2, (4.10)

According to this definition, a higher degree of concentration goes along with a higher concentration

parameter c! A high value of c means a halo is more concentrated than one with a lower value of c.

We find that this definition of concentration parameter is unbiased, since it depends only on the

position of the particles within the halo. rmsep will give more weight to the outer particles, while rH

gives more weight to the inner ones. A particular virtue of the mean harmonic radius is that it is a

good measure of the effective radius of the gravitational potential of the halo. Both radius have the

advantage that they are independent of the definition of cluster center. Note that as halos become more

concentrated, the mean harmonic radius will become smaller. The same will happen with the mean

separation distance, but this decrease will be slower.

4.3.2 Individual halos

Figure 4.3 — Evolution of the concentration parameter for two different clusters in four different cosmologies.

Left: a halo that underwent a major merger. Right: a calmly evolving and accreting halo.

Fig. 4.3 shows the evolution of two different clusters in four cosmological models. We identify

one halo in one cosmology (the SCDM in this case) and then we see to which halo corresponds in the

other models. The halo in the lefthand frame has undergone a major merger, while the one in the right

hand frame did evolve quiescently by gradual accretion.

Noticeable differences between these halos can only be observed in the SCDM cosmology. In that

situation we find that the halo’s concentration is substantially stronger for the merger halo than the

accretion-only halo: over nearly the whole expansion range the concentration parameter c is lower in

the first one than in the latter. The merger halo does appear to become more concentrated in time.

With the exception of the early epochs, presumably beset by resolution issues, we cannot find major

differences in halo concentration in the other cosmologies. In contrast to the SCDM merger halo, we

cannot detect a real change in halo concentration for the merger halos in the ΛCDMO2, ΛCDMF2 and

ΛCDMC2 cosmologies. However, the quiescently accreting halos all seem to attain a higher level of

concentration, be it moderate, in all four depicted cosmologies.


4.3.3 Mass dependence

Figure 4.4 — Concentration parameter c as a function of mass M for all halos in the four simulations of halo

formation in SCDM (top left frame), ΛCDMO2 (top right frame), ΛCDMF2 (bottom left frame) and ΛCDMC2

(bottom right frame) cosmologies. The plots are scatter plots, with each point representing a halo in the simulation.

The density of the points in the scatter plot can be inferred from the superimposed density grey scale plot.

In an attempt to get an impression of more global and generic behaviour of the halo concentration

in the various cosmologies we investigate its dependence on the mass of the halos.

Fig. 4.4 shows the concentration parameter as a function of mass for the four indicated cosmolo-

gies. The plots are scatter plots, with each point representing a halo in the simulation. The density of

the points in the scatter plot can be inferred from the superimposed density grey scale plot.

In all four cosmologies we find the largest range of halo concentration amongst the low mass

halos. There appears to be a trend that the range of halo concentration decreases as the mass of the

halos becomes larger, although the average concentration is almost independent of mass. Comparing

the average concentration as a function of mass in Fig. 4.4 we do not find any mass dependence in the

SCDM halos, but a quite systematic and rather strong dependence in the ΛCDMO2 cosmology: high

mass halos are more concentrated than the low mass ones. The ΛCDMF2 and ΛCDMC2 cosmologies

display similar trends, although much less prominent.

Comparing the halo concentrations in the different cosmologies we find that the halos in the open

model are less concentrated than in the higher Ωm models. In fact, Fig. 4.4 shows a systematic trend

going from SCDM to ΛCDMC2 to ΛCDMF2 and finally to ΛCDMO2 in finding less concentrated

halos. This seems to contradict the results of Huss et al. (1999). On the other hand, it seems to tie in

with the recent work of Maccio et al. (2007), who claims that there is an environmental effect akin to

the assembly bias results of Gao & White (2007) and Sheth & Tormen (2004). It certainly must be

related to the fact that halos in low Ωm Universes and Universes with a higher Λ have formed earlier

and do not show much evolution anymore at the present epoch.


Figure 4.5 — Concentration parameter c as a function of mass M for all halos in the four simulations of halo

formation in SCDM (top left frame), ΛCDMO2 (top right frame), ΛCDMF2 (bottom left frame) and ΛCDMC2

(bottom right frame) cosmologies.

4.3.4 Evolution of the concentration parameter

The fact that in SCDM halos over the entire mass range retain an active evolutionary lifestyle and keep

on growing in mass apparently leads to a substantially higher concentration (see Fig. 4.5).

In order to investigate this issue we have studied the evolution of the average concentration param-

eter of the halo population. Fig. 4.6 shows the evolution of the concentration parameter for the four

indicated Universes. The development of c is shown for the entire halo sample (right hand panel) and

for the most massive cluster mass halos (left hand panel).

The differences between the three ΛCDM Universes on the one hand and the SCDM on the other

are quite strong. The concentration parameter in the three ΛCDM cosmologies is not only of com-

parable strength but also remains so over the entire expansion history of the Universe. After a ∼ 0.3,

the halos gradually but weakly evolve towards a higher concentration, probably reflecting both steady

accretion and virialization. The effect may be some what stronger for the clusters than for the full

sample of halos.

The difference with the SCDM halos is striking. While the SCDM halos seem to start out as

less concentrated mass clumps, after a ∼ 0.3 they increase in concentration much stronger than in the

Figure 4.6 — Evolution of the concentration parameter c as a function of the expansion factor for the cosmolo-

gies depicted. Left panel: evolution of c for the sample of cluster halos. Right panel: evolution of c for the entire

sample of dark matter halos.


other cosmologies. The continous growth of the SCDM halos apparently implies a continous increase

towards a more concentrated mass configuration. This may explain why at the present epoch we do

find them to be much more concentrated over the entire mass range (see Fig.4.4).

We also investigated whether there were differences in concentration evolution between halos that

form by steady accretion only or ones that at least once underwent a major merger. In none of the

cosmologies we did find any significant difference between these two samples. This suggests that the

way in which halos absorb mass does not influence the final configuration. However, this should be

investigated in more detail in future work.

4.4 Morphology in different cosmologies: dependence on mass,

redshift and formation

Figure 4.7 — Five different halos of different masses at z= 0 in theΛCDMF2 model. There is a mass dependence

on the shape of the halo.

Fig. 4.7 shows examples of various halos in the ΛCDMF2 cosmology. Going from left to right, we

see an example of a massive cluster mass halo and subsequently a set of lighter halos. The axis ratio.

To investigate the shapes and morphology of the clusters as a function of their evolutionary stage and

the cosmological background we have determined the shapes of the identified simulation halos along

the lines described in section 4.2.2.

On the basis of the eigenvalues of the inertia tensor of every dark matter halo in our halo samples

we calculated the average halo axis ratios 〈b/a〉 and 〈c/a〉. These are tabulated in Table 4.2 for the

present epoch a = 1. One immediate observation is that the average triaxial shape of the halos tends

towards a prolate shape. Also, the impression is that this is true independent of the cosmology: 〈b/a〉and 〈c/a〉 are remarkably similar in each of our simulations.

The numbers that we find in Table 4.2 are also similar to the ones found in others studies, although

they do differ in some details. The axis ratios found in this study are quite similar to the ones found

by Dubinski & Carlberg (1991). They found that in a SCDM Universe halos have mean axis ratios of

〈b/a〉=0.71 and 〈c/a〉=0.50, prodding them to remark that halos are triaxial and very flat. van Haarlem

& van de Weygaert (1993) seem to have found much more elongated halos than ours, while Kasun &

Evrard (2005) found more roundish halos. van Haarlem & van de Weygaert (1993) quotes values of

〈b/a〉=0.58 and 〈c/a〉=0.48 in a set of simulations in SCDM and scale-free Universes, while Kasun &

Evrard (2005) obtains values of (b/a,c/a) = (0.76,0.64) for halos more massive than 3× 1014h−1M⊙in a ΛCDM model and a τCDM model. Perhaps the largest differences are with the values quoted by

Katz (1991), who found b/a values ranging from 0.84 to 0.93 and c/a values of 0.43 to 0.71. This may

perhaps be related to the fact that they limited themselves to purely isolated halos. These halos could

not interact with their surroundings, and were therefore strongly constrained with respect to accretion,

let alone merging, with the surrounding mass distribution.

The question is whether the apparent universal average shape that we find is independent of the

mass of the halo or of the cosmic epoch.

4.4. MORPHOLOGY IN DIFFERENT COSMOLOGIES: DEPENDENCE ON MASS,

REDSHIFT AND FORMATION 75


Fig. 4.8 shows the sphericity s = c/a of halos as a function of mass in the four cosmological models:

SCDM, ΛCDMO2, ΛCDMF2 and ΛCDMC2. Each of the lines in the graph depict the average halo

sphericity in a given mass bin. Low mass limits are due to the fact that the smallest group in each

cosmology has 100 particles. We see a clear variation in sphericity between the halos in different

cosmologies.

Figure 4.8 — Sphericity s = c/a of halos as a function of mass for four cosmological models.

In every cosmology, low-mass halos appear to be more spherical than their massive peers. Inter-

estingly, in the primordial density field the higher peaks are more spherical than the low mass ones

(Bardeen et al. 1986). It is therefore clearly an effect of the subsequent nonlinear evolution. The more

massive halos have been involved with much more mass accretion than the low mass halos and this

apparently remains to have a strong effect on their shape (van Haarlem & van de Weygaert 1993). In

addition, there must be a substantial environmental component in this shape distribution. Low den-

sity regions, such as voids and walls, merely contain low mass halos, while the high mass clumps are

mainly concentrated in filaments and even more so in compact dense clusters. Interactions between

the halos is considerably more frequent and strong in those high density regions, apparently resulting

in more aspherical shaped halos (see Aragon-Calvo (2007)).

The variation in shape between halos of different mass is most outstanding in the SCDM cosmol-

ogy. This must tie in with the continuing evolution of the matter distribution in this cosmology. This

is supported by the same systematic trend in the ΛCDMC2 and the ΛCDMF2 cosmologies, be it at a

more moderate level.

Model 〈b/a〉〈c/a〉 Model 〈b/a〉〈c/a〉SCDM 0.70 ± 0.16 0.55 ± 0.14 ΛCDMO3 0.70 ± 0.15 0.55 ± 0.13

OCDM01 0.73 ± 0.15 0.57 ± 0.13 ΛCDMF2 0.70 ± 0.15 0.55 ± 0.13

OCDM03 0.71 ± 0.15 0.56 ± 0.13 ΛCDMC1 0.70 ± 0.15 0.54 ± 0.13

OCDM05 0.70 ± 0.16 0.55 ± 0.14 ΛCDMF3 0.70 ± 0.15 0.55 ± 0.13

ΛCDMO1 0.72 ± 0.15 0.57 ± 0.13 ΛCDMC2 0.70 ± 0.15 0.54 ± 0.13

ΛCDMO2 0.72 ± 0.15 0.56 ± 0.13 ΛCDMC3 0.69 ± 0.15 0.54 ± 0.13

ΛCDMF1 0.71 ± 0.14 0.55 ± 0.13

Table 4.2 — Mean values of the axis ratios of every group found in each cosmological model at z = 0.


Figure 4.9 — Evolution of the shapes of objects present in four cosmological models. Color contours shows the

1-sigma level at every redshift.

4.4.2 Shape Evolution

To appreciate the evolutionary trends in the shape distribution of halos, we have plotted the halos in

a scatter diagram of axis ratio c/a versus b/a (see Fig. 4.9). In this diagram we have indicated where

one can find prolate, oblate and spherical halos. Instead of plotting the individual halo shapes, we have

plotted the contour of the 32% percentile around the average shape of the halos in a simulation. In

each of the panels we plotted these contours for a sequence of timesteps, with the lightest contour level

corresponding to z = 3 and the darkest one to the present epoch z = 0.

In all cosmologies we see a distint and similar trend of the shape distribution. Although the halos

tend to retain a rather prolate shape, we also see them shifting towards a more spherical shape. This

must be a reflection of the continuing virialization and relaxation of the halos in our simulations, in

this respect it is perhaps indicative that the change in shape is far less strong in SCDM than in e.g.

ΛCDMO2.

We also assesed whether there are any significant differences between halos that underwent a major

4.5. ANGULAR MOMENTUM 77

Model Mergers Accretion

〈b/a〉〈c/a〉〈b/a〉〈c/a〉SCDM 0.64 ± 0.18 0.49 ± 0.15 0.72 ± 0.15 0.56 ± 0.13

ΛCDMO2 0.66 ± 0.16 0.50 ± 0.14 0.73 ± 0.14 0.57 ± 0.12

ΛCDMF2 0.65 ± 0.17 0.50 ± 0.14 0.71 ± 0.14 0.56 ± 0.13

ΛCDMC2 0.64 ± 0.17 0.49 ± 0.14 0.71 ± 0.15 0.55 ± 0.13

Table 4.3 — Axis ratios of the complete sample of dark matter halos in the four indicated cosmologies at z = 0.

merger or ones that had a more quiescent life. In Table 4.3 we listed the average axis ratios b/a and

c/a of an accretion sample and a merging sample in four cosmologies. With the exception of perhaps

some minor trend of accreting halos being somewhat more spherical, we can not find any major effect.

This result is apparently related to similar finding by Allgood et al. (2006).

4.5 Angular Momentum

An interesting though often discarded aspect of the structure and kinematics of galaxy clusters is

their rotation. Nonetheless, their mass distribution will surely be affected by the angular momentum

imparted on them by the surrounding large scale structure. Angular momentum has been extensively

studied in the context of angular momentum growth of dark halos and the origin of rotation of galaxies.

According to the Tidal Torque Theory (TTT), collapsing bodies in the Universe acquire angular

momentum as a result of the tidal torque imparted on them by the surrounding inhomogeneous mat-

ter distribution. The original idea by Hoyle (1949) got worked out by Peebles (1969); Doroshkevich

(1970); White (1984). Its success within the cosmological context has been demonstrated by numerous

N-body simulations, starting with the first seminal papers by Efstathiou & Jones (1979); Jones & Efs-

tathiou (1979); Barnes & Efstathiou (1987). With the availability of simulations of far larger dynamic

range, necessary to probe the large scale tidal field over a sufficiently large spatial range, our insight

into the origin of galaxy rotation has grown impressively so that a reasonable accurate picture of both

dark matter angular momentum and gaseous galaxy rotation has been formed (e.g. Warren et al. 1992;

Catelan & Theuns 1996; Lemson & Kauffmann 1999; Porciani et al. 2002; Bullock et al. 2001; van

den Bosch et al. 2002; Bett et al. 2007; Aragon-Calvo 2007). Most of the angular momentum is gained

during the linear phase of collapse, proportional to the time (White 1984).

It is thought that when the collapse becomes nonlinear and when the halo decouples from the

cosmic expansion, the growth of the angular momentum gets suppressed. During the nonlinear hi-

erarchical buildup of a halo, we may identify several important aspects in the growth of its angular

momentum. Even by its own, its angular momentum will grow as a result of the tidal torque exerted by

the surrounding mass distribution. After turnaround, while the halo shrinks in size, the magnitude of

the tidal torque will decrease accordingly. Meanwhile, the halo will interact with neighboring clumps

and will accrete and absorb matter and clumps from its immediate surroundings. Given that the an-

gular momentum is dependent on the relative orientation of the halo spin with respect to the angular

momentum vector of the accreted matter, merger and accretion greatly affects the angular momentum

history of dark matter halos. As a result, its angular momentum will increase because it will absorb the

specific angular momentum of the accreted halos and the orbital angular momentum related to their

motion around the common mass center.

Here we will study in how far cluster halos do follow the same behavior as the smaller galaxy sized

halos with respect to the angular momentum that they acquire. This will surely involve differences.

The generating mass fluctuations, in the case of clusters this will be fluctuations on scales up to a few

hundreds Mpc, will probably be less prominent than the ones that make the galaxies spin. For most

part this concerns the question of the initial power spectrum for the cosmological scenario at hand.


Figure 4.10 — Evolution of the total angular momentum, J, as a function of the expansion factor for two

different set of clusters. Left panel: clusters formed via merger. Right panel: clusters formed via accretion.

Angular momentum in arbitrary units.

Figure 4.11 — Evolution of the spin parameter λ (upper panels) and the mass history (bottom panels) as a

function of the expansion factor for two sets of clusters. Left panels: clusters formed via merger. Right panels:

clusters formed via accretion.


Although our results will provide insight into this issue, we also should note that our simulations will

not be able to probe over all relevant scales. Our simulations boxes are simply not large enough.

Before addressing the generic behaviour, we first study the angular momentum growth of a few

particular cluster halos.

4.5.1 Angular momentum in single halos

We identified two halos on the basis of the SCDM simulations. We extracted them and their equivalents

from each of the simulations. One of the halos is a massive one that underwent a massive merger, the

other one is a less massive halo that only evolved through a quiescently accreting lifestyle. Note that

we define a “merger” as a halo that absorbed an infalling clump with a mass of at least 30% of its

own mass. Once selected in the SCDM simulation, we also identified their equivalents in the other

simulated cosmologies.

Fig. 4.10 shows the evolution of the total angular momentum J as a function of the expansion

factor. We find that the evolution of the angular momentum is erratic and correlates strongly with the

growth in mass of the halo (see bottom panels). On the basis of individual halos it is hard to find any

systematic differences in angular momentum growth between accreting and merging halos.

Taking account of the growing mass and potential well of the growing halo, we may obtain a more

direct view of the growth of the specific angular momentum by assessing the evolution of the spin λ

(see Eqn. 4.4). The same figure shows the evolution of the spin parameter for the same halos, along

with the evolution of their mass accretion history.

Along with the angular momentum J, we observe a similar erratic development of the spin param-

eter λ. Its correlation with the mass growth of the halo, and in particular the events involving mass

jumps, stands out even more clear than when looking at J. For example, when we see the “merging”

SCDM halo absorbing a substantial mass clump at a ∼ 0.3, we immediately see a peak in λ. This is

even more outstanding upon its merger with a massive neighbour at a ∼ 0.9. Also, the accreting halo

has some marked events in its λ history. When the SCDM halo absorbs a clump at a ∼ 0.5, we see this

reflected in a rise of λ. Also note that after some time the effect of the infalling clump subsides as λ

returns to a more equilibrium value of around λ ∼ 0.05: there is no systematic increase of λ. Note that

this is true for these individual halos. We find a somewhat different trend for the generic population of

halos (see section 4.5.2).

4.5.2 Generic angular momentum growth

Despite the erratic variations of the angular momentum when individual halos are considered, we ob-

serve a steady and systematic increase when considering the average for the whole halo population.

This can be seen in Fig. 4.12, which shows the median angular momentum J as a function of ex-

pansion time. The median was determined from the distribution function of J, which we discuss in

section 4.5.5. We also look at the time evolution in two different ways: as a function of cosmic time t

(top panels) and as a function of cosmic expansion factor a.

One immediate observation is that the absolute value of the angular momentum J is related to

the cosmic matter density. Halos in a cosmology with a lower value of Ωm consistently have a lower

angular momentum J. This relates to the fact that the mass of the clusters are less massive in low Ωm.

Apart from the dependence of J’s amplitude on Ωm we see that the growth of J is more or less the

same for each of the cosmological models.

Turning to the generic evolution of the spin parameter λ, see Fig. 4.13, we find that in all cosmolo-

gies it shows the same decreasing trend. The spin parameter slowly but gradually goes from around

λ ∼ 0.06 at a ∼ 0.2 to λ ∼ 0.04 at the current epoch. The only difference is the situation for ΛCDMO2.

Given that low number of clusters in its simulations we still recognize the erratic evolution of λ of the

individual halos.

Knowing that λ is affected strongly at each major infall, we do suspect a systematic difference

between the evolution of λ of accreting halos and of merging halos. This indeed can be seen in the


Figure 4.12 — Evolution of the median of the logarithmic of the angular momentum J as function of time (in

units of H0, top panel) and as a function of the expansion factor for clusters halos.

Figure 4.13 — Evolution of the spin parameter as a function of the expansion factor of the cluster size halos for

the SCDM (top left panel), ΛCDMO2 (top right panel), ΛCDMF2 (bottom left panel) and the ΛCDMC2 (bottom

right panel) cosmologies. Shown is the evolution when considering halos that grew by merger (dashed lines), that

grew by accretion (dotted lines) and when considering both samples together (solid line).


same figure. There is an upward trend of λ for merging halos after a ∼ 0.5 at least for the SCDM,

ΛCDMO2, ΛCDMF2 and ΛCDMC2 cosmologies. We will shortly touch unpon this issue in section

4.5.4.

4.5.3 Angular momentum and cosmological constant

For the purpose of distinguishing possible influences of the cosmological constant, in Fig. 4.14 we

consider the growth of the angular momentum J in four different cosmologies of which all have the

sameΩm = 0.3 but a different valueΩΛ of the cosmological constant: OCDM03, ΛCDMO3 ΛCDMF2

and ΛCDMC1.

When comparing the J growth as a function of time (top panels in Fig. 4.14, we see that it is the

same for all four models: we can not detect any dynamical influence of Λ. Note that the amplitude

of the halo angular momenta in the different cosmologies is different because of the differences in

large scale power between these cosmological scenarios (see chapter 2). Also note the contrast to the

situation in which we compare cosmologies with different Ωm. In the top panel of Fig. 4.15 we see that

there is also a horizontal shift of the angular momentum growth curves: in a higher Ωm Universe the

more rapid growth of structure goes along with stronger tidal torques, thus affecting an earlier growth

of angular momentum.

Figure 4.14 — Evolution of the median of the logarithmic of the angular momentum as function of time (in

units of H0, top panel) and as a function of the expansion factor for clusters formed via merger (upper curves) and

via accretion (lower curves).

4.5.4 Angular momentum, merging and accretion

Fig. 4.15 is the same as Fig. 4.12, depicting the growth of angular momentum J as a function of

cosmic time t (upper panel), and cosmic expansion factor a (lower panel). However, in this case we

have distinguished the growth for samples of halos that formed by quiescent accretion and samples of



units of H0, top panel) and as a function of the expansion factor for clusters formed via merger (upper curves) and

via accretion (lower curves).

merging halos. This distinction is somewhat artificial, in that we brand a halo as a merger when at least

once in its lifetime it experienced the infall of a massive clump with at least 30% of its mass.

While at early times there is no difference between the merging and accreting halo populations,

there is a marked and clear deviation between the angular momentum growth of both populations as

time progresses. The merging halos on average acquire a larger share of angular momentum, undoubt-

edly related to the gain in angular momentum as they absorb the specific and orbital angular momentum

of the infalling clumps. This seems to be true in most cosmologies, with the possible exception of the

open ΛCDMO2 cosmology. For low mass halos this was also found by Peirani et al. (2004).

Perhaps the most interesting and qualitative difference concerns the evolution of the spin parameter

λ. As we observed before, we see that merging halos do not share in the trend of the overall halo

population to have a decreasing λ. Instead, in the SCDM, ΛCDMF2 and ΛCDMC2 cosmology we

find that after a ∼ 0.5, λ increases systematically for the mergers. This is clearly related to the jumps

of the spin parameter λ of individual halos following the infall of massive clumps (see Fig. 4.13).

4.5.5 The spin distribution

Several authors have pointed out that the distribution of the spin parameter follows a lognormal distri-

bution (Vitvitska et al. 2002; Knebe et al. 2002; Peirani et al. 2004)

p(λ)dλ =1

σλ√

2πexp

− ln2 (λ/λ0)

2σ2λ

dλ

λ, (4.11)

with a median value of λ ∼ 0.05. Note that the λ0 parameter of the lognormal distribution is not equal

to the mean of λ.


Figure 4.16 — The distribution of the spin parameter and its corresponding lognormal distribution for the SCDM

(top left frame), ΛCDMO2 (top right frame),ΛCDMF2 and the ΛCDMC2 cosmologies.

In Fig. 4.16 we have plotted the spin parameter distribution of the cluster halos in our SCDM,

ΛCDMO2, ΛCDMF2 and ΛCDMC2 cosmologies. Superimposed are the fits following the lognormal

distribution in Eqn. 4.11. Clearly, in all cases it provides a more than satisfactory description. This

ties in with a similar claim by Peirani et al. (2004) that the lognormal distribution is independent of

cosmology. Moreover, we see that we can hardly find any cluster with a spin parameter λ > 0.2.

On the basis of the fits of the lognormal distribution to the λ distribution of the halos in our simu-

lations, we have inferred the values of the parameters λ0 and σλ. The inferred parameters for cluster

halos are shown in Table 4.4. We also calculated the fitted parameters for the entire sample of dark

matter halos, but we did not find any significant differences.

The table does not show any systematics dependence of λ0 and σλ on cosmological parameters,

and certainly not on the cosmological constant.


Model λ0 σλ Model λ0 σλ

SCDM 0.035 0.815 ΛCDMF1 0.033 0.977

OCDM01 0.019 0.617 ΛCDMF2 0.027 0.770

OCDM03 0.027 0.792 ΛCDMF3 0.031 0.789

OCDM05 0.030 0.804 ΛCDMC1 0.029 0.852

ΛCDMO1 0.029 0.611 ΛCDMC2 0.032 0.791

ΛCDMO2 0.029 0.509 ΛCDMC3 0.031 0.760

ΛCDMO3 0.026 0.712

Table 4.4 — Inferred parameters of the lognormal distribution for λ.

4.6 Conclusions

We have studied individual properties of dark matter cluster halos in thirteen cosmological models.

These individual properties are: the angular momentum, morphology and density profiles. The cos-

mological models that we studied included a set of open, flat and closed Universes with a reange of

matter density parameter Ωm and cosmological constant ΩΛ.

The cluster halo sample are obtained from a set of N-Body simulations in each of the cosmologies.

These simulations concerns 2563 particles in a box of 200h−1Mpc side. We set up the initial conditions

in such a way that the phases of the Fourier components of the primordial density field are the same in

all simulations. In this way, the same objects can be identifed in each of the different simulations.

We ran the simulation from expansion factor a= 0.2 to the present epoch a= 1 using the GADGET2

code. We used HOP to identify and extract the cluster halos. Of each of these cluster halos we

calculated several properties: angular momentum, shape and concentration. We studied this properties

as a function of the underlying cosmology.

The main conclusions from our study are:

• The average halo density profile has the same appearence in each of the simulated cosmologies.

The differences concern the amplitude and extent of the halo.

• We find a significant dependence of the halo concentration on the cosmology. The halos in the

SCDM cosmology are much more concentrated than in e.g. the low Ωm ΛCDMO2 cosmology.

• There is a difference in dependence of halo concentration on mass in the different cosmologies.

Halos in the SCDM cosmology are equally strongly concentrated over the whole mass range,

while in lower Ωm Universes we find a stronger concentration of the massive halos than of the

low mass halos.

• We do not find any indication for a dependence of halo concentration on cosmological constant

Λ.

• We find that accretion and/or merging process do not strongly influence the cluster halo concen-

tration.

• Dark matter halos have a triaxial shape tending towards prolate. The axis ratios are typically

between 0.55 and 0.75.

• There is no marked difference of average halo shape at the current epoch between the different

cosmologies.

• The sphericity of halos as a function of mass does reveal interesting differences between the

different cosmologies. In particular, high mass halos in high Ωm Universes tend to be less spher-

ical. This probably is related to their more active lifestyle involving numerous recent infalling

clumps.

4.6. CONCLUSIONS 85

• In all cosmologies, halos evolve towards an increasingly spherical configuration. There is some

difference of this trend between the different cosmologies.

• The angular momentum of halos increases erratically but steadily, and is intimately related to

the increasing mass of the halos.

• The spin parameter, on the other hand, has a radically different behaviour. It reacts immediately

and strongly upon the infall of a massive clump into the halo. Subsequently, it subsides and

almost returns to its original value as the halo returns to a state of equilibrium and virialization.

• On average, the spin parameter of the halo population decreases from λ ∼ 0.06 at a = 0.5 to

λ ∼ 0.04 at the present epoch. This is seen in nearly all cosmologies.

• The evolution of the spin parameter is different for cluster halos that grew by violent mergers

and for clusters that grew by steady accretion. This evolution is strongly correlated with the

mass accretion history of halos. This is reflected in a substantial dependence on Ωm, but not on

ΩΛ.

• The difference between merging and accreted halos also comes forward from the evolution of

the average spin parameter λ. Halos that underwent at least one massive merger show a gradual

and steady increase of λ, and thus differ systematically from the average behaviour.

• The lognormal distribution of the spin parameter appears to be universal, and does not show

major differences between different cosmological models.


5Cluster Halo Scaling Relations †

We explore the effects of dark matter and dark energy on the dynamical scaling properties of

galaxy clusters. We investigate the cluster Faber-Jackson, Kormendy and Fundamental Plane

relations between the mass, radius and velocity dispersion of cluster size halos in cosmological N-body

simulations. These relations are a profound expression of the virial state of these objects. The sim-

ulations span a wide range of cosmological parameters, representing open, flat and closed Universes.

Independently of the cosmology, we find that the simulated clusters do indeed lie on a Fundamental

Plane, and are close to a perfect virial state. The one outstanding influence is that of Ωm. While

the Fundamental Plane parameters are hardly dependent on cosmology, we do find a noticeable trend

when looking at its width. The FP in low Ωm Universes tend to converge early on to a tight and well

defined plane. The redshift evolution of the scaling parameters do reveal some interesting facts. The

Fundamental Plane turns out to be almost universal in nature, its only change concerning a gradual de-

crease of its width as the halos become more virialized. The Kormendy and Faber-Jackson parameters

do change radically. The Faber-Jackson parameter increases continuously, while the Kormendy one

initially increases rapidly after which it stabilizes or slightly decreases its value. On the basis of this,

we conclude that the cluster population’s evolution, constrained to the Fundamental Plane, involves

a a weak structural and dynamical homology. Our suspicion of a possible effect of the cosmology in

terms of their different power spectrum slope at cluster scale is not confirmed. Related to this we also

investigated the differences between clusters that quietly accreted their mass and those that underwent

massive mergers. The latter have a less well defined plane as their virial state is severely disrupted in

the merging process.

†Pablo A. Araya-Melo, Rien van de Weygaert & Bernard J.T. Jones, 2008, MNRAS, to be submitted.

88 Chapter 5: Cluster Halo Scaling Relations

5.1 Introduction

Recent observations of distant supernovae (Riess et al. 1998; Perlmutter et al. 1999) suggest that we

are living in a flat, accelerated Universe with a low matter density. This accelerated expansion has

established the possibility of a dark energy component which behaves like Einstein’s cosmological

constant Λ. A positive cosmological constant solves the apparent conflict suggested by the old age of

globular clusters stars and the estimated amount of it (Spergel et al. 2003, 2007) appears sufficient to

yield a flat geometry of our Universe.

The role of Λ in the process of structure formation is not yet fully understood. Although its influ-

ence can be noticed when looking at the global evolution of the Universe its role in the dynamics of

the structures is not clear. Its most direct impact will be that via its influence on the amplitude of the

primordial perturbation power spectrum and, perhaps most direct, via its influence on the cosmic and

dynamical time scales. Its direct dynamical influence is probably minor: we do know that in the linear

regime it accounts for a mere ∼ 1/70th of the influence of matter perturbation (Lahav et al. 1991).

Most viable theories of cosmic structure formation involve hierarchical clustering. Small structures

form first and they merge to give birth to bigger ones. The rate and history of this process is highly

dependent on the amount of (dark) matter present in the Universe. In Universes with a low Ωm,

structure formation ceases at much early times than that in cosmologies with high density values.

Within this hierarchical process, clusters of galaxies are the most massive and most recently formed

structures in the Universe. Their collapse time is comparable to the age of the Universe. This makes

them important probes for the study of cosmic structure formation and evolution. The hierarchical

clustering history from which galaxy clusters emerge involves a highly complex process of merging,

accretion and virialization. In this chapter we investigate in how far we can get insight into this history

on the basis of the internal properties of the clusters. This involves characteristics like their mass and

mass distribution, their size and their kinetic and gravitational potential energy. In particular, we are

keen to learn whether these do show any possible trace of a cosmological constant.

One particular profound manifestation of the virial state of cosmic objects is via scaling relations

that connect various structural properties. Scaling relations of collapsed and virialized objects relate

two or three fundamental characteristics. The first involves a quantity measuring the amount of mass

M, often in terms of the amount of light L emitted by the object. The second quantity involves the

size of the object, while the third one quantifies its dynamical state. Since the mid 70s, we know

that elliptical galaxies do follow such relations. The Faber-Jackson relation (Faber & Jackson 1976)

relates the luminosity L and the velocity dispersion σ of an elliptical galaxy. The Tully-Fisher relation

(Tully & Fisher 1977) is the equivalent for spiral galaxies. A different, though related, scaling is that

between the effective radius re and the luminosity of the galaxy. This is known as the Kormendy

relation (Kormendy 1977). These two relations turned out to be manifestations of a deeper scaling

relation between three fundamental characteristics, which became known as the Fundamental Plane

(Djorgovski & Davis 1987; Dressler et al. 1987). Not only do this scaling relations inform us about

the dynamical state of the objects, they also function as important steps in the extragalactic distance

ladder.

Similar scaling relations are also found for clusters of galaxies. This suggests that stellar systems

and galaxy clusters have similar formation process, favoring the hierarchical scenario. These clusters

scaling relations were first studied by Schaeffer et al. (1993). They studied a sample of 16 galaxy

clusters, concluding that theses systems also populate a Fundamental Plane. Adami et al. (1998) used

the ESO Nearby Abell Cluster Survey to study the existence of a Fundamental Plane for rich galaxy

clusters, finding that it is significantly different from that of elliptical galaxies. Later, Lanzoni et al.

(2004) addressed the question of for high mass halos, which are thought to host clusters of galaxies.

On the basis of 13 simulated massive dark matter halos in a ΛCDM cosmology they found that the

dark matter halos follow the FJ, Kormendy and FP relations.

In this chapter we specifically address the question whether we can trace an influence of a positive

cosmological constant in the scaling relations for simulated clusters. We use a set of dissipationless N-

body simulations involving open, flat and closed Universes. All the simulations are variants of the cold

5.2. SCALING RELATIONS 89

dark matter (CDM) scenario, representing different cosmological models with Ωm < 1 and ΩΛ , 0.

The organization of this chapter is as follows. In the next section we present a general description

of scaling relations, addressing the relation between the Fundamental Plane and the Virial Theorem.

In section 5.3 we describe the simulations and the definitions of the various parameters we use. We

investigate the scaling relations of galaxy clusters in different cosmologies at z = 0 in section 5.4. In

section 5.5 we study the evolution of the scaling relations as a function of redshift and cosmic time.

We also investigate the dependence of merging and accretion on the scaling relations in section 5.6.

Conclusions are presented in section 5.7.

5.2 Scaling Relations

From observations of elliptical galaxies we have learned that there are tight relations between a few of

their fundamental structural properties (see e.g. Binney & Merrifield (1998)). These properties are the

total luminosity L of a galaxy, its characteristic size Re and its velocity dispersion σv.

The first of this relations is the Faber-Jackson relation Faber & Jackson (1976) between the lumi-

nosity of the galaxy and its velocity dispersion,

L ∝ σγv , (5.1)

where the index γ ∼ 4. A similar relation, known as the Tully-Fisher relation (Tully & Fisher 1977),

holds for HI disks of spiral galaxies. According to this relation, the galaxies’ rotation velocity is tightly

correlated with the absolute magnitude of the galaxy.

Another relation was established by Kormendy (Kormendy 1977). He found that there is a strong,

not entirely unexpected, correlation between the luminosity and effective radius of the elliptical galax-

ies:

L ∝ Rαe , (5.2)

where the index α ∼ 1.5.

It turns out that these relations between two structural characteristics should be seen as projections

of a more fundamental and tight relations between all three structural quantities, the Fundamental

Plane (FP). The Fundamental Plane of elliptical galaxies was first formulated by Djorgovski & Davis

(1987) and Dressler et al. (1987). When we take the three-dimensional space defined by the radius Re

of the galaxy, its surface brightness Ie (with total luminosity L ∝ IeR2e) and velocity dispersion σv, we

find that they do not fill space homogeneously but instead define a thin plane. In logarithmic quantities,

this plane may be parametrized by

logRe = α logσv + β log Ie + γ . (5.3)

For example, Jørgensen et al. (1996) found that a reasonable fit to the Fundamental Plane is given by

logRe = 1.24log σ − 0.82log Ie + γ . (5.4)

While nearly all galaxies, ranging from giant ellipticals to compact dwarf ellipticals, appear to lie on

the FP (also see e.g. Jørgensen et al. 1995; Bernardi et al. 2003; Cappellari et al. 2006; Bolton et al.

2007) it is interesting to note that diffuse dwarf ellipticals do not (Kormendy 1987): they seem to be

fundamentally different objects.

The galaxy scaling relations are of great importance for a variety of reasons. First of all, they must be a

profound reflection of the galaxy formation process (Robertson et al. 2006). Also, they have turned out

to be of substantial practical importance. Because they relate a intrinsic distance independent quantity

like velocity dispersion to a distance dependent one like Le, they can be used as cosmological distance

indicators.


5.2.1 Fundamental Plane and Virial Theorem

In its pure form, the Fundamental Plane is a direct manifestation of the Virial Theorem. As such, it is

expected to hold for any virialized clump of matter, be it a galaxy or a cluster of galaxies.

In order to appreciate this, we first look at the expected relationship between the mass M of a virial-

ized halo, its size R and its velocity dispersion σv = 〈v2〉1/2. The kinetic energy K and the gravitational

potential energy U of this halo are given by

K =M〈v2〉

2, U = −GM2

R. (5.5)

The Virial Theorem establishes the following strict relationship between U and K,

2K +U = 0 , (5.6)

which implies that

〈v2〉 =GM

R. (5.7)

This would imply the following scaling relation between M, R and σv,

log M = 2logσv + logR + γM , (5.8)

in which γM is a constant.

In hierarchical scenarios of structure formation halos build up by subsequent merging of smaller

halos into larger and larger halos. Some of these mergers involves sizeable clumps, most of it consists

of a more quiescent accretion of matter and small clumps from the surroundings. This process is

certainly leaving its mark on the phase-space structure of the halos. Indeed, these dark halo streams

are a major source of attention in present day studies of the formation of our Galaxy (Helmi & White

1999; Helmi 2000). It remains an interesting question whether we can also find the influence of these

merging events on the Fundamental Plane. For example, Gonzalez-Garcıa & van Albada (2003) did

look into the effects of major mergers on the Fundamental Plane. They did find that the Fundamental

Plane does remain largely intact in the case of two merging ellipticals. However, what the effects will

be of an incessant bombardment of a halo by material in its surroundings has not been studied in much

detail. Given that this is a sensitive function of the cosmological scenario, we will study the influence

on FP parameters and width in more detail.

5.2.2 Fundamental Plane Evolution:

Mass-to-Light ratio and Galaxy Homology

In case the object of mass M has a luminosity L, the observed Fundamental Plane not only provides

information on the dynamical state of the object but also on the evolution of its stellar content. Encap-

sulating the relation between M and L in terms of the mass-to-light ratio,

M =

(

M

L

)

L , (5.9)

we find that the radius R, σ and the mean surface brightness I = L/4πR2 are related according to

R ∝ 〈v2〉I−1(

M

L

)−1

, (5.10)

which yields the following theoretical Fundamental Plane relation

logR = 2logσv − log I − log(M/L) + Cs , (5.11)

5.2. SCALING RELATIONS 91

in which Cs is a constant dependent on the structure of the object. Although the measured Fundamental

Plane of elliptical galaxies is rather thin, it does have an intrinsic scatter. The latter is not completely

explained and may be a manifestation of the formation process.

When looking at the observed parameter values for elliptical galaxies (see 5.4), we do find a dif-

ference with the “predicted” Fundamental Plane: this it not only due to a (constant) mass-to-light ratio

M/L, but also involves an angle between the two planes known as the tilt of the Fundamental Plane

(Gonzalez-Garcıa 2003; Robertson et al. 2006).

Several explanations have been given to account for this tilt. One is homology of the galaxies,

i.e. that is all galaxies are structurally equivalent, in combination with a mass-dependent M/L ratio.

It would imply a formation process involving a tight fine tuning of M/L. It might also be that there

is no such systematic variation of M/L, implying that elliptical are not homologous systems so that

the tilt would be due to variations in the structure parameters of the galaxy. Recent semi-analytical

modelling of galaxy formation do suggest a more complex relation between the mass-to-light ratio and

luminosity, involving a minimum M/L for galaxies with M ≈ 1011−1012h−1M⊙.

Following the first view, the parameters inferred by Jørgensen et al. (1996) (Eqn. 5.4) would imply

a mass-to-light ratio

(M/L) ∝ M0.25 , (5.12)

using M ∝ σ2vRe and L ∝ IeR2

e (see e.g. Faber (1987)). By tracing the possibly systematic evolution of

the galaxy FP, e.g. amongst a population of rich clusters spanning a range of redshifts (van Dokkum &

Franx 1996), one might be able to infer the evolution of the mean M/L ratio. A range of physical pro-

cesses may underlie such systematic shifts. For example, Jørgensen et al. (1995) suggested a relation

with the metallicity of the galaxies.

5.2.3 Cluster Fundamental Plane

Instead of assessing the FP of galaxies in clusters, one might also investigate the question in how far

clusters themselves do follow a Fundamental Plane relation. After all, they are fully virialized objects.

The first to address this question were Schaeffer et al. (1993). They inferred an FP relation from a

sample of 29 Abell clusters, L ∝ R0.89e σ1.28

v , which would translate into an equivalent relation

logRe = 1.15logσv − 0.90log Ie , (5.13)

in which Ie is the hypothetical mean surface brightness of the cluster. The corresponding FJ relation

is L ∝ R1.87e and the Kormendy relation is L ∝ R1.34

e . Similar numbers were inferred by Lanzoni et al.

(2004), L ∝ R0.90e σ1.31

v . Adami et al. (1998) also found a FP relation for a sample of ENACS Clusters,

although their inferred parameters do show some marked differences, L ∝ R0.87σ0.70.

In studies of simulated galaxy clusters, which are known to be dark matter dominated, we may

wonder whether scaling relations similar to those inferred from observable quantities do also exist for

the dark matter host. To infer these relations we base ourselves on the mass M of the object. If the

selected objects have the same average density, we would expect an equivalent Kormendy relation

given by

M ∝ R3e . (5.14)

Any difference in slope should be ascribed to a dependence of mean density 〈ρ(Re)〉 on the size Re of

the object. The equivalent Fundamental Plane relation will be that of Eqn. 5.8, while the Faber-Jackson

relation would then be

M ∝ σ3v . (5.15)

Note that this is based on the assumption of constant mean density ρ of the selected objects, in line

with HOP overdensity criterion.

In line with the above, Lanzoni et al. (2004) analyzed the scaling relations on the basis of a sample

of 13 massive dark matter halos identified in a high resolution ΛCDM N-body simulations. They did



SCDM 1.0 0 0 9.31 13.23 1323 177.65 177.65

OCDM01 0.1 0 0.9 12.55 1.32 132 978.83 97.88

OCDM03 0.3 0 0.7 11.30 3.97 397 402.34 120.70

OCDM05 0.5 0 0.5 10.53 6.62 662 278.10 139.05

ΛCDMO1 0.1 0.5 0.4 14.65 1.32 132 838.30 83.83

ΛCDMO2 0.1 0.7 0.2 15.96 1.32 132 778.30 77.83

ΛCDMF1 0.1 0.9 0 17.85 1.32 132 715.12 71.51

ΛCDMO3 0.3 0.5 0.2 12.70 3.97 397 358.21 107.46

ΛCDMF2 0.3 0.7 0 13.47 3.97 397 339.78 101.93

ΛCDMC1 0.3 0.9 -0.2 14.44 3.97 397 320.79 96.237

ΛCDMF3 0.5 0.5 0 11.61 6.62 662 252.38 126.19

ΛCDMC2 0.5 0.7 -0.2 12.17 6.62 662 241.74 120.87

ΛCDMC3 0.5 0.9 -0.4 12.84 6.62 6622 30.85 115.43






confirmed the existence of FP relations for the dark matter clusters but also found that these have a

significant different slope. On the basis of this, they suggest a mass dependent cluster M/L ratio

(M/L) ∝ M0.8 . (5.16)

Interestingly, this is markedly different from that inferred for early types galaxies.

In this study, we systematically investigate this issue within the context of a range of different

cosmologies. These concern both different values for the mass density Ωm, for dark energy ΩΛ and for

the implied power spectrum of density perturbations and the related merging and accretion history of

the clusters.

5.3 The Simulations


















Figure 5.1 — Comparison between the mean harmonic radius and the half-mass radius of the halos in ΛCDMF2

scenario.



















this.


In our study, we limit ourselves to cluster-like halos. A galaxy cluster is defined as a dark matter

halo with a mass M> 1014h−1M⊙. Of these clusters, we measure three quantities and test their scaling

relation.

Scaling relations of collapsed and virialized objects relate two or three fundamental characteristics

of those objects. The first involves a quantity measuring the amount of mass, often in terms of the

amount of light emitted by the object. The second quantity involves the size of the object, while the

third one quantifies its dynamical state.

• Mass: defined as the number of particles multiplied by the mass per particle present in each


M ∝ rah

M ∝ σb M ∝ rchσd

Model Ωm ΩΛ a

√

χ2

(n−1)b

√

χ2

(n−1)c d

√

χ2

(n−2)

SCDM 1 0 2.27 0.37 2.85 0.27 1.05 1.82 0.12

OCDM01 0.1 0 2.01 0.57 2.36 0.34 0.94 1.79 0.14

OCDM03 0.3 0 2.33 0.41 2.73 0.29 1.09 1.78 0.13

OCDM05 0.5 0 2.27 0.40 2.81 0.29 1.03 1.84 0.11

ΛCDMO1 0.1 0.5 1.87 0.58 2.28 0.38 0.96 1.72 0.12

ΛCDMO2 0.1 0.7 1.91 0.52 2.47 0.38 1.00 1.74 0.13

ΛCDMF1 0.1 0.9 1.97 0.57 2.39 0.36 0.98 1.79 0.11

ΛCDMO3 0.3 0.5 2.35 0.41 2.72 0.30 1.12 1.76 0.11

ΛCDMF2 0.3 0.7 2.40 0.39 2.76 0.29 1.14 1.76 0.11

ΛCDMC1 0.3 0.9 2.50 0.39 2.72 0.28 1.16 1.76 0.11

ΛCDMF3 0.5 0.5 2.38 0.40 2.75 0.27 1.06 1.82 0.11

ΛCDMC2 0.5 0.7 2.42 0.40 2.76 0.26 1.06 1.84 0.11

ΛCDMC3 0.5 0.9 2.43 0.41 2.74 0.27 1.08 1.82 0.11

Table 5.2 — Scaling relations parameters: inferred parameters for the Kormendy relation, the Faber-Jackson

relation and Fundamental Plane for the galaxy clusters in each of the simulated cosmological models.

group:

M = npartmpart , (5.17)

where npart is the number of particles in the halo and mpart is the mass of each particle. The

mass of the particle is different for each cosmology.

• Velocity dispersion: computed as

σ2 =2K

npartmpart

, (5.18)

where K is the kinetic energy of the halo.

As a measure for the size of the halos, we have explored two options: the virial radius and the mean

harmonic radius.

• Virial radius: rvir is the size of the entire virialized clump identified by HOP. It does suffer from

several artifacts, such as occasional unbound particles that happen to fly by or bridges between

groups. Also, it suffers from the precise location of the outermost particles.

• Half-mass radius: rhal f is the radius that encloses half of the mass of the clump. This radius is

closest in definition to the half-light radius used in observational studies.

• Mean harmonic radius: rh is defined as the inverse of the mean distance between all pairs of

particles in the halo:1

rh

=1

N

∑

i< j

1

|ri j|, N =

npart(nnpart −1)

2, (5.19)

where ri j is the separation vector between the ith and the jth particle. The great virtue of this

radius is that it is a good measure of the effective radius of the gravitational potential of the

clump, certainly important when assessing the virial status of the clump. Also, it has the practical

advantage of being independent of the definition of the cluster center. To some extent, it is also


Figure 5.2 — Kormendy Relation. Each panel plots the relation between mass M and mean harmonic radius

rh of the cluster-sized dark halos in the simulations corresponding to one particular cosmology. Going from

top left to bottom right these are: SCDM, ΛCDMO2, ΛCDMF2 and ΛCDMC2. In each of the panels we have

superimposed the fitted Kormendy relation, the style of each fit is given in the top left panel.

an indicator of the internal structure of the halo because it put extra weight to close pairs of

particles.

Most of the results presented in this chapter refer to the mean harmonic radius of the halos. We

have also compared the results which follow for the half-mass radii of the halos, and in a few cases we

have also looked at the virial radius. We did find that indeed the mean harmonic radius is physically

a better defined and therefore preferred radius. In Fig. 5.1 we plot the mean harmonic radius vs. the

half-mass radius of the halos in the ΛCDMF2 model. In particular, for small size halos there can

be a considerable difference between the two radii. This reflects strongly in the different Kormendy

relations, but turned out to be less bearing on the inferred Fundamental Plane parameters. Also we

found that the mean harmonic radius relations are more solid, beset by less scatter and noise.

5.3.3 Determination of Scaling Relations

Given the inferred mass M (Eqn. 5.17), velocity dispersion σ (Eqn. 5.18), from here onward we write

σ = σv when writing the velocity dispersion) and the mean harmonic radius rh (Eqn. 5.19) of the

cluster halos, we find the scaling relation parameters by linear fitting of the relations. In order, these


are the Kormendy relation

log M = a logr + Ca . (5.20)

For the Faber-Jackson relation,

log M = b logσ + Cb . (5.21)

And finally, for the Fundamental Plane,

log M = c logr + d logσ +C f p . (5.22)

The perpendicular distance of the cluster halos to the fitted plane we take as a measure for its width

w f p:

wfp =

√

∑

D2⊥

N, (5.23)

where N is the number of cluster halos in the sample and D⊥ is the perpendicular distance of a point

to a plane

D⊥ =c logrh+d logσ+C f p− log M

(c2+d2+1)1/2. (5.24)

5.4 Scaling Relations in Different Cosmologies: z=0

We first investigate our cosmological models at the current epoch, z = 0, for the viability of the scaling

relations of the cluster dark matter halos and for possible systematic differences between the parameter

values and FP width as a function of the cosmology.

The parameters of the resulting linear fits, to be discussed in the ensuing subsections, are listed in Table

5.2.

5.4.1 Kormendy Relation

Fig. 5.2 shows the relation between the mass M of each cluster halo and their mean harmonic radius

rh. Each of the four panels depicts the relation for the halos in one particular simulated cosmology.

The top left panel concerns the SCDM cosmology, the top right one, the ΛCDMO2, the bottom left

one, the ΛCDMF2 and the bottom right one, the ΛCDMC2.

A visual comparison between the SCDM (top left panel) and the ΛCDMF2 (bottom left panel)

also shows that clusters of comparable mass have a larger size in the low Ωm cosmology than in the

ones with a higher density value. In other words, clusters are more compact in the SCDM cosmology.

Perhaps not unexpected, in higher Ωm models we find objects of a higher density.

Fig. 5.2 clearly shows that in each cosmology there is a strong and systematic almost linear relation

between log M and logrh: the Kormendy relation appears to be a good description for all situations.

There is a difference in the width of this relation: the spread in the SCDM cosmology is far more

substantial than that in the case of the other cosmologies, with the possible exception of the closed

ΛCDMC2 cosmology. This hints at a systematic trend as a function of the cosmological density

parameter Ωm. Along with this, we discern a widening of the relation as we go to smaller masses.

When fitting the plotted point distributions, we infer the parameter values listed in Table 5.2. In

each of the panels in Fig. 5.2 we plotted the linear fits for all of the four depicted cosmologies. We

can immediately see systematic differences. While the high density cosmologies – SCDM, ΛCDMF2

and ΛCDMC2 – have slopes in the order of 2.4, the low density ΛCDMO2 has a considerable flatter

Kormendy relation with a in the order a∼ 1.9. This we find to be the case for allΩm = 0.1 cosmologies.

This seems to imply that the mean density 〈ρ(rh)〉 ∝ r−0.6h

in the higher density Universes: larger halos

have a proportionally lower average density. The contrast is much stronger in the open model, where

〈ρ(rh)〉 ∝ r−1.1h

. This indicates a strong dependence of halo concentration as a function of its mass (also

see chapter 3). If this is indeed so, it is a strong evidence of weak, instead of a perfect, homology

5.4. SCALING RELATIONS IN DIFFERENT COSMOLOGIES: Z=0 97

amongst the halos in all cosmologies. This might be in line with some of the findings of Lanzoni et al.

(2004).

We investigated the dependence of the Kormendy parameter a on the cosmology. In Fig. 5.3 we

have plotted a as a function of the average mass density parameter Ωm (top left panel), as a function of

the cosmological constant ΩΛ (top right panel) and as a function of the cosmic curvature, in terms of

Ωtotal = Ωm+ΩΛ.

The one outstanding influence is the dependence of a on Ωm. We find the marked difference in a

for all low density Universes. There seems to be a significant change going fromΩm = 0.1 toΩm = 0.3.

It is likely that this is a consequence of the fact that nearly all existent structures and objects in these

Universes did form at very high redshift. As a result, nearly all objects have experienced a very

quiescent life since these early epochs. In an ongoing study we are investigating this in detail.

There is no evidence for any systematic trends of the Kormendy parameter as a function of the

value of the cosmological constant. This is a good argument for the absence of any influence of ΩΛon the internal structure of the halos. The picture is somewhat confusing for the impact of cosmic

curvature and we do not feel enabled to draw any firm conclusions with respect to its influence.

5.4.2 Faber-Jackson Relation

Fig. 5.4 shows the Faber-Jackson relation, the relation between the mass M and the velocity dispersion

σ of the cluster halos. Like in Fig. 5.2, each of the four panels corresponds to one particular simulated

cosmology. And as in Fig. 5.2, these are the SCDM cosmology (top left panel), the ΛCDMO2 (top

right panel), the ΛCDMF2 (bottom left panel) and the ΛCDMC2 (bottom right panel).

For comparison, in each of the panels we show the four lines corresponding to the linear fits of

this relation in each of the depicted four cosmologies. The M−σ relation is clearly well fitted by the

Faber-Jackson like relation. It is considerably tighter than the equivalent Kormendy relation. Also the

spread around the relation seems to be less dependent on the mass of the halo.

Figure 5.3 — Inferred parameters for the Kormendy relation in the cosmological models as a function of Ωm

(top left panel), ΩΛ (top right panel) and Ωm +ΩΛ (bottom panel).


Also interesting is to note that the inferred FJ relation does not vary significantly as a function of

the underlying cosmology: the slope b in all cases is in the order of ∼ 2.75 (see Table 5.2). As may be

inferred from Fig. 5.5, the only possible deviant cases are the low Ωm Universes. However, even this

is hardly significant while any dependence on ΩΛ or Ωtotal can be immediately discarded.

A likely explanation for the fact that b ∼ 2.75 in most situations, instead of the expected value

of b = 3 for virialized perfectly homologous systems (see Eqn. 5.15), is that halos do form a weakly

homologous population along the lines described in, e.g., Bertin et al. (2002).

5.4.3 Fundamental Plane

The Kormendy relation and the Faber-Jackson relation are two dimensional projections of an intrinsi-

cally three dimensional relation between mass M, size rh and velocity dispersion σ of the halos. By

implication, the spread of the Fundamental Plane relation should be less than that of each of the two

previous relations.

We have illustrated the Fundamental Plane relation in Fig. 5.6, for the same cosmologies as in

Fig. 5.2 and 5.4 (SCDM, ΛCDMO2, ΛCDMF2 and ΛCDMC2 going from top left to bottom right

panel). In each of the frames we have plotted the mass M of the halos against the quantity rchσd. The

Figure 5.4 — Faber-Jackson relation. Each panel plots the relation between mass M and the velocity dispersion

σ of the cluster-sized dark halos in the simulations corresponding to one particular cosmology. Going from top

left to bottom right these are: SCDM, ΛCDMO2, ΛCDMF2 and ΛCDMC2. In each of the panels we have

superimposed the fitted Faber-Jackson relation, the style of each fit is given in the top left panel.

5.4. SCALING RELATIONS IN DIFFERENT COSMOLOGIES: Z=0 99

parameters c and d in the latter quantity, combining velocity dispersion σ and mean harmonic radius

rh of each halo, are the best fit FP parameters for the corresponding cosmology.

The galaxy clusters in each cosmology do indeed seem to populate a tightly defined plane. The

point clouds in each of the frames confirm our expectation that they do have a much lower scatter

around the plane than in the case of the Kormendy and Faber-Jackson relation.

From Table 5.2 we find a surprising level of consistency between the Fundamental Planes that we

find in each of the cosmologies. We find that the inferred parameters are close to the one theoretically

expected for perfectly homologous virialized clusters halos, M ∝ rhσ2. In particular, the scaling pa-

rameter c for the radius rh is very close to unity. The difference is somewhat larger for the parameter

d, showing that the velocity dispersion scaling has a difference of ∼ 0.15− 0.25 from the theoretical

value of 2.

As can be seen in both Table 5.2 and Fig. 5.6, there is hardly any variation between the FP relations

in each of the cosmologies: they almost all coincide. This is indeed true when it concerns the FP

parameters c and d. The two top panels of Fig. 5.7 do confirm the impression that there is no systematic

difference as a function of Ωm and/or ΩΛ. This in itself is a strong argument against differences in the

scaling relations parameters being due to a partial or incomplete level of virialization, as was claimed

by Adami et al. (1998).

One outstanding difference in Fundamental Plane between the different cosmologies concerns its

width. Inspection of Fig. 5.6 does suggest a somewhat larger width of the FP for high density Uni-

verses. This turns out to be a very systematic effect: the lower left hand panel of Fig. 5.7 shows a

nearly linear increase of the FP width with Ωm, in combination with a near independence of cosmo-

logical constant ΩΛ. We may try to understand this difference in terms of the ongoing evolution of

the cluster population. In low Ωm Universes all clusters formed at high redshift and have since had

ample time to reach full virialization. In high Ωm Universes, clusters still undergo a substantial levels

of merging and accretion, both of which may affect the virial state of the cluster. We investigate this

question in more detail in section 5.6.

Figure 5.5 — Inferred parameters for the FJ relation according to three different parameters: Ωm (top-left panel),

ΩΛ (top-right panel) and Ωm +ΩΛ (bottom panel).


Figure 5.6 — Fundamental Plane. Each panel plots the relation between mass M and the quantity rchσd , the

combination of mean harmonic radius rh and velocity dispersion σ, with rh to the power c of the FP linear fit

and σ to the power d of the FP linear fit. The dots in the top left panel represent the cluster halos in the SCDM

simulation, in the top right panel the ones in the ΛCDMO2 simulation, in the bottom left panel the ones in the

ΛCDMF2 simulation while the bottom right panel contains the ones in the ΛCDMC2. The superimposed lines

in each panel represent the fitted Fundamental Plane for the four different cosmologies (see insert top left panel).

Note that by definition each of these lines has slope unity and only differs in amplitude.

Finally, if we relate the Fundamental Plane M − rh −σ of our simulated cluster samples to the

observationally measure one L−Re −σ, e.g. L ∝ R0.90e σ1.31

v found by Lanzoni et al. (2004), we may

try to see whether the difference can be solely ascribed to a mass dependent mass-to-light ratio M/L.

We do find a significant dependence in that more bright clusters would have a larger M/L. However,

in each of our cosmologies it is not possible to get an agreement only on the basis of a mass dependent

M/L. Although our results suggest that M/L ∼ Lα with α ∼ 0.2− 0.4, other factors of structural or

dynamical origin do seem to be of importance.

5.4.4 Halo size: radius definition and scaling relation

Apart from the mean harmonic radius that we have used as a measure of halo size in the previous sec-

tions, we have also assessed the viability of the scaling relations in case of alternative size definitions.

In Table 5.3 we list the resulting parameters for the Kormendy relation and the Fundamental Plane in

the case of using 1) half-mass radius rhal f and 2) virial radius rvir.

The parameters for the Kormendy relation and the Fundamental Plane are significantly different

from the ones inferred on the basis of the mean harmonic radius. This is true for both half-mass

5.5. EVOLUTION OF SCALING RELATIONS 101

radius and virial radius. The difference is most stark for the low Ωm ΛCDMO2 cosmology. Note

the substantial increase in the width of the Fundamental Plane for the SCDM cosmology. The other

cosmologies do not seem to yield substantially wider FP relationships. The Kormendy relation also

does not appear to widen.

Interestingly, the differences nearly all concern the scaling parameter for the size while the scaling

for the velocity dispersion σ in the Fundamental Plane remains well behaved, and seems to yield a

value d closer to the ideal virial value d = 2.

The change in scaling parameter values may be ascribed to the use of quantities that probe different

aspect of the structure and dynamics of the halos. In an extreme situation, this might have disrupted

the scaling relations. However, our finding shows that scaling relations do still hold but in a slightly

different disguise. It may be an indication for our contention that halos do not form a perfectly homol-

ogous population. Size measures sensitive to different aspects of the halos’ internal mass distribution

may then result in somewhat different scaling properties. In this respect, we agree with the conclusions

of Adami et al. (1998) and Lanzoni et al. (2004).

5.5 Evolution of Scaling Relations

In the previous sections we have extensively studied the scaling relations at the current cosmic epoch

z= 0. We have also noted that there are differences between the scaling relation parameters that we find

in our simulations and those for perfect virialized and homologous systems. This makes it interesting

to trace the evolution of the different scaling relations.

In this section we have investigated the evolution of the scaling relations as a function of redshift

and as a function of cosmic time. Cosmic time is the time that has passed since the Big Bang. While

observers usually think in terms of redshift, it is good to realize that a given redshift corresponds to

an entirely different dynamical epoch in different cosmologies. Given the same Hubble parameter, the

Figure 5.7 — Top panels: inferred parameters c and d of the Fundamental Plane relation as a function of Ωm

(top left) and as ΩΛ (top right). Bottom panels: rms scatter of the FP relation as a function of Ωm (bottom left)

and as ΩΛ (bottom right).


Figure 5.8 — Top: Kormendy relation of dark halos in a SCDM (left) and ΛCDMO2 (right) cosmology, using

the virial radius rvir (upper row) and the half-mass radius rhal f (lower row). Each of the panels plots mass versus

radius of the halos. The lines represent the best fit relations: SCDM, solid line, and ΛCDMO2, dashed line.

Bottom: Fundamental Plane relation of dark halos in the SCDM (left) and ΛCDMO2 (right) cosmology, using

the virial radius rvir (upper row) and the half-mass radius rhal f (lower row). Plotted are mass M versus the FP

quantity rcσd . The lines represent the best fit FP relations.


age of the Universe is a sensitive function of the cosmic density parameter Ωm and even more so of

the cosmological constant. As for the latter, we have to realize that the change in cosmic time as a

function of the cosmological constant is the most important influence of Λ. To give an impression

of the differences in cosmic time for a given redshift between the different cosmologies, we refer to

Table 5.4.

We have probed the scaling relations over a range of redshifts from z = 4 to z = 0 and over a range

of cosmic time going from 1−10 Gyr. The evolution of the obtained scaling parameters as a function

of redshift is shown in the left column of Fig. 5.10. The corresponding evolution as a function of

cosmic time can be found in the right hand column. The Kormendy parameter a is shown in the top

panels, the Faber-Jackson parameter b in the center panels and the FP parameters c and d in the bottom

panels. Each different cosmology is represented by a different linestyle, listed in the insert of the top

left hand frame.

The Kormendy relation clearly evolves as a function of redshift, at least for the three high density

cosmologies. Only in the case of the low Ωm ΛCDMO2 we can not discern any significant increase

of a, partially due to the large uncertainties in the calculated parameter as a result of the low number

of halos in this simulation. The evolution of the Kormendy parameter in the other cosmologies is

considerably more interesting. Up to a redshift of z ∼ 1, we find a continuous increase of a towards a

value of a ∼ 2.5. At later times the impression is one of a mild decline in the value of a in the case of

SCDM. In the ΛCDMF2 and ΛCDMC2 the Kormendy parameter seems to remain constant. This is

particularly clear when assessing the evolution in terms of the cosmic time, as can be seen in the top

right panel.

Evolutionary trends for the Faber-Jackson relation are comparable to that seen in the Kormendy

relation. No discernible trends are found in the open cosmology, while all of the other high density

Universes do show a uniform increase over the full redshift range 4 to 0. At the most recent redshifts

b reaches a value in the range b > 2.5, from a value in the order of b ∼ 2 at higher redshifts. There

is an indication that the increase of b is slowing down at more recent epochs. This is the impression

obtained when looking at the evolution of b as a function of cosmic time (see center right panel).

Interestingly, there is absolutely no trend whatsoever in the values of the Fundamental Plane pa-

rameters. When it comes to the evolution of the Fundamental Plane, it mainly or even exclusively

concerns the width of the fitted plane. In Fig. 5.9 we show the development of the FP width as a

function of cosmic expansion factor a(t) = 1/(1+ z). We see a systematic increase of FP width over

the whole cosmic evolution in the case of the high Ωm SCDM cosmology. While we do see a rise of

the FP width before a < 0.5 in the ΛCDMF2 and ΛCDMC2 cosmologies, after that time this increase

M ∝ ra M ∝ rcσd

Model Ωm ΩΛ Radius a

√

χ2

(n−1)c d

√

χ2

(n−2)

Half-mass 1.88 0.52 0.72 2.13 0.16SCDM 1 0 Virial 2.09 0.40 0.80 1.89 0.25

Half-mass 1.49 0.69 0.69 2.15 0.19ΛCDMO2 0.1 0.7 Virial 1.56 0.62 0.62 1.72 0.37

Half-mass 2.04 0.53 0.82 2.06 0.16ΛCDMF2 0.3 0.7 Virial 2.21 0.60 0.76 2.01 0.27

Half-mass 2.00 0.54 0.75 2.13 0.14ΛCDMF2 0.5 0.9 Virial 2.23 0.46 0.71 2.01 0.27

Table 5.3 — Scaling relations parameters: inferred parameters for the Kormendy relation and the Fundamental

Plane relation for the galaxy clusters when considering half-mass radius and virial radius.


z

Cosmic Time SCDM ΛCDMO2 ΛCDMF2 ΛCDMC2

2.36 1.49 4 2.71 2.21

3.26 1.01 2.92 1.98 1.60

4.06 0.74 2.35 1.56 1.24

5.07 0.50 1.83 1.19 0.93

9.31 0 0.71 0.38 0.24

Table 5.4 — Cosmic times in Gyr and its corresponding redshift for each cosmological model.

levels off and may even flatten completely. It may be of relevance that these simulations do not attain

sufficient halo mass resolution at higher redshifts: in these cosmologies halos still are low mass objects

at these epochs. The one outstanding cosmology is that of the low Ωm Universe ΛCDMO2. Except for

a rather abrupt and sudden jump in FP width at a ∼ 0.3, there is no noticeable change at later epochs.

By a = 0.3 nearly all its clusters are in place and define a Fundamental Plane that does not undergo any

further evolution.

On the basis of their study of galaxy merging, Nipoti et al. (2003) argued that the origin of the

Fundamental Plane is more due to a structural non-homology than a dynamical non-homology. Our

finding that the development of the Kormendy and FJ relations somehow appear to compensate each

other in reproducing the same Fundamental Plane at every redshift, but with a stronger and more sys-

tematic evolution of the Kormendy relation, seems to point at the same conclusion. The systematic

evolution of the Kormendy and the Faber-Jackson relation that we find, in combination with the un-

changing character of the Fundamental Plane certainly hints at the role of the structural and dynamical

properties of halos in defining the Fundamental Plane.

The latter clearly indicates that although the halo population remains solidly within the Fundamen-

tal Plane, its location within the plane clearly shifts as time proceeds. Apparently the evolution of halos

consists of a gradual shift along an almost universal Fundamental Plane. This is clearly illustrated in

Figure 5.9 — Evolution of the width of the Fundamental Plane for four different cosmologies. Note the almost

consistently tighter FP for the low Ωm Universe and the modest decrease of FP width in the other cosmologies.


Figure 5.10 — Evolution of the obtained scaling relations parameters as a function of redshift (left column) and

as a function of cosmic time (right column). Top: Kormendy parameter a. Center: Faber-Jackson parameter b.

Bottom: FP parameters c and d.


fig. 5.11. It shows that the halo population seems to evolve from a rather scattered and loose one into

a tightly elongated point cloud at later epochs, providing interesting clues towards understanding the

cluster virialization process.

5.6 Merging and accretion dependence

In the above, we have studied the scaling relations as a manifestation of the virial state of the cluster

halos in each of the cosmologies. In the real world of hierarchical structure formation scenarios the

formation and evolution of halos is hardly a quiescent and steadily progressing affair. Instead, halos

grow in mass by steady accretion of matter from its surrounding as well through the merging with

massive peers. Even the accretion is not a continuous and spherically symmetric process: most matter

flows in in a strongly anisotropic fashion through filamentary extensions into the neighboring large

scale matter distribution. As a result, we do expect that numerous halos will not have settled in a

Figure 5.11 — Shifting location of the cluster halo population within the Fundamental Plane. The depicted

halo sample is the one in ΛCDMF2 cosmology, and is shown at four different redshifts: z = 2.61 (top left panel),

z = 1.61 (top right panel), z = 0.89 (bottom left panel) and z = 0 (bottom right panel). The abscissa and ordinate

axis are arbitrarily chosen axes within the FP plane log M− logrh − logσ.

5.6. MERGING AND ACCRETION DEPENDENCE 107

Figure 5.12 — Width of the Fundamental Plane when considering accretion (dotted lines) or mergers (solid

lines).

perfect virial state. This will certainly be the case when it recently suffered a major merger with one

or more neighboring clumps.

The detailed accretion and merging history is a function of the underlying cosmology. Low density

cosmologies or cosmologies with a high cosmological constant will have frozen their structure for-

mation at early epochs. The halos that had formed by the time of that transition will have had ample

time to settle into a perfect virialized object. Also, there is a dependence on the power spectrum of

the corresponding structure formation scenario. Power spectra with a slope n < −1.5 (at cluster scales)

will imply a more homologous collapse of the cluster sized clumps, less marked by an incessant bom-

bardment by smaller clumps. It may be clear that a more violent life history of a halo will usually be

reflected in a substantial deviation from a perfect virial state.

In order to investigate the implications of a difference in accretion or merging history of halos,

we have splitted the samples of cluster halos in each of our cosmologies into a merging sample and

a accretion sample. The mergers were those halos which suffered a merger with another halo that

contained at least 30% of its mass. Possible differences in virial state should be reflected in the quality

of the scaling relations, in particular that of the width of the Fundamental Plane.

In Fig. 5.12 we show the evolution of the width of the Fundamental Plane for each of the two

samples in the four indicated cosmologies. Note that our simulations do not have sufficient resolution

for reconstructing the precise merging or accretion history before a = 0.3− 0.4, so that we may not

draw conclusions on the rise of the FP width up to that epoch.


In the more recent history we do find some significant differences between merging and accretion-

only halos in the different cosmologies. In the case of the ΛCDMO2 scenario we do not have enough

cluster halos to detect any systematics differences between the merging and accreting halos. There

does not seem to be a systematic difference between these groups in the ΛCDMF2 cosmology. The

story is quite different for the ΛCDMC2 and SCDM cosmology. While the cluster halos that undergo

a major merger do reveal a constantly growing FP width, their accretion-only clusters do not display

such a systematic increase. Instead, their FP width remains lower and levels off. In other words,

accretion halos (dotted lines) do on average display a tighter FP relation. This is particularly true at

the current epoch. Apparently, the absence of violent mass gain in the case of accretion halos implies

them to have more time to relax and virialize.

5.7 Conclusions

We have studied three structural scaling relations of galaxy clusters in thirteen cosmological models.

These relations are the Kormendy relation, the Faber-Jackson relation and the Fundamental Plane.

Their validity and behavior in the different cosmological models should provide information on the

general virial status of the cluster halo population. The cosmological models that we studied involved

a set of open, flat and closed Universes with a range of matter density parameter Ωm and cosmological

constant ΩΛ.

The cluster samples are obtained from a set of corresponding N-body simulations in each of the

cosmologies. These simulations concerns a box of 200h−1Mpc with 2563 dark matter particles. The

initial conditions were set up such that the phases of the Fourier components of the primordial density

field are the same for all simulations. In this way, we have simulations of a comparable morphological

character: the same objects can be recognized in each of the different simulations (be it at a different

stage of development).

After running the simulations from z = 4 to the current epoch, with the help of the GADGET2

code, we used HOP to identify the cluster halos. Of each halo population we investigated whether

it obeyed a mass-radius relation akin to the Kormendy relation, a mass-velocity dispersion relation

similar to the Faber-Jackson relation and a two parameter family between mass, radius and velocity

dispersion that resembles a Fundamental Plane relation. We studied the dependence of the obtained

scaling parameters as a function of the underlying cosmology and investigated their evolution in time.

Our results can be summarized as follows:

• In each cosmological model we do recover Kormendy, Faber-Jackson and Fundamental Plane

relations for the population of cluster halos. This is a strong indications for the virialized state

of the halos, as we do expect in hierarchical clustering scenarios.

• There are significant differences between the measured parameters of the various scaling rela-

tions and those seen in the observational reality. Given the different behavior of the Kormendy

relation, Faber-Jackson relation and Fundamental Plane relation, we tend to agree with claims

that these differences are not only due to simple difference in mass-to-light ratio M/L but that

we are dealing with a weakly homologous population of objects. This maybe more due to a

structural non-homology to a dynamical non-homology.

• We find that the parameters of the Kormendy and Faber-Jackson relations are mildly sensitive to

the value ofΩm: a and b are somewhat larger in high density Universes. By far, the largest impact

is that of Ωm on the width of the Fundamental Plane. The width is almost directly proportional

to the value of Ωm. There is no indication for any influence of ΩΛ on the scaling relations.

• The Fundamental Plane parameters do not show any sign of evolution: the scaling parameters

c and d remain constant over the investigated redshift range. However, the width of the Funda-

mental Plane does evolve significantly. In all high density Universes we see a uniform but mild

5.7. CONCLUSIONS 109

decrease of width as time proceeds. This reflects the gradually virializing tendency of the cluster

population.

• With the exception of low Ωm Universes, we do find a systematic increase of the Kormendy and

Faber-Jackson parameters a and b from z = 4 up to z = 1. From z = 1 to the present epoch, the

Faber-Jackson parameter b continues to grow, while the situation is less clear for a. In a SCDM

cosmology a shows a slight decrease, while in the other models seems to remain constant.

• The combination of evolution of the Kormendy and Faber-Jackson with the apparent universal

nature of the Fundamental Plane reveals that the evolution of the cluster population is constrained

to paths within the FP.

• Given our expectation that there is a difference in virial state between quiescently accreting

clusters and those experiencing massive mergers, we have investigated the evolution of the Fun-

damental Plane width for samples of merging clusters and samples of accreting clusters. We

find that accreting clusters at recent epochs do appear to be better virialized than the merging

population in that the FP width is smaller in the former.


6Galaxy Clusters into the Future †

We explore the effects of future evolution on several properties of galaxy cluster halos in cosmo-

logical simulations. These properties include the morphology, angular momentum, virialization

and scaling relations. We also explore some global properties, such as the mass function and the mass

accretion history. The simulations span a wide range of cosmological parameters, representing open,

flat and closed Universes. The simulations are run into the far future when the Universe is 60-70 Gyrs

old. This timespan is long enough for halos to essentially reach dynamical equilibrium. We find that

the there is no significant increase of low mass halos, while large mass objects continue to evolve until

a f ∼ 1.85. As a consequence of the latter, the mass accretion history of cluster halos does not show

any significant difference between cosmologies. We also find that in the far future, halos will become

nearly spherical a a consequence of virialization. Also, the evolution of the angular momentum is

constant. As a consequence, the spin parameter shows that halos will slow down. In contrast with

a f = 1, halos in the far future will become more virialized, with a virial state |U | ∼ 1.8− 1.9K. This

is reflected in the scaling relations of galaxy cluster halos. The Kormendy and the Faber-Jackson are

closer to the theoretical relations. The Fundamental Plane shows that in the far future, all galaxy clus-

ter halos will be virialized, an aspect that is reflected in its width: it is similar in every cosmology. In

the far future, it will become increasingly difficult to discriminate in which type of Universe we live

in.

†Pablo A. Araya-Melo, Bernard J.T. Jones & Rien van de Weygaert , 2008, in preparation.

112 Chapter 6: Galaxy Clusters into the Future

6.1 Introduction

The existence (or lack) of a cosmological constant has profound consequences in the future evolution

of the Universe. Although there seems to be an agreement in the range of values that both Ωm and ΩΛcan have in order to match observations of high-redshift objects such as quasars and be in accordance

with the age of globular clusters, there has not been a detailed study of how the interplay between

these two cosmological parameters affects the future evolution of the large scale structure, particularly

galaxy clusters. The effect of a nonzero cosmological constant drives the Universe towards unbounded

exponential expansion (see Carroll et al. 1992, for a detailed review of the cosmological constant),

while a zero cosmological constant, makes the expansion to decelerate. Previous studies (Nagamine

& Loeb 2003; Busha et al. 2003; Dunner et al. 2006; Hoffman et al. 2007, and chapter 7 of this

thesis) have shown that for a ΛCDM Universe with present day cosmological parametersΩm = 0.3 and

ΩΛ = 0.7 structure formation and halo growth come to a rapid end, at approximately a f ∼ 3. At this

time, the cosmological constant is nearly 1. The exponential expansion causes mergers and accretion

to stop and pushes halos further away from each other. From a f ∼ 3 and beyond, no considerable

matter is left on the surrounding of the halos to accrete, and they start to relax towards equilibrium in

effective isolation.

For different cosmological models as the one stated before, the time at which structure formation

and halo growth stops will be different because of the different values of the cosmological constant.

The values of Ωm and ΩΛ have a large effect on the time scales of the Universe. The cosmological

constant either aids or resists the attraction of matter. This can lead to younger or older Universes,

respectively, with the consequent effect on the dynamical evolution of galaxy clusters.In some cases,

the cosmological constant already took over the expansion of the Universe and some structures are

already growing in isolation. In others, it is just beginning. But the final scenario will be the same:

objects grow in complete isolation. In all cases, the internal dynamics of clusters of galaxies will be

affected by this gravitational expansion. However, the question that remains is how they will behave.

In order to study the role of the cosmological constant in the evolution towards the future of galaxy

clusters and how this influences their final dynamical structure, we extract information of the future

gravitational growth of the large scale structure of the Universe and of physical quantities such as

shape, angular momentum, kinetic and potential energy and scaling relations from a set of thirteen

cosmological simulations. Each of these models describe different cosmological models with different

values of curvature. The physical quantities under study will tell us when and how galaxy clusters reach

dynamical equilibrium, and will allow us to determine the importance of the cosmological constant in

the fate of the Universe.

The chapter is organized as follows. In section 6.2 we describe the numerical simulations that we

carried out. Power spectrum evolution is described in 6.3. Mass function towards the future of the

different cosmogonies are studied in section 6.4. The mass accretion history of four galaxy clusters of

one particular cosmology is presented in section 6.5, and then they are used as a base to compare with

the remaining models. Section 6.6 study the shape evolution of this same four models, together with

the evolution of the entire sample. A comparison is made between all thirteen models. Dynamical

quantities are studied in the remaining sections. Section 6.7 look at the angular momentum of galaxy

clusters, while in section 6.8 we present analysis on the virialization. In section 6.9 we study how

the mass, velocity dispersion and radius correlate with the virialization by looking into the scaling

relations of galaxy clusters. Finally, in section 6.10 we present our conclusions.

6.2 Numerical Experiments

6.2.1 The simulations



6.2. NUMERICAL EXPERIMENTS 113










The initial conditions are evolved from a f = 1 until a f = 54, assuming that at late epochs structure

formation will decrease significantly, so no major changes will be seen from then on (see Nagamine

& Loeb (2003)). We use the massive parallel tree code GADGET2 (Springel 2005). The Plummer-

equivalent softening was set at ǫpl = 15h−1kpc in physical units. For each cosmological model we

wrote the output of 14 snapshots, equally spaced in log(a). The only exceptions were the SCDM

and OCDM05 cosmologies. They were stopped at a f ∼ 3.5 because the power at small scales was

increasing dramatically, with the result that the high clustering affected the simulations.










cosmology at a f = 0.

Figure 6.1 — Evolution of a single cluster in the ΛCDMF2 cosmolgy in comoving (upper panels) and physical

(lower panels) coordinates. Box sizes are of 14×14Mpch−1.




Model Ωm ΩΛ Ωk Age at a f = 1 Age at a f = 54 mdm mcut

SCDM 1.0 0 0 9.31 58.76 13.23 1323

OCDM01 0.1 0 0.9 12.55 791.46 1.32 132

OCDM03 0.3 0 0.7 11.30 884.81 3.97 397

OCDM05 0.5 0 0.5 10.53 49.571 6.62 662

ΛCDMO1 0.1 0.5 0.4 14.65 90.01 1.32 132

ΛCDMO2 0.1 0.7 0.2 15.96 81.21 1.32 132

ΛCDMF1 0.1 0.9 0 17.85 76.36 1.32 132

ΛCDMO3 0.3 0.5 0.2 12.70 88.55 3.97 397

ΛCDMF2 0.3 0.7 0 13.47 79.07 3.97 397

ΛCDMC1 0.3 0.9 -0.2 14.44 73.21 3.97 397

ΛCDMF3 0.5 0.5 0 11.61 87.97 6.62 662

ΛCDMC2 0.5 0.7 -0.2 12.17 78.13 6.62 662

ΛCDMC3 0.5 0.9 -0.4 12.84 71.89 6.62 662



in Gyr since the Big Bang at the present epoch, the age of the Universe in Gyr at a f = 54, the mass per particle

in units of 1010h−1M⊙ and the mass cut of the groups given by HOP in units of 1010h−1M⊙. For the SCDM and

OCDM05, the simulations were stopped at a f ∼ 3.4, hence, the age shown is the correspondent to that expansion

factor.

simulations. As a result, SCDM does not have groups with masses lower than 1013M⊙. We have




this.

Fig. 6.1 shows the evolution at a f = 1, 1.85, 6.3 and a f = 54 of a single cluster in both comoving

(upper panels) and physical (lower panels) coordinates. The difference between both coordinates is

clear: while in physical coordinates it has nearly the same size throughout its history, in comoving

coordinates it shrinks, to the point it is almost invisible. By looking the evolution in physical coordi-

nates, we see that the cluster gains its mass via a mergers process. This can be seen at a f = 1, where it

has many substructure. By a f = 1.85 the merging process is almost finished, and then it starts to relax

(a f = 6.3, until it becomes an isolated compact object (a f = 54).


For each group in every output we compute the mass, the kinetic energy, the potential energy, the

angular momentum and the inertia tensor. All properties are computed using only the particles within

the group, i.e., no post-processing is done to any of the groups. In some situations we limit ourselves

to those dark matter halos that would correspond to rich galaxy clusters. A galaxy cluster is defined to

have a dark matter mass M of M> 1014h−1M⊙.

Given a halo of N particles, we compute the above quantities as follows:

• Mass: the number of halo particles multiplied by the mass per particle present in each halo:

M = N mpart , (6.1)

where mpart is the mass per particle.

• Shape: in order to calculate the shape of the halo, we calculate the inertia tensor using all

6.2. NUMERICAL EXPERIMENTS 115

particles inside the region of interest:

Ii j =∑

xix j . (6.2)

We chose our coordinate system with respect to the center of mass of the halo. By diagonalizing

the matrix, we obtain eigenvalues a1 > a2 > a3. The eigenvalues of the inertia tensor are a

quantitative measure of the degree of symmetry of the distribution of particles. The axis ratios

follow from the ratio of the eigenvalues

b

a=

√

a2

a1,

c

a=

√

a3

a1, (6.3)

with a f > b > c the axes of the object.

• Angular momentum: defined as

J =

N∑

i=0

miri×vi , (6.4)

where ri and vi are the position and velocity of the ith particle with respect to the center of mass

of the group.

It is often useful to define the spin parameter, a dimensionless quantity which relates the angular

momentum and the energy of a group (Peebles 1971),

λ =J√|E|

GM5/2; (6.5)

where J is the angular momentum of the group (see eqn. 6.4), E is the total energy, M its mass

and G is the gravitational constant. Note that its dependence on the total energy of the system is

rather weak. The spin parameter is essentially the ratio of the angular momentum of an object

to that required for rotational support. A value of λ = 0.05, for example, implies very little

systematic rotation and negligible rotational support.

• Kinetic Energy: the total sum of the particle kinetic energies (with respect to the center of the

halo):

K =1

2

N∑

i=1


where vi is the physical velocity of particle i and vcenter the physical velocity of the halos’ center

of mass.

• Potential Energy: defined as

U = −N

∑

i=1

N∑

j=i+1

Gmim j

|ri− r j|, (6.7)

where ri and rj are the locations of particles i and j.

• Mean harmonic radius: rh is defined as the inverse of the mean distance between all pairs of

particles in the halo:1

rh

=1

N

∑

i< j

1

|ri j|, N =

npart(nnpart −1)

2, (6.8)

where ri j is the separation vector between the ith and the jth particle. The great virtue of this

radius is that it is a good measure of the effective radius of the gravitational potential of the

clump, certainly important when assessing the virial status of the clump. Also, it has the practical

advantage of being independent of the definition of the cluster center. To some extent, it is also

an indicator of the internal structure of the halo because it put extra weight to close pairs of

particles.


6.3 Power Spectrum Evolution

Fig. 6.2 shows the evolution of the power spectrum for three cosmological models: ΛCDMO2,

ΛCDMF2 and ΛCDMC2. We also plot the linear power spectrum for comparison. The measured

power spectrum at high-k modes (small length scale) have become strongly nonlinear, while the low-k

modes (large length scale) are still following the linear power spectrum. As explained in chapter 2, the

shape of the (linear) power spectrum depends on the value of Ωm, while the amplitude depends on ΩΛ.

The power spectrum is plotted in comoving space, making that the expansion of the Universe transfers

power to larger scales with no real change in the shape.

For the open model case, we see that there is hardly any growth from a f = 1 to a f = 54, caused by

the freezing of structure in an expanding Universe (it can be seen on both the linear and the measured

spectrum). On the other hand, ΛCDMC2 shows an increase in the growth from a f = 1 to a f = 1.85,

and freezes at late times. ΛCDMF2 is the case in between. The growth from a f = 1 to a f = 1.85 is less

than in the ΛCDMC2 but more than in the ΛCDMO2. These results are in agreement with the ΛCDM

simulations studied by Busha et al. (2007).

The “break” from linearity occurs at different k modes depending on the cosmology. For the

ΛCDMO2 case, it happens at k ∼ 0.1 hMpc−1, while for the ΛCDMF2 and the ΛCDMC2, at k ∼ 0.2

Figure 6.2 — The evolution of the power spectrum for three different cosmological models. Each panels shows

four different times, from bottom up: a f = 1, a f = 1.85, a f = 6.3 and a f = 54.

6.4. MASS FUNCTIONS 117

hMpc−1 and k ∼ 0.3 hMpc−1, respectively. This means that the non linear evolution of the power

spectrum increases the power on small scales more rapidly in the low Ωm Universes than in the high

density cosmologies.

6.4 Mass Functions

The mass function is the number density of objects of a given mass. Fig. 6.3 shows the evolution of

the cumulative number of dark matter halos for the ΛCDMO2, ΛCDMF2 and ΛCDMC2 cosmologies.

The mass functions are shown at expansion factors a f = 1, a f = 1.85, a f = 6.3 and a f = 53.

The first noteable difference is the halt of structure formation after a f = 1.85: the mass function

freezes in every cosmology. In the ΛCDMO2 cosmology we see a small increase of large mass objects

from a f = 1 to a f = 1.85, but for the rest, there is no increase of the mass function. This is tied in with

the fact that in low Ωm, structure growth comes to a halt at high redshift. The increase of large mass

objects between a f = 1 and a f = 1.85 is moderate in the ΛCDMF2 model, while in the ΛCDMC2 it is

higher.

Figure 6.3 — Evolution of the mass function for the ΛCDMO2, ΛCDMF2 and the ΛCDMC2 cosmology at four

different epochs: a f = 1, a f = 1.85, a f = 6.3 and a f = 54.

The Jenkins mass function is derived from the Press-Schechter formalism (see chapter 2). Its form

is given bydnJ

dM= A

ρ

M2

dlnσ(M)

d ln Me(−| lnσ−1+B|)C . (6.9)

where a f = 0.315, B = 0.61 and C = 3.8 are the original values given by Jenkins et al. (2001). a f

sets the overall mass fraction in collapsed objects, eB plays the role of a (linearly evolved) collpase

perturbation threshold (similar to the parameter δc in the Press-Schechter model) and C is a stretch

parameter that provides the correct shape of the mass function (Evrard et al. 2002).

Fig. 6.4 shows the Jenkins mass function (JMF) (Jenkins et al. 2001) as dashed lines at two expan-

sion factors, a f = 1 and a f = 54, for the ΛCDMO2, ΛCDMF2 and the ΛCDMC2 model.

At a f = 1, we use the original parameters of the JMF. At a f = 54 we use the parameters given by

Evrard et al. (2002): a f = 0.199, B= 0.76 and C = 3.90. This parameters were derived for models with

ΩΛ = 1 and/or Ωm = 0. We find that the JMF is consistent at both epochs, although there is a slight

overestimation of large mass halos in the ΛCDMO2 model. At a f = 54, the original values of the JMF

do not manage to reproduce the simulated mass functions.


Figure 6.4 — Mass functions for the ΛCDMO2, ΛCDMF2 and the ΛCDMC2 cosmology at a f = 1 (left panel)

and a f = 54 (right panel). Dashed lines represent the fit of the Jenkins mass function.

Although not shown, we also fitted the Press-Schechter mass function (PSMF) (Press & Schechter

1974) and the Sheth & Tormen mass function (STMF) with published values (Sheth & Tormen 1999).

At a f = 1, both the PSMF and STMF agrees well with the mass functions. At a f = 54, PSMF shows a

slight mass excess for every cosmology. This may be corrected by a precise value of δc. STMF shows

a significant mass excess at this expansion factor.

6.5 Mass Accretion Histories

The freezing of growth that structures in some Universes will suffer in the far future will have an effect

in their mass accretion history (MAH). Structures which are bound at the present cosmic epoch a f = 1

Figure 6.5 — Average mass accretion histories for galaxy cluster halos in three cosmological models: ΛCDMO2

(top frame), ΛCDMF2 (bottom left frame) and ΛCDMC2 (bottom right frame). The shaded are denotes the

corresponding standard deviations.

6.5. MASS ACCRETION HISTORIES 119

will merge, while others will separate from each other and will grow in isolation. To investigate the

MAH of cluster sized halos we construct their merging tree history. We do this by identifying the

particles that belong to cluster halos at a f = 54 and we trace them back in history, checking for the

most massive progenitor.

We determine the general mass accretion history by averaging all individual mass accretion his-

tories of the halos in each cosmological model. The average MAH of these galaxy cluster halos is

shown in Fig. 6.5. A few facts can be immediately inferred. Galaxy clusters in high density Universes

(ΛCDMC2) did absorbe some amount of matter during their evolution from a f = 1 to a f = 54. The

opposite effect can be observed for the low Ωm ΛCDMO2 model: galaxy cluster halos had at a f = 1

∼80% of the mass at a f = 54. We find this behavior in every high density simulated cosmology. The

spread of the MAHs are similar in the depicted cosmologies. Nevertheless, there are hardly any sig-

nificant differences on the mass accretion history between the depicted cosmologies. The impression

is that in the far future it will become increasingly difficult to infer in what cosmoloy we live on.

Figure 6.6 — Evolution of the shapes of objects present in the ΛCDMO2 (top-left panel), ΛCDMF2 (top-right

panel) and ΛCDMC2 model (bottom panel).


6.6 Shapes in the far future

In chapter 4 we studied the shape of dark matter halos as a function of the expansion factor. We found

that, although the halos tend to retain a rather prolate shape, there was a shift towards a more spherical

shape.

In Fig. 6.6 we have plotted the halos in a scatter diagram of axis ratio c/a versus b/a. We have

indicated where one can find prolate, oblate and spherical halos. Instead of plotting individual halo

shapes, we have plotted the contour of the 32% percentile around the average shape of the halos in

a simulation. In each of the panels we plotted these contours for a sequence of timesteps, with the

lightest contour level corresponding to a f = 1 and the darkest one to a f = 54.

The trend found in chapter 4 is confirmed when appreciating Fig. 6.6: halos become more and

more spherical when evolving into the far future. They have had enough time to virialize and relax.

6.7 Angular Momentum and Spin Parameter into the far future

In chapter 4 we studied the angular momemtum of dark matter halos as a function of the expansion

factor for a variety of cosmologies. We found that the angular momentum of halos increases erratically

but steadily, and that it is intimately related to the increasing mass of the halos. It is interesting, then,

to investigate the behavior of the angular momentum in the far future, where halos stop increasing in

mass.

Fig. 6.7 shows the evolution of the median of the angular momentum J as a function of expansion

time. We look at the time evolution in two different ways: as a function of cosmic time t (top panel)

and as a function of expansion factor a f . Unlike the evolution of the angular momentum J from early

times to the present epoch, where we saw a steady increase, we now see that it remains constant from


units of H0, top panel) and as a function of the expansion factor for clusters halos.

6.7. ANGULAR MOMENTUM AND SPIN PARAMETER INTO THE FAR FUTURE 121

Figure 6.8 — Evolution of the spin parameter as a function of the expansion factor of the cluster size halos for

the ΛCDMO2 (solid line), ΛCDMF2 (dotted line) and the ΛCDMC2 (dashed line) cosmologies.

a f = 1 to a f = 54.

The evolution of the spin parameter λ is perhaps more revealing (see Fig. 6.8). In chapter 4 we saw

that it showed a decreasing trend, from λ ∼ 0.06 at early times to λ ∼ 0.04 at present epoch. We now

see that towards the far future it continues to decrease, from λ ∼ 0.04 to λ ∼ 0.03 at a f = 54.

In order to see possible differences between different cosmological models, we fit the lognormal

distribution

p(λ)dλ =1

σλ√

2πexp

−ln2 (λ/λ0)

2σ2λ

dλ

λ, (6.10)

to the sample of cluster halos. In Fig. 6.9 we have plotted the spin parameter distribution of the

cluster halos in the ΛCDMO2, ΛCDMF2 and ΛCDMC2 cosmologies at a f = 54. Superimposed are

the fits following the lognormal distribution in Eqn. 6.10. Clearly, it provides a more than satisfactory

description. This ties in with the findings of chapter 4, where we saw that the lognormal distribution

is independent of cosmology (see also Peirani et al. (2004)). On the basis of the fits of the lognormal

distribution to the λ distribution of the halos in our simulations, we have inferred the values of the

parameters λ0 and σλ. These are shown in Table 6.2. In contrast with the values at a f = 1 (Table 4.4

of chapter 4), we see a slight decrease on the value of λ0. This is due to the slow down on the spin

rotation of cluster halos in the far future.

Model λ0 σλ Model λ0 σλ

SCDM 0.038 0.777 ΛCDMF1 0.020 0.827

OCDM01 0.019 1.123 ΛCDMF2 0.021 0.698

OCDM03 0.020 1.064 ΛCDMF3 0.025 0.753

OCDM05 0.031 0.744 ΛCDMC1 0.021 0.638

ΛCDMO1 0.022 0.763 ΛCDMC2 0.024 0.741

ΛCDMO2 0.025 0.711 ΛCDMC3 0.023 0.721

ΛCDMO3 0.021 0.735

Table 6.2 — Inferred parameters of the lognormal distribution for λ.


Figure 6.9 — The distribution of the spin parameters and its corresponding lognormal distribution for the

ΛCDMO2 (top left panel), ΛCDMF2 (top right panel) and ΛCDMC2 (bottom panel) cosmologies.

6.8 Virialization towards the future

After collapsing, any given halo will virialize. During this processm the internal energy of the dark

matter halo is distributed such that it reaches a perfect equilibrium configuration.

In chapter 3 we investigated the virialization as a function of expansion factor of dark matter halos

until the present epoch a f = 1. We found that there was a considerable amount of unvirialized halos

and also a large number of unbound halos.

Dark matter halos have had sufficient time to evolve from a f = 1 to a f = 54. We wish to see, then,

what would be their virialization stage in the far future.

6.8. VIRIALIZATION TOWARDS THE FUTURE 123

Model µ Model µ

SCDM 1.81 ΛCDMO3 1.86

OCDM01 1.81 ΛCDMF2 1.83

OCDM03 1.80 ΛCDMC1 1.81

OCDM05 1.80 ΛCDMF3 1.86

ΛCDMO1 1.87 ΛCDMC2 1.83

ΛCDMO2 1.84 ΛCDMC3 1.82

ΛCDMF1 1.80

Table 6.3 — Virial ratio µ of the halo sample in each cosmology.

6.8.1 Virial Theorem

Any self gravitating system has a kinetic energy given by:

K =1

2

N∑

i=1


where the sum is over all particle velocities within any given region, and mi is the mass of the i particle.

The gravitational potential energy of the system is given by

U = −N

∑

i=1

N∑

j=i+1

Gmim j

|ri− r j|, (6.12)

where the sum of the distances is over all particles pairs. The system is virialized if it fulfills the

condition

2K +U = 0 . (6.13)

This is known as the Virial Theorem for a perfect isolated self gravitating system.

A good measure for the virial state of an object may be obtained from the virial ratio

V =2K

|U |. (6.14)

For virialized systems,V→ 1. The system is bound if 1 <V < 2, while it is unbound ifV > 2.

We first investigate how the entire sample of dark matter halos fits the relation between potential

and kinetic energy predicted by the virial theorem. To do so, we model the virial relation in the form

of the linear “virial line” |U | −K,

|U | = µK +λ, (6.15)

where a perfect, isolated virialized object, µ = 2 and λ = 0 (see also chapter 3. In order to find µ, we

fitted a line through the point cloud. We fitted the best virial line to the complete set of dark matter

halos present in every simulation. Table 6.3 shows the inferred virial ratios for the complete set of

cosmological simulations. We see that in all the cosmologies the halo sample are quite close to the

virial state. The inferred virial ratio µ is higher than the one at a = 1. However, we do not see any

significant difference between the cosmologies.

In Fig. 6.10 we plot the virial ratio |U |/K as a function of the mass of the halos. The plots are

scatter plots with each point representing a dark matter halo in the simulation. We have superimposed

a density grey scale plot in order to identify the density of the points. The horizontal lines represent

the perfect virial state |U |/K = 2 and the criterion for boundness |U |/K = 1. In each of the three

cosmologies we see that over the entire mass range the majority of halos lie between the virial ratio of

|U |/K ∼ 1.8−1.9, higher than the virial ratio of |U |/K ∼ 1.5−1.6 found at a f = 1 (see chapter 3).


Figure 6.10 — The virial ratio |U |/K as a function of the mass of the halos in our simulated cosmologies. The

three cosmological models represented are: ΛCDMO2 (top left panel), ΛCDMF2 (top right panel) and ΛCDMC2

(bottom panel). The plot are scatter plots, with each point representing a halo in the simulation. The density

of the points in the scatter plot can be inferred from the superimposed density grey scale plot. In each panel we

indicate by means of horizontal lines the perfect virial state |U |/K = 2, and the criterion for a gravitationally bound

configuration |U |/K = 1.

The spread in the case of low mass halos is narrower than the spread seen at a f = 1. It appears that

the amount of highly virialized halos has increased slightly, while those that were not even bound at

a f = 1 became bound and achieve a virialization. An interesting question would be that of following

the history of the halos hat were not bound at a f = 1 and see if they merge to some larger halos or grew

in isolation, with the result of relaxing and virializing. This aspect is subject of an ongoing study.

6.8.2 Virialization of Galaxy Clusters in the Far Future

By studying the virialization of of dark matter halos that we brand as cluster sized halos with a mass

larger than M > 1014h−1M⊙, we do not find any significant difference with respect to the complete

sample of dark matter halos. Galaxy cluster halo also center around the virial ratio |U |/K ∼ 1.8− 1.9

(see Fig. 6.11). We do not find any significant difference between cosmologies. There is also no

significant spread around the virial line. This is a consequence of the high virial state galaxy cluster

halos attain in the far future.

6.9. SCALING RELATIONS IN THE FAR FUTURE 125

Figure 6.11 — The virial ratio |U |/K as a function of the mass of the cluster halos in three cosmologies:

ΛCDMO2 (top left panel), ΛCDMF2 (top right panel) and ΛCDMC2 (bottom panel). The plots are scatter plots,

with each point representing a halo in the simulation. In each panel we indicate by means of horizontal lines the

perfect virial state |U |/K = 2, and the criterion for a gravitationally bound configuration |U |/K = 1.

6.9 Scaling Relations in the Far Future

A complete description of scaling relations can be found in chapter 5. We will derive here the scaling

relations for galaxy cluster halos.

Basing ourselves on the mass M of a given cluster halo, and assuming that the selected objects

have the same average density, we expect an equivalent Kormendy relation given by

M ∝ R3 , (6.16)

where R is the size of the object. Any deviation in the slope may be understood as a dependence of the

mean density 〈ρ(Re)〉 on the size R of the object.

Every selected object have a kinetic and potential energy given by

K =Mσ2

2, U = −GM2

R, (6.17)

where σ = 〈v2〉1/2 is the velocity dispersion. If halos are in virial equilibrium, i.e., 2K +U = 0, then

we can express the former equations as

σ2 =GM

R. (6.18)


M ∝ rah

M ∝ σb M ∝ rchσd

Model Ωm ΩΛ a

√

χ2

(n−1)b

√

χ2

(n−1)c d

√

χ2

(n−2)

SCDM 1 0 2.16 0.23 3.35 0.19 1.01 1.89 0.11

OCDM01 0.1 0 2.47 0.30 3.04 0.15 0.87 2.11 0.02

OCDM03 0.3 0 2.37 0.23 3.19 0.12 0.87 2.12 0.02

OCDM05 0.5 0 2.27 0.29 3.11 0.19 0.92 2.00 0.07

ΛCDMO1 0.1 0.5 2.34 0.22 3.23 0.15 0.98 1.98 0.03

ΛCDMO2 0.1 0.7 2.37 0.20 3.19 0.14 1.00 1.93 0.03

ΛCDMF1 0.1 0.9 2.47 0.22 3.07 0.16 1.06 1.86 0.04

ΛCDMO3 0.3 0.5 2.57 0.21 3.09 0.12 0.95 2.02 0.02

ΛCDMF2 0.3 0.7 2.62 0.19 3.07 0.11 0.97 2.00 0.02

ΛCDMC1 0.3 0.9 2.63 0.18 3.08 0.10 0.97 2.00 0.02

ΛCDMF3 0.5 0.5 2.51 0.21 3.12 0.12 0.95 2.01 0.02

ΛCDMC2 0.5 0.7 2.55 0.19 3.11 0.11 0.97 1.99 0.02

ΛCDMC3 0.5 0.9 2.57 0.18 3.11 0.10 0.98 1.98 0.03

Table 6.4 — Scaling relations parameters: inferred parameters for the Kormendy relation, the Faber-Jackson

relation and Fundamental Plane for the galaxy clusters in each of the simulated cosmological models.

This would imply the following scaling relation between M, R and σ

log M = 2logσ+ logR+C f p , (6.19)

where C f p is a constant. This equation is known as the Fundamental Plane (FP). From Eqns. 6.17 and

6.19 we find the Faber-Jackson relation

M ∝ σ3 , (6.20)

6.9.1 Kormendy Relation

Fig. 6.12 shows the relation between the mass M of each cluster halo and their mean harmonic radius

rh. Each of the three panels depicts the relation for the halos in one particular cosmology: ΛCDMO2

(top left panel), ΛCDMF2 (top right panel) and ΛCDMC2 (bottom panel).

A visual comparison shows that halos of comparable masses have more or less the same mean

harmonic radius in every cosmology. In other words, in the far future (a f = 54) cluster halos have the

same compactness independent of cosmology.

The Kormendy relation appears to be a good description for the M − rh relation. Also interesting

to note is that the M− rh relation is much tighter at a f = 54 than at a f = 1. The spread of the relation is

similar in every model, thus looking for a systematic trend as a function of either Ωm or ΩΛ is difficult.

The linear fits for three cosmologies are also plotted in Fig. 6.12. The inferred parameters are listed

in Table 6.4. Unlike the situation at a f = 1 (see chapter 5), we do not see any clear differences between

the cosmologies. High density Universes (ΛCDMF2 and ΛCDMC2) have slopes in the order of ∼ 2.6,

while the ΛCDMO2 has a slope in the order of ∼ 2.4. This seems to imply that the dependence of

halo concentration in halos with Ωm = 0.1 is less dependent of its mass, as it was the case at a f = 1.

We also see that the inferred parameters are far from the perfect theoretical relation of M ∝ r3h. In

order to investigate this, we checked the Kormendy relation with the virial radius (the size of the entire

virialized halo), and found that the Kormendy parameter a is higher than the inferred parameter using

the mean harmonic radius, in contrast with the findings of chapter 5. This tells us that, in the far future,


Figure 6.12 — Kormendy relation. Each frame plots the relation between mass M and mean harmonic radius

rh of the cluster halos in the simulations corresponding to one particular cosmology. From top left to bottom

these are: ΛCDMO2, ΛCDMF2, ΛCDMC2. In each of these frames we have superimposed the fitted Kormendy

relation. The linestyle of each fit is given in the top left frame.

HOP is identifying halos that are more spherical than in the present epoch due to their isolation and

relaxation. The fact that the Kormendy parameter is lower when using the mean harmonic radius is

just a reflection of the weak homology cluster halos present. If they were perfect homologous, the

Kormendy relation will be the same independent of the radius.

We do not find any systematic trend on the influence of Ωm, ΩΛ or Ωtotal on a.

6.9.2 Faber-Jackson Relation

Fig. 6.13 shows the Faber-Jackson relation between the mass M and the velocity dispersion σ of the

cluster halos. Each panel correspond to one particular cosmology: ΛCDMO2 cosmology (top left

panel), ΛCDMF2 cosmology (top right panel) and ΛCDMC2 cosmology (bottom panel).

For comparison, in each of the panels we show the three lines corresponding to the linear fits of

this relation in each of the three depicted cosmologies. The M−σ relation is well fitted by the Faber-

Jackson relation and it is tighter than in the cosmic epoch a f = 1 (see Fig. 5.4 of chapter 5). It is also

close to the theoretical relation M ∝ σ3 and somewhat slightly tighter than the Kormendy relation.

The inferred FJ relation does not vary significantly as a function of the underlying cosmology.

However, there is an evolution from a f = 1 to a f = 54. While at a f = 1 the slope b was in all cases in

the order of ∼ 2.75, at a f = 54 it is of the order of ∼ 3.10 (with the exception of the SCDM cosmology,


Figure 6.13 — Faber-Jackson relation. Each frame plots the relation between mass M and velocity dispersion

σ of the cluster halos in the simulations corresponding to one particular cosmology. From top left to bottom

these are: ΛCDMO2, ΛCDMF2, ΛCDMC2. In each of these frames we have superimposed the fitted Kormendy

relation. The linestyle of each fit is given in the top left frame.

see Table 6.4). There is no clear dependence on Ωm, neither on ΩΛ nor on Ωtotal. In other words,

galaxy cluster halos have evolved from a f = 1 to a f = 54 achieving a high degree of virialization.

As with the Kormendy relation, we do not find any influence of the cosmological parameters on

the Faber-Jackson parameter b at a = 54.

6.9.3 Fundamental Plane

Fig. 6.14 shows the Fundamental Plane relation for the same three cosmologies: ΛCDMO2 (top left

panel), ΛCDMF2 (top right panel) and ΛCDMC2 (bottom panel). In each of the panels we plotted the

mass M of the halos against the quantity rchσd.

As with the Kormendy and Faber-Jackson relation, the Fundamental Plane relation at a f = 54 is

tighter than at a f = 1. The inferred parameters c and d are close to the ones theoretically expected

for perfect virialized halos, c = 1 and d = 2. This just confirms what it was found in chapter 5: the

Fundamental Plane is almost independent of cosmology and of expansion factor.

The noteable difference is with respect to the width of the Fundamental Plane. While in chapter 5

we found that the width was depedent of cosmology, i.e., high density Universes having larger width,

at a f = 54 we see that every cosmology have a more or less similar width, with the exception of the

SCDM and OCDM05 models which were stopped at a ∼ 3.4. If at a f = 1 we found that clusters in


Figure 6.14 — Fundamental Plane. Each frame plots the relation between mass M and the quantity rchσd , the

combination of mean harmonic radius rh and velocity dispersion σ, with rh to the power of c and σ to the power

of d of the FP linear fit. The dots in each frame represent the cluster halos in the ΛCDMO2 cosmology (top left

frame), in the ΛCDMF2 cosmology (top right cosmology) and in the ΛCDMC2 cosmology. The superimposed

lines in each frame represent the fitted Fundamental Plane for the four different cosmologies (see insert in the top

left frame).

high density Universes still undergo substantial levels of merging and accretion, they have had the

necessary time to relax and virialized in the far future.

Figure 6.15 — Inferred parameters for the FJ relation according to three different parameters: Ωm (top left

panel), ΩΛ (top right panel) and Ωm +ΩΛ (bottom panel).


6.10 Conclusions

We have evolved thirteen different cosmological simulations into the far future (a f = 54). From these

set of simulations, we have identified the dark matter cluster halos and studied several individual

properties in order the investigate how they compare to the ones at the present cosmic epoch. The

main conclusion are:

• The mass function of every cosmological models freezes after a f = 1.85. This effect is observed

before for low Ωm Universes due to their early formation epoch.

• There is hardly any differences between the mass accretion history of halos in any given cos-

mology. They all show an slight increase of mass between a f = 1 and a f = 3, and after, halos do

not accrete significant mass clumps.

Interestingly, the spread of the MAHs are similar in the depicted cosmologies. The impression

is that in the far future it will become increasingly difficult to infer in what cosmoloy we live on.

• In all cosmologies, halos evolve towards a nearly spherical shape.

• The angular momentum remains constant between the present epoch and a f = 54. This is a

reflection of the mass accretion history: halos do not gain angular momentum due to the absence

of mass gain.

• On average, the spin parameter decreases from λ ∼ 0.04 at a f = 1 to λ ∼ 0.03 at a f = 54. Dark

matter halos slow down due to their isolation.

• Halos show a high degree of virialization, with a relation |U | ∼ 1.8 − 1.9K, higher than the

relation found at a f = 1 (|U | ∼ 1.5−1.6K), but still not a perfect virial state |U | = 2K.

• In each cosmological model we recover the Kormendy, Faber-Jackson and Fundamental Plane

relation in the far future.

• These relations are much tighter than the ones at a f = 1, showing that in the far future halos are

highly virialized.

• The width of the Fundamental Plane at a f = 54 is thinner than at a f = 1 and nearly the same for

every cosmological model: galaxy cluster halos in every cosmology have had enough time to

virialized and reach dynamical equilibrium.

• Due to the above finding, it will become increasingly difficult in the far future to identify in

which Universe we live in.

7Future Evolution of Superclusters in an

Accelerating Universe†

W contrast the large scale evolution of the population of marginally bound supercluster-like ob-

jects in an accelerating Universe with their continuing internal development while they collapse

gravitationally. We identify these objects in a large ΛCDM cosmological simulation of 5123 particles

in a cube of 500h−1Mpc side length, on the basis of the binding density criterion introduced by Dunner

et al. (2006). We construct the “supercluster” mass function at a = 1 and at a = 100 in order to measure

the accuracy of the criterion. The recovered mass functions are in good agreement with the theoretical

predictions of the Press-Schechter formalism, and the Sheth-Tormen and the Jenkins approximation.

According to these mass functions, we may expect to find up to two Shapley-like superclusters in a

volume comparable to that of the Local Universe (z < 0.1). Our simulations do recover one massive

supercluster with a mass of ∼ 8× 1015h−1M⊙ containing 15 cluster members, slightly larger than the

Shapley supercluster. The three manifestations of the internal evolution which we investigated in some

detail are the shape of the bound objects, their compactness and density profile, and their substructure

in terms of the supercluster multiplicity function. While most superclusters are prolate at a= 1, we find

them to evolve towards a spherical shape at a= 100. We also find that they become highly concentrated

in the far future while their substructure merges and produces one compact massive object.

†Pablo A. Araya-Melo, Andreas Reisenegger, Andres Meza, Rien van de Weygaert, Rolando Dunner & Hernan Quintana,

2008, MNRAS, to be submitted.

132 CHAPTER 7: Future Evolution of Superclusters in an Accelerating Universe

7.1 Introduction

The evidence for an accelerated expansion of the Universe has established the possibility of a ‘dark

energy’ component. In this scenario, the universe is entering into an accelerating phase, which is in-

creasingly dominated by the dark energy. It is assumed that this dark energy, in its simplest form,

behaves like Einstein’s cosmological constant. The possibility of an exponential expansion has moti-

vated the concept of ‘island universes’: the largest bound structures will grow isolated as the expansion

accelerates (Chiueh & He 2002; Nagamine & Loeb 2003; Busha et al. 2003; Dunner et al. 2006). In

periods when the matter density was larger than the dark energy, structures formed by gravitational

instability. Nowadays, the universe is already starting to accelerate, meaning that structure formation

is virtually finished. At this stage, those structures that are much denser than the dark energy are

not affected by the latter and remain bound, separating from each other at an accelerating rate which

prevents them from joining larger structures. Therefore, at the present cosmological time, the largest

bound structures are just forming.

Dunner et al. (2006, hereafter Paper I) presented a criterion to determine the limits of bound struc-

tures, defining superclusters as the biggest gravitationally bound structures that will be able to form.

The criterion defined a density contrast over which a spherical shell will remain bound to a spherically

distributed overdensity. Here we use this criterion to identify superclusters in a large cosmological box

and study their abundance and shape at the present time and in the future. The abundance or mass

function serves as a good indicator of the growth of structure of a cosmological model, while the shape

allows us to study their internal evolution and compare with real data.

In this study we aim at contrasting the large scale evolution of structure in an accelerated Universe

with that of the internal evolution of bound objects. While clusters of galaxies are the most massive and

most recently fully collapsed and virialized structures, superclusters should be seen as the largest bound

but not yet fully evolved objects in our Universe. In the future they may develop into genuine “island

Universes”. We study the mass function of these objects in order to see the largest “island Universes”

that will form. On the other hand, we study three aspects of the continuing internal evolution of the

superclusters. Their shape will be one of the most sensitive probes of their continuing contraction

and collapse. In addition, their collapse will produce a continuous sharpening of their internal mass

distribution and density profile. Finally, we also study their evolving and diminishing substructure, in

terms of the supercluster multiplicity function, as its subclumps merge during the collapse process.

The study of future evolution has already been addressed by different authors Chiueh & He (2002)

solved the spherical collapse equations numerically, obtaining a theoretical criterion for the mean

density enclosed in the last gravitationally bound shell. Busha et al. (2003) followed the internal

evolution of the density and velocity structures of bound objects, while Nagamine & Loeb (2003)

focused on the evolution of the Local Universe. Dunner et al. (2007) used the criterion of Paper I

to study the limits of bound structures in redshift-space. Hoffman et al. (2007) repeated the study of

Nagamine & Loeb (2003), adding the dependence on dark matter and dark energy to the fate of the

Local Universe. Later, Busha et al. (2007) studied the effects of small-scale structure on the formation

of dark matter halos in two different cosmologies. Of the previous works, Nagamine & Loeb (2003),

Hoffman et al. (2007) and Busha et al. (2007) studied the mass functions of objects present in their

simulation. They all found that after the present time, the mass function hardly changes.

The definition of superclusters have presented a problem because they are not virialized and their

overdensity is very low, making it difficult to impose a limit. Using different definitions, shape of

superclusters have been studied both using real data (e.g. Plionis et al. 1992; Basilakos et al. 2001;

Einasto et al. 2007) and in N-Body simulations (e.g. Basilakos et al. 2006; Wray et al. 2006; Einasto

et al. 2007). The different studies have concluded that superclusters have a prolate shape.

This chapter is organized as follows. In section 7.2, we present a review of the spherical collapse

model and a prescription for deriving an analytical solution for the spherical collapse equations. Sec-

tion 7.3 describes the simulation and the group finder algorithm that we employ when determining the

mass functions. Section 7.4 presents the mass functions of the bound structures at a = 1 and a = 100

and a comparison with the ones obtained by the Press-Schechter formalism and its variants. Shapes of

7.2. SPHERICAL COLLAPSE MODEL 133

the structures are studied in section 7.5. In section 7.6 we look into the mass distribution and density

profiles. Section 7.7 presents the study of the supercluster multiplicity function. Finally, in section 7.8,

we present our discussions.

7.2 Spherical Collapse Model

The spherical collapse model (Gunn & Gott 1972; Lilje & Lahav 1991) describes the evolution of a

spherically symmetric mass density perturbation in an expanding Universe. It is still a powerful tool

for understanding how a spherical patch of homogeneous overdensity forms a bound system through

gravitational instability. Since we assume sphericity, the non-linear dynamics of a collapsing shell is

determined by the mass interior to it.

We present here a short description of the spherical collapse model following Dunner et al. (2006).

Consider a flat Universe with a cosmological constant (Λ). In this model, a given mass shell with

radius r(t) contains a fixed mass M and satisfies the energy equation (Peebles 1984):

E =1

2

(

dr

dt

)2

−GM

r−Λr2

6= constant , (7.1)

where Λ = 3H2ΩΛ, r is the shell’s radius, M is the total mass enclosed and the constant E is the total

energy per unit mass of the shell. Introducing the following dimensionless variables:

r =

(

Λ

3GM

)13

r , (7.2)

t =

(

Λ

3

)12

t , (7.3)

we can rewrite 7.1 as

E =1

2

(

dr

dt

)2

− 1

r− r2

2, (7.4)

where

E =

(

G2M2Λ

3

)− 13

E . (7.5)

7.2.1 Criterion for Bound Structures

In this section, we briefly describe the steps required in order to obtain an analytical solution for the

spherical collapse equations for a bound structure. A full description can be found in Paper I.

We are looking for a critical shell that will asymptotically stay at the limit between expanding

forever or recollapsing into the structure. In order to find the energy of such a shell, we maximize the

(dimensionless) potential energy defined as:

V = −1

r−

r2

2(7.6)

The maximum of this potential occurs at r∗ = 1, so E∗ = V(r∗ = 1) = − 32

is the maximum possible

energy for a shell to remain attached to the structure.

Integrating Eqn. 7.4 for a critical shell (with E = − 32), we get

t =

∫ r

0

√rdr

(1− r)√

r+2. (7.7)


The solution to this integral is given by

t =1

2√

3ln

(

1+2r+√

3r(r+2)

1+2r−√

3r(r+2)

)

− ln (1+ r+√

r(r+2)) . (7.8)

Knowing that in a flat Universe with cosmological constant the vacuum energy density parameter

varies with time as (Peebles 1980)

ΩΛ ≡Λ

3H2= tanh2

(

3t

2

)

, (7.9)

we can replace Eqn. 7.8 in the latter, obtaining

ΩΛ(r) =

[

χ(r)−1

χ(r)+1

]2

, (7.10)

where

χ(r) =

[

1+2r+√

3r(r+2)

1+2r−√

3r(r+2)

]

√3

2

×(

1+ r+√

r(r+2))−3. (7.11)

Eqns. 7.10 and 7.11 allow us to determine the radius of a critical shell to remain bound in a ΛCDM

Universe. If we evaluate Eqn. 7.10 for our preferred cosmology (Ωm = 0.3,ΩΛ = 0.7), we get r0 = 0.84,

meaning that a critical shell has a present radius that is 84% of the maximum radius it will reach as

t→∞.

It is convenient to express the critical condition as the minimum enclosed mean density needed by

a shell to stay bound. The critical density of the Universe is given by

ρc =3H2

8πG, (7.12)

and the average mass density enclosed by any given shell is

ρs =3M

4πr3. (7.13)

The ratio between the enclosed density and the critical density is then given by:

ρs

ρc

=2ΩΛ

r3. (7.14)

The condition for a shell to be bound is then

ρs

ρc

≥ρcs

ρc

=2ΩΛ

r3cs

. (7.15)

where the subscript ‘cs’ stands for critical shell. For ΩΛ = 0.7, we get that this ratio is ρcs/ρc = 2.36,

where we have used the fact that r0 = 0.84. Evaluating at ΩΛ = 1 (t→∞), we get ρcs/ρc = 2.

This criterion has the property that it has an analytical relation with ΩΛ, allowing us to evaluate it

easily at any cosmological time. Verifying this criterion through simulations in Paper I it was found

that, on average, 72% of the mass enclosed by the estimated radius is really bound to the structure,

while 0.3%, although bound to the structure, is not enclosed by the radius.

7.2.2 Determining the linear overdensity for a marginally bound object

In view of our intention to identify the bound superclusters at any redshift, and their mass spectrum,

we also need a proper estimate for the linear critical density value for bound and collapsing mass

7.2. SPHERICAL COLLAPSE MODEL 135

halos. Among others, this is crucial for a comparison with the theoretical mass functions predicted by

Press-Schechter theory (see 7.A). Here we derive the corresponding value.

We know that δ(a) = δ0D(a), where δ is the density contrast given by Eqn. A-1 (see 7.A) and D(a)

is the amplitude of the growing mode, given by (Heath 1977; Peebles 1980):

D(a) =5ΩmH2

0

2H(a)

∫ a

0

da′

a′3H(a′)3

= ag(a) , (7.16)

where g(a) is the linear growth factor. An accurate approximation formula for g(a) is (Carroll et al.

1992):

g(a) ≈5

2Ωm(a)

[

Ωm(a)4/7−ΩΛ(a)+

(

1+Ωm(a)

2

)(

1+ΩΛ(a)

70

)]−1

. (7.17)

We need to find δ0 for the critical shell. In order to achieve that, we consider its initial evolution

(a≪ 1).

The density of a spherical region of mass M and radius r is given by Eqn. 7.13 and the back-

ground density can be computed at any time simply by doing ρb = ρc,0Ωm,0/a3, where ρc,0 and Ωm,0

are the critical density and the matter density parameter, respectively, at the present time, and a is the

expansion factor. Using Eqn. 7.2, we can rewrite Eqn. A-1 as:

1+δ =ρ

ρb

=2ΩΛ0

Ωm,0

(

a

r

)3

, (7.18)

where ΩΛ,0 denotes the present value of the cosmological density parameter.

For the problem at hand we make the approximation that the expansion of the Universe is still

reasonably approximated by an EdS expansion

a(t) =

(

t

t∗

)2/3

, (7.19)

where t is some generic time and t∗ is a characteristic constant time. Replacing this equation in Eqn.

7.18 we get

t(r) ≈ t∗

(

1+δ

2

)

(

Ωm,0

2ΩΛ,0

)1/2

r3/2 , (7.20)

were we have kept terms of order lower than δ2. Since we are interested in objects that will remain

bound, we can take Eqn. 7.7 and replace the integral by a derivative by doing

dt

d√

r=

√r

(1− r)√

r+2·2√

r (7.21)

We can expand Eqn. 7.21 (keeping low order terms) and integrate. This gives:

t(r) ≈√

2

3Λr3/2+

3

10

√

3

2Λr5/2 , (7.22)

Comparing with Eqn. 7.20 we get

t∗ =

[

4ΩΛ,0

3ΛΩm,0

]1/2

=1

(6πGρm,0)1/2, (7.23)

which is the well known result for the time-density relation in a matter dominated Universe, and, in

the limit a→ 0,

δ0 g(a→ 0) =δ

a=

9

10

(

2ΩΛ,0

Ωm,0

)1/3

= 1.504 . (7.24)

Evaluating Eqn. 7.24 for our preferred cosmology (Ωm = 0.3, ΩΛ = 0.7) and using Eqn. 7.17, we

get δ(a = 1) = 1.17. This is the present linear density contrast for marginally bound structures that will

collapse when a→∞.


7.3 The Numerical Simulation

We simulate one cosmological model, assuming a standard flat Lambda Cold Dark Matter (ΛCDM)

Universe. The cosmological parameters in the simulation are Ωm,0 = 0.3, ΩΛ,0 = 0.7, and h = 0.7,

where the Hubble parameter is given by H0 = 100h km s−1Mpc−1, and the normalization of the power

spectrum is σ8 = 1. In order to have a large sample of bound objects, the simulation box has a side

length of 500h−1 Mpc and contains 5123 dark matter particles of mass mdm = 7.75×1010h−1M⊙. The

initial conditions were generated at expansion factor a = 0.02 (redshift z = 49), and were evolved until

a = 100 using the massive parallel tree N-Body/SPH code GADGET-2 (Springel 2005). The Plummer-

equivalent softening was set at ǫPl = 20 h−1kpc in physical units from a = 1/3 to a = 100, while it was

taken to be fixed in comoving units at higher redshift. Given the mass resolution and the size of the

box, our simulation allows us to reliably identify massive superclusters with ∼ 80 000 particles. The

simulation was performed on the Beowulf Cluster at the University of Groningen.

We took snapshots at the present time (a = 1) and in the far future (a = 100). We do not expect

major large scale evolution anymore beyond a = 1, while a = 100 is representative for epochs at which

the internal evolution of all bound objects has also been completed.

The top two panels of Fig. 7.1 show a slice of 30h−1Mpc width of the particle distribution projected

along the z axis, both at a = 1 and a = 100. At a = 1, the large-scale structures of the cosmic web are

well established, and, as was pointed out by Nagamine & Loeb (2003), they hardly change thereafter.

The lower left panel depicts the region shown in the box of the top-left panel in physical coordinates at

a = 1, centered on a massive structure following the criterion of Eqn. 7.15. The radius of the circle is

that of the bound region. We see how it is well connected with the surrounding structures. The lower

right panel has the same physical size, and shows the same object, but now at a = 100. We see that

the size of it (the circle shown) is nearly similar at both expansion factor. Although the accelerated

expansion of the Universe freezes in comoving coordinates, in physical coordinates the separation of

structures grows exponentially in time. Objects grow in complete isolation, as it can be seen from the

figure.

7.3.1 HOP halos

In order to find the objects present in our simulation, we use HOP (Eisenstein & Hut 1998). This

algorithm first assigns a density estimate at every particle position by smoothing the density field with

an SPH-like kernel using the ndens nearest neighbors of a given particle. In our case, we use ndens = 64.

Subsequently, particles are linked by associating each particle to the densest particle from the list of

its nhop closest neighbors. We use nhop = 16. The process is repeated until it reaches the particle

that is its own densest neighbor. The algorithm associates all particles to their local maxima. It often

happens that the local maxima causes groups to fragment. To correct this, groups are merged if the

bridge between them exceeds some chosen density thresholds. Three density thresholds are defined as

follows (Cohn et al. 2001):

• δout is the required density for a particle to be in a group.

• δsaddle is the minimum boundary density between two groups so that they can be merged.

• δpeak is the minimum central density for a group to be independently viable.

Eisenstein & Hut (1998) claim that the method works best is the values are in the ratio

δouter:δsaddle:δpeak=1:2.5:3. We follow their recommendation and use this relation.

7.3.2 Virialized groups

To identify the bound groups present at each output time, we will first identify virialized groups. For

this purpose, we associate δpeak with the corresponding virial density, ∆vir(z).

7.3. THE NUMERICAL SIMULATION 137

Figure 7.1 — Top panels: 30h−1Mpc slice through the z axis in comoving coordinates, at a= 1 (left) and a = 100

(right). We see that structures are well defined at a = 1 and change little thereafter. The panel on the lower left

depicts the region enclosed in the upper panel in physical coordinates, centered on a massive bound structure. The

panel on the right has the same physical coordinates, but now at a = 100. It is also centered on the same massive

bound structure. The radius of the circles are of the size of the structure. We see how the structures surrounding

the object at a = 1 have detached from it, making it grow in complete isolation.

The value of ∆vir(z) is taken from the solution to the collapse of a spherical top-hat perturbation

under the assumption that the object has just virialized. Its value is 18π2 for a critical Universe, and it

has to be obtained numerically for other cases. A good approximation for the case of a flat Universe

(Ωm+ΩΛ = 1) was found by Bryan & Norman (1998):

∆vir(z) ≈18π2+82x−39x2

1+ x(7.25)

where x=Ωm(z)−1. This relation is accurate in the rangeΩm(z)= 0.1−1. For the cosmology described

here, ∆vir ≈ 337. Note that this value is with respect to the background density of the Universe, not the

critical density.

At a = 100, Ωm = 4.3× 10−7, so we can not apply Eqn. 7.25. We need to find the characteristic

radius that will correspond to the virial radius. The energy at that time is Evir = Kvir +Vvir. From the


virial theorem we know that K = V/2. Then, the (dimensionless) energy at virialization is given by

Evir =1

2VG,vir +2VΛ,vir , (7.26)

where VG is the potential energy due to gravity and VΛ is the potential energy due to the cosmolog-

ical constant. By energy conservation, the energy at maximum expansion is equal to the energy at

virialization. We can then write

r2+1

2r=

3

2. (7.27)

This is a cubic equation with solutions r = 1 (maximum radius of the shell), r ≈ −1.366 (unphysical

solution) and r ≈ 0.366, which corresponds to the virial radius. Note that Eqn. 7.27 is Eqn. 26 of Lahav

et al. (1991) but using our dimensionless variables.We use the corresponding value of r in Eqn. 7.14,

and get ρ/ρc ≈ 40.8. Then, in order to get physically virialized objects with HOP at a = 100, we use

this value divided by Ωm at that time.


To extract the groups present in the simulation, we take a subsample of 2563 particles (1/8 of the total

particle number), both at a = 1 and a = 100. We do this because of computer limitation. HOP found

∼ 20 600 independently virialized groups at a = 1 and ∼ 18 000 at a = 100 with more than 50 particles,

i.e., total masses m ≥ 3.1×1013h−1M⊙.

Having found the groups, we apply the density criterion of Eqn. 7.15 at a = 1 and a = 100. We

do this in the following way. We first find the center of mass of the groups given by HOP using

the subsampled particles. We first take the densest particle of the group as a guess for the center of

mass. We grow a sphere around this center, with the radius being increased until the mean overdensity

(density of the sphere with respect to the critical density of the Universe) reaches a value of 300. This

value is chosen in order to find the densest core of the structure. We then calculate the center of mass

of this sphere and repeat the process, iterating until the shift in the center between successive iterations

is less than 1% of the previous value. Once we have the center of mass, we proceed to define the

objects present in the simulation. Since we used the subsampled particles to find the center of mass,

we proceed to find again the center of mass, but this time considering all the particles in the simulation.

We follow the same procedure as described before. Having found the center of mass of the groups, we

apply the criterion for a structure to remain bound as given by Eqn. 7.15, i.e., we calculate the density

ratio of concentrical spheres with increasing radius until the desired condition is reached.

Now, objects defined by applying Eqn. 7.15 will have larger radii than the groups found by HOP.

Since we are constructing bound objects from virialized structures, it is to be expected that we may be

counting the same object twice since two or more virialized objects might be bound to each other. We

take care of this issue by checking if there is another center of mass inside the bound radius, starting

from the most massive group. Any lower mass group whose center of mass is inside a more massive

one is taken out of the sample. Doing this gives ∼ 15 000 bound groups. Repeating the procedure at

a = 100 does not change the number of bound groups. This indicates that at a = 1 bound groups are

well identified.

7.3.4 Sample completeness

Identifying structures at a = 1 in this way poses another problem. Since we are selecting bound struc-

tures that have a virial group in its center, we are forcing our structures to have one virial group with

a mass equal or higher to the mass cut we gave HOP (50 particles). This means that we will not

detect bound groups with masses lower than 3.1× 1013h−1M⊙. This is not a problem for the most

massive groups, but we may have an incomplete sample for lower mass groups. Fig. 7.2 (left panel)

shows a scatter plot of the virial groups found by HOP and the bound structure constructed from it.

As expected, the lower right region is empty: HOP groups will always be less massive than the bound

7.3. THE NUMERICAL SIMULATION 139

Figure 7.2 — Contour maps of the scatter plot of HOP groups and their corresponding bound structures, at a = 1

(left panel) and a = 100 (right panel).

groups. There is a correlation between both masses, but with a high scatter. By cutting at 50 particles,

we are eliminating many bound structures with masses that reach 2×1014h−1M⊙. It is fair to say, then,

that our sample of bound structures at a = 1 is complete for masses greater than 2× 1014h−1M⊙. We

do the same analysis for objects at a = 100. We see from the right panel of the same figure that the

correlation is better and that our sample will be complete for masses greater than 6×1013h−1M⊙.

Figs. 7.3 and 7.4 show four structures, two at each epoch (a = 1 and a = 100). At both epochs we

show a massive object and a less massive one. At a = 1 (Fig. 7.3), the presence of substructure in the

massive object is evident. The massive central structure is the virial object from which we constructed

the final object. On the other hand, the less massive object does not show substructure, just the central

region and a few particles within the bound radius. At a = 100, the less massive object is quite similar

to that at a = 1 in terms of mass and appearance. The massive object is highly concentrated: all

Figure 7.3 — Bound objects at a = 1. The panel on the left depicts one of the most massive bound groups; the

panel on the right, one of the less massive ones. The presence of substructure is evident on the massive one, but

not in the less massive one.


Figure 7.4 — Bound objects at a= 100. The left panel shows a massive bound group, on the right, a less massive

structure. We see that both structures are highly concentrated.

substructure within its radius will have collapsed, and will form one big mass concentration.

7.4 Supercluster Mass Functions

Perhaps the most outstanding repercussion of the slowdown of large scale structure formation in hier-

archical cosmological scenarios is the fact that new objects will no longer condense out of the density

field. This should be reflected in the mass spectrum of the objects that were just on the verge of forma-

tion around the time of the cosmological transition. Hence, here we investigate the mass distribution

of bound but not yet fully virialized structures (we also investigate a sample of virialized halos). In our

Universe these are the superclusters.

7.4.1 Press-Schechter mass functions

Fig. 7.5 shows the evolution of the mass function predicted by the Press-Schechter and the Sheth-

Tormen formalism for our cosmology. The plot on the left follows the evolution at z = 4,2,1,0.5 and

0. It is clear that, as the Universe evolves, structures grow in mass and in number. The panel on the

right shows evolution into the future for a = 1,2,4,10 and 100. We see that after a = 1 the number of

low mass objects does not change substantially, and after a = 4 the evolution stops. In fact, the curves

for a = 10 and a = 100 overlap. We also see that the Sheth-Tormen approximation predicts a lower

number of less massive objects than Press-Schechter.

7.4.2 Mass functions in Simulations

Considering two epochs (a = 1 and a = 100), and three different identification criteria (HOP, virial, and

bound), we obtain six different mass functions from our simulation data. Two of them are the ones

given by HOP. Two others correspond to virialized objects, this is, objects that in their radius enclose

100 times the critical density at a = 1 and 40.8 times the critical density at a = 100. The other two mass

functions are the ones defined by the bound objects as given by the criterion described in section 4.2.

Given the way we identified objects (see section 7.3), both the HOP and the virial mass functions are

similar. For convenience, we will use the HOP mass functions in our analysis.

Fig. 7.6 shows the mass functions of the objects found by HOP (left panel) and of the groups that

satisfy the criterion given by Eqn. 7.15 (right panel), both at a = 1 and at a = 100. We see that objects

7.4. SUPERCLUSTER MASS FUNCTIONS 141

at a = 100 are more massive than those at a = 1, but not substantially, confirming the freezing of the

mass functions shown by Nagamine & Loeb (2003) and the theoretical results shown in Fig. 7.5. Since

some of the groups at a = 1 will end up together at a = 100, at this expansion factor there are more

massive groups than at a = 1, and fewer low mass groups.

Although the binding criterion found in Paper I predicts that the mass for bound objects should be

the same at both expansion factors, we can see from the right panel of Fig. 7.6 that there is a slight

difference. This was also noted in Paper I, where the authors found that, on average, 72% of the mass

enclosed today by the critical radius will end up in a bound structure at a = 100.

As done in Paper I we also marked the particles that belong to a group at a = 1 and follow them

to a = 100. In order to check the amount of mass that remains bound. We identified which groups at

a = 1 correspond to the ones at a = 100. Having done this, we get that the fraction of mass that will

remain bound is 72%, with a standard deviation of 13%. Note that meanwhile accretion did not enrich

the halo with more than 1% of its final mass. Both are in agreement with the findings in Paper I.

Fig. 7.7 shows the same as the right panel of the previous figure, but now we plotted the “reduced”

mass function at a = 1, this is, we reduced all the masses at a = 1 by a factor of 0.72. We see there is a

good overlap with the mass function at a = 100, as expected.

If we consider the entire sample of bound groups at a = 100, we get a total mass of 2.83 ×1018h−1M⊙. Since the mass of our simulated Universe is 1.04× 1019h−1M⊙, 27% of the total mass

ends up in bound structures.

Using the criterion given by Eqn. 7.15 as a physical definition of superclusters, we estimate the

most massive supercluster present in the local Universe (z < 0.1) to have a mass of ∼ 8×1015h−1M⊙.

This mass is slightly larger than the one of Shapley, which we found to be ∼ 7×1015h−1M⊙ (Dunner

et al. 2008). According to the mass functions, we may find up to two Shapley-like superclusters in our

Local Universe.

7.4.3 Comparison of simulated and theoretical mass functions

Fig. 7.8 shows the cumulative mass function of the objects found by HOP at a = 1 (left panel) and

at a = 100 (right panel), together with the three theoretical mass functions. At a = 1, we can see that

the Jenkins (Jenkins et al. 2001) mass function is the one that fits best (right). The value of δc was

Figure 7.5 — Evolution of the mass function predicted by Press-Schechter and Sheth-Tormen. On the left, from

left to right, evolution for z = 4,2,1,0.5 and 0. On the right, evolution for a = 1,2,4,10 and 100.


Figure 7.6 — Mass functions for HOP objects (left panel) and for bound groups (right panel), both at a = 1 and

a = 100.

Figure 7.7 — Mass functions of bound objects, now showing the “reduced” mass function at a = 1.

calculated according to Eqn. A-9, where for our cosmology we get δc = 1.675. Governato et al. (1999)

argues that, for their simulation (they also have a box of 500h−1Mpc in a side and Ωm = 0.3) a better

value for δc is 1.775 (at z = 0). We find that this value does make a small difference, specially with the

Sheth-Tormen approximation, where the fit improves. For the Press-Schechter mass function, we see

a good fit for lower masses, but it underestimates higher masses. The Jenkins approximation does not

change, since it is independent of δc.

For the a = 100 mass function, we used the fitting parameters for Ωm = 0 listed in Evrard et al.

(2002) for the Jenkins mass function. With these parameters, it agrees very well with the HOP mass

function, although it slightly overestimates the number of lower mass objects. However, for Jenkins’

original parameter values, it would lead to a significant overabundance of objects with respect to the

ones found in the simulations. Also note that neither the pure Press-Schechter nor the Sheth-Tormen

7.4. SUPERCLUSTER MASS FUNCTIONS 143

Figure 7.8 — The mass function of HOP objects at a = 1 (left panel) and at a = 100 (right panel) with the three

theoretical mass functions.

functions manage to fit the mass spectrum over the entire mass range. PS does agree at the high

mass end while ST results in a better agreement at lower masses. This is an indication for the more

substantial role of external tidal forces on the evolution of the low mass halos.

Fig. 7.9 shows the mass functions for bound objects at a = 100. We did not plot the objects at a = 1

since it gives a similar plot. We use the value δc = 1.17 derived in section 7.2.2. Since the Jenkins

approximation is independent of δc it does not fit the simulated mass function. We see that the best fit

is the provided by the PS mass function. This is may be due to the fact that the original implementation

of PS was based on spherical objects, which are the type of objects that we find at a = 100.

Figure 7.9 — The mass function of bound objects at a = 100 and the three theoretical mass functions. With our

calculated value of δc, the PS approximation is the best fitting.


7.5 Shapes of Bound Structures

Arguing that the large scale formation and evolution of structure comes to a halt once the Universe

starts to accelerate, while the internal evolution of overdense patches continues, one particular mani-

festation of this will be the changing shape of these collapsing objects.

The issue of shapes of superclusters of galaxies has already been addressed, both using real data

(e.g., Plionis et al. 1992; Basilakos et al. 2001) and in N-body simulations (e.g., Basilakos et al. 2006;

Wray et al. 2006; Einasto et al. 2007). In every study, the authors state that the dominant shape of

superclusters at the present time is prolate. Their definition of superclusters is done using percolation

analysis. This makes it into a somewhat arbitrary definition since a supercluster will depend on the

chosen percolation radius. By contrast, our definition of a supercluster – including its size, mass and

shape – is based upon on a strict physical criterion.

7.5.1 Definitions

In order to determine the shape, we calculate the inertia tensor using all particles inside the region of

interest:

Ii j =∑

xix jm . (7.28)

Since the matrix is symmetric, it is possible to find a coordinate system such that the the matrix

representing the tensor has elements only along the diagonal. Our coordinate system will be chosen

with respect to the center of mass of our bound objects. It is therefore possible to diagonalize the

matrix, obtaining the eigenvalues a1, a2 and a3. The eigenvalues of the inertia tensor give a quantitative

measure of the degree of symmetry of the distribution. For example, if a1 is a far larger number than

a2 and a3, the distribution is not spherically symmetric. The two axis ratios are given by

b

a=

√

a2

a1,

c

a=

√

a3

a1, (7.29)

where a1 > a2 > a3. This means that, if both ratios b/a and c/a are close to one, the object is almost

spherical.

7.5.2 Shape evolution

Fig. 7.10 shows the distribution of axis ratios of the groups that satisfy the binding criterion (Eqn.

7.15) at present time. Objects must lie in the lower portion of the plot. Spherical groups should be at

(1,1), oblate groups should tend to b/a = 1 and prolate groups should lie near the diagonal b/a = c/a.

There is a big empty region for lower values of c/a, meaning that there are no pancake-like structures

(a pancake shape is defined such that c/a→ 0). We also see that there are almost no nearly-spherical

objects. This is to be expected given the fact that in reality there are no spherical peaks in the initial

density field (Bardeen et al. 1986).

The distribution of supercluster shapes is a combination of at least two factors. One is the shape

of the proto supercluster in the initial density field. The second factor is the evolutionary state of the

supercluster. We know that objects collapse anisotropically and proceed from a pancake-like shape via

filamentary configuration towards a fully collapsed triaxial object. While it is still in the process of full

collapse one may still recognize the internal substructure of its constituent clusters.

While the Universe is entering into an accelerating phase, the bound structures become increasingly

isolated. No major mergers between structures happen after a= 1. The substructures that are within the

bound radius will merge with each other, giving rise to spherical concentrated structures. Nagamine

& Loeb (2003) ran an N-Body simulation that resembles the Local Universe, and found that the Local

Group will get detached from the rest of the Universe, and the physical distance from it to the other

systems that are not bound will increase exponentially.

We see this process reflected in Fig. 7.11. The left hand frame shows the shape distribution at

a = 1, the one on the right hand side the resulting distribution at a = 100. At a = 1 the mean value of

7.5. SHAPES OF BOUND STRUCTURES 145

Figure 7.10 — Distribution of axis ratios for bound objects at a = 1 (left panel) and at a = 100 (right panel). The

shape evolution towards less elongated objects is clearly visible.

the axis ratios is (〈b/a〉,〈c/a〉)= (0.69,0.48), with a standard deviation of (σb/a,σc/a)=(0.13,0.11). The

figure shows that there is a wide spread of shapes with a tendency of the majority of groups to lie near

the diagonal: bound groups have a prolate shape.

If we now turn our attention to the groups at a = 100, the distribution of group axis ratios in Fig.

7.10 is clearly different from the one at a = 1. The mean values of the axis ratios for all bound groups

are (〈b/a〉,〈c/a〉)=(0.94,0.85), with standard deviations of (σb/a,σc/a)=(0.03,0.05). We see that for

the majority of objects there is a predominance of nearly spherical shape. This demonstrates that,

although there is virtually no large scale structure evolution after a = 1, there is an internal evolution

of each individual group. It manifests itself in the object’s shape to evolve from prolate to more or less

spherical.


One potentially relevant issue concerns the possible dependence of shape on the mass shape of bound

structures. In order to investigate this, we divide our sample into three mass ranges. We will

consider massive groups (superclusters-like structures) with M > 1015h−1M⊙, massive clusters with

5×1014h−1M⊙ < M ≤ 1015h−1M⊙, and less massive clusters with M ≤ 5×1014h−1M⊙.

Figure 7.11 shows a contour map of the distribution of axis ratios for the three mass ranges de-

scribed. The first mass ranges has ∼ 530 objects (upper left panel); the second, ∼ 1020 (upper right)

and the third, ∼ 3330 (lower panel). We see a predominance for prolate shape in every mass range.

Keeping in mind that our groups are bound but perhaps not virialized, it is instructive to contrast

our objects to galaxy clusters, which are virialized by definition. Usually, superclusters are defined

using different percolation radii, but clusters are defined using Eqn. 7.25, which, for our cosmology,

has a value of ∼ 101. By definition, these clusters are virialized. In our case, although the masses

are comparable, the radius are not, since our bound groups have lower densities than clusters. Hence,

they have larger radius, which will make them to have more substructure. Shapes of galaxy cluster

halos in N-Body simulations have already been addressed by many authors (e.g., Dubinski & Carlberg

1991; Katz 1991; van Haarlem & van de Weygaert 1993; Jing & Suto 2002; Kasun & Evrard 2005;

Paz et al. 2006; Allgood et al. 2006), and all of them agree that they tend to be more prolate as the halo

mass increases. Dubinski & Carlberg (1991) found that halos are “strongly triaxial and very flat”, with

mean axis ratios of 〈b/a〉=0.71 and 〈c/a〉=0.50, Katz (1991) found, for simulations of isolated halos


Figure 7.11 — Contour maps of the distribution of axis ratios at a = 1 according to the three mass ranges

described in the text. We see a predominance of prolate shapes.

with different power spectra indices, values ranging from 0.84 to 0.93 for b/a and 0.43 to 0.71 for c/a,

while Kasun & Evrard (2005) found for massive clusters peak values of (b/a,c/a) = (0.76,0.64).

The bound superclusters in our study also tend to be more prolate in shape, although their radii are

different (larger) than the virial radius. We found that the three mass ranges have similar mean axis

ratios (see Table 7.1). The shapes are quite close to the ones mentioned in the previous paragraph.

This is indeed very interesting given that cluster mass virialized objects are quite different from the

less dense supercluster objects we have studied. This immediately raises the following question: if the

axis ratios for bound objects are similar to the ones of virialized objects, what would be the values for

our virial objects?

To this end, we calculated the moment of inertia and its eigenvalues for the virial objects in

# Objects 〈b/a〉〈c/a〉 σb/a σc/a

M > 1015h−1M⊙ 533 0.70 0.50 0.13 0.11

5×1014h−1M⊙ < M ≤ 1015h−1M⊙ 1015 0.70 0.49 0.13 0.10

2×1014h−1M⊙ < M ≤ 5×1014h−1M⊙ 3333 0.69 0.48 0.13 0.11

Table 7.1 — Number of objects and average values of the axis ratios with their standard deviation according to

mass range for bound objects in a = 1.

7.5. SHAPES OF BOUND STRUCTURES 147

our simulation at a = 1. The obtained values are (〈b/a〉,〈c/a〉)=(0.83,0.71) with standard deviation

(σb/a,σc/a)=(0.09,0.09). This means that also the virial halos are prolate, somewhat more pronounced

than those quoted in other studies.

Figure 7.12 — Same as in Fig. 7.11 but for bound objects at a = 100. For clarity, the third and fourth mass

ranges are plotted with thinner dots due to the amount of points. We see that in all four mass ranges there is a

predominance of spherical shapes.

When we turn towards the situation in the future, we find that, in all mass ranges, objects attain

a predominantly spherical morphology. Fig. 7.12 clearly shows this (see also Table 7.2). It is good

to realize that bound objects grow in complete isolation and that all substructures within the binding

radius merge into one, single, compact spherical object.

# Objects 〈b/a〉〈c/a〉 σb/a σc/a

M ≥ 1015h−1M⊙ 347 0.93 0.84 0.05 0.06

5×1014h−1M⊙ ≤ M < 1015h−1M⊙ 727 0.94 0.85 0.04 0.05

1014h−1M⊙ ≤ M < 5×1014h−1M⊙ 6436 0.94 0.86 0.03 0.05

M < 1014h−1M⊙ 5048 0.93 0.85 0.04 0.05

Table 7.2 — Number of objects and average values of the axis ratios with their standard deviation according to

mass range for bound objects in a = 100.


7.6 Density Profiles of Bound Structures

Along with the evolution of shape of objects, studied in the last section, the contracting superclusters

will develop into a much more compact object. In order to assess this aspect of their internal evolution

we study the evolution of the density profile and mass distribution of the halos in our sample.

Figure 7.13 — Density profiles of two objects, both at a = 1 and a = 100.

Fig. 7.13 shows the density profile of two objects at two expansion factors, a = 1 and a = 100.

We choose one massive object (Object 8) and a less massive one (Object 98) at a = 1, and track

them into the far future. We check the density profile of the other structures, and they show the same

behavior. These two objects represents a fair example. The profiles were constructed using equal-size

logarithmic bins.

We see from the figure that the profiles show essentially the same form: a high density at the center,

due to the virial structure it has, and then it becomes less dense. The only difference is at the outer

boundary, stretched in order to match the density of the background Universe (see also Busha et al.

2003). We see that in Object 98 the outer boundary starts to stretch before in comparison with object

8. This is because the virial structure of object 8 is similar in mass to the virial mass it will end up

having at a = 100, which is not the case of object 98, whose virial structure is less massive than the

final structure at a = 100.

Figs. 7.14 and 7.15 show a clearer picture of the internal changes objects go through. Both figures

show the particle distribution at a= 1 and at a= 100 (top panels). Object 8 (Fig. 7.14) is one of the most

massive objects identified at a = 1, and it ends up as one of the most massive at a = 100. We see that

the virial structure from which it was constructed,depicted by the dashed-dot circle, has almost half of

the mass of the object, depicted by the solid circle. At a = 100, the object has become more compact

and denser at the center, as can be deduce by looking at the position of the half-mass radius and the

virial radius. Along with the substantial rearrangement of its internal mass distribution into a more

compact and spherical concentration, we also note a radical change of its cosmic surroundings. At

a = 1 this is still marked by outstanding neighboring inhomogeneities as the objects are embedded and

connected with the Cosmic Web (see Figs. 7.14 and 7.15, left frames). At a = 100 these supercluster

concentrations have turned into isolated islands.

This evolution is directly reflected in the radial mass distribution. The bottom panel of these figures

show the cumulative mass distribution as a function of radius. For Object 8, we see that, at a = 1, there

is a high concentration of mass at small radii, indicating that the virial structure at the center has

almost half of the mass of the object. There is then a slight increment in mass, reflecting that there are

not many particles. At a = 100, half of the mass is within a small radius, comparable to that at a = 1.

However, we see that at large radii, beyond r = rb, the object does not gain more mass. This is reflected

in the flattening of the curve.

On the other hand, the cumulative mass distribution of Object 98 (Fig. 7.15) shows the same trend,

but a different behavior at a= 1. Half of the mass of the object is enclosed at large radii (see the particle

distribution on the top-left panel). The particle distribution shows that the virial object at the center

is not as massive as that of Object 3. The mass increases almost uniformly until the half-mass radius,

7.6. DENSITY PROFILES OF BOUND STRUCTURES 149

Figure 7.14 — Upper panels: particle distribution at a = 1 (left) and a = 100 (right) of object 8 in physical

coordinates. The half-mass radius (solid circle) and the virial radius (dashed-dotted circle) are also drawn for

comparison. Bottom panel: Cumulative mass distribution of the object at a = 1 (solid line) and at a = 100 (dotted

line). Both quantities have been normalized to their corresponding final values. The solid lines departing from the

axis show the values of the half-mass radius.

and then there is an abrupt rise until the final radius, which is due to the object at the border. As in the

previous object, if it goes further of r = rb, its mass will continue growing.

The cumulative mass distribution at a = 100 shows the same behavior as that of Object 8: half of

the mass is enclosed at small radii, and at large radii will stop gaining mass. These cumulative mass

distributions are a fair representation of the relaxation of the objects.

We see that objects are highly concentrated at a = 100, as was suggested also by their almost

spherical shape (section 7.5). We define a concentration parameter as

c =rhm

rb

, (7.30)

where rhm is the radius that encloses half of the mass and rb is the radius of the object. Fig. 7.16 shows

the distribution of c as a function of mass at a = 1 (left panel) and a = 100 (right panel).

As expected, objects at a = 100 are highly concentrated, with c = 0.16 and σ = 0.02, while objects

at a = 100, the concentration parameter has a wider distribution, with c = 0.35 and σ = 0.14. To some

extent, this is due to the presence of substructure.


Figure 7.15 — Same as Fig. 7.14 but now for object 98.

Figure 7.16 — Concentration parameter as a function of mass for bound objects at a = 1 and at a = 100.

7.7. SUPERCLUSTER MULTIPLICITY FUNCTION 151

7.7 Supercluster Multiplicity Function

As a final aspect of the internal evolution of the superclusters, we turn towards their substructure.

In particular, we wish to investigate their multiplicity, i.e., the number of clusters they contain. The

hierarchical development of the supercluster involves the gradual merging of its constituent subclumps

into one condensed object. Therefore, we expect superclusters containing several to dozens of clusters

ultimately to end up as a object of unit multiplicity.

We have to realize that the method of supercluster identification has some influence on the defini-

tion of the multiplicity function. For example, while superclusters are often identified on the basis of

a percolation criterion, the multiplicity will depend on the used percolation radius. Our supercluster

identification involves a more physical criterion which automatically defines the radius and, therefore,

we hope yields a more natural multiplicity function.

At the current epoch, we found ∼ 4 900 structures that satisfy Eqn. 7.15, but not all of these are

superclusters. We assume a lower mass limit of 5×1015h−1M⊙ to define a supercluster. Seventeen ob-

jects satisfy the criterion. The low mass threshold of clusters members was taken to be 4×1013h−1M⊙,

which correspond to the lowest mass of virial groups found at a = 1.

Fig. 7.17 shows the multiplicity function for the 17 objects. We see that half of the superclusters

have 10 or more members. The mean mass of clusters in the total sample is M = 9.4× 1013h−1M⊙,

which is substantially lower than the average mass of clusters in superclusters, M = 3.6×1014h−1M⊙.

This indicates that superclusters contain more massive clusters than the mean.

According to Fig. 7.17, the largest supercluster in the Local Universe would have 15 members, and

a Shapley-like supercluster would have a radius of ∼ 14h−1Mpc and host between 10 to 15 members,

almost half of that estimated by Quintana et al. (2000). In the observational reality of our Local

Universe (z < 0.1), we find 5 superclusters with 10 or more members, the largest one containing 12

members.

If we compare our results with the work of Wray et al. (2006), we notice there is a difference in the

number of clusters members. In some cases,they obtain superclusters with more than 30 members (see

their Fig. 3). This is due to the choice of the linking length when defining a supercluster. The maximum

size of superclusters they find ranges from ∼ 150h−1Mpc to ∼ 30h−1Mpc for different linking lengths,

which are much larger than the superclusters we find according to our physical definition.

Fig. 7.18 shows the multiplicity as a function of mass (left panel) and as a function of radius (right

Figure 7.17 — Integrated multiplicity function for superclusters at a = 1.


Figure 7.18 — Multiplicity vs. mass (right panel) and multiplicity vs. bound radius (right panel).

panel). As expected, larger and more massive superclusters have more members. Again, if we compare

these results with those of Wray et al. (2006), we get that our superclusters are smaller in size (see their

Fig. 5), but the correlation between the number of members to size is the same.

Finally, we may point out that all superclusters in our sample by a = 100 have evolved into single,

compact objects. By definition, they all have multiplicity one.

7.8 Conclusions

In this work, we studied several properties of bound structures, such as their mass function, shape and

density profile. These bound structures were defined by the density criterion given in Paper I (Eqn.

7.15), and identified from a 500h−1 Mpc cosmological box with 5123 dark matter particles in aΛCDM

(Ωm = 0.3, ΩΛ = 0.7 and h = 0.7) Universe. We ran the simulation up to a = 100, which is a time where

structures have stopped forming. We use HOP in order to identify independently virialized structures,

both at a = 1 and a = 100. Our main results can be summarized as follows:

• The marginally bound objects that we study resemble the superclusters in the observed Universe.

While clusters of galaxies are the most massive and most recently fully collapsed and virialized

objects in the Universe, superclusters are the largest bound –but not yet collapsed– structures in

the Universe.

• The superclusters are true island Universes: as a result of the accelerating expansion of the

Universe, no other, more massive and larger, structures will be able to form.

• While the superclusters collapse between a= 1 and a= 100, their surroundings radically change.

At the present epoch solidly embedded within the Cosmic Web, at a = 100 they have turn into

isolated cosmic islands.

• The mass functions in the simulations are generally in good agreement with the theoretical

predictions of Press-Schechter, Sheth-Tormen and Jenkins mass functions. At a = 1, the Sheth-

Tormen prescription provides a better fit. At a = 100, the pure Press-Schechter function seems

to be marginally better. This may tie in with the more anisotropic shape of superclusters at a = 1

in comparison to their peers at a = 100.

7.8. CONCLUSIONS 153

• While the large scale evolution of superclusters comes to a halt as a result of the cosmic accel-

eration, their internal evolution continues at least up to a = 100.

• As a result of their collapse, the shape of the bound objects appears to change from prolate at a=

1 into almost spherical at a = 100. We find that at a = 1 their mean axis ratio are (〈b/a〉,〈c/a〉)=(0.69,0.48). At a = 100, they have mean axis ratios of (〈b/a〉,〈c/a〉)= (0.94,0.85).

• The change in the internal mass distribution and that in the surroundings is directly reflected by

the radial density profile. The inner density profile steepens substantially when the inner region

of the supercluster is also still contracting. On the other hand, when at a = 1 it has already

developed a substantial virialized core, the inner density profile hardly changes.

• The mass profile in the outer realms of the supercluster always changes radically from a =

1 to a = 100. At a = 1 it shows an irregular increase as a function of radius, reflecting the

surrounding inhomogeneous mass distribution of the Cosmic Web. By a = 100 the superclusters

have developed a smooth, regular and steadily increasing mass profile.

• At the current epoch the superclusters still contain a substantial amount of substructure. Par-

ticularly interesting is the amount of cluster mass objects within its realm, expressed in the

so called multiplicity function. Restricting ourselves to superclusters with a mass larger than

5×1015h−1M⊙, of which we have 17 in our simulation sample, we find a multiplicity of 5 to 15

at the current epoch. By contrast, all these have evolved into concentrated singular mass clumps

of unit multiplicity.

• In a region of a volume comparable to the Local Universe (z < 0.1) we find that the most massive

supercluster would have a mass of ∼ 8×1015h−1M⊙. This is slightly bigger than the mass of the

Shapley Supercluster given in Dunner et al. (2008). Also, we find 2 Shapley-like superclusters.

These host between 10 to 15 members, almost half of those estimated by Quintana et al. (2000).


7.A Press-Schechter formalism and its variants

Mass functions given by numerical simulations are good approximations. Yet they cover a limited

volume, given by the size of the simulation box. There is an excellent analytic description, the Press-

Schechter formalism (Press & Schechter 1974). It provides a simple but powerful way to calculate the

number density of objects of a given mass and at any redshift. The Press & Schechter (PS) formalism

and its extensions has been extensively studied and compared to numerical simulations (see the works

by, e.g., Bond et al. (1991),Lacey & Cole (1993), Sheth & Tormen (1999), Jenkins et al. (2001)).

It considers the emergence of collapsed objects from a primordial Gaussian random density field.

Such primordial circumstances have been confirmed by the mapping of the temperature fluctuations

in the Cosmic Microwave Background (e.g., Spergel et al. 2003). Moreover, inflationary theories of

cosmology do predict such primordial fluctuations.

Let us consider spheres of radius r = (3M/4πρb)1/3, where ρb is the mean density of the Universe

and M is some mass scale of interest. We also define the overdensity as

δ =ρ−ρb

ρb

. (A-1)

The fractional rms mass fluctuation is

σ(M) =

√

〈δM2〉M

. (A-2)

Recall that the primordial density perturbations are assumed to be Gaussian fluctuations. Thus,

the phases of the waves that make up the density distribution are random, and the distribution of the

overdensities δ in spheres of radius r can be described by a Gaussian function

p(δ) =1

√2πσ(M)

exp

[

−δ2

2σ2(M)

]

. (A-3)

At a given time, the fraction of points which are surrounded by a sphere of radius r within which the

mean overdensity exceeds some density threshold δc is given by

F(δc,M) =1

√2πσ(M)

∫ ∞

δc

exp

[

−δ2

2σ2(M)

]

dδ . (A-4)

Changing variables, u = δ/(√

2σ), we can express the latter equation in the form

F(δc,M) =1√π

∫ ∞

δc√2σ(M)

e−u2

du =1

2erfc

(

δc√2σ(M)

)

. (A-5)

From Eqn. A-5, we can obtain the comoving number density of halos of mass M, the Press-Schechter

mass function:

dn

dM= 2ρb

M

∣

∣

∣

∣

∣

∂F

∂M

∣

∣

∣

∣

∣

=

√

2

π

ρb

M2

δc

σ(M,z)

∣

∣

∣

∣

∣

dlnσ(M,z)

d ln M

∣

∣

∣

∣

∣

e−δ2c

2σ2(M,z) . (A-6)

Note that the (notorious) factor 2 was inserted in order to account for the matter in underdense regions,

which also eventually falls into overdense ones. Later, the factor has been physically explained within

the extended Press-Schechter formalism proposed by Bond et al. (1991). Since, most theoretical work

along these lines has followed their excursion set formalism.

The quantity δc/σ represents how many standard deviations away from the mean amplitude a

positive density perturbation must lie in order to collapse, ρb is the mean background density, and δcis the effective linear overdensity required for the collapse.

7.A. PRESS-SCHECHTER FORMALISM AND ITS VARIANTS 155

In order to estimate the mass function from the Press-Schechter formalism, we need to specify

σ(M) and δc. We can express the variance of the density fluctuations in terms of the power spectrum

P(k) of the linear density field,

σ2(M) = 4π

∫ ∞

0

P(k)ω(kr)k2dk , (A-7)

where ω(kr) is the Fourier space representation of a real-space top-hat filter enclosing a mass M in a

radius r at the mean density of the Universe, which is given by

ω(kr) = 3

[

sin(kr)

(kr)3− cos(kr)

(kr)2

]

. (A-8)

The value of δc has a weak dependence on Ωm. A good numerical approximation is given by Navarro

et al. (1997):

δc(Ωm) =

0.15(12π)2/3Ω0.0185m if Ωm < 1 and ΩΛ = 0,

0.15(12π)2/3Ω0.0055m if Ωm+ ΩΛ = 1.

(A-9)

For a similar expression for the critical linear virial density see Bryan & Norman (1998).

The Press-Schechter mass function has been extensively tested against N-body simulations and

was shown to be in reasonable agreement with them (e.g., Efstathiou et al. 1988; Lacey & Cole 1994;

Governato et al. 1999).

The standard (extended) PS formalism assumes perfectly spherical collapse. However, we know

even on purely theoretical grounds that there are no spherical primordial density peaks (Bardeen

et al. 1986). Sheth & Tormen (1999) improved the PS formalism by taking into account the implied

anisotropic collapse of density peaks on the basis of the ellipsoidal collapse model (e.g., Icke 1973).

Implicitly, this also involves the anisotropic tidal stresses imparted by external mass concentrations.

They showed that this implies a more fuzzy “moving collapsed barrier”. The resulting mass function,

dnS T

dM= A

√

2a

π

[

1+

(

σ(M)2

aδ2c

)p]ρb

M2

δc

σ(M)

∣

∣

∣

∣

∣

dlnσ(M)

d ln M

∣

∣

∣

∣

∣

e−aδ2c

2σ2(M) , (A-10)

with a = 0.707, p = 0.3 and A ≈ 0.322, gives a substantially better fit to the mass functions obtained

in N-body simulations. In comparison with the standard PS mass function, ST predicts a higher abun-

dance of massive objects and a smaller number of less massive ones. Later, Jenkins et al. (2001)

reported a small disagreement with respect to N-body simulations: underpredictions for the massive

halos and overpredictions for the less massive halos. They suggested the alternative expression:

dnJ

dM= Aρb

M2

dlnσ(M)

d ln Me(−| lnσ−1+B|)ǫ . (A-11)

with A = 0.315, B = 0.61 and ǫ = 3.8. Note, however, that their expression does not depend explicitly

on δc. They showed that “for a range of CDM cosmologies and for a suitable halo definition, the

simulated mass function is almost independent of epoch, of cosmological parameters, and of initial

power spectrum”.


Nederlandse Samenvatting

Vroege beschavingen hebben zich al afgevraagd waar alles vandaan kwam en hoe alles is begonnen.

Eeuwenlang heeft de mensheid zich afgevraagd wat zijn positie is in dit onmetelijke Heelal waar-

in we leven. Nauw verbonden met deze fundamentele vraag is, is de vraag hoe het Heelal is ontstaan.

Vroege civillizaties hadden een mythische voorstelling van het Heelal. Bijvoorbeeld de Mesopota-

mische beschavingen, de Sumerische en Babylonische, hadden een kosmogonie waarin de Aarde een

schijf is, die is omgeven door de ondergrondse wateren van Apsu en de onderwereld van de doden, om-

geven door de sterren. Ondanks het hoge niveau van de astronomische kennis in de neo-Babylonische

wereld en zijn erfgenamen, is het tentijde van de Oude Grieken dat de astronomie en de kosmologie

in een wetenschappelijk context wordt gezet. Voortbouwend op de nauwkeurige waarnemingen en ar-

chieven van de Babyloniers worden in het Hellinistische Griekenland de eerste geometrische modellen

ontwikkeld die in staat zijn kwantitatieve voorspellingen te doen. Zoals Eratosthenes die de omtrek

van de Aarde mat en Aristarchus met zijn bepaling van de afmeting en afstand van de Aarde en de

Maan. De laatste kwam zelfs met de suggestie dat de Zon in het centrum moest worden gezet, een

wereldbeeld die pas met Copernicus en Galilei in de 15de en 16de eeuw werd geaccepteerd.

Het is de Wetenschappelijke Revolutie van de 16de en de 17de eeuw die de ontwikkeling van een

echt wetenschappelijk model mogelijk maakt. Met Nicolai Copernicus in 1543 verliest de Aarde uit-

eindelijk zijn centrale positie in het Heelal. Gebruik makend van het werk van Copernicus, Brahe,

Kepler en Galilei, was het Isaac Newton die hier voor het eerst de wetten van de zwaartekracht en

mechanica in wist te ontdekken. Deze vormen de fundamenten van de klassieke fysica. Echter, deze

wetten beschrijven een statische wereldbeeld met een zwaartekracht die op afstand werkt, en het is

onmogelijk om hiermee het huidige dynamische kosmologisch wereldbeeld te beschrijven. Pas met de

ontwikkeling van de Algemene Relativiteits Theorie door Einstein, kon de kosmologie in een weten-

schappelijke context worden geplaatst. In deze metrische theorie van gravitatie is de ruimte-tijd een

dynamisch medium, waarin gravitatie voortkomt uit de kromming van de ruimte-tijd. Spoedig reali-

seerde wat voor implicaties dit had voor het Heelal. Het was geen statisch geheel maar kon krimpen

en/of expanderen. Friedmann en Lemaıtre zijn de eersten die dit hebben uitgewerkt voor een homo-

geen en isotroop expanderend Heelal. De bevestiging van deze theoretische ideeen volgde al vlug met

de fundamentele ontdekking van Edwin Hubble in 1929 dat melkwegstelsels van ons afbewegen. De

wet van Hubble stelt dat de expansiesnelheid van een ver melkwegstelsels proportioneel is met zijn

afstand, dit is nog steeds de grondslag voor het huidige kosmologische onderzoek.

Big Bang Theorie en Inflatie

Het was Lemaıtre die de volledige gevolgen begreep van een uitdijende Heelal. Het vroege Heelal

zou een stuk kleiner moeten zijn, en daarom een stuk dichter en heter dan het huidige Universum. Dit

wordt met de term Big Bangtheorie/Oerknaltheorie aangeduid, wat doelt op het feit dat het Heelal op

een zeker moment in het verleden is ontstaan. De leeftijd van het Heelal wordt op dit moment op 13.7

miljard jaar geschat. Ondanks het overweldigende succes van de standaard Big Bangtheorie, kan het

niet de oorsprong van structuur in het Heelal verklaren.

158 Nederlandse Samenvatting

Een mogelijke oplossing ligt in de aanname dat het vroege Heelal een phase van exponentiele ex-

pantie is ondergaan. Tijdens deze kosmische inflatie, 10−34 second na de Big Bang, werd het Heelal in

een zeer kort tijd een factor 1060 groter. Inflatie kan een verklaring geven voor de platte geometrie van

het Heelal, de uniformiteit van het stralingsveld en de oorsprong van kleine schommeling in dichtheid

van materie. Deze verstoringen (oorspronkelijk op een subatomische schaal) zijn de kiemen waaruit

sterren, melkwegstelsels en clusters van melkwegstelsels zijn ontstaan.

Figuur 1 — The Big Bangtheorie. Een schematische voorstelling van het Heelal sinds de Big Bang.

Kosmische AchtergrondStraling

Terwijl de eerste structuren zich begonnen te vormen, waren materie en straling nog aan elkaar gekop-

peld. Het Universum was donker door de sterke interacties tussen fotonen en baryonen. Dit stadium

duurde ongeveer 379000 jaar voort, en eindigde toen het Heelal koud genoeg was om stabiele atomen

te vormen. Het Heelal, toen afgekoeld tot ongeveer 3000 Kelvin, werd ineens transparant; fotonen

waren plots in staat om zich ongehinderd te verplaatsen. Deze fotonen kunnen nog steeds worden

waargenomen, en vormt een stralingsveld dat zich in het gehele Heelal bevindt. Het staat bekend als

de Kosmische Achtergrond Straling, en heeft een spectrum van een perfecte zwarte lichaamstraler met

een temperatuur van T = 2.755 K. Deze achtergrondstraling is een van de belangrijkste bewijzen voor

de Big Bang.

Donkere Materie

Toen het Universum was geexpandeerd tot het de omvang had van 1/1090 keer de huidige grootte,

begon materie de dynamica van het Heelal te domineren. De gravitatiekracht van de donkere materie

is verantwoordelijk voor de vorming van structuren in het Heelal. Deze bijzondere vorm van materie

is ongevoelig voor elektromagnetische krachten, waardoor het vrijwel onzichtbaar en dus moeilijk te

bestuderen is. De huidige waarnemingen doen vermoeden dat ongeveer 85% van al de materie uit

donkere materie bestaat. Het is een grote uitdaging voor de astronomie om deze getals verhoudingen

te verklaren.

Donkere Energie

Een nog vreemdere component van het Heelal is de aanwezigheid van een hele bijzondere vorm van

energie; de donkere energie. Het feit dat het een afstotende gravitatie-kracht heeft, is het enige wat er

momenteel van bekend is. Waarnemingen wijzen uit dat 73% van de hoeveelheid energie voor rekening

van de donkere energie komt. Dit aandeel is precies genoeg om er voor te zorgen dat het Heelal een


platte geometrie heeft. De waargenomen versnelde expansie, veroorzaakt door de donkere energie, kan

het best worden verklaard met behulp van een kosmologische konstante, die voorkomt in Einstein’s

Relativiteits Theorie. Wanneer de donkere energie de dynamica van het Universum domineert, zijn

structuren niet in staat om nog verder door te groeien. In het Heelal is dit op een expansiefactor van

a ∼ 0.7 gebeurd.

Grote Schaal Structuur van het Heelal

De Big Bangtheorie alleen beantwoordt wel de vraag over de oorsprong van het Heelal, maar geeft

geen antwoord op hoe structuren in het Heelal zijn ontstaan. De groei van deze structuren is het

resultaat van ineenstorting van kleine dichtheidsschommelingen onder invloed van de zwaartekracht.

Als zo’n schommeling een grotere dichtheid heeft dan zijn omgeving is het in staat om in elkaar te

vallen. Deze kleine klompjes begonnen geleidelijk samen te klonteren om steeds grotere objecten te

vormen. De eerste objecten in het Heelal waren van subgalactische schaal. Zij hebben zich ontkoppeld

aan de Hubble expansie en konden toen ineenstorten. De samenklontering van deze objecten leidde

uiteindelijk tot de vorming van de eerste sterrenstelsels, die op hun beurt weer verder samenklonteren

om clusters van melkwegstelsels te vormen. In dit hierarchisch proces zijn clusters van sterrenstelsels

de laatste en meest massieve ineengestorte objecten. Dus de tijd die nodig is voor clusters om in elkaar

storten is ongeveer van dezelfde orde grootte als de leeftijd van het Heelal. Dit maakt clusters van

sterrenstelsels tot essentiele onderdelen om de evolutie en vorming van het Universum te bestuderen.

Clusters van Sterrenstelsels

De grootste stabiele objecten in het Heelal zijn clusters van melkwegstelsels. De typische kenmerken

van clusters zijn:

• Ze bevatten 50 tot 1000 melkwegstelsels, heet gas en grote hoeveelheid donkere materie.

• Ze hebben een totale massa van ongeveer ∼ 1014−1015h−1M⊙. 1.

• Hun afmeting is van orde grootte ∼2-6 h−1Mpc2.

• De sterrenstelsels in clusters hebben een snelheidsdispersie van ca. 800-1000 km/s.

Sterrenstelselclusters spelen een sleutelrol in het verkrijgen van onze huidige kennis en begrip van de

Grote SchaalStructuur. De aanwezigheid van donkere materie werd voor het eerst in clusters ontdekt.

Ook zijn ze bijzonder heldere X-ray bronnen. Dit wordt uitgestraald door het ijle hete gas, dat zich

in de clusters bevindt en een typische temperatuur heeft van ongeveer T ∼ 107 −108 K. Verder zijn de

clustersterrenstelsels van een zeer bijzondere soort, waardoor ze belangrijke informatie opleveren over

de evolutie van melkwegstelsels.

Clusters zien er optisch uit als een verzameling van melkwegstelsels die bij elkaar worden ge-

houden door hun onderlinge zwaartekracht. Echter, de gemeten snelheden van de stelsels zijn veel te

groot, en de zwaartekracht van de stelsels alleen zou nooit genoeog zijn om het bij elkaar te houden.

Dit impliceert dat er dus nog meer massa moet zijn die we niet zien of dat er nog een extra kracht is

die ervoor zorgt dat clusters niet uit elkaar vliegen. Gebleken is dat sterrenstelsels slechts ∼5% van

de totale massa uit maken, ∼10% komt voor rekening van het X-ray stralende gas en het overgrote

gedeelte bestaat uit donkere materie.

Aangezien cluster uitvoerig zijn bestudeerd en daardoor goed begrepen zijn, kunnen we ons afvra-

gen of de aanwezigheid van donkere energie via clusters valt aan te tonen. Dit is de hoofddoelstelling

van dit proefschrift.

11M⊙=1.989×1030 kg, is de massa van de Zon21 Mpc=3.086×1022 mts.


In dit proefschrift

In dit proefschrift hebben we gekeken naar de invloed die de donkere energie heeft op de vorming en

evolutie van clusters. Daaruit is gebleken dat de kosmologische constante niet of nauwelijks invloed

heeft op de globale en individuele eigenschappen van clusters van sterrenstelsels. Wel heeft de total

hoeveelheid materie in het Heelal een significante invloed op de evolutie van clusters.

Kosmologische Simulaties en Massa Distributies

Hoofdstuk 2 biedt een uitgebreide beschrijving van de kosmologische simulaties die we voor dit proef-

schrift hebben gebruikt. Deze nummerieke simulaties omvatten zes open, vier kritische en drie geslo-

ten Universa, zowel met als zonder kosmologische constante. De afmeting (200 Mpch−1) en het aantal

deeltjes (2563) is voor elke periodieke simulatiekubus hetzelfde gehouden, dit geldt tevens voor de

Hubble-expansie en de normalisatie van het spectrum. De simulaties zijn gestart op een roodverschui-

ving van 49, en liepen door tot de huidig kosmische tijd.

Om een onderlinge vergelijking mogelijk te kunnen maken tussen de simulaties die verschillende

kosmologieen representeren, zijn de Fourier-fases van het beginveld hetzelfde gehouden. Hierdoor

zijn we ervan verzekerd dat de structuren op grote schaal in alle simulaties hetzelfde zijn.

Figuur 2 laat deze eigenschap zien. We tonen doorsnedes door het midden van de kubus voor drie

verschillende modellen; ΛCDMF2 (een kritisch model, boven), ΛCDMO2 (een open model, midden)

en ΛCDMC2 (een gesloten model, onder). Op grote schaal zien we hetzelfde patroon. De verschillen

tussen de modellen zijn echter te zien in de mate van clustering. De open modellen, dus met een lage

dichtheid, bevatten veel minder details dan de andere twee. Dit is waarschijnlijk het gevolg van de

extreem lage materiedichtheid in combinatie met een kosmologische constante. Hierdoor stopte de

groei van structuur al op een zeer vroeg tijdstip. De uitvergrotingen aan de linker zijde van de figuur

benadrukken dit verschil. Modellen met een hoge materie dichtheid laten een meer geevolueerde

structuur zien, met een hogere mate van clustering.

We hebben ook de massaverdeling van objecten in de simulaties bestudeerd. Deze verdeling re-

presenteert het aantal objecten per volume-eenheid voor een gegeven massa. Uit dit onderzoek blijkt

dat de massaverdeling inderdaad gevoelig is voor de totale hoeveelheid materie in het Heelal. Echter,

de kosmologische constante heeft op de huidige verdeling geen significante invloed. Wel vinden we

duidelijke verschillen op hogere roodverschuiving. De oorzaak hievan is dat een verandering van de

kosmologische constante zorgt voor andere dynamische tijdschalen.

Massa groei en virialisatie van clusters

In Hoofdstuk 3 beschrijven we de vorming en virializatie van clusterobjecten. We hebben eerst geke-

ken naar de vormingsgeschiedenis van een aantal identieke clusters door te kijken naar het verloop in:

roodverschuiving, terugkijktijd, en kosmische tijd. De gevonden verschillen zijn volledig te verklaren

door de verandering van de materiedichtheid in de verschillende modellen. Net als de massaverdeling

is de kosmologische constante alleen merkbaar via zijn invloed op de kosmische tijd.

Een belangrijke eigenschap van clusters is de accretiegeschiedenis. We hebben dit onderzocht

door zowel te kijken naar een aantal individuele clusters als wel naar de gemiddelde eigenschappen

van clusters in verschillende modellen. Als we kijken naar een enkele object dan blijkt dat accretie-

en samenklonteringsprocessen duidelijk van invloed zijn op de ontwikkeling van een cluster. Tot op

zekere hoogte wordt dit gereguleerd door de hoeveelheid materie die een heelalmodel bevat. Model-

len met een lage dichtheid hebben de meeste groei in het vroege heelal via zeer zware mergers. In

tegenstelling tot universums met een hoge dichtheids, waar zulke mergers op het huidige tijdstip vaker

voorkomen.

De spreiding rond de gemiddelde accretiegeschiedenis is zeer groot. Daardoor is het mogelijk dat

de trends in de accretiegeschiedenis uit verschillende simulaties met elkaar overlappen. Het zal daarom

zeer moeilijk zijn om de subtiele invloed van de kosmologische constante hierin te ontdekken.


Figuur 2 — Drie verschillende simulaties: ΛCDMF2 (boven), ΛCDMO2 (midden) en ΛCDMC2 (onder). De

inzet rechts toont een uitvergroting van het gebied links.


We hebben ook gekeken naar de virialisatie van halo’s en in het bijzonder clusterhalo’s. Het blijkt

dat de halopopulatie in alle modellen op het huidige tijdstip nagenoeg volledig is gevirializeerd. Echter,

de gevonden waarden zijn niet perfect in overeenstemming met de theoretische relatie voor bolvormi-

ge objecten. Met name lichtere halo’s vertonen een grote spreiding ten opzichte van deze relatie. De

spreiding bij zwaardere clusters halos is daar en tegen veel minder. Onafhankelijk van de kosmolo-

gische achtergrond volgen alle clusters dezeldfe viriaalrelatie. Wel heeft de kosmologie invloed op

de spreiding die we vinden ten opzichte van deze viriaalrelatie, die is groter voor modellen met een

hogere masssadichtheid.

Fysische eigenschapen van Clusters

De fysische eigenschappen van clusters zijn in Hoofdstuk 4 onderzocht. Hier hebben we gekeken naar

de invloed van een positieve kosmologische constante op de interne verdeling van de massa, de vorm,

de morphologie en het hoekmoment. De interne ruimtelijke verdeling van de massa kan worden be-

schreven door middel van een dichtheidsprofiel. Wij hebben gevonden dat dit profiel hetzelfde gedrag

vertoont in alle simulaties. Wel blijken de halo’s in hogere dichtheids modellen meer geconcentreerd

te zijn. Deze mate van concentratie is niet afhankelijk van de waarde van de kosmologische constante.

De morfologie van halo’s kan goed beschreven door een triaxiale vorm, waarbij er een voorkeur is

voor een sigaarvormige verhouding (twee korte assen en een lange as). De mate waarin een object van

een bepaalde massa afwijkt van een bolvorm is afhankelijk van de kosmologie. Clusters in een heelal

met een hoge dichtheid hebben een grotere afwijking en zijn dus minder bolvormig. Tot slotte vinden

we dat het hoekmoment van halo’s toeneemt, wat sterk gerelateerd is aan een toename van de massa.

Schalingsrelaties

De structuur en dynamica van (bijna) gevirialiseerde objecten als clusters van sterrenstelsels valt te

verklaren uit een aantal fundamentele schalingsrelaties tusssen; massa, afmeting en kinematica (snel-

heidsdispersie). In hoofdstuk 5 kijken we naar de Kormendy, de Faber-Jackson en de Fundamental

Plane relaties. Alhoewel deze relaties eerst werden ondekt voor elliptische melkwegstelsels Schaeffer

et al. (1993), blijken ze ook voor clusters goed te gelden. Wij vinden dat de clusters in alle simulaties

voldoen aan de Kormendy, Faber-Jackson en Fundamental Plane relaties. De Kormendy en de Faber-

Jackson vertonen geen afhankelijkhid van de kosmologische constante, wel zijn ze in mindere mate

afhankelijk van de totale dichtheid van het Heelal. In de Fundamental Plane relatie is deze afhanke-

lijkheid nog beter terug te vinden. De breedte van de Fundamental Plane schaalt bijna perfect met de

materiedichtheid van het Heelal. Dit blijkt een directe afspiegeling te zijn van de mate van virialisatie

van clusters. De evolutie van de Kormendy en de Faber-Jackson relatie geeft een indicatie waar de

clusters zich in het Fundamental Plane bevinden.

Toekomstige evolutie van het Heelal

N-deeltjes simulaties zijn een noodzakelijk gereedschap voor het onderzoek naar structuur in het Heel-

al. Ze representeren realistische beschrijvingen van de vorming en evolutie van structuren in het Heel-

al. Ze stellen ons ook in staat om de invloed te bestuderen die de kosmologische constante heeft op

de toekomstige evolutie in het Heelal. In hoofdstuk 6 bestuderen we de toekomstevolutie van clusters

van sterrenstelsels. Figuur 3 laat zien hoe een object in een expanderend Heelal er uit zal komen te

zien. In de bovenste afbeelding laten we de evolutie in comoving coordinaten zien, en in de onderste

de evolutie in fysische coordinaten. De fysische coordinaten laten zien dat het object nagenoeg even

groot blijft, terwijl in comoving coordinates het object krimpt, tot het uiteindelijk niet meer zichtbaar

zal zijn door de expansie van het Heelal.

De massaverdeling van objecten (het aantal objecten van een bepaalde massa) zal in de nabije

toekomst constant blijven aangezien er geen groei van structuur is ’uitgevroren’. Gevolg is dat er nau-

welijks nog verschil is tussen de mass-accretiegeschiedenis in de verschillende modellen. De evolutie


in de toekomst beperkt zich tot het steeds bolvormiger worden van de objecten, waarin ze een pefect

bolvormige stadium bereiken in de verre toekomst. Doordat er geen of nauwelijk nog massa-accretie

plaats vindt, blijft het hoekmoment ongeveer constant. Bovendien zullen de clusters hierdoor in hoge

mate gevirialiseerd zijn. Dit valt ook te zien in de schalingsrelaties, waarin veel minder spreiding is

te zien. De breedte van de Fundamental Plane is smaller dan op het huidige kosmologische tijdstip

en is nagenoeg hetzelfde voor elke simulatie. Dit duidt er op dat clusters genoeg tijd hebben gehad

om te virializeren en in dynamische evenwicht te komen. We concluderen hieruit dat het in de verre

toekomst bijzonder moeilijk zal zijn om na te gaan in wat voor een Heelal we leven.

Figuur 3 — Evolutie van een enkele in een kritisch Heelal. Boven: evolutie in comoving coordinaten. Onder:

evolutie in fysische coordinaten.

Supercluster of galaxies

Identificatie van superclusters van melkwegstelsels is bijzonder moeilijk. Deze objecten zijn de groot-

ste structuren in de hierarchie van het Heelal, en zijn pas net aan het vormen. Hier passen we een cri-

terium toe waarin superclusters worden geidentificeerd met gravitationeel gebonden objecten, Dunner

et al. (2006). We gebruiken dit als een fysische definitie voor superclusters van sterrenstelsels. Met be-

hulp van hun gemeten massaverdeling en de aanname dat dit de meest massieve structuren zijn, zou het

meest massieve object in het Lokale Universum een massa van ∼ 8×1015h−1M⊙ moeten hebben. Dit

is slechts iets massiever dan de gemeten massa van de Shapely Supercluster (de grootste concentratie

van melkwegstelsels in het Lokale Universum), zie ook Dunner et al. (2008). Op basis van ons model

zouden we 2 objecten in het Lokale Universum verwachten met dezelfde massa en grootte als de Sha-

pely Supercluster. De morfologie van superclusters wordt gekemerkt door een zeer veel substructuur,

en een globale structuur die langgerekt van vorm is.

Conclusies

In dit proefschrift hebben we de invloed onderzocht die de kosmologische constante heeft op de vor-

ming en evolutie van clusters van sterrenstelsels. Hiertoe, hebben we gebruik gemaakt van een grote

set van kosmologische simulaties die uit drie geometrische modellen kunnen bestaan: open, kritische

en gesloten. De simulaties bestaan telkens uit twee versies waarbij in de ene wel en in de andere geen

kosmologische constante voorkomt. Op deze manier hebben we geprobeerd beter te achterhalen wat


de invloed is van de kosmologische constante op de structuur, dynamica en evolutie van clusters van

melkwegstelsels.

Verschillende onderwerpen over de globale en individuele eigenschappen zijn hierbij onderzocht.

Hieruit hebben we geconcludeerd dat de kosmologische constante geen invloed heeft op de vorming

en evolutie van clusters. Het enige aantoonbaar effect is de invloed die het heeft op de kosmische tijd:

het verlengt of verkort de beschikbare dynamische tijd voor evolutie van clusters.

Resumen en Espanol

Cvilizaciones antiguas se han estado preguntando de donde viene todo y como todo comenzo. Por

siglos, la humanidad se ha estado preguntando cual es nuestro lugar en el extenso espacio que

nos rodea. Unido a esto estaba la pregunta fundamental de como el mundo se formo. Las antiguas

civilizaciones tenıan una vision mıstica del cosmos. Por ejemplo, las civilizaciones Mesopotamicas de

Sumeria y Babilonia tenıan una cosmologıa en la cual la Tierra era un disco rodeado por las aguas

subterraneas del Apsu y el bajo mundo de los muertos, con los cielos de las estrellas rodeando todo. A

pesar del nivel altamente sofisticado de la astronomıa en el mundo neo-Babilonico y sus descendientes,

fue con los Griegos que la astronomıa y la cosmologıa alcanzaron el nivel de investigacion cientıfica.

Basados en las cuidadosas observaciones obtenidas y archivadas cuidadosamente por los Babilonios,

los Griegos helenısticos fueron los primeros en utilizar sus modelos geometricos para predecir la rea-

lidad observacional, creando un modelo tanto cuantitativo como cualitativo del Universo. Eratostenes

midio la circunferencia de la Tierra, mientras que Aristarco midio el tamano y la distancia de la Tie-

rra y la Luna. El incluso sugirio que el Sol estaba en el centro de nuestro mundo, una vision que se

acepto con Copernico y Galileo en el siglo XV y XVI.

Fue con la Revolucion Cientıfica del siglo XVI y XVII que el verdadero modelo cientıfico del

Universo se formo. Con Nicola Copernico en 1543, la Tierra finalmente perdio su privilegiada posicion

central en el cosmos. Fue Isaac Newton, apoyado en los hombros de Copernico, Brahe, Kepler y

Galileo, quien enmarco las leyes de la gravedad y la mecanica. Estas fueron las bases de la fısica

clasica.Su vision del espacio-tiempo estatico y de la fuerza gravitacional como resultado de la accion

a distancia, hizo imposible ampliar la vision del dinamico mundo cosmolgico que hoy tenemos. Fue

la Teorıa General de la Relatividad de Einstein la que genero el descubrimiento final que convirtio la

cosmologıa en una inquietud cientıfica. Su teorıa metrica de la gravedad, convirtio el espacio-tiempo en

un medio dinamico en el cual la gravedad es una manifestacion de la curvatura espacio-tiempo. Pronto

se entendio que esto implica que el Universo no podrıa ser estatico sino en expansion o contraccion.

Firedmann y Lemaıtre fueron los primeros en entender las soluciones de expansion para un Universo

homogeneo e isotropico. Su ideas teoricas pronto fueron confirmadas con el descubrimiento de Edwin

Hubble, en 1929, del sistema en expansion de galaxias que nos rodea. Su “Ley de Hubble”, que descrbe

el hecho de que galaxias distantes se estan alejando con velocidades proporcionales a su distancia,

sigue siendo la base fundamental de la cosmologıa actual.

Teorıa del Big Bang e Inflacion

Fue Lemaıtre quien noto las tremendas implicaciones de este descubrimiento. En el pasado, el Universo

pudo haber sido mucho mas pequeno, mucho mas denso y mucho mas caliente que el Universo actual.

Esto dio lugar a la Teorıa del Big Bang, indicando el hecho de que el Universo se formo en un cierto

punto finito en el pasado. Ahora sabemos que el cosmos se formo hace 13.7 mil millones de anos en

un mar de radiacion y materia. Aun a pesar de sus enormes exitos, la Teorıa del Big Bang no explica

el origen de la estructura del Universo. Esto se puede solucionar si el Universo experimento una fase

inflacionaria de expansion exponencial. Durante esta inflacion cosmica, 10−34 segundos despues del

166 Resumen en Espanol

Big Bang, el Universo crecio en un factor de 1060.

La inflacion puede explicar la planariedad del Universo, la uniformidad de su radiacion y el ori-

gen de la inhomogenidad primitiva de la materia. Estas inhomogeneidades (inicialmente a escalas

subatomicas) fueron las semillas de donde estrellas, galaxias y cumulos de galaxias se formaron.

Figura 1 — La Teorıa del Big Bang. Una representacion grafica de la historia del Universo desde el Big Bang.

Cortesıa de M. Norman.

Fondo Cosmico de Microondas

Cuando las estructuras se comenzaron a formar, materia y radiacion estaban acopladas. El Universo

era oscuro como consecuencia de la fuerte interaccion entre fotones y bariones. Este estado se mantuvo

durante los siguientes 379 000 anos. Finaliz’øcuando la temperatura del Universo se enfrio a unos 3000

K. En este momento, el Universo se hizo transparente y los fotones pudieron viajar libres a traves del

Universo. Estos fotones aun pueden observarse hoy, como una radiacion que ocupa todo el Universo.

Esto es conocido como el Fondo Cosmico de Microondas (CMB, por sus siglas en ingles). Su espectro

es de un cuerpo negro, con una temperatura de T = 2,755 K. El CMB es una de las evidencias mas

importantes del Big Bang.

Materia Oscura

Cuando el Universo tenia sim1/1090 de su tamano actual, la materia comenzo a dominar la dinami-

ca del Universo. La influencia gravitacional de una misteriosa materia oscura es responsable de la

formacion de las estructuras que vemos hoy. Esta extrana forma de materia es insensible a la fuerza

electromagnetica, haciendola invisible y, por lo tanto, muy difıcil de estudiar. Se piensa que represen-

ta mas del 85 % de materia en el Universo. Esta gran cantidad es uno de los mayores desafıos de la

cosmologıa actual.

Energıa Oscura

Aun mas misteriosa es la presencia de otro tipo de energia, la energia oscura. Todo lo que podemos

decir es que la energıa oscura tiene una fuerza gravitacional repulsiva. Las observaciones indican

que representa casi el 73 % de la energıa del Universo. Es responsable de la acelerada expansion del

Universo y asegura su geometria plana. Las observaciones tambien tienden a senalar que la energıa

oscura esta en la forma de una constante cosmologica, que aparecio en la Teorıa de la Relatividad de

Einstein. Cuando la constante cosmologica asume el control de la evolucion del Universo, que sucede

en un redshift de z ∼ 0,7, el crecimiento de estructura se congela.


Estructura a Gran Escala del Universo

La Teorıa del Big Bang puro responde la pregunta del origen del Universo, pero no provee una res-

puesta sobre como las estructuras que vemos hoy se formaron. El crecimiento de las estrcturas es el

resultado del crecimiento gravitacional de minusculas fluctuaciones primordiales (aparecidas inmedia-

tamente despues de la inflacion) en el Universo primordial. Una fluctuacion dada, en la que la densidad

es mas alta que su alrededor, colapsara. Los pequenos grupos materia se combinan y crecen gradual-

mente mientras que se ensamblan en estrcturas aun mas grandes. Los objetos subgalacticos pequenos

son los primeros en formarse en el Universo. Se desacoplan de la expansion del Universo y colapsan.

Estos pequenos objetos se combinan y dan a luz galaxias. El proceso continua, conduciendo al ensam-

blaje de halos del tamano de galaxias, en cumulos de galaxias. Dentro de esta evolucion jerarquica, el

cumulo de galaxia sobresale como el mas grande y como los objetos en el Universo que han colapsado

mas recientemente. La mayorıa se encuentra actualmente en un estado de relajacion y de virializacion.

Su tiempo de colapso es comparable con la edad del Universo. Estas propiedades hacen de los cumulos

de galaxias laboratorios indispensables para el estudio de la evoluci’on y formaci’on del Universo.

Cumulos de Galaxias

Cumulos de galaxias son las estructuras estables mas grandes del Universo. Las propiedades tıpicas de

los cumulos de galaxias incluyen:

• Contienen 50 a 1000 galaxias, gas caliente y grandes cantidades de materia oscura.

• Tienen masas totales de ∼ 1014−1015h−1M⊙1.

• Sus radios son del orden de ∼2-6 h−1Mpc2.

• Galaxias miembros tienen velocidades de dispersion en el orden de ∼800-1000 km/s.

Cumulos de galaxias han sido objetos astrofısicos claves en el desarrollo de nuestra actual per-

cepcion de la estructura a gran escala del Universo. Fue en los cumulos de galaxias donde primero se

detecto la materia oscura. Cumulos son tambien fuentes de rayos X muy luminosos, emitido por un

tenue gas intra-cumulo extremadamente caliente con una temperatura de T ∼ 107 − 108 K. El hecho

que contengan una mezcla atıpica de galaxias los hace importantes centros de pruebas para el estudio

de evolucion de galaxias.

Cuando son observados visualmente, los cumulos de galaxia aparecen como una coleccion de

galaxias ligadas por su atraccion gravitacional mutua. Sin embargo, sus velocidades son demasiado

grandes para que sigan ligadas gravitacionalmente por su atraccion mutua. Esto implica que debe haber

un componente de masa invisible adicional o una fuerza atractiva adicional ademas de la gravedad. La

mayorıa de la masa de los cumulos de galaxias esta en forma de gas caliente, que emite en rayos X. En

un cumulo tıpico, tal vez solo un ∼5 % de la masa total esta en forma de galaxias, ∼10 % en forma de

gas caliente emitiendo en rayos X y el resto esta en forma de materia oscura.

Dado que los cumulos de galaxia son los objetos mejor estudiados y entendidos en nuestro cosmos,

podemos preguntarnos si podemos encontrar un rastro de la energıa oscura cosmica dominante. Este

es el objectivo primario de esta tesis.

En esta tesis

En esta tesis hemos investigado la influencia de la energ’ia oscura en la formaci’on y evoluci’on de

cumulos de galaxias en varios Universos de materia oscura fr’ia (CDM por sus siglas en ingles). Hemos

mostrado que hay un impacto imperceptible de la constante cosmologica positiva en varias propiedades

11M⊙=1.989×1030 kg, es la masa del Sol.21 Mpc=3.086×1022 mts.


individuales y globales que marcan la vida de un cumulo de galaxia. Hay, sin embargo, una influencia

considerable del contenido de materia en el Universo en la evolucion de cumulos de galaxias.

Simulaciones cosmologicas y funciones de masa

En el Capitulo 2, hemos descrito extensamente las simulaciones cosmologicas que usamos a lo largo

de esta tesis. Estas simulaciones numericas incluyen seis Universos abiertos, cuatro modelos planos

y tres cosmologıas cerradas, con o sin una constante cosmologica. Cada simulacion consiste de 2563

partıculas de materia oscura, en una caja de tamano 200h−1Mpc con condiciones periodicas. Todas las

simulaciones tienen el mismo parametro de Hubble, h = 0,7, y la misma normalizacion del espectro de

potencia. Las simulaciones se iniciaron a un redshift de z = 49, y se corrieron hasta la epoca cosmica

actual.

Para facilitar la comparaci”on entre cada una de las simulaciones en las diferentes cosmologıas,

se asumio que los campos primordiales de densidad Gaussiana tienen las mismas fases para cada uno

de los componentes de Fourier. Haciendo esto, nos aseguramos de tener los mismos patrones de gran

escala en nuestro completo conjunto de simulaciones.

Esta propiedad se ve en la Fig. 2, en donde mostramos rebanadas por el centro de la caja de tres mo-

delos differentes: ΛCDMF2 (un modelo plano, panel superior), ΛCDMO2 (un modelo abierto, panel

central) y ΛCDMC2 (un modelo cerrado, panel inferior). Los patrones de la estructura a gran escala

son similares. Las diferencias entre ellos se manifiestan en los diferentes niveles de aglomeracion de

masa. El modelo abierto con baja cantidad de materia contiene menos estructura que los otros. Esto es

resultado de un extremadamente bajo contenido de materia en esta cosmologıa, posiblemente ligado a

la presencia de una constante cosmologica. Como resultado, el crecimiento de estructura se detuvo en

un epoca significativamente temprana. Esto se reafirma en las regiones ampliadas, donde es posible ob-

servar que Universos con mayo densidad de materia tienen patrones mas desarrollados, caracterizados

por un nivel mas alto de aglomeracion de masa.

Tambien hemos investigado la funcion de masa de objetos en cada una de las simulaciones. La fun-

cion de masa es la densidad por numero de objetos de una masa dada. Encontramos que las funciones

de masa se distinguen entre los Universos debido a la cantidad de materia presente en ellos. Sin embar-

go, en el present, no pudimos encontrar ninguna influencia significativa de la constante cosmologica.

Encontramos algunos efectos notables a diferentes redshifts. Esto es el resultado de los diferentes

tiempos dinamicos, como consecuencia de los diferentes valores de la constante cosmologica.

Crecimiento de masa y virializacion de cumulos de galaxias

En el Capıtulo 3 investigamos la historia de la formacion y virializacion de halos de cumulos. Prime-

ro, miramos la historia de aglomeracion de unos pocos cumulos identicos en tres escalas de tiempo

diferentes: redshift, tiempo de lookback y tiempo cosmico. Encontramos que casi todas las diferencias

tienen que ser atribuıdas a la diferencia en densidad de materia de la cosmologıa. Como con la fun-

cion de masa, la unica influencia notable de la constante cosmologica es vıa su impacto en el tiempo

cosmico.

Una caracterıstica importante en la evolucion de cumulos de galaxias es en su historia de acrecion

de masa. Investigamos esto mirando unos pocos cumulos en varias cosmologıas y en el promedio de

cada una de las cosmologıas simuladas. Al mirar en halos individuales, encontramos que efectos de

merging o acrecion claramente influyen en la vida de un halo. En cierto grado, esto pareciera estar

regulado por la cantidad de materia presente en cada una de las cosmologıas. En aquellas cosmologıas

en donde la cantidad de materia es baja, gran parte de la evoluci’on sucede tempranamente, a menudo

acompanado por mergers masivos a redshifts altos. Por el contrario, en Universos con alto contenido

de materia, tales mergers son mas frecuentes en epocas recientes.

La dispersion alrededor de la historia de acrecion de masa pareciera ser sustancial. Como resultado,

la banda correspondiente de la historia de acrecion de masa en una cosmologıa tiende a traslaparse con


Figura 2 — Tres diferentes simulaciones cosmologicas: ΛCDMF2 (arriba), ΛCDMO2 (centro) y ΛCDMC2

(abajo). A la drecha, la region ampliada mostrada a la izquierda.


varias otras. Esto hace un poco difıcil obtener conclusiones sobre efectos cosmologicos sutiles, tales

como los relacionados con la constante cosmologica.

Estudiamos tambien la virializacion de los halos de materia oscura, y en particular halos de cumu-

los de galaxia. En la presente epoca cosmica, pareciera que la poblacion de halos en todas las cosmo-

logıas se aproxima a un estado virial. Sin embargo, no tienen un estado virial perfecto . Mientras los

halos de baja masa muestran una gran dispersion, los halos de cumulos son los que tal vez presentan

un comportamiento mas apropiado. Casi todos ellos obedecen la misma relacion virial, independiente

de la cosmologıa. La unica influencia notable de la cosmologıa es a traves de la dispersion alrededor

de la relacion virial. En Universos con mayor cantidad de materia es mayor que en aquellos con menor

materia.

Caracterısticas fısicas de cumulos de galaxias

En el Capitulo 4 nos enfocamos en las propiedades fısicas internas de los cumulos de galaxias. Ex-

ploramos la influencia de una constante cosmologica positiva en la distribucion interna de la masa,

la forma y morfologıa, y el momentum angular. La distribucion interna de la masa esta caracterizada

por el perfil de densidad de la muestra de halos. Encontramos que estos perfiles de densidad tienen

la misma apariencia en cada uno de los Universos simulados. Encontramos, sim embargo, que halos

en Universos de alta densidad de materia, estan mas concentrados. No hay indicacion clara de de-

pendencia de la concentracion del halo en la constante cosmologica. Con respecto a la morfologıa,

encontramos que los halos en todas las cosmologıas tienen forma triaxial tendiendo a prolato. Cuando

estudiamos la esferecidad de los halos como funcion de la masa, encontramos que los halos en Univer-

sos de alta densidad tienden a ser menos esfericos. Finalmente, encontramos que el momentum angular

de los halos aumenta constantemente, relacionado ıntimamente con el aumento de masa de los halos.

Relaciones de Escala

La estructura y dinamica de los objetos (casi) virializados como los cumulos de galaxias se traduce

en algunas profundas relaciones de escala entre masa, tamano y cinematica (velocidad de dispersion)

de estos objetos. En el Capıtulo 5 investigamos las relaciones de Kormendy, Faber-Jackson y Plano

Fundamental entre la masa, el radio y la velocidad de dispersion de los halos de tamano de un cumulo.

Aunque estas relaciones se asociaron primero a galaxias elıpticas, Schaeffer etãl. (1993) encontra-

ron que tambien se relacionan a los cumulos de galaxias. En cada una de las cosmologıas simuladas

recuperamos las relaciones de Kormendy, Faber-Jackson y Plano Fundamental. Encontramos que las

relaciones de Kormendy y Faber-Jackson son levemente sensibles a la densidad de materia del Uni-

verso mas que a la constante cosmologica. El mayor impacto de la densidad de materia es en el ancho

del Plano Fundamental: este es casi directamente proporcional a la densidad de materia del Univer-

so.Nosotros encontramos que este ancho es el reflejo del estado virial de los halos de los cumulos. La

evolucion de la relacion Kormendy y Faber-Jackson es una indicacion de la ubicacion de los halos de

cumulos en el Plano Fundamental.

Evolucion futura del Universo

Las simulaciones de N-Body se han convertido en una herramienta necesaria para la investigacion de

la evolucion de estructuras en el Universo. Ellas representan descripciones realistas de la formacion

y evolucion de estructuras en el Universo. Ellas tambien nos permiten investigar la influencia de la

constante cosmologica en la evolucion futura de estructuras en el Universo. En el Capıtulo 6 miramos

la evolucion futura de los cumulos de galaxias. En la Fig. 3 vemos como un objeto en un Universo en

expansion se vera. En los paneles superiores, vemos la evolucion en coordenadas comoviles, mientras

que en los paneles inferiores la evolucion es vista en coordenadas fısicas. En coordenadas fısicas, el

tamano del objeta es casi el mismo a lo largo de su historia, mientras que en coordenadas comoviles se

encoje, al punto que es casi invisible, debido a la expansion del Universo. Encontramos que en el futuro


cercano, la funcion de masa de objectos (densidad de numero de objetos de una masa dada) se congela:

no hay crecimiento de estructura. Como consecuencia de esto, casi no hay diferencia entre la historia

de acrecion de masa de halos en cualquier cosmologıa. A medida que los halos evolucionan hacia el

futuro, se hacen cada vez mas y mas esfericos en forma, alcanzando una morfologıa casi perfectamente

esferica en el futuro lejano. Dado que los halos apenas ganan masa in su evolucion hacia el futuro, el

momentum angular permanece casi constante. Tambien alcanzan un alto grado de viriliazacion. Esto

tambien se refleja en las relaciones de escala, las cuales encontramos son mas ajustadas que en la

actualidad. El ancho del Plano Fundamental es mas delgado que en el presente, y es casi el mismo para

todas las cosmologıas. Esto es una indicacion de que los cumulos de galaxias han tenido el tiempo

suficiente para virializarse y alcanzar equilibrio dinamico. Estos resultados nos indican que en el futuro

lejano sera difıcil identificar en que Universo vivimos.

Figura 3 — Evolucion hacia el futuro de un cumulo en un Universo plano. Paneles superiores: evolucion en

coordenadas comoviles. Paneles inferiores: evolucion in coordenadas fısicas. Observamos como en coordenadas

comoviles, como consecuencia de la expansion del Universo, comienza a crecer aislado.

Supercumulos de galaxias

Identificar supercumulos de galaxias es una tare muy dif’icil. Como son las estructuras mas grandes

en la jerarquıa del Universo, recien se estan comenzando a formar. Aplicamos el criterio derivado en

Dunner etãl. (2006), en el cual son identificados como estructuras ligadas gravitacionalmente. Utili-

zamos esto como una definicion fısica de supercumulos de galaxias. Construimos la funcion de masa

de estos objetos ligados, y, asumiendo que los supercumulos son los mas masivos de esta muestra,

encontramos que en un volumen comparable al Universo Local el supercumulo mas masivo tendrıa

una masa de ∼ 8×1015h−1M⊙. Esto es levemente mas mas grnade que la masa encontrada para el Su-

percumulo de Shapley (la concentracion de galaxias mas grande en el Universo Local) dado en Dunner

etãl. (2008). Tambien encontramos que en el Universo Local podrıamos encontrar 2 supercumulos si-

milares en tamano y masa al Supercumulo de Shapley. En cuanto a la morfologıa, encontramos que los

supercumulos actualmente contienen una gran cantidad de subestructura, tenindo una forma triaxial,

tendiendo hacia prolato.


Conclusiones finales

En esta tesis hemos investigado la influencia de una constante cosmologica en la formacion y evolucion

de cumulos de galaxias. Con este fin, hemos utilizado un amplio set de simulaciones cosmologicas

que incluyen las tres posibles geometrıas del Universo: abierto, plano y cerrado. Cada una de estas

simulaciones incluıa o no incluıa una constante cosmologica. De esta manera, buscamos aprender mas

sobre la influencia de la constante cosmologica en la estructura, dinamica y evolucion de cumulos de

galaxias.

Hemos realizado varios estudios de caracterısticas fısicas globales e individuales de cumulos de

galaxias. Concluimos que la constante cosmologica no influye en la formacion y evolucion de cumu-

los de galaxia. La unica influencia notable es en el tiempo cosmico: estira o comprime los tiempos

dinamicos disponibles para la evolucion de los cumulos.

English Summary

Early civilizations have been wondering where everything came from and how everything began.

For centuries, humanity has been wondering what our place in the vast world surrounding us is.

Tied in with this was the fundamental question of how the world came into being. Early civilizations

had a mythical view of the cosmos. For example, the Mesopotamian civilizations of Sumer and Baby-

lon had a cosmology in which Earth was a disk surrounded by the underground waters of the Apsu

and the underworld of the dead, with the heavens of the stars surrounding it all. Despite the highly

sophisticated level of astronomy in the neo-Babylonian world and its inheritants, it is with the ancient

Greeks that astronomy and cosmology entered the level of scientific inquiry. Building upon the ob-

servations carefully obtained and archived by the Babylonians, the Hellenistic Greeks were the first to

use their geometric models to predict the observational reality, creating a quantitative as well as a qual-

itative model of the Universe. Eratosthenes measured the circumference of Earth, while Aristarchus

measured size and distance of Earth and Moon. He even forwarded the suggestion that the Sun was at

the center of our world, a view which only became the accepted view with Copernicus and Galilei in

the 15th and 16th century.

It is with the Scientific Revolution of the 16th and 17th century that a truly scientific model of the

Universe came into being. With Nicolai Copernicus in 1543 Earth finally lost its privileged central

position in the cosmos. Standing on the shoulders of Copernicus, Brahe, Kepler and Galilei, it was

Isaac Newton who managed to frame the laws of gravity and mechanics. This formed the foundations

of classical physics. His view of a static space-time and a gravitational force resulting from action

at distance made it impossible to open the view on the dynamic cosmological world view which we

presently hold. It was Einstein’s General Theory of Relativity that formed the final breakthrough

towards turning cosmology into a scientific inquiry. His metric theory of gravity turned space-time

into a dynamic medium in which gravity is a manifestation of the curvature of space-time. Soon it

was realized that this implies that the Universe could not be static and instead should be expanding

or contracting. Friedmann and Lemaıtre were the first who worked out the expanding solutions for

a homogeneous and isotropic Universe. Their theoretical ideas were soon confirmed in the seminal

discovery in 1929 by Edwin Hubble of the expanding system of galaxies around us. His “Hubble Law”,

describing the fact that distant galaxies are receding with velocities proportional to their distance, is

still the fundamental basis for present day cosmology.

Big Bang Theory and Inflation

It was Lemaıtre who realized the tremendous implications of this finding. At earlier times, the Universe

would have been a lot smaller, a lot denser and much hotter than the present Universe. This gave rise to

the Big Bang Theory, stating the fact that the Universe came into being at some finite point in the past.

We now know that the cosmos came into being 13.7 billion years ago in a seething sea of radiation

and matter. Even despite its tremendous successes, the standard Big Bang theory does not explain

the origin of structure of the Universe. This may be solved if the Universe underwent an inflationary

exponential expansion phase. During this cosmic inflation, 10−34 seconds after the Big Bang, the

174 English Summary

Universe blew up by a factor of 1060.

Inflation can account for the flatness of the Universe, the uniformity of its radiation and the origin

of the primeval matter inhomogeneity. These inhomogeneities (initially in subatomic scales) were the

seeds from which stars, galaxies and cluster of galaxies formed.

Figure 1 — The Big Bang theory. An schematic picture of the history of the Universe since the Big Bang.

Cosmic timline from M. Norman.

Cosmic Microwave Background

When structures started to form, matter and radiation were coupled. The Universe was dark as a

consequence of the tight interaction between photons and baryons. This state lasted 379 000 years. It

ended when the temperature of the Universe was cold low enough so as to allow the creation of stable

atoms. The Universe had cooled down to 3000 K. In this moment, the Universe became transparent

and the photons could travel freely through the Universe. This photons can still be observed today, as

a radiation that occupies the entire Universe. This is known as the Cosmic Microwave Background

(CMB) radiation. Its spectrum is that of a blackbody, with a temperature of T = 2.755 K. The CMB is

one of the most important evidences of the Big Bang.

Dark Matter

When the Universe was ∼ 1/1090 of its actual size, matter started to dominate the dynamics of the

Universe. The gravitational influence of a mysterious dark matter component is responsible for the

formation of structures we see today. This rare form of matter is insensitive to the electromagnetic

force, making it invisible and, therefore, very difficult to study. It it thought to represent more than

85% of matter in the Universe. This large presence is one of the major challenges for present day

cosmology.

Dark Energy

Even more mysterious is the presence of another kind of energy, the dark energy. All we can say is

that dark energy has a repulsive gravitational force. Observations indicate that it represents nearly 73%

of the Universe’s energy. It is responsible for the accelerated expansion of the Universe and assures

its flat geometry. Observations also tend to indicate that dark energy is in the form of a cosmological

constant, which appeared in Einstein’s Relativity Theory. When the cosmological constant takes over

on the evolution of the Universe, which happened at a redshift of z ∼ 0.7, growth of structure freezes.

English Summary 175

Large Scale Structure of the Universe

The pure Big Bang theory answers the question of the origin of the Universe, but it does not pro-

vide us with an answer of how the structures that we see today formed. The growth of structures is

the result of the gravitational growth of tiny primordial fluctuations (appeared immediately after in-

flation) in the primordial Universe. A given fluctuation whose density is higher than its surrounding

will collapse. Small clumps of matter gradually merge and accrete while assembling into ever larger

structures. Small subgalactic objects are the first to form in the Universe. They decouple from the

expansion of the Universe and collapse. These small objects then merge and give birth to galaxies.

The process continues, leading to the assembly of galaxy-sized halos into clusters of galaxies. Within

this hierarchical evolution, galaxy cluster stand out as the most massive and most recently collapse

objects in the Universe. Most find themselves at the moment in a state of relaxation and virialization.

Their collapse time is comparable to the age of the Universe. These properties make galaxy clusters

indispensable laboratories for the study of the evolution and formation of the Universe.

Galaxy Clusters

Galaxy clusters are the largest stable structures in the Universe. Typical properties of galaxy clusters

include:

• They contain 50 to 1000 galaxies, hot gas and large amounts of dark matter.

• They have total masses of ∼ 1014−1015h−1M⊙1.

• Their radius are in the order or ∼2-6 h−1Mpc2.

• Galaxy members have velocity dispersions in the order of ∼800-1000 km/s.

Galaxy clusters have been key astrophysical objects in the development of our current understand-

ing of the large scale Universe. It was in galaxy clusters that dark matter was first detected. Clusters

are also very luminous X-ray sources, emitted by a tenuous extremely hot intracluster gas with a tem-

perature of T ∼ 107−108 K. The fact that they contain an atypical mixture of galaxies makes them into

important probes of the study of galaxy evolution.

When observed visually, cluster of galaxies appear to be collections of galaxies held together by

mutual gravitational attraction. However, their velocities are too large for them to remain gravitation-

ally bound by their mutual attraction. This implies that there must be an additional invisible mass

component or an additional attractive force besides gravity. Most of the mass of galaxy clusters is in

the form of hot gas, which emits in X-ray. In a typical cluster perhaps only ∼5% of the total mass is

in the form of galaxies, ∼10% in the form of hot X-ray emitting gas and the rest is in the form of dark

matter.

Because galaxy clusters are on the best studied and understood objects in our cosmos, we may

wonder whether we can find a trace of the cosmic dominant dark energy. This is the primary goal of

this thesis.

In this thesis

In this thesis we have investigated the influence of dark energy on the formation and evolution of cluster

of galaxies in several cold dark matter (CDM) Universes. We have shown that there is a neglegible

impact of a positive cosmological constant on several global and individual properties that marks the

life of a galaxy cluster. There is, however, a considerable influence of the content of matter in the

Universe on the evolution of cluster of galaxies.

11M⊙=1.989×1030 kg, is the mass of the Sun.21 Mpc=3.086×1022 mts.

176 English Summary

Cosmological simulations and mass functions

In Chapter 2 we have extensively described the cosmological simulations we used throughout this

thesis. These numerical simulations include six open Universes, four flat models and three closed

cosmologies, with or without a cosmological constant. Each simulation consist of 2563 dark matter

particles in a box of size 200h−1Mpc with periodic boundary conditions. Every simulation has the

same Hubble parameter, h = 0.7, and the same normalization of the power spectrum. The simulations

were started at a redshift of z = 49, and runned until the present cosmic epoch.

In order to facilitate comparison between each of the simulations in the various cosmologies, the

primordial Gaussian density fields were assumed to have the same phases for each of the Fourier

components. By doing this, we ensure then to have the same large scale patterns in our complete set

of simulations.

This property is seen in Fig. 2, were we show slices through the center of the box of three different

models: ΛCDMF2 (a flat model, top panel), ΛCDMO2 (an open model, middle panel) and ΛCDMC2

(a closed model, bottom). The patterns of the large scale structure are similar. The differences between

themselves manifest in the different levels of clustering. The open, low matter density model contains

less structure than the other ones. This is a result of the extremely low matter content in this cosmology.

Possibly, in connection with the presence of a cosmological constant. As a result, growth of structure

came to a halt at a significantly earlier epoch. This fact is strengthen in the zoom-in regions, where it

is possible to see that Universes with high matter density have more evolved patterns, characterized by

a higher level of clustering.

We have also investigated the mass function of objects in each of the simulations. The mass

function is the number density of objects of a given mass. We find that mass functions do distinguish

between the amount of matter in the Universe of the different cosmologies. However, at present epoch,

we could not find any significant influence of the cosmological constant. We do find some noticeable

effects at different redshifts. This is a result of the different dynamical timescales, a consequence of

the different values of the cosmological constant.

Mass growth and virialization of galaxy clusters

In Chapter 3 we investigate the formation history and virialization of cluster halos. We first looked into

the assembly history of a few identical clusters in three different time scales: redshift, lookback time

and cosmic time. We found that nearly all differences have to be ascribed to the difference in matter

density of the cosmological background. As with the mass function, the only noticeable influence of

the cosmological constant is via its impact on the cosmic time.

An important characteristic in the evolution of galaxy cluster halos is their mass accretion history.

We investigate this by looking into a few single cluster in various cosmologies and at the average in

each of the simulated cosmologies. When looking at individual halos, we find that merging or accretion

effects clearly influences the life of a halo. To some extent this seems to be regulated by the amount of

matter present in each of the cosmologies. In those cosmologies where the matter content is low, most

of the evolution takes place at early times, often accompanied by massive mergers at high redshifts.

By contrast, on Universes with high matter content, such mergers are more frequent at recent epochs.

The spread around the average mass accretion history appears to be substantial. As a result, the

corresponding band of mass accretion history in one cosmology tends to overlap with several others.

This will make rather difficult to draw any conclusions on subtle cosmological effects such that of the

cosmological constant.

We also study the virialization of dark matter halos, and in particular galaxy cluster halos. At

present cosmic epoch, it appears that the halo population in every cosmology is close to a virial state.

However, they do not attain the perfect virial relation for spherical matter clouds. While low mass halos

show a large spread, cluster halos (high mass halos) are perhaps the most well behave halos. Nearly

all of them obey the same virial relation, independent of cosmology. The only noticeable influence

of cosmological background is through the spread around the virial relation. In Universes with high

English Summary 177

Figure 2 — Three different cosmological simulations: ΛCDMF2 (top),ΛCDMO2 (center) and ΛCDMC2 (bot-

tom). On the right, the zoomed-in region depicted on the left.

178 English Summary

matter content it is larger than in low matter content.

Physical characteristics of galaxy clusters

In Chapter 4 we turned to the internal physical properties of galaxy clusters. We explore the influence

of a positive cosmological constant on the internal mass distribution, the shape and morphology, and

the angular momentum. The internal mass distribution is characterized by the density profile of the

halo sample. We find that these density profiles have the same appearance in each of the simulated

Universes. We do find, however, that halos in high matter density Universes are more concentrated.

There is no clear indication for a dependence of halo concentration on cosmological constant. Regard-

ing the morphology, we found that halos in every cosmology have a triaxial shape tending towards

prolate. When studying the sphericity of halos as a function of mass, we find that halos in high density

Universes tend to be less spherical. Finally, we find that the angular momentum of halos increases

steadily, intimately related to the increasing mass of halos.

Scaling relations

The structure and dynamics of (near) virialized objects like cluster of galaxies translate into some pro-

found scaling relations between the mass, size and kinematics (velocity dispersion) of these objects. In

Chapter 5 we investigate the Kormendy, the Faber-Jackson and Fundamental Plane relations between

the mass, radius and velocity dispersion of cluster sized halos. Although these relations were first

related to elliptical galaxies, Schaeffer et al. (1993) found that they also relate to galaxy clusters. In

each of the simulated cosmologies we recover the Kormendy, Faber-Jackson and Fundamental Plane

relation. We find that the Kormendy and Faber-Jackson relation are mildly sensitive to the density

matter of the Universe rather to the cosmological constant. The largest impact of the matter density is

on the width of the Fundamental Plane: it is almost directly proportional to the matter density of the

Universe. We find that this width is a reflection of the virial state of cluster halos. The evolution of the

Kormendy and Faber-Jackson relation is an indication as to where in the Fundamental Plane cluster

halos lie.

Future evolution of the Universe

N-body simulations have become a necessary tool for the investigation of the evolution of structures

in the Universe. They represent realistic descriptions of the formation and evolution of structure for-

mation in the Universe. They also allow us to to investigate the influence of the cosmological constant

on the future evolution of structures in the Universe. In Chapter 6 we look into the future evolution

of galaxy clusters. In Fig. 3 we see how an object in an expanding Universe will look like. In the top

panels, we see the evolution in comoving coordinates, while in the lower panels the evolution is seen

in physical coordinates. In physical coordinates, the size of the object is almost the same throughout

its history, while in comoving coordinates it shrinks, to the point it is almost invisible, due to the ex-

pansion of the Universe. We find that in the near future, the mass function of objects (number density

of objects of a given mass) freezes: there is no growth of structure. As a consequence of this, there is

hardly any differences between the mass accretion history of halos in any given cosmology. As halos

evolve towards the future, they become more and more spherical in shape, achieving a nearly perfect

spherical morphology in the far future. Given that halos hardly gain mass in their evolution towards the

future, the angular momentum remains almost constant. They also reach a high degree of virialization.

This is also reflected on the scaling relations, which we found to be much tighter that at present time.

The width of the Fundamental Plane is thinner than at present cosmic epoch, and is nearly the same for

every cosmology. This is an indication that galaxy cluster halos have had enough time to virialize and

reach dynamical equilibrium. These findings tell us that in the far future it will be difficult to identify

in which Universe we live in.

English Summary 179

Figure 3 — Evolution of a single cluster in a flat Universe towards the future. Top panels: evolution in comov-

ing coordinates. Lower panels: evolution in physical coordinates. We see how in comoving coordinates, as a

consequence of the expansion of the Universe, starts to grow in isolation.

Supercluster of galaxies

Identifying supercluster of galaxies is a very difficult task. As they are the largest structures in the

hierarchy of the Universe, they are just starting to form. We apply the criterion derived in Dunner

et al. (2006), in which they are identified with gravitationally bound structures. We use this as a

physical definition of supercluster of galaxies. We construct the mass function of this bound objects,

and, assuming that superclusters are the most massive of this sample, we find that in a region of

a volume comparable to the Local Universe the most massive supercluster would have a mass of

∼ 8× 1015h−1M⊙. This is slightly bigger than the mass derived for the Shapley Supercluster (the

largest concentration of galaxies in the Local Universe) given in Dunner et al. (2008). We also find

that in the Local Universe we would be able to find 2 superclusters that are similar in size and mass to

the Shapley Supercluster. As for the morphology, we find that superclusters at present time contain a

substantial amount of substructure, having a triaxial shape, tending towards prolate.

Final Conclusions

In this thesis we have investigated the influence of a cosmological constant in the formation and evo-

lution of galaxy clusters. To this end, we used a wide set of cosmological simulations that included the

three possible geometries of the Universe: open, flat and closed. Each of these simulations did or did

not include a cosmological constant. In this way, we sought to learn more about the influence of the

cosmological constant on the structure, dynamics and evolution of cluster of galaxies.

We have carried out several studies of global and individual physical properties of galaxy clusters.

We conclude that the cosmological constant does not influence the formation and evolution of galaxy

clusters. The only noticeable influence is on the cosmic time: it either stretches or compresses the

available dynamical timescales for the cluster evolution.

180 English Summary

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Acknowledgements

I would like to thank my supervisor, Rien van de Weygaert. Dear Rien, thank you for your patience,

for pushing me to see beyond the results and for teaching me to be independent and critical in my

resarch. Thank you for your help in the writing of this thesis, specially in the last stages.

Bernard, much of what I accomplished in the thesis is because of your continuous support. You

are not just a collaborator but also a friend. I really enjoyed our talks about science, life and football.

I really hope we can keep collaborating.

I would also like to thanks Thijs for his help and support during my entire PhD thesis. Thank you

for pushing me at the end to get the thesis out.

Thanks to the computer group of the Kapteyn Institute, in particular Wim. I am still waiting for

my black case! Thanks to the secretaries, specially Hennie and Jackie. Hennie, thank you for your

patience towards me in the last months and for the nice conversations we had. Jackie, thank you for

translating me all those letters in dutch, I really appreciate it!

I would like to thank the Kapteyn Institute and the Leids Kerkhoven Bosscha Fonds (LKBF) for

their financial support during my PhD.

Emilio, thanks for your friendship since the first day I arrived in Groningen. I really enjoyed our

many talks about everything. Thank you for your help during my entire PhD thesis. No matter where

you were (Groningen, Israel and USA), you were always there to answer my many questions.. Thank

you for taking me to Teziutlan and meeting your family. Muchas gracias por todo Emilio.

Miguel, what can I say? How can I forget the many trips we took, either for conferences or for fun.

Getting lost in Belgium, not knowing where to sleep. Or discussing dianetica in Florence. I think they

are still waiting to give us our tunics! Gracias por tu amistad Miguelın.

Erwin, I appreciate your help with the various things I came to you for (programming, science).

Also, thank you for helping me with the translation of the english summary. Sorry for the short notice!

I wish you the best for the rest of your PhD.

Isa and Angel, thank you very much for letting me be your housemate, specially in the last stages

of my thesis. I enjoyed our breakfasts and dinners, talking and discussing about everything. Living

with you really made me feel like at home.

Cesar, I really enjoyed the lunches and afternoon breaks at Hanze. Thank you for your support.

Good luck in Antwerpen! Carol, thank you for your friendship. Your continuous support and help

through my PhD period are invaluable. I really enjoyed the late saturday movies. You are a nice

person and I consider myself very fortunate to share your friendship. Muchas gracias por todo. Jesus,

we have to continue making “recorridos”. I really appreciate your friendship. I enjoyed our continous

talks, about different topics. Thank you for opening Pandora’s box - yes, it does work!. Our trip to

Prague, getting lost at 6 o’clock in the morning. I am still missing the 1400 rupias! We are definitely

going to be laughing about everything in a year. Dr. Patricio, thank you for the lunches and afternoon

breaks. I wish you the best in your adventure in Chile. I look forward meeting you in Santiago. To

my many friends in Groningen, thank you for making me feel like at home. Thank you to the Mamio

team, I really enjoyed playing football with you guys.

No puedo olvidar a mis amigos en Chile. Pato, Jaime, Chino, gracias por su apoyo. Espero que

nos podamos volver a ver pronto!

186 Acknowledgements

Gracias a mi familia. Gracias a mis hermanos por sus continuas muestras de carino y apoyo.

Gracias a mis padres por todo el apoyo y amor que me han dado. Mucho de lo que soy hoy en dıa y lo

que he logrado se los debo a ustedes. Los quiero mucho.

Pablo Andres Araya Melo, Groningen, April 2008