Aachen University of Technology - RWTHI. Physics Institute BSommerfeldstr. 14D-52074 Aachen

Dark matter - new experiments on particle physics and astrophysics

Structure Formation in the Universe

Colin Barschel

July 13, 2007

Adviser Prof. Dr. W. Wallraff

This article summarizes the presentation on structure formation given during the seminar entitled “DunkleMaterie - Neue Experimente zur Teilchen- und Astroteilchenphysik” (Dark matter - new experiments onparticle physics and astrophysics). The seminar is organized by the I. Physics Institute B of the Aachenuniversity of technology (RWTH). The presentation about structure formation in the Universe was held the16 th of April 2007 by the author and advised by Prof. Dr. W. Wallraff.

The content outlines the current understanding and status of structure formation in the Universe. Thetext is intended to be readable with limited mathematical and physical background and should be accessibleto non-physicist readers. The theoretical section can be understood with the text only and important aspectsare also explained without equations.

Contents

1 Introduction 4

2 First Principles 5

2.1 Distances and redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 The parsec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 The Hubble parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 The redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Hierarchy of structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 How did large structures grow? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 The need for dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Velocity distribution of stars in galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.2 Kinematic of cluster of galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.3 Gravitational lensing by galaxy clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Dark energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5.1 The missing mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5.2 Brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5.3 The expansion of the Universe is accelerating . . . . . . . . . . . . . . . . . . . . . . . 112.5.4 What is dark energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Fundamentals roundup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Theory of Structure Formation 13

3.1 Linear theory of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.1 Solutions for dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Jeans instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.1 Instability of baryonic matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.2 Instability of dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Nonlinear evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Models and Simulations 19

4.1 Lambda-CDM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.1 Parameters and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.2 Outlook for the future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 N-body simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 The millennium simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Observational Tests 23

5.1 Sloan Digital Sky Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.1.1 Technical information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.1.2 Some results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Dark Matter Scaffolding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2.1 Observation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Conclusion 27

3

1 Introduction

Understanding how the Universe’s large-scale structures like galaxies and galaxy clusters form remainslargely an unsolved problem in cosmology today. The Big Bang theory describes the origin and evolution ofour Universe and is supported by observations like the expansion of the Universe [1], the abundance of thelight elements [2] and the cosmic microwave background [3]. However the big bang theory does not explainhow cosmic structures like stars, galaxies and galaxy clusters form nor what are their origin. In this contextstructure formation refers to many unanswered questions and to the elaboration of new theories. Thus thequestion to answer is how did today’s cosmic structures form?

We are helped to answer this question with new observation methods, specifically space and large groundbased telescopes, large CCD sensors and automated observation routines coupled with powerful computersfor analysis and simulation. Those tools and new observations brought tremendous progress in our under-standing of structure formation in the last ten years. Indeed, as we will discover, the actual concordancemodel not only agrees with observations, but its precision improves with the flow of measurements and alsoreinforces the theory.

Some basic principles will be introduced first and we will discover the need for dark matter and darkenergy. Afterward we will obtain a first answer to how did large structures grow. The linear theory ofperturbations, which is a simple approach to the evolution of cosmic structures, will be introduced and usedto outline the possible models. The linear theory then leads to the actual concordance model which servestoday as a basis for simulations and is also compared with observations. Finally we will look into two recentobservations reflecting the latest research status in cosmic structures.

We will see that the theory is in good agreement with observations and that most processes seems to bewell understood, even though the theoretical model relies on many open questions. Furthermore we will seethat the main goal to develop such a theory, that is to understand the underlying physics at work, has stillto be reached.

4

2 First Principles

Some new tools and definitions are necessary beforewe tackle the problem of structure formation. In thischapter we will define the standard unit of distance,the Hubble constant and its related redshift. Usingthose definitions we will introduce the hierarchy ofstructures and then provide a first simplified view ofhow the large structures came to exist.

2.1 Distances and redshift

2.1.1 The parsec

The standard unit of length in astronomy is the par-sec. The parsec (pc) is an acronym for “parallax ofone second of arc” as it is based on a trigonomet-ric parallax method. The parsec is the distance atwhich one astronomical unit (AU) subtends an angleof one arc-sec ( 1/3600 ) [4]. The astronomical unitis the semi-major axis of the earth’s orbit around thesun which is about 150 million kilometers. Using thisdefinition we can express the parsec in SI units withsimple trigonometry based on figure 2.1, therefore theparsec is

1 pc =1 AU

tan 1

3600

≈ 149.598 × 109 mπ

180×3600

= 3.086 × 1016 m = 206265 AU,

(2.1)

and defined with the speed of light,

1 pc = 3.36 light years,

where a light year is the distance traveled by light orone photon in one year.

To put the parsec into perspective, lets considerthe following examples. The distance between theearth and the sun (1 AU) equals 0.000005 pc, or 1pc ≈ 206265 AU. Pluto’s orbit is 40 AU from thesun and voyager 1 which is the furthest of any man-made object took 30 years to reach 100 AU. Usingthe standard prefix in units, one thousand parsecsis a kiloparsec (kpc) and one million parsec is themegaparsec (Mpc). The typical separation of starsin a galaxy is in the order of one pc and the typicaldistance between large galaxies is of the order or one

Earth

Sun1 parsec

1”= 1

3600

1 AU

Figure 2.1: Schematic view of the parsec definition. Theparsec is related to the AU unit and the second of arc.(The image is not to scale)

Mpc. For example our nearest star Proxima Centauriis 1.3 pc away and our nearest galaxy, the AndromedaGalaxy is 0.77 Mpc away.

2.1.2 The Hubble parameter

Extragalactic distances are difficult to measure di-rectly and one usually use the Hubble law to esti-mate large distances. The Hubble law states thatthe redshift from light emitted from distant galaxiesis proportional to their distance [5]. After nearly adecade of observation Hubble formulated this law in1929. He discovered that galaxies are receding witha velocity v proportional to the distance d from theobserver:

v = H0d (2.2)

where H0 the Hubble constant expressed in km s−1Mpc−1

and corresponds to the Hubble parameter H(t) whichis time dependent. Since the Hubble constant is diffi-cult to measure accurately, it is useful to parametrizeH0 in term of a dimensionless value h, where

h =H0

100 km s−1 Mpc−1. (2.3)

Using this notation, distances are expressed in unitsof h−1 Mpc. The current evaluation of the Hubbleparameter is about

H(t0) = H0 ' 74 ± 4 km s−1 Mpc−1[6].

2.1.3 The redshift

The redshift is an increase in wavelength of an elec-tromagnetic radiation, for instance in astronomy a

5

2 FIRST PRINCIPLES 2.2 HIERARCHY OF STRUCTURES

redshift occurs when light is shifted toward the redend of the spectrum. A redshift occurs when thesource of the radiation moves away from the observerand is therefore useful to deduce the receding velocityof a luminous object. Together with the Hubble law(2.2) we see that the distance of the object is there-fore directly connected with its redshift. For smallredshift (10−2 ≤ z ≤ 10−1), that is relatively nearbygalaxies, the distance d is simply approximated by

d ' c

H0

z ' 3000 h−1 z Mpc−1. (2.4)

The redshift z of a luminous source is defined as

z =λ0 − λe

λe(2.5)

where λe is the wavelength of the radiation emittedby the source and λ0 is the observed wavelength. Ob-jects at larger distances need a better approximationwhich uses the luminosity distance as a function ofredshift in the Friedmann Model [7][5]:

dL =c

H0

1

q20

q0z + (q0 − 1)[−1 +√

1q0z + 1]

' c

H0

[z +1

2(1 − q0)z

2],

(2.6)

with q0 the deceleration parameter defined as

q0 = − a(t0)a0

a(t0)2(2.7)

and a(t) the cosmic scale factor is defined with Hub-ble law, basically the Hubble parameter H(t) is de-fined as the relative expansion rate:

H(t) =a(t)

a(t). (2.8)

As the light has a finite velocity, looking at an objectwith a high redshift is like looking back in time, simi-larly we can use the redshift as a time scale. Howeverwe must take the expansion of the Universe into ac-count. For this purpose, astronomers have definedthe “look-back time” τ(z) [8] which is how long agothe observed light was emitted. The look-back timeis defined as [9]

τ(t) ≡ t0 − t(z)

t0(2.9)

and has an explicit value with a flat Universe Ω = 1of

τ(t) = 1 − 1

(1 + z)2/3. (2.10)

For nearby objects, the look-back time is identicalto the distance, however because the Universe is ex-panding, distant objects are further away than thelook-back time. For instance a photon emitted fromthe edge of the observable Universe has a value for τof about 13 billion years, but due to the expansion ofthe Universe, the distance to the horizon is now over47 billion light-years away [10].

2.2 Hierarchy of structures

We have now defined the standard distance unit andhave linked it with the redshift and the Hubble pa-rameter. But how are the structures defined, whatare they? First the smallest unit for the study of thestructure formation is the galaxy. Furthermore thestandard unit of mass is the solar mass M and isequal to the mass of the sun with M = 1.989× 1024

kg [11]. Galaxies are grouped in three basic types:spirals, elliptical and irregular. Without going intothe details of the classification, elliptical galaxieshave a large mass distribution extending from 105

to 1012 M. Spiral galaxies have a smaller mass vari-ation and usually have a typical mass of 1011 M

[5].

Our galaxy, the Milky Way, is gravitationallybound with around twenty galaxies, most of whichare small dwarf galaxies and form the so called localgroup as illustrated in figure 2.2. The Andromedaspiral galaxy M31 and also M33 are the other twolarge members of this group.

The Local Group is itself part of a larger structure:the Virgo cluster at about 10 h−1 Mpc. The Virgocluster is pulling the Local Group toward itself andcontains 1500 to 2000 galaxies. There are severallarge clusters further away within 100 h−1 Mpc, withthe Coma cluster being the largest with thousands ofgalaxies.

The largest structures known today are the CfA2Great Wall and the newly discovered Sloan GreatWall. Both structures can be see in figure 2.3. TheCfA2 Great Wall also includes the Coma cluster withan “O” shape at the center. The Sloan Great Wallextends from the top left to the bottom right cornerand is split in two filaments in the center. With atotal length of about 433 Mpc [14], it is nearly threetimes longer than the CfA2 Great Wall. Howeverthe Sloan Great Wall is not exactly a structure as its

6

2 FIRST PRINCIPLES 2.3 HOW DID LARGE STRUCTURES GROW?

SextansN3109/

Group

NGC 6822

And II

M33IC 10

M31

NG

C 1

85

WLM

Car

Fo

r

SexDra

UM

i

Cet

An

d I

IIA

nd

I

Le

o II

Leo A

Tuc

Milk

y W

ay

500 kpc

500 kpc

NGC 3109Antlia

Sex B

Sex A

And V

LG

S 3

NGC 147

NGC 205

Cas dSph

And VI

Peg dI

IC 1

613

Sg

rS

cl

SM

C &

LM

C

Phe

Leo I

DD

O 2

10

SagD

IG

SgrGroup

Local Group

1 Mpc

2Mpc

Figure 2.2: Three dimensional schematic view of the LocalGroup. The plane indicated by solid lines passes throughthe Milky Way and is composed of 500 kpc squares. TheLocal Group with the Milky Way and Andromeda extendsup to 3 Mpc but most galaxies are located within 300 kpcof the Milky Way and M31. The two largest galaxies arethe Milky Way and Andromeda which are spiral galaxies.(Source: “The Local Group” by Grebel, Eva K. [12]).

galaxies are not all gravitationally bound to the samepotential well.

2.3 How did large structures grow?

Let us first examine the current theory of galaxy andstructure formation. The basic rationale is that twokind of matter exists. The first matter, which we alsocall the normal matter, is composed of atoms and isthe building block of our known world and also of our-self. We call this matter the baryonic matter becauseits constituents, the neutron and proton are baryons(baryons are made of three quarks). The particles ofbaryonic matter interact with electromagnetic radi-ation, that is the matter will interact with photonsand therefore absorb or emit photons and will thusalso scatter with radiations. This interaction is thereason why matter, can emit, absorb or reflect lightand is therefore visible. The second kind of matter iscalled dark matter because it does not interact withradiation and is thus invisible to us, even though itrepresents up to 85 percent of all matter. Despitethe lack of radiation interaction, dark matter is alsobound gravitationally like normal matter.

Right after the big bang as the Universe wasstrongly dominated by radiation, dark matter imme-diately began to clump as it was not scattered by

Figure 2.3: From “A Map of the Universe” [13]. The SloanGreat Wall compared to CfA2 Great Wall at the samescale in co-moving coordinates. The equivalent distancesare indicated in cz as well as the total numbers of galaxies.

electromagnetic radiations. We will see in later chap-ters that a small fluctuation in the matter density willcreate a gravitation instability and a small accumu-lation will attract more and more matter over time.Although dark matter immediately started to formspherical blobs, or so called “halos”, baryonic mat-ter, due to its strong interaction with radiation wasnot free to condensate and was kept in equilibrium asa hot thermal plasma [15]. Then, as the Universe ex-pands and cools down, at one time, electrons will beable to bind with nucleus and begin to form hydrogenand helium atoms. Specifically, during this so calledrecombination epoch at about 300000 years after thebig bang, the hot plasma recomposes to atoms andthe photons are now decoupled from the matter andcan evolve independently. In the meantime the Uni-verse becomes transparent at the recombination andthe decoupled photons, now free to travel, are seen to-day in the form of the cosmic microwave background(CMB). Accordingly the CMB is like an imprint ofthe fluctuation at the recombination epoch and is amap of the density fluctuation of the Universe at thatepoch.

Now that Universe became transparent, the bary-onic matter is also free to evolve as it is not scatteredby the radiation anymore, the tiny density perturba-tions which are visible in the cosmic microwave back-ground will also evolve into a gravitation instability.Indeed due to the additional gravitational potentialdenser region will grow denser and thinner region will

7

2 FIRST PRINCIPLES 2.4 THE NEED FOR DARK MATTER

get thinner. Eventually in some region the matterwill reach a density high enough to form the firststars, then quasars and small galaxies. However thefirst stars and galaxies did not materialize in somerandom location, but in the center of the dark mat-ter halos which had a head start to form and alreadytook shape (2.4).

Den

sity

contr

ast

TimeBB radiation epoch matter epoch

z'10000 z'1200 recombination

dark matter

baryonic matter

linear growth

Figure 2.4: Growth of matter density contrast. After therecombination the baryonic is free to evolve and fall intothe dark mater halos which formed earlier. Afterwardboth matter grow together.

In conclusion the fluctuation seeds visible in thecosmic microwave background at the recombinationepoch will follow the gravitational wells of the darkmatter halos and eventually collapse to form the firstgalaxies. The dwarf galaxies will merge together toform larger one and groups of galaxies will later formclusters of galaxies with up to thousands of galaxiessurrounded by wast voids.

2.4 The need for dark matter

We have seen in the previous section that the theoryof structure formation is based on the existence ofdark matter [16]. Furthermore an important prop-erty of dark matter is that it does not interact withradiation and therefore will not emit or absorb light.Not only is the dark matter invisible but its composi-tion is unknown, consequently one should ask ourselfwhat makes us think that this dark matter exists inthe first place.

Dark matter is invisible with electromagnetic ra-diation, it does however interact with gravity andthus reveals it presence through gravitation pull onthe baryonic matter. This means that its density isdetected indirectly from the motions of astronomi-cal objects like galaxy rotation and observations ofclusters and super clusters [17]. The search for dark

matter has not only dominated the field of cosmol-ogy for half a century, but is still ongoing today. TheSwiss astronomer Fritz Zwicky detected a ”matterwe cannot see” [18][19] in 1933 when studying thedynamics of the Coma cluster which holds about 800galaxies. He calculated the necessary mass neededfor the cluster to hold the galaxies together despitetheir large peculiar velocity and found out that a 400fold of the mass was missing. His assumption thatthe missing mass was a kind of invisible dark matterwas rejected by the scientific community at the time,but is largely accepted today.

Without diving into all the latest observationmethods to find dark matter, only three simple ar-guments explaining its influence are presented here.Other methods like the X-Ray emission of clustersand the temperature fluctuation in the cosmic mi-crowave background have usually findings in the sameorder of magnitude. In fact researchers in a recentmeasurement using the Chandra X-ray Observatoryin orbit claim to have found a direct proof of darkmatter [20]. Furthermore the root of the theory ofstructure formation is based on the existence of darkmatter. During the theoretical approach in chapter3 we will see what influence the dark matter had onthe formation of structures and why its presence isnecessary to explain today’s structures.

2.4.1 Velocity distribution of stars in galaxies

According to Newtonian mechanics one would expectmatter like gas and stars within a galaxy to orbit thecenter of the galaxy in the same way as planets doin our solar system. Therefore the velocity of starsat the outside border of the galaxy is expected todecrease according to Keplerian dynamics when bothcentrifugal and gravitation forces are at equilibriumwith

mv2

r=

GmM(r)

r2(2.11)

where M(r) is the mass of the galaxy within the ra-dius r and G the gravitational constant. This equa-tion leads directly to the rotation velocity of a keple-rian system like our solar system

v ∝ 1√r. (2.12)

However Vera Rubin and W. K. Ford first measuredthe rotation of the Andromeda galaxy in 1970 [21]and later studied a set of spiral galaxies [22] and

8

2 FIRST PRINCIPLES 2.4 THE NEED FOR DARK MATTER

found out that the rotation speed of stars with largeradii was constant:

v → const. (2.13)

This means that stars located farther away from thecenter had the same rotational velocity as the onenear the core of the galaxy. But the stars can notkeep this high velocity without flying away from thegalaxy, there has to be something to keep them ontheir orbit. A typical galaxy rotation curve can beseen in figure 2.5.

Figure 2.5: Rotational velocities for four galaxies as a func-tion of distance from the center. Measured by RubinThonnard and Ford in 1978 [23]. The typical flat curvefor larger radii is an indication of dark matter.

The observed constant velocity can be explainedwith the presence of a large dark matter halo with adensity distribution of ρ ∝ 1/r2 and a much largercircumference as the visible galaxy. Furthermore thishalo must have more than 2/3 of the whole mass ofthe galaxy [24]. Today the mass of dark matter haloshas been estimated for thousands of galaxies with thismethod.

On the other hand, could there be enough baryonicmatter in the galaxy to shape its rotational prop-erty and simply be invisible? Our first candidatewould be gas spread out far away from the visiblepart of the galaxy. However cold gas density is di-rectly measured with the 21cm line absorption [25]and is not distributed as a halo. And hot gas emitsX-rays and its observations are used to map hot gasin galaxies and clusters [26], but it can not accountfor the missing mass. If it is not gas, it could beStellar Remnants like black holes, white or browndwarfs which are called “MAssive Compact Halo Ob-jects” or MACHOS. Nevertheless those candidatesare also ruled out by micro-lensing measurements[27][28], even though they account for a small partof the invisible mass. There are other baryonic mat-ter candidates, however the big bang nucleosynthe-

sis theory which is accurately confirmed by measure-ments [2][29][30] restricts the total amount of bary-onic matter and can only explain a small part of themissing mass.

2.4.2 Kinematic of cluster of galaxies

As introduced before, Fritz Zwicky studied the mo-tion of galaxies in the Coma cluster to calculate thetotal mass. Clusters contain many large galaxies withold stars and thus seem to be very stable. If we con-sider that galaxies in a cluster are moving in equilib-rium in the cluster gravitational potential well, thevirial theorem provides a general equation relatingthe galaxy velocity with the total cluster mass. Thevirial theorem states that in a system in equilibrium[31], the average total kinetic energy 〈T 〉 equals halfthe total potential energy 〈Vtot〉:

2〈T 〉 = −〈Vtot〉. (2.14)

Additionally the galaxies have a velocity distributionmeasured with the redshift of [32]

σ2v = 〈v2

‖〉 − 〈v‖〉2 (2.15)

with v‖ the velocity component parallel to the obser-vation direction. The typical velocity dispersion areof the order σv ∼ 1000 km/s.The cluster mass can now be estimated. The totalpotential energy is the gravitational potential and weneed to take into account that the velocity dispersionis measured along one spacial direction only and musttherefore by multiplied by 3. Thus the virial theorem(2.14) yields

2m

2(3σ2

v) =GMm

R(2.16)

and the mass estimation for the cluster is

M ' 3Rσ2v

G. (2.17)

This mass is approximately ten times the visible massin galaxies and is a first hint that dark matter ac-counts for a substantial amount of matter in theUniverse [33][34]. Without dark matter the galax-ies would quickly escape the cluster as the observedmass with its gravitational potential is insufficient tokeep the galaxies inside the cluster.

Using similar arguments as with the galaxy rota-tion curve one should ask ourself if the missing mat-ter could not simply be invisible baryonic matter.

9

2 FIRST PRINCIPLES 2.5 DARK ENERGY

Interestingly the baryonic mass of the intraclustermedium, that is the superheated gas in the centerof the cluster, exceeds the total visible cluster stellarmass by a factor of 5 [35]. This means that the largestpart of the baryonic matter in a galaxy cluster is con-centrated in the intracluster medium and not in thegalaxies. This gas mass can be accurately measuredwith X-ray observations [36] and it only accounts fora fraction of the total mass. Moreover the missingmass could not be cold gas either because a cold gascan not coexist with the very hot gas. Finally onlyfew Stellar Remnants like black holes or MACHOScould escape galaxies [37] and their mass is negligibleon a cluster scale.

In conclusion the only viable explanation for thestability of galaxy clusters and the fact that the veryhot intracluster medium does not escape the clusteris the presence of dark matter.

2.4.3 Gravitational lensing by galaxy clusters

According to Einstein’s Theory of General Relativity,the gravity from a massive object will bend a lightpath. As a cluster of galaxies is so massive, lightpassing through it are defected by the large gravita-tional field. In the same way as an optical lens, thegalaxies lying far beyond the cluster are brightenedand distorted in so-called arcs. If the lensing objectis exactly lined up with the background source, thenthe arcs trace a circle with the Einstein Radius ΘE

of the cluster. Without mathematical derivation, themass of the cluster can be deduced from the Einsteinradius [38][32], the requirement is that the mean clus-ter convergence equals one:

〈κ〉 =M(ΘE)

π(DdΘE)2· 1

Σcr

!= 1 (2.18)

with Dd the angular diameter distance to the clusterand Σcr the critical surface mass density. Accordinglythis equation can be inverted to determinate the massof the cluster with

M(ΘE) = πD2dΘ

2EΣcr (2.19)

The equation for the total mass responsible for thelens effect can be further transformed to include adirect relationship with the classical optical distances

to the lens and the object:

ΘE =

(

4GMDLS

DLDS

)1/2

=

(

M

1011.09M

)1/2 (

DLDS/DLS

Gpc

)−1/2

arcsec.

(2.20)

Where DL is the distance to the “lens” e.g the darkmatter halo or cluster, DS is the distance to the dis-torted object and DLS is the distance between thelens and the object.

Figure 2.6: The massive and compact cluster Abell 2218bends the light from galaxies which are far away behindit [39][40]. The galaxies observed by the astronomer onthe left are focused and distorted due to the gravity of thedark matter. (Source: Richard Massey [41]).

The mass estimated with the gravitational lens-ing method is in agreement with the other observa-tion and is a useful tool to map dark matter halos.This method is considered to be reliable and has agood accuracy because it is sensitive to any mass field[42]. Interestingly the gravitational lensing methodwas first proposed by Zwicky himself in 1937 [43].

Even though dark matter remains today unprovenby experiment, the evidence for its presence is over-whelming.

2.5 Dark energy

We have seen that dark matter accounts for most ofthe matter in the Universe. Additionally not only itspresence is required for the actual structure forma-tion model, but many observation methods convergeto the same conclusion and make it possible to esti-mate the total mass of dark matter in the Universe.

10

2 FIRST PRINCIPLES 2.5 DARK ENERGY

Furthermore, the standard model states that the ge-ometry of the Universe is determined by its curvaturewhich can be open, closed or flat. On the other handThe curvature is governed by the total energy, thatis the total matter and energy content. For instancethe Universe is only flat if the total density equalsexactly the so-called critical density ρcr. Otherwisethe Universe is closed if the total density is largerthan the critical density or open for a smaller den-sity. We will now see that this critical density is acrucial parameter.

2.5.1 The missing mass

The latest results from the cosmic microwave back-ground experiments such as WMAP [44] indicatesthat the Universe is spatially flat [45] or at leastextremely close of being flat. And this curvature,or lack of, can be measured with great accuracy.However different independent observations such asgalaxy surveys like the 2dF [46] or 6dF projects orthe study of cluster abundances like the SDSS project[47][48], and also measurements of the baryon densityinferred from Big Bang Nucleosynthesis (BBN) [49]all come to the same stunning conclusion. That isthe total matter in the Universe including dark andbaryonic matter only accounts for a maximum of 30percent of the total energy. Hence the matter bud-get cannot “close” the Universe, which means thereis more than 70 percent of the energy in the Universewhich is unaccounted for and also totally unknown.We call this energy the dark energy with its densityρΛ. It is useful at this point to describe the contentsof the Universe in units of ρcr with the cosmologicalparameter Ω (or Ωtot).

The parameter for the dark energy is ΩΛ := ρΛ/ρcr,the total matter which includes dark matter andbaryons is defined as Ωm := ρm/ρcr and finally Ωb :=ρb/ρcr is the baryonic matter component. Conse-quently as seen before a flat Universe implies thatΩΛ + Ωm = 1 and if Ωm = 0.3, where are the missingΩΛ = 0.7?

2.5.2 Brief history

The history of dark energy started in 1917 when Ein-stein added a constant called Λ (lambda) to his equa-tion of general relativity to allow a static Universe toexist [50][51]. With the general relativity the tempo-ral development of the Universe is described by the

Friedmann equation [52]

H2 =

(

a

a

)2

=8πGρ

3− Kc2

R2+

Λ

3(2.21)

where K and Λ are constants. However Hubble madehis famous discovery shortly thereafter that the Uni-verse is in fact expanding and is not static. Thus theconstant Λ was rejected by Einstein and forgotten fornearly 70 years [53]. Nevertheless after the develop-ment of the quantum theory, Λ could be interpretedas a vacuum energy and although the term dark en-ergy is more fashionable, we can also interpret it asa vacuum energy [54]. In this case the cosmologicalconstant Λ is not zero but very small.

2.5.3 The expansion of the Universe is

accelerating

The dark matter quantity seems to be tightly relatedto the flatness of the Universe, still this missing en-ergy problem can be tackled from an other side. Aseemingly simple question is how did the Universe ex-pansion rate change over time. Due to the attractivegravity force one would expect the expansion to slowdown with time. Indeed the gravitational force ofthe matter could even stop the expansion and couldalso reverse it. Moreover the change in the expansionrate can be measured with so-called standard candleswhich are supernovae of type 1A with a well definedluminosity.

According to measurements [55] with supernovaeof type 1A, astronomers discovered that the expan-sion rate of the Universe is in fact accelerating! Theycould therefore also calculate the dark energy contri-bution which acts as a pressure force against grav-ity. They found out that the dark energy componentwas identical to the missing energy to form a flatUniverse. Later observations [56][57][58][59] also ob-tained the same results. A summary of the findingsfrom the “Supernova Cosmology Project” [60][57] isshown in figure 2.7. The supernovae redshift (z) areplotted against their peak B-magnitude (mb) whichis a measure of their effective luminosity. The bestfit for the high redshift supernovae provide a value ofΩm = 0.25 and ΩΛ = 0.75 which is consistent withthe latest Wilkinson Microwave Anisotropy Probe(WMAP) observations [62][63][64]. In conclusion thedark energy is responsible for the accelerated expan-sion rate and for the flat curvature of the Universe.

11

2 FIRST PRINCIPLES 2.6 FUNDAMENTALS ROUNDUP

0.01 0.02 0.1

26

24

22

20

18

16

14

22

21

20

0.2 0.4 0.6 1.0

Supernova Cosmology Project

High-Z Supernova Search

Calan/TololoSupernova Survey

effec

tive

mb

z

Figure 2.7: Hubble diagram of the observed magnitudeof supernovae of type 1A. The effective luminosity of thesupernovae is plotted against their redshift. (Source: S.Perlmutter, Physics Today, April 2003 [61]). The best fitfor high redshift supernovae is consistent with the cosmol-ogy parameter of a flat Universe and 75% dark energy. Seealso the article of Robert Knop and others [57].

2.5.4 What is dark energy

We have seen that the dark energy is the missing en-ergy filling the gap in the cosmological parameters toobtain the observed flat Universe. Moreover this gapis filled by the energy necessary to explain the accel-eration of the Universe [56] which we would expectnormally to slow down due to the gravity force. In-terestingly only the negative pressure of dark energyis responsible for the expansion acceleration and notits energy.

Even though we still don’t know what this dark en-ergy is, we can describe its characteristics [51]. Mostimportantly dark energy acts as anti-gravitation orrepulsive force. Additionally by its very nature, itdoes not clump like matter but is spread smoothlyeverywhere [65]. We can estimate its density to beabout 10−26 kg/m3.

Although dark energy has no consequence for oursolar system or even our Milky Way, with time andlarge distances, it adds up to a powerful force. Overtime dark energy influenced the formation of galaxiesand cluster of galaxies and is therefore an importantplayer in the theory of structure formation as we willsee in the next chapter.

2.6 Fundamentals roundup

Let’s summarize the ingredients of structure forma-tion we gained so far before tackling the theory. To-day’s structures have grown from small density per-turbations originated during the inflation just afterthe big bang. Moreover there is two types of matterin the Universe, the baryonic matter which we aremade of and the dark matter. The baryonic matteraccounts only for one-sixth of the total matter andis gravitationally trapped within spherical dark mat-ter halos. Finally an important player is the darkenergy (or vacuum energy) which makes 74% of thetotal energy in the Universe.

Recent observation suggest that the Universe is al-most flat [66] with 0.98 ≤ Ωtot ≤ 1.08 and the distri-bution of energy is

Ωb ' 0.04 Ωm ' 0.22 ΩΛ ' 0.74 (2.22)

The discussed energy composition of the Universeaccording to the latest results from the Chandra X-ray Observatory [67] is illustrated in figure 2.8.

73 % dark energy

23 % dark matter

3.6 % intergalactic gas

0.4 % stars andheavy elements

Figure 2.8: Energy Distribution of the Universe accordingto the measurements of the Chandra X-ray Observatory[67].

12

3 Theory of Structure Formation

The cosmological principle states that the Universeis homogeneous and isotropic when averaged oververy large scales. In other words, the Universe looksthe same in every direction and from any other place.Even though this principle is actually an assumption,it is successfully tested. However on smaller scalesthe Universe looks like a sponge composed of largestructures formed by galaxies and cluster of galaxies.Because those structures are also stretched by thecosmic expansion, one can expect to find the root ofthose fluctuation early in the Universe. In fact thesmall fluctuation measured in the uniform cosmic mi-crowave background (CMB) are the best evidence forthe existence of those early perturbations and are theinitial imprint of cosmic structures.

Our goal is to describe the growth of the fluctua-tion, to understand their evolution and which physi-cal processes are required [68]. Furthermore since theUniverse is still evolving, the initial fluctuation can’tgrow exponentially like most physical processes do.

The primeval fluctuations are created according tothe inflation theory [69]. According to this model,quantum fluctuations provide the seeds to classicalfluctuations in the early universe (10−37 s). Theoriginal quantum vacuum fluctuations which are 1020

times smaller than a proton are amplified in a shortperiod of exponential expansion from t = 10−35 tot = 10−32 seconds to a sphere of 10 cm [70]. Further-more it is assumed that that the amplitude whereidentical on all physical scales.The classical fluctuations then evolved under grav-ity to the large structures observed today. Our firstapproach is to describe the evolution of the fluctu-ation with a linear approximation [71]. We will seethat the linear solutions can produce good results forvery large structures. A full relativistic treatment[9][72] can produce similar results but is longer toderive. For smaller scales, and specially to explainthe formation of galaxies, a non-linear theory whichcan only be solved numerically is necessary.

3.1 Linear theory of perturbations

The evolution of the perturbations are described witha Newtonian approximation. This approach is onlyvalid if the perturbations are small which imply weakfields. Nevertheless we expect very small amplitudesin the inhomogeneities so this approach should hold.Since the cosmological principle describes the smoothbackground as a perfect fluid, and the CMB is like auniform sea, we use the equation for a perfect fluid togovern the motion of gravitating particles. Also wemust treat the baryonic and dark matter differently,as the dark matter is expected to be made of collision-less particles.The baryonic matter is treated as a collisional fluidwith the speed ~v, the density ρb, the Pressure p andthe gravity potential field Φ. The equations of motionare therefore:

∂ρ

∂t= −∇ · (ρ~v) Continuity equation (3.1)

∂~v

∂t+ (~v · ∇)~v = −1

ρ∇p −∇Φ Euler equation

(3.2)∆~Φ = 4πGρ0 Poisson equation (3.3)

The continuity equation describes the conservationof mass, that is the mass can flow from one elementto an other, but there is no loss. The Euler equa-tion describes the conservation of momentum and thePoisson equation binds the gravity variation with thedensity variation.For a smooth background without perturbation, theabove equations are simply the zero order [73]:

∂ρ0

∂t+ (~v0 · ∇)ρ0 = −ρ0∇ · ~v0 (3.4)

∂~v0

∂t+ (~v0 · ∇)~v0 = −∇p0

ρ0

−∇Φ0 (3.5)

∆Φ0 = 4πGρ0 (3.6)

Now lets consider small perturbations around theequilibrium of density ρ0, speed ~v0, pressure p0 andpotential Φ0:

ρ = ρ0 + δρ , ~v = ~v0 + δ~v , p = p0 + δp , Φ = Φ0 + δΦ

13

3 THEORY OF STRUCTURE FORMATION 3.1 LINEAR THEORY OF PERTURBATIONS

Substituting the above equations into equations(3.1), (3.2) and (3.3). Also we can ignore termshigher that the first order and then subtract the zeroorder equations. This procedure finally yields thefirst order perturbation:

d

dt

(

δρ

ρ0

)

=dδ

dt= −∇ · δ~v (3.7)

∂δ~v

∂t+ (δ~v · ∇)~v0 = −∇δp

ρ0

−∇δΦ (3.8)

∆δΦ = 4πGδρ (3.9)

where δ = δρ/ρ0 is the density contrast.Those three equations have four variables, we canhowever link the pressure variation with the densityvariation. We assume that a change in pressure isonly due to a change of density and not with a changein entropy. Accordingly the perturbations are adia-batic and the pressure is related to the density with

c2s =

δp

δρ(3.10)

where c2s is the speed of sound.

Those equation are valid for small perturbations, butthey are based on a static Universe and described inEulerian coordinates. To take the background expan-sion into account, it is simpler to express the aboveequation in so-called comoving coordinates. The cos-mic expansion factor a(t) is used to define the comov-ing position ~r with the Eulerian position ~x:

~x = a(t)~r (3.11)

and thus a perturbed position becomes

δ~x = ~rδa(t) + a(t)δ~r. (3.12)

Therefore the speed is composed of two factors:

~v =δ~x

δt= a(t)~r + a(t)

d~r

dt. (3.13)

The first term describes the Hubble expansion andthe second term is the deviation from the expansionwhich we define as

δ~v = a(t)~u (3.14)

where ~u is the comoving peculiar velocity. Addition-ally we must also convert the nabla operator usingthe subscript p for the proper (Eulerian) coordinatesand c for the comoving coordinates:

∇p =∂

∂x=

∂

a∂r=

∇c

a. (3.15)

Now we can rewrite the equations (3.7) - (3.9) incomoving coordinates. Finally with ~g = −∇pδΦ and~v0 = H~x the equation of motion for baryonic matterin comoving coordinates are:

δ = −∇c~u (3.16)

~u + 2H~u =~g

a− ∇cδp

a2ρ0

(3.17)

∆2cδΦ = 4πGρ0δa

2 (3.18)

We are interested in the evolution of the densitywhich describes the evolution of structures. For thispurpose we eliminate the velocity field in the Eulerequation (3.17) with the continuity equation (3.16)and its time derivative.

δ = −∇c~u = −∇c

(

−2H~u +~g

a− ∇2

cδp

a2ρ0

)

(3.19)

and with the Poisson equation (3.18) we obtain thegrowth equation

δ + 2Hδ = 4πGρ0δ +c2s

a2∇2

cδ (3.20)

This is a linear wave equation in comoving coordi-nates with a source and damping term 2Hδ whichis caused by the expansion. Therefore we use theansatz of a plane wave δ(~r, t) = δk exp i(~kc · ~r) withthe wave vector ~k = a(t)~kc which yields the followingfundamental equation for the amplitude δk(t):

δk+2Hδk = ∆k(4πGρ0+~k2cc

2s) for baryonic matter

(3.21)We have seen that our Universe is dominated by

dark matter which is collision-less. Consequentlythe evolution of δk for dark matter is identical withthe above equation (3.21) without the pressure term.The evolution of the amplitude δk(t) for dark matteris therefore:

δk + 2Hδk = 4πGρ∆k for dark matter (3.22)

The general solutions for baryonic matter are toocomplex to show here. General solutions and fitsfor the growth of fluctuations where calculated byPeacock and Dodds [74]. However solutions for darkmatter are easier to derivate and provide the first useof the theory.

14

3 THEORY OF STRUCTURE FORMATION 3.2 JEANS INSTABILITY

3.1.1 Solutions for dark matter

For a flat Universe, the equation (3.22) is transformedwith 4πGρ0(k = 0) = 3/2H2 in

D + 2H(t)D − 3

2H2(t)D = 0 (3.23)

With δk(t) = D+(t)δ(k)+D−(t)δ(k) = 0 as two inde-pendent solutions. The equation has two solutions,a decaying solution D−(t) = H(t) = 2/(3t) and agrowing solution which can be found with the ansatzD(t) ∝ tn and the constrain [68]

n(n − 1) +4

3n − 2

3= 0. (3.24)

This condition has two solutions: n = 2/3 and n =−1. Therefore the modes will not grow exponentially,but only with

D+(t) ∝ t2/3 ∝ a(t) =1

1 + z(3.25)

It is interesting to note that in a flat Universe theFriedmann equation (2.21) yields a(t) ∝ t2/3 thusD+(t) grows as the same rate as a(t) [75].A general solution which takes the Friedmann equa-tion (2.21) into account is

D+(t) =5ΩmH2

0

2H(z)

∫ z

∞

1 + z′

H ′3(z′)dz′ (3.26)

Now the linear growth of the perturbations can beobserved for different cosmological models as seen infigure 3.1 with three general models.

• The solid red line is the flat standard model(ΛCDM) with Ωm < 1, ΩΛ = 1 − Ωm.

• The black long dashed line is the Einstein-deSitter model with Ωm = 1, ΩΛ = 0. This modelis usually referred as the standard CDM model(SCDM).

• The blue dotted line is the open model withΩm < 1, ΩΛ = 0.

In the SCDM model the structures can still grow atsmaller redshift, whereas in the ΛCDM model thestructures are almost frozen at z=1. This means thataccording to the ΛCDM model, the galaxy clustershad to be well pronounced at this time. Furthermorein the open model, with Ωm < 1, ΩΛ = 0, largestructures cease to grow even earlier.

z

D(z

)

Ωm = 1.0 ΩΛ = 0.0

Ωm = 0.3 ΩΛ = 0.0

Ωm = 0.3 ΩΛ = 0.7

1

1

0.10.1

0.01 10

Figure 3.1: Linear growth D+ in function of the redshiftfor three different cosmological models. The cosmologicaldensity fluctuations are normalized for the present time(z=0). (Source: Max Camenzind [68]).

This linear analysis can be used for the early Uni-verse when the perturbations are assumed to besmall. However when the gravitation instability be-gins to form structures, this approximation is notvalid anymore. Nevertheless on a large scale, thatis with small modes (k < 0.1 h Mpc−1) the lineartheory is reasonably accurate. For small structures(large k-modes) a non-linear approach is necessary.

3.2 Jeans instability

The Jeans instability describes how or under whichconditions a gas cloud will collapse due to its owngravity. We will use this theory to get a first under-standing of the instability conditions.Lets consider the growth function for the baryonicmatter (3.21). Without the expansion term, thatis with a(t) = 0 and with a linear wave ansatzδ = δ0 exp i(~k · ~x − ωt), the equation yields the lin-earized dispersion relation.

ω2 = c2sk

2 − 4πGρ0. (3.27)

This equation has two solutions, an oscillation and agrowth mode depending of the sign of the right term.If c2

sk2 > 4πGρ0 the perturbations are oscillations

and the solution is a sound wave. Stable oscillationsare only possible if the wavelength is smaller than theso-called Jeans-wavelength λJ which is defined as

λJ =2π

kJ= cs

√

π

Gρ0

(3.28)

15

3 THEORY OF STRUCTURE FORMATION 3.2 JEANS INSTABILITY

This result defines the instability of a self gravitatingsystem with pressure. As long as the scale is smallthe pressure can balance the gravitation force result-ing in sound waves. However on larger scales thegravitation overwhelms the pressure and the systemcollapse under its own weight. Therefore the Jeanswavelength defines the maximum stable mass as asphere with diameter λJ . This critical mass is calledthe Jeans mass and can be expressed with the Jeanslength as

MJ =4π

3ρ

(

λJ

2

)3

. (3.29)

This instability parameter can now be used to inves-tigate the behavior of baryonic and dark matter inthe early Universe. We will restrain the analysis toonly two examples here to demonstrate the applica-tion possibilities of the theory.

3.2.1 Instability of baryonic matter

We have seen that the Jeans mass is the smallest fluc-tuation which is able to grow. Before the recombina-tion, the baryons are strongly coupled with radiationby Thomson scattering. It is possible to calculate thejeans mass in this condition with cs = c/

√3 [76]:

MJ ' 9 × 1016(Ωoh2)−2M.

In this case MJ is of the order of a super-cluster.Just after the recombination when the Universe be-comes transparent, the baryonic matter is no longerscattered by radiation. A this point the matter be-haves like a mono atomic gas with mp the protonmass and with a speed of

cs =

(

5kBT

3mp

)1/2

.

The recombination temperature being T ' 3000 K,the resulting Jeans mass is now

MJ = 1.3 × 106(Ωoh2)−1/2M

and dropped severely, but is still of the order of mag-nitude of a globular cluster.

An additional effect called Silk damping [77] occursclose to the recombination when the photon meanfree path gets longer. In this case photons will dis-perse more easily in over-densities and thus dampfluctuation. Specifically the smallest mass survivingthe damping at the recombination is given by [78]

MD ' 2 × 1012(Ωo/Ωb)3/2(Ωoh

2)−5/4M.

In conclusion a model based on baryonic mass onlyis difficult to sustain as the minimal fluctuation massat the recombination is too large and small scales aredamped.

3.2.2 Instability of dark matter

We have seen that a fundamental property of darkmatter is that it does not interact with radiationand therefore Silk damping has no dissipative effect.Furthermore dark matter is expected to be collision-less and is consequently instable by nature. Howeverwhen the particle are free to move in the matter dom-inated epoch, the fluctuations are damped due to theso-called free-streaming. In fact as long as the parti-cles are relativistic, they are free to cross the Hubbleradius which is the age of the Universe multiplied bythe speed of light and they will wash out all fluctua-tions smaller than the horizon. The dark matter par-ticles will become non relativistic when the temper-ature drops below the mass of the particles and thefree streaming effect will stop at this time [79]. Thusthe shape of the fluctuations are fixed by the size ofthe horizon at this epoch. Therefore it is clear thatthe size of the fluctuations in a dark matter model ishighly dependent of the particles velocity.

A neutrino with an non zero mass is a good candi-date for hot dark matter (HDM). A massive neutrinowith a mass mν becomes non relativistic at an esti-mated redshift of zν ' 6×104(mν/30eV ), which cor-responds to the Jeans wavelength for free-streaming

λν,FS ' 28 Mpc( mν

30eV

)−1

(3.30)

with the Jeans mass

Mν,FS ' 4 × 1015M

( mν

30eV

)−2

. (3.31)

Consequently all fluctuations smaller than λν,FS aredamped, thus all structure with a mass M < Mν,FS

are erased. This means that after the recombina-tion only structures of the order of a super-clustercould survive. Accordingly this model is referredas a “top-down” structure formation where galaxiesoriginated via the fragmentation of original super-clusters. However a HDM model has multiple prob-lems to overcome and the formation of structures inthe order of galaxies is difficult to explain.

Dark matter is said to be cold (CDM) when itsparticles becomes non-relativistic at a time when the

16

3 THEORY OF STRUCTURE FORMATION 3.4 NONLINEAR EVOLUTION

mass contained within the horizon is smaller than atypical galaxy. In this case the particles have a lowvelocity.

3.3 Transfer function

The different effects discussed before which are de-pendent of the matter can be generalized in a transferfunction. The effect of those processes is to modifythe shape of their original power spectrum

P (k) = Akn (3.32)

with A the amplitude and n the power index whichcan be set to n = 1. The power spectrum for eachmodel describes which power is carried for a specificfrequency or wavelength and can be seen as a “powerper wavelength” relation. Accordingly a higher powerresults in stronger fluctuation growth.

The spectrum changes with time as the discussedprocesses take effect. Specifically this change of thespectrum is described with a transfer function. Infact the transfer function relates the processed powerspectrum P (k) to its initial form P0(k) via

P (k, t) = a2(t)P0(k)T 2(k, a(t)) (3.33)

with P0(k) = Akn and a(t) the scale factor (2.8).The physical process is therefore encapsulated intothe transfer function plotted in figure 3.2. Large wavevectors k which correspond to small scales are erasedfor HDM and baryons. On the other hand smallscales in the CDM model are not dissipated. Thusthose two alternative reflect two scenarios for struc-ture formation. HDM is a “top-down” model wheresuper-clusters are created first and subsequently frag-ment into smaller structures to finally produce galax-ies. In contrast CDM is a “bottom-up” procedurewhere small structures like dwarf galaxies are createdfirst and later merge to form larger structures [80].

Event though the linear theory discussed here isquickly limited as we will see in the next section,however on larger scale the theory can have many ap-plications. In this particular case we have seen thatthe growing mode solutions are highly sensitive to thenature of dark matter and observations can be com-pared with properties of the power spectrum. Colddark matter is today the standard assumption andthe model gives excellent fits (3.3) to observations.

baryons HDM CDM

|Tk|

k/Ωh2(Mpc−1)

1

1

0.1

0.1

0.01

0.01

10

10

10−

3

Figure 3.2: Adiabatic transfer function for baryons, hot(HDM) and cold dark matter (CDM). Small scales arestrongly damped in the baryons and HDM models whereassmall scales are not dissipated for CDM. Oscillations forbaryons are produced by acoustic oscillation when pres-sure is in equilibrium with gravitation. (Source: JohnPeacock “The Universe at late time” [6]).

3.4 Nonlinear evolution

The existence of todays structures can not be ex-plained with a linear theory only. As soon as grav-itational instability begins to form structures andthe fluctuation contrast reaches δρ/ρ ∼ 1, the lin-ear theory ceases to be valid. At this moment thedescription of the process needs to include both hy-drodynamical processes and gravitational dynamics.Furthermore dissipative hydrodynamical effects alsotake place and are complicated to account for. Theusual approach is to solve the problem in so-calledN-body simulations which we will shortly see in thenext chapter.

Nevertheless in the light of the complications thatarise in the non-linear regime, the linear method is agood starting point for small scales and it is there-fore interesting to know its limitations and uses. Thelinear hydrodynamic theory gives excellent results atearly times when the fluctuations are small and thusstill linear [5]. Today large scales or small modesk ≤ 0.1hMpc−1 can be described with the above the-ory, however smaller scales are not. Specifically largeredshift survey like CfA2 and SDSS are well suited totest the linear theory [81]. Compared to figure 3.3,the linear theory would be valid for the largest scales

17

3 THEORY OF STRUCTURE FORMATION 3.4 NONLINEAR EVOLUTION

and up to 500 million light-years when the densityfluctuation is still well below unity.

Figure 3.3: Density fluctuations of the Universe on differ-ent places on scales of millions of light-years. The blackdots from the SDSS [82] survey are the most accurate den-sity measurements to date. The solid blue curve is cal-culated for a Universe composed of 5% atoms, 25% darkmatter and 70% dark energy and is in accordance with ob-servations from various sources. (Source: Max Tegmark[83]).

18

4 Models and Simulations

The first approach with the linear theory madeclear that not only the proportion of baryonic matterto dark matter are important to describe the growthof structures, but also the type of dark matter hasa crucial role to play. Moreover the shape and sizeof the initial fluctuations, the rate of expansion andmost importantly the value of the cosmological con-stant Λ, or dark energy, also influence the evolutionof structures. Accordingly there are many attemptsto construct a model describing the growth of struc-tures including various parameters.

Currently the only viable model is the Lambda-CDM (ΛCDM) model [84][85]. This model is used asbasis for simulations and is compared with observa-tions.

4.1 Lambda-CDM model

The Lambda-CDM model (ΛCDM) is the most suc-cessful model of cosmological structure formation.The model is in agreement with the cosmic microwavebackground [64][47], the large scale structures [86][83]and also the accelerating expansion of the universe[55][56][87][57]. Accordingly it can be refereed to asthe concordance model of cosmology [88].

As seen in section 2.5, Λ is the cosmological con-stant which describes the dark energy term. AndCDM stands for cold dark matter as the model as-sumes the dark matter to have a low velocity, that isto be non-relativistic.

4.1.1 Parameters and assumptions

The ΛCDM assumes a spatially flat Universe com-posed of approximately 4% baryonic matter, 20%dark matter and 76% dark energy depending on themeasurements. The cosmological constant is the can-didate for dark energy. According to the model, theUniverse contains two fluids, that is dark matter anddark energy. Although matter and energy are usu-ally interchangeble concepts in physics, dark matterand dark energy are not the same here. Further-more it is assumed that the initial fluctuations are

adiabatic, Gaussian and also nearly scale-invariant[89][87][6](section “The cosmological parameters”).

The model has six parameter which are providedin table 4.1 for completeness, but we will not inves-tigate them further here, the first three are alreadyknown. More interesting are some derived parame-

Table 4.1: Parameters for the ΛCDM model according tothe WMAP latest measurements [64] and a combinationof all datasets [6]

Baryon Density Ωbh2 0.0220+0.0006

−0.0008

Matter Density Ωmh2 0.131+0.004−0.010

Hubble Constant h 0.71+0.01−0.02

Amplitude A 0.67+0.04−0.05

Optical Depth τ 0.069+0.026−0.029

Spectral Index ns 0.938+0.013−0.018

ters. specifically due to the flatness of the Universe,the dark energy represents 75% of the total energydensity. Also the age of the Universe is calculatedto be 13.7+0.1

−0.2 billion years old and finally the crit-ical density is ρcrit = 0.94 × 10−26kg/m3. One canunderstand the critical density as the mean Universedensity. Interestingly this density is very small, itcorresponds to less than one half of a carbon atomper cubic meter.

Even though the model contains two physical pa-rameters, dark matter and dark energy, which are notyet verified by a laboratory experiment and whichcompose more than 95% of the energy in the Uni-verse, the ΛCDM model provides the best fit to thelatest observations and is widly accepted by cosmol-ogists.

4.1.2 Outlook for the future

The cosmological parameters are today measuredwith increasing accuracy over a wide range of cosmicepochs and the concordance model is well established.Future developments could either improve and con-firm the current paradigm, or it could be necessary toadd new parameters. For instance many discoveries

19

4 MODELS AND SIMULATIONS 4.3 THE MILLENNIUM SIMULATION

could bring the model into difficulties, like a mas-sive neutrino, scale variant or non-Gaussian initialperturbations and also variations in the dark energydensity. However today a fundamental revision seemsunlikely [6].

Future survey like the Planck [90] and Polariza-tion experiments, supernova measurements and manyground-based survey will address physical questionsbeyond the concordance model and could also test thealternative modified theories of gravity. Neverthelessthe underlying physics which govern the Universe’sevolution are still a mystery. Specifically there is noconsensus about the nature of dark matter and darkenergy remains fully unknown.

4.2 N-body simulations

We have seen in section 3.4 that the linear theory istoo limited to describe the whole evolutions of fluctu-ations and no analytical solution exist either. How-ever we can assume that there is no dissipative forceson scales larger that a typical galaxy. Therefore thedynamics can be solved by analyzing the gravitationinteraction only. In this context numerical N-bodysimulations are the best tool to understand the na-ture of non-linear dynamics [75]. In principle everyparticle’s trajectory is determined by gravitationalforces and its initial conditions like fluctuations andvelocity. The particles will then evolve into a non-linear gravitational clustering.

The reliability of the simulation to represent afaithful clustering depends on many factors like thenumber of particles, the mass resolution, the box size,the number of steps, the initial conditions and manyothers. However the computing power and costs ul-timately sets the upper limits and requires to adoptcompromises. Accordingly there are various meth-ods to reduce the processing complexity. The sim-plest code is so-called particle-particle and integratesthe forces on a particle based on all others and iter-ate them for all particles. This approach offers thegreatest precision but the complexity scales with N2

and can only be used with simulations up to 104 par-ticles. An other method called “particle-mesh” (pm)is used for simulation with a large number of parti-cles. In this case the box is divided by a grid andthe forces are not calculated for every particle, butonly on a grid scale. This method greatly reduces

the complexity, but the resolution is too poor to de-scribe small scales clumping. A combination of thetwo methods called “particle-particle-particle mesh”improves the resolution of the pm method [76] andalso scales to large particle numbers.

Today N-body simulation are an important toolto compare theoretical models like the ΛCDM modelwith observations. A typical N-body simulation isshown in figure 4.1. The simulation employed hydro-dynamic theory to study the formation of stars in the“dark age” with a redshift starting at z ' 20 [91].

Figure 4.1: N-body simulation of structure formation withgas and dark matter particles to study the formation ofstars. The simulation box has a side of 100 Mpc/h withperiodic boundary conditions. The study is based on theΛCDM model. The redshift values are: z=6, z=2, andz=0 from left to right. (Source: Volker Springel [91]).

4.3 The millennium simulation

The millennium simulation is so far the largest of itkind to simulate the structure formation based onthe ΛCDM model [92]. The simulation is part of theprogram of the Virgo Consortium [93] and was car-ried out on the Max Planck Societys supercomputercenter in Garching [94] in the summer of 2004. Theexpanding box with a side of 500 h−1Mpc contains1010 particles with a mass of 8.6×108h−1M and ranfrom a redshift of z = 127 until z = 0.

The end result at z = 0 is shown in figure 4.2where the complex filament-like formation of clusteris visible.

20

4 MODELS AND SIMULATIONS 4.3 THE MILLENNIUM SIMULATION

Figure 4.2: The Millennium Run at z = 0. The color-codeddark matter density is displayed in slices with consecutiveenlargements by factors of four. The thickness in eachpanel is 15h−1Mpc. (Source: Volker Springel and others[95]).

The bottom panel appears homogeneous on largescale while the zoom on a large cluster in the toppanel has hundreds of dark matter halos orbiting thelarge cluster.

Quasars are the most luminous objects in the Uni-verse and can be detected at large distances. Todaythe most distant quasar has been detected with aredshift of z = 6.43 [96]. Quasars a thought to har-bor a supermassive black hole, but it is still an openquestion what kind of object such a first quasar is.However since every bright galaxy appears to hosta supermassive black hold, one could assume thatthe quasars are hosted by the largest galaxies. Thelargest dark matter filament which is a candidate fora first quasar is illustrated in figure 4.3. The centralobject has the largest mass in the entire simulation atz = 6.2 with a mass of 6.8×1010h−1M. The simula-tion can therefore locate the descendants of the firstquasars, the authors found out that all quasar can-didates ended up in the center of the most massiveclusters.

Finally the evolution over time of the same sliceis shown in figure 4.4. The dark matter grows likea sponge into filaments and the clustering get denser

Figure 4.3: A first quasar candidate at redshift z = 6.2and the same region at redshift z = 0. The two panels onthe left show the dark matter density distribution whilethe two panels on the right show the baryonic matter.The spheres correspond to galaxies and their volume isproportional to the stellar mass. The galaxies representthe light distribution and the dark matter (grayed out)is invisible. The blue spheres are galaxies with ongoingstar formation, while red spheres have little or no starformation. (Source: Volker Springel and others [95]).

with time.

Because the simulation analyzes the clusteringgrowth over time, it is possible to compare not onlythe final result with observations, but also the evo-lution over time. The CfA Redshift Survey and thenew Sloan Digital Sky Survey detected millions of ce-lestial objects and measured their spectra and theirdistance from us. Therefore an almost direct com-parison is possible with the millennium simulation asshown in figure 4.5. The SDSS and 2dFGRS subre-gions are slices through the three-dimensional galaxydistribution. The surveys reveal a group and clusterslinked within filaments like a web or sponge. Thesimilarity with the simulation based on the ΛCDMmodel is striking.

More detailed informations about the simulationand the post-processing can be found in the arti-cle “The large-scale structure of the Universe” by V.Springel [98].

21

4 MODELS AND SIMULATIONS 4.3 THE MILLENNIUM SIMULATION

Figure 4.4: Time evolution of slices through the densityfield with a thickness of 15 Mpc/h. The redshift fromleft to right is z=18.3 (t = 0.21 Gyr), z=5.7 (t = 1.0 Gyr),Redshift z=1.4 (t = 4.7 Gyr), Redshift z=0 (t = 13.6 Gyr)today. (Source: Gerard Lemson [92]).

Figure 4.5: Observation of galaxy distribution comparedwith the simulation. The top pie is identical to figure 2.3and shows the CfA2 great wall with the coma cluster inthe center. The top slice shows the “Sloan great wall”containing over 10,000 galaxies. The left wedge shownone half of the 2dFGRS [46][97] with 220000 galaxies anda depth of 2 billion years. The bottom and right wedge aresimulated galaxy survey from the millennium simulation.(Source: Volker Springel [98]).

22

5 Observational Tests

Observations always have the last word against atheory, and cosmology is no exception, indeed theo-ries evolve as the observation become more detailedthan ever. For instance the strong evidence for darkmatter and more recently the observational evidencefor dark energy have entailed the concordance ΛCDMmodel. This model has later been validated by theWMAP and SDSS experiments.

The field of cosmology and structure formation hasgrown significantly in the last ten years and so havethe observation experiments. The number of projectsand observations methods in action today is too largeto handle here. Nevertheless we dedicate this chapterto two recent observation methods. The Sloan DigitalSky Survey (SDSS) [82] provided significant resultsfor the theory of structure formations and is there-fore shortly introduced. We will then look at one ofthe results of the “Cosmological Evolution Survey”(COSMOS) [99] which derived a three-dimensionalmap of dark matter.

5.1 Sloan Digital Sky Survey

The Sloan Digital Sky Survey (SDSS) uses a dedi-cated 2.5 meter telescope to map objects and mea-sure their spectra and hence the redshift. The com-bined data is used to create a three-dimensional mapof the observed objects. The results are used in awide range of astronomical topics. Specifically inthe domain of structure formation the study of thefirst quasars, the distribution of galaxies and com-parison with simulations are invaluable. The surveyprovides an excellent complement to the cosmic mi-crowave background measurements from the WMAPsatellite. For instance the large-scale structure ob-servations with the sheer amount of data providedby SDSS came to the same evidence for dark mat-ter and dark energy as the WMAP observations andtherefore both observations tighten the parameters.

5.1.1 Technical information

The survey uses a dedicated 2.5 m f/5 telescopemodified Ritchey-Chretien wide-field telescope at the

Apache Point Observatory in New Mexico [100]. Thetelescope is equipped with a photometric/astrometricmosaic camera with 120 million pixels and five filtersto take images of the objects, like a classic photo-graph. Additionally spectrograph fed with 640 fiberspositioned for each object analyze the same portionof the sky. The spectroscopy analysis is later used toclassify the elements and deduce the redshift.

5.1.2 Some results

So far in the last fifth data release [101] the Sloan sur-vey recorded over a million spectra including 675000galaxies and 90000 quasars and mapped about one-quarter of the entire sky. See for example figure 2.3 inchapter 2. An other example is the study of the powerspectrum using a sample of over 200000 galaxies [83]from the SDSS database. The results are useful to es-timate the cosmological parameters because the threedimensional power spectrum gained from a galaxysurvey are a good complementary method to CMBobservations. The latest results are therefore com-bined with the WMAP observation of the CMB andmaximally confine the cosmological parameters. Theanalysis of Tegmark, Max et al. 2006 [86] is based on58360 so-called luminous red galaxies (LRGs) whichare farther away and also 285804 main galaxies (seefigure 5.1).

The SDSS data sets can provide very accurate re-sults because it is possible to measure the galaxy clus-tering on large scale. Indeed, as we have learned, thelinear theory is still valid on large scales and the non-linear clustering can be better approximated. TheSDSS Luminous Red Galaxy sample offers an opti-mal data set to measure the clustering and its powerspectrum. The results are displayed in figure 5.2. It isinteresting to note that the non-linear corrections arestarting to take effect like expected at k > 0.09h/Mpcmarked by the vertical dashed line. This confirms notonly the validity of the linear theory, but also showshow a galaxy survey like SDSS can measure clusteringon large scales and thus produce robust constrains.The cosmological parameters measured from SDSSLRG and WMAP are presented in table 5.1. Noticehowever that only the parameters discussed previ-

23

5 OBSERVATIONAL TESTS 5.2 DARK MATTER SCAFFOLDING

-1000 -500 0 500 1000

-1000

-500

0

500

1000

Figure 5.1: Distribution of luminous red galaxies (LRGs)and main galaxies from SDSS. All 38893 galaxies (dots)lie within 1.25 of the equator plane and are displayed in apie diagram. The LRGs (black dots) are farther away, themain galaxies (green) are closer. This sample of galax-ies is used to compare the power spectrum on variousscales. Additionally the boundaries for the subsamplesare marked with NEAR, MID and FAR. (Source: MaxTegmark [86]).

ously in this article are presented here. Interestinglythe neutrino density is measured to be an order ofmagnitude smaller than cold dark matter.

Table 5.1: Cosmological parameters based on the SDSSLRG analysis and coupled with the latest WMAP results.(Source: Max Tegmark [86]).

Par. Value Meaning

Ωtot 1.003+0.010−0.009 Total density/ρcrit

ΩΛ 0.761+0.017−0.018 Dark energy parameter

Ωm 0.239+0.018−0.017 Matter density/ρcrit

Ωb 0.0416+0.0019−0.0018 Baryon density/ρcrit

Ωc 0.197+0.016−0.015 CDM density/ρcrit

Ων < 0.024 Neutrino density/ρcrit

Ωk −0.0030+0.0095−0.0102 Spacial curvature

h 0.730+0.019−0.030 Hubble parameter

5.2 Dark Matter Scaffolding

Most of the current scientific knowledge is based onbaryonic matter only, thus a direct comparison of the

Figure 5.2: Measurement of the power spectrum for allgalaxies. The solid red curves curve satisfy a linear ΛCDMfitted to the latest WMAP data. The dashed lines includenon-linear corrections. (Source: Max Tegmark [86]).

theory with dark matter is of great importance. How-ever the nature of dark matter is still a mystery todaymostly because it does not interact with electromag-netic radiation and can therefore not be observed di-rectly. Nevertheless, as we have seen, dark mattercan be observed indirectly due to its gravitationalinfluence on visible baryonic matter. Indeed as dis-cussed in 2.4.3 gravitational lensing is an accuratemethod to measure its presence. Furthermore themethod is reliable because it is not only sensitive toall matter, but it is also based on Einstein’s generalrelativity only and does not rely on further astrophys-ical assumptions. Because dark matter has a funda-mental influence on the formation of stars and cosmicstructure and also lies on the root of the theory, thestudy of its growth over time is of great interest.

This section is dedicated to the Cosmic EvolutionSurvey (COSMOS) and a related recent article “Darkmatter maps reveal cosmic scaffolding” [41] publishedin Nature on January 2007. The article explains howscientists derived a first high resolution three dimen-sional map of dark matter.

5.2.1 Observation methods

The Cosmic Evolution Survey (COSMOS) [99] is de-signed to measure the evolution of galaxies and large

24

5 OBSERVATIONAL TESTS 5.2 DARK MATTER SCAFFOLDING

scale structures (LSS) over a redshift range z > 0.5to 6, thus over time [102]. The major scientific ob-jective of the survey also include nuclear activity ingalaxies and gravitational lensing for the mapping ofdark matter. The survey covers an area of two squaredegree located near the celestial equator to enable anobservation from both hemisphere. The area is ninetimes the area of the moon (figure 5.3).

Figure 5.3: Sky coverage of the COSMOS survey comparedto the moon. Image credit: NASA, ESA and Z. Levay(STScI) [103]

COSMOS is the largest survey the Hubble SpaceTelescope (HST) has ever done and took nearly1000 hours of observation in 640 orbits. Addition-ally the major space telescopes (Spitzer, GALEX,XMM, Chandra) where also involved in the survey.The project also received the commitments from thelargest ground based telescopes like ESO-VLT inChile, Subaru in Hawaii, VLA radio telescope in NewMexico, USA and others [102]. In a sense this smallsky field received the full mankind observing powerfrom X-ray to millimeter wavelength.

The high resolution and depth of HST images aremainly used to map dark matter with the weak lens-ing method. Moreover the HST images enable thestudy of galaxy evolution from their morphologies.Deep imaging are recorded by the large ground basedtelescopes and the ESO’s VLT (Very Large Tele-scope) provided most of the spectra. At completionthe survey detected more than two million galaxiesand also quasars with a redshift z > 6 going back

when the Universe was only one billion years old[104].

5.2.2 Results

The combination of weak lensing method with visibleand X-ray imaging enables to compare directly thedistribution of dark matter with the baryonic matteras seen in figure 5.4. The results support our the-oretical understanding that baryonic matter is builtwithin a scaffolding of dark matter. A major sur-

Figure 5.4: Comparison of the distribution of baryonicmatter (left) with dark matter (right). The colors havebeen inverted to provide a better contrast. The colorstrength is proportional to the density of mass. Credit:NASA, ESA and R. Massey (California Institute of Tech-nology) [41].

vey objective is to study the evolution of large scalestructures and dark matter over time. The overlap-ping redshift datasets provide an accurate spectra forthousands of galaxies and allow a faithful observa-tion of the evolution. That is the galaxies are splitin discrete redshift slices and the dark matter can bemapped Together with the weak lensing data. Ac-cordingly the dark matter growth over time can bevisualized (figure 5.5). The galaxies in red and thestellar mass in yellow accumulate into the densest re-gion of dark matter and offer a strong evidence to thetheoretical predictions.

A full three dimensional map is obtained by com-bining all discrete slices and is displayed in figure5.6. The map extends halfway back to 7.7 billionyears ago and shows how dark matter evolved andgrew more clumpy. This result is in agreement withthe theoretical predictions discussed in the previouschapters. Moreover this direct mass exploration withweak lensing has the potential to bind observationswith the theories. Because the model is mainly basedon the evolution of dark matter under gravity.

25

5 OBSERVATIONAL TESTS 5.2 DARK MATTER SCAFFOLDING

Figure 5.5: Evolution of the of dark matter distributionover time. The panels from left to right are taken at aredshift of ∼0.3, ∼0.5, ∼0.7. The original colors have beeninverted to provide a better contrast. The black solid linedelimit the contour of the lensing convergence in steps of0.33%. The red shapes show the distribution of galaxiesand yellow is the stellar mass. (Source: Richard Massey[41]).

Figure 5.6: This three-dimensional map of dark matterspans over three billion years. The time is increasing fromleft to right. Credit: NASA, ESA and R. Massey (Cali-fornia Institute of Technology) [41][103].

26

6 Conclusion

Today’s cosmic structures find their seeds in initial fluctuations generated during the inflationary epoch.Dark matter, the dominating mass in the Universe grew into filaments and clusters like a “cosmic web” andis organized into spherical halos. Baryonic matter then collapsed into the dark matter gravitational scaffoldwhere stars and galaxies could later form. The fluctuations are growing first according to a linear theoryand later to a non-linear collapse model. Small galaxies then merged to form larger lens and spiral galaxiesin a bottom-up process. And finally galaxies shaped clusters and super clusters in a process dictated by thedark matter presence. Moreover the mysterious dark energy not only closed the energy budget to explainour flat Universe but its strange negative pressure is now responsible for the cosmic expansion’s acceleration.

The standard concordance model or ΛCDM model describes this general perspective and is constrainedby ever more accurate observations. Indeed the model is so well established that there seems to be littleroom to prove this paradigm wrong. However many fundamental questions remain unanswered, namely whyis the density parameter so close to one? Why is the Universe dominated by dark matter and not normalmatter? Where did the primeval fluctuations came from as the inflation theory is highly speculative? Andmost importantly what is dark matter and what is this pervasive dark energy?

The last decade brought remarkable progress in our understanding of structure formation to a point wherewe call this era “precision cosmology”. On one side it seems that the Universe’s evolution is fully understood,at least within 10%, but on the other side 95% of the underlying physics are fully unknown! Indeed neitherdark matter nor dark energy have been verified by laboratory experiments. On that account this quote fromFeynman is certainly appropriate:”It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are – if it doesn’t agree withexperiment, it’s wrong.” – R.P. Feynman

Of course time will tell and the existence of many known particles today where predicted long beforethey could be directly observed. Additionally new satellites like Planck (the WMAP successor), JWST (theHST successor) and large ground based ten meters optical telescopes will be able to look at ever fainterobjects and bring a precision able to settle or break the model. Nevertheless key questions remain open andthe actual model could be fragile in the light of future surveys and might require us to rethink the currentparadigm.

27

List of Figures

2.1 Schematic view of the parsec definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Three dimensional schematic view of the Local Group . . . . . . . . . . . . . . . . . . . . . . 72.3 The Sloan Great Wall compared to CfA2 Great Wall . . . . . . . . . . . . . . . . . . . . . . . 72.4 Growth of matter density contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Rotational velocities for four galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Gravitational lensing with the massive and compact cluster Abell 2218 . . . . . . . . . . . . . 102.7 Hubble diagram of the measurement of supernovae of type 1A . . . . . . . . . . . . . . . . . . 122.8 Energy Distribution of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Linear growth D+ in function of the redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Adiabatic transfer function for baryons, HDM and CDM . . . . . . . . . . . . . . . . . . . . . 173.3 Density fluctuations of the Universe on different scales . . . . . . . . . . . . . . . . . . . . . . 18

4.1 Simulation of structure formation with gas and dark matter particles . . . . . . . . . . . . . . 204.2 The Millennium Run at z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 A first quasar candidate at redshift z = 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Time evolution of slices through the density field . . . . . . . . . . . . . . . . . . . . . . . . . 224.5 Observation of galaxy distribution compared with the simulation . . . . . . . . . . . . . . . . 22

5.1 Distribution of luminous red galaxies (LRGs) and main galaxies from SDSS . . . . . . . . . . 245.2 Measurement of the power spectrum for all galaxies . . . . . . . . . . . . . . . . . . . . . . . 245.3 Sky coverage of the COSMOS survey compared to the moon . . . . . . . . . . . . . . . . . . 255.4 Comparison of the distribution of baryonic matter with dark matter . . . . . . . . . . . . . . 255.5 Evolution of the of dark matter distribution over time . . . . . . . . . . . . . . . . . . . . . . 265.6 Three-dimensional map of dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

28

Bibliography

[1] Hubble, Edwin: A relation between distance and radial velocity among extra-galactic nebulae. In:Proc. Nat. Acad. Sci. 15 (1929), S. 168–173

[2] Steigman, Gary: Primordial nucleosynthesis: Successes and challenges. In: Int. J. Mod. Phys. E15(2006), S. 1–36. astro-ph/0511534

[3] Peebles, P. J. E.: The Cosmological Tests. In: ArXiv Astrophysics e-prints (2001).astro-ph/0102327

[4] Unsold, Albrecht ; Baschek, Bodo: Der neue Kosmos. Einfuhrung in die Astronomie und Astro-physik. 7. A., korr. Nachdr. Springer, Berlin, 2004. – ISBN 3540421777

[5] Coles, Peter ; Lucchin, Francesco: Cosmology. The Origin and Evolution of Cosmic Structure. 2ndedition. Wiley & Sons, 2002. – ISBN 0471489093

[6] Yao, W. M. et al.: Review of particle physics. In: J. Phys. G33 (2006), 1-1232.http://www.iop.org/EJ/article/0954-3899/33/1/001/g_33_1_001.html

[7] Harrison, Edward: The redshift-distance and velocity-distance laws. In: Astrophys. J. 403 (1993),January, S. 28–31. http://dx.doi.org/10.1086/172179. – DOI 10.1086/172179

[8] Krisciunas, Kevin: Look back time, the age of the Universe, and the case for a positive cosmologicalconstant. In: J. Roy. Astron. Soc. Canada 87 (1993), S. 223. astro-ph/9306002

[9] Padmanabhan, T.: Structure Formation in the Universe. 1st edition. Cambridge University Press,1993. – ISBN 0521414482

[10] Lang, Harald: How Far Out in the Universe Can We See?http://www.math.kth.se/~lang/distance.htm. Version: 2007

[11] Demtroder, Wolfgang: Experimentalphysik. Bd.4 : Kern-, Teilchen- und Astrophysik. 2., uberarb.A. Berlin Heidelberg New York : Springer, Berlin, 2004. – ISBN 3540214518

[12] Grebel, Eva K.: The Local Group. In: ASP Conference Series (2000). astro-ph/0008249

[13] Gott, III J. R. J. Richard et al.: A Map of the Universe. In: Astrophys. J. 624 (2005), S. 463.astro-ph/0310571

[14] Deng, Xin-Fa et al.: The Sloan Great Wall from the SDSS Data Release 4. In: Acta Physica PolonicaB 38 (2007), jan, S. 1–256

[15] Peebles, P. J. E.: Recombination of the Primeval Plasma. In: Astrophys. J. 153 (1968), S. 1

[16] Primack, Joel R.: Dark matter and structure formation. In: ArXiv Astrophysics e-prints (1997).astro-ph/9707285

[17] Silk, Joe: Dark Matter essay. In: University of California - Berkeley Astronomy Department (2007).http://astro.berkeley.edu/~mwhite/darkmatter/essay.html

[18] Zwicky, Fritz: Spectral displacement of extra galactic nebulae. In: Helv. Phys. Acta 6 (1933), S. 110

29

Bibliography Bibliography

[19] Zwicky, Fritz: On the Masses of Nebulae and of Clusters of Nebulae. In: Astrophys. J. 86 (1937), S.217

[20] Clowe, Douglas et al.: A direct empirical proof of the existence of dark matter. In: Astrophys. J.(2006). astro-ph/0608407

[21] Rubin, Vera C. ; Ford, W. K.: Rotation of the Andromeda Nebula from a Spectroscopic Survey ofEmission Regions. In: Astrophys. J. 159 (1970), S. 379–403

[22] Rubin, V. C. ; Thonnard, N. ; Ford, Jr. W. K. W. K.: Rotational properties of 21 SC galaxies witha large range of luminosities and radii, from NGC 4605 /R = 4kpc/ to UGC 2885 /R = 122 kpc/. In:Astrophys. J. 238 (1980), S. 471

[23] Rubin, V. C. ; Thonnard, N. ; Ford, W. K. Jr.: Extended rotation curves of high-luminosityspiral galaxies. IV - Systematic dynamical properties, SA through SC. In: Astrophys. J. 225 (1978),November, S. L107–L111. http://dx.doi.org/10.1086/182804. – DOI 10.1086/182804

[24] Kolanoski, Hermann: Einfuhrung in die Astroteilchenphysik. In:Lehrstuhl fur Experimentelle Elementarteilchenphysik (DESY) (2006), 171-188.http://www-zeuthen.desy.de/~kolanosk/astro0607/

[25] Briggs, F. H.: Cold gas, the HI 21cm line and evolving galactic potentials. In: ArXiv Astrophysicse-prints (2000). astro-ph/0001070

[26] Fukazawa, Yasushi ; Makishima, Kazuo ; Ohashi, Takaya: ASCA Compilation of X-Ray Propertiesof Hot Gas in Elliptical Galaxies and Galaxy Clusters: Two Breaks in the Temperature Dependences.In: ArXiv Astrophysics e-prints (2004). astro-ph/0411745

[27] Tisserand, Patrick ; Milsztajn, Alain: Preliminary results on galactic dark matter from the com-plete EROS2 microlensing survey. In: ArXiv Astrophysics e-prints (2005). astro-ph/0501584

[28] Alves, David R. et al.: MEGA: Microlensing Exploration of the Galaxy and Andromeda. In: ArXivAstrophysics e-prints (2003). astro-ph/0312167

[29] Schramm, David N. ; Turner, Michael S.: Big-bang nucleosynthesis enters the precision era. In:Rev. Mod. Phys. 70 (1998), S. 303–318. astro-ph/9706069

[30] Steigman, Gary: Big bang nucleosynthesis: Probing the first 20 minutes. In: ArXiv Astrophysicse-prints (2003). astro-ph/0307244

[31] Schnakenberg, Jurgen: Thermodynamik und Statistische Physik. Einfuhrung in die Grundlagen derTheoretischen Physik. 2nd edition. Wiley-VCH, 2002. – ISBN 3527403620

[32] Bartelmann, Matthias: Lectures on Cosmology. In: Institut fur Theoretische Astrophysik - Univer-sitat Heidelberg (2007). http://msb.ita.uni-heidelberg.de/Lectures/cosmology

[33] Lokas, Ewa L. ; Mamon, Gary A.: Dark matter distribution in the Coma cluster from galaxykinematics: breaking the mass-anisotropy degeneracy. In: Mon. Not. Roy. Astron. Soc. 343 (2003), S.401. astro-ph/0302461

[34] Horner, Donald J. ; Mushotzky, Richard F. ; Scharf, Caleb A.: Observational Tests of the Mass-Temperature Relation for Galaxy Clusters. In: Astrophys. J. 520 (1999), S. 78–86. astro-ph/9902151

[35] Donahue, Megan: Origins of Clusters and the Intracluster Medium. In: Woodward, C. E. (Hrsg.); Shull, J. M. (Hrsg.) ; Thronson, H. A. Jr. (Hrsg.): Origins Bd. 148, 1998 (Astronomical Societyof the Pacific Conference Series), S. 109–+

30

Bibliography Bibliography

[36] Zhang, Yu-Ying: The Structure of Galaxy Clusters and their Intracluster Mediumas Observed with XMM-Newton, Ludwig-Maximilians-Universitat Munchen, Diss., July 2005.http://edoc.ub.uni-muenchen.de/archive/00004628/. – 174 S.

[37] Durrell, Patrick R. ; Ciardullo, Robin ; Feldmeier, John J. ; Jacoby, George H. ; Sigurdsson,Steinn: Intracluster Red Giant Stars in the Virgo Cluster. In: Astrophys. J. (2002), May, S. 119–131.astro-ph/0201219

[38] Wambsganss, Joachim: Gravitational lensing in astronomy. In: Living Rev. Rel. 1 (1998), S. 12.astro-ph/9812021

[39] Gravitational Lens Helps Hubble and Keck Discover Galaxy Building Block.http://hubblesite.org/newscenter/archive/releases/2001/32/image/b. Version: 2001

[40] Kneib, J. P. ; Ellis, R. S. ; Smail, I. ; Couch, W. J. ; Sharples, R. M.: Hubble Space Telescope Ob-servations of the Lensing Cluster Abell 2218. In: Astrophys. J. 471 (1996), S. 643. astro-ph/9511015

[41] Massey, Richard et al.: Dark matter maps reveal cosmic scaffolding. In: Nature 445 (2007), January,S. 286. astro-ph/0701594

[42] Van Waerbeke, Ludovic et al.: Cosmic Shear Statistics and Cosmology. In: Astron. Astrophys. 374(2001), S. 757–769. astro-ph/0101511

[43] Zwicky, Fritz: Nebulae as Gravitational Lenses. In: Phys. Rev. 51 (1937), Feb, Nr. 4, S. 290.http://dx.doi.org/10.1103/PhysRev.51.290. – DOI 10.1103/PhysRev.51.290

[44] Hu, Wayne ; Dodelson, Scott: Cosmic Microwave Background Anisotropies. In: Ann. Rev. Astron.Astrophys. 40 (2002), S. 171–216. astro-ph/0110414

[45] Cornish, Neil J. ; Spergel, David N. ; Starkman, Glenn D. ; Komatsu, Eiichiro: Constrainingthe Topology of the Universe. In: Phys. Rev. Lett. 92 (2004), S. 201302. astro-ph/0310233

[46] Peacock, John A. et al.: A measurement of the cosmological mass density from clustering in the2dF Galaxy Redshift Survey. In: Nature 410 (2001), S. 169–173. astro-ph/0103143

[47] Tegmark, Max et al.: Cosmological parameters from SDSS and WMAP. In: Phys. Rev. D69 (2004),S. 103501. astro-ph/0310723

[48] Zehavi, I. et al.: Galaxy clustering in early SDSS redshift data. In: Astrophys. J. 571 (2002), S.172–190. astro-ph/0106476

[49] Burles, S. ; Nollett, K. M. ; Turner, Michael S.: What is the BBN prediction for the baryondensity and how reliable is it? In: Phys. Rev. D63 (2001), S. 063512. astro-ph/0008495

[50] Carroll, Sean M.: Why is the universe accelerating? In: ECONF C0307282 (2003), S. TTH09.astro-ph/0310342

[51] Peebles, P. J. E. ; Ratra, Bharat: The cosmological constant and dark energy. In: Rev. Mod. Phys.75 (2003), S. 559–606. astro-ph/0207347

[52] Perkins, Donald H.: Introduction to High Energy Physics. 4th edition. Cambridge University Press,2000. – ISBN 0521621968

[53] Krauss, Lawrence M. ; Turner, Michael S.: The cosmological constant is back. In: Gen. Rel. Grav.27 (1995), S. 1137–1144. astro-ph/9504003

[54] Carroll, Sean M.: The cosmological constant. In: Living Rev. Rel. 4 (2001), S. 1. astro-ph/0004075

31

Bibliography Bibliography

[55] Perlmutter, S. et al.: Measurements of Omega and Lambda from 42 High-Redshift Supernovae.In: Astrophys. J. 517 (1999), S. 565–586. astro-ph/9812133

[56] Riess, Adam G. et al.: Observational Evidence from Supernovae for an Accelerating Universe and aCosmological Constant. In: Astron. J. 116 (1998), S. 1009–1038. astro-ph/9805201

[57] Knop, Robert A. et al.: New Constraints on ΩM , ΩΛ, and w from an Independent Set of Eleven High-Redshift Supernovae Observed with HST. In: Astrophys. J. 598 (2003), S. 102. astro-ph/0309368

[58] Ballinger, W. E. ; Peacock, J. A. ; Heavens, A. F.: Measuring the cosmological constant withredshift surveys. In: ArXiv Astrophysics e-prints (1996). astro-ph/9605017

[59] Perlmutter, Saul ; Schmidt, Brian P.: Measuring Cosmology with Supernovae. In: Weiler,K. (Hrsg.): Supernovae and Gamma-Ray Bursters Bd. 598, 2003 (Lecture Notes in Physics, BerlinSpringer Verlag), S. 195–217. astro-ph/0303428

[60] Supernova Cosmology Project. http://panisse.lbl.gov. Version: 2007

[61] Perlmutter, Saul: Supernovae, Dark Energy, and the Accelerating Universe. In: Physics Today116 (2003), April, 53-60. http://panisse.lbl.gov/

[62] Spergel, D. N. et al.: First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observa-tions: Determination of Cosmological Parameters. In: Astrophys. J. Suppl. 148 (2003), S. 175.astro-ph/0302209

[63] Scranton, Ryan et al.: Physical Evidence for Dark Energy. In: ArXiv Astrophysics e-prints (2003),July. astro-ph/0307335

[64] Spergel, D. N. et al.: Wilkinson Microwave Anisotropy Probe (WMAP) three year results: Implica-tions for cosmology. In: ArXiv Astrophysics e-prints (2006), March. astro-ph/0603449

[65] Conselice, Christopher J.: The Universe’s Invisible Hand. In: Scientific American 272 (2007), S.74–79

[66] Padmanabhan, T.: Dark Energy: Mystery of the Millennium. In: AIP Conf. Proc. 861 (2006), S.179–196. astro-ph/0603114

[67] NASA ; CXC: Galaxy Clusters and Dark Energy. In: Chan-dra X-ray Observatory Harvard-Smithsonian Center for Astrophysics (2004).http://chandra.harvard.edu/photo/2004/darkenergy/

[68] Camenzind, Max: Einfuhrung in die Astronomie und Astrophysik Iund II. In: Landessternwarte Ruprecht-Karls-Universitat Heidelberg (2003).http://www.lsw.uni-heidelberg.de/projects/theory/resource.htm

[69] Liddle, Andrew R. ; Lyth, David H.: Cosmological Inflation and Large-Scale Structure. 2nd edition.Cambridge University Press, 2000. – ISBN 0521575982

[70] Howarth, Ian: Lecture Notes - Cosmology. In: Astrophysics Group at the University College London(2006). http://zuserver2.star.ucl.ac.uk/~idh/1B23/

[71] Padmanabhan, T.: Advanced topics in cosmology: A pedagogical introduction. In: AIP Conf. Proc.843 (2006), S. 111–166. astro-ph/0602117

[72] Bertschinger, Edmund: Cosmological Perturbation Theory and Structure Formation. In: ArXivAstrophysics e-prints (2001). astro-ph/0101009

32

Bibliography Bibliography

[73] Francis, Caroline: Structure formation in the linear regime. In: Institute for Astronomy of the Uni-versity of Edinburgh (2006). http://www.roe.ac.uk/ifa/postgrad/pedagogical_seminars.html

[74] Peacock, J. A. ; Dodds, S. J.: Non-linear evolution of cosmological power spectra. In: Mon. Not.Roy. Astron. Soc. 280 (1996), June, S. L19–L26. astro-ph/9603031

[75] Springek, Volker: On the Formation and Evolution of Galaxies, Ludwig-Maximilians-UniversitatMunchen, Diss., July 1999. http://deposit.ddb.de/cgi-bin/dokserv?idn=959597069. – 273 S.

[76] Borgani, Stefano: The Large-Scale Structure of the Universe. In: Theo-retical and Practical Elementary Aspects of High Energy Physics (2001), 75-+.http://fisica.usac.edu.gt/public/curccaf_proc/borganihtml

[77] Silk, Joseph: Cosmic Black-Body Radiation and Galaxy Formation. In: Astrophys. J. 151 (1968),February, S. 459–+

[78] Efstathiou, G. ; Silk, Joseph: The formation of galaxies. In: Fundamentals of Cosmic Physics 9(1983), November, S. 1–138

[79] Sahni, Varun: Dark matter and dark energy. In: Lect. Notes Phys. 653 (2004), S. 141–180.astro-ph/0403324

[80] Coles, Peter: Large-scale Structure with the SKA. In: ArXiv Astrophysics e-prints (2004), August.astro-ph/0408440

[81] Dodelson, Scott: Modern Cosmology. 1st edition. Academic Press, 2003. – ISBN 0122191412

[82] Sloan Digital Sky Survey. http://www.sdss.org/. Version: 2004

[83] Tegmark, Max et al.: The 3D power spectrum of galaxies from the SDSS. In: Astrophys. J. 606(2004), S. 702–740. astro-ph/0310725

[84] Kurek, Aleksandra ; Szydlowski, Marek: The LambdaCDM model on the lead – a Bayesiancosmological models comparison. In: ArXiv Astrophysics e-prints (2007). astro-ph/0702484

[85] Turner, Michael S.: The case for Lambda-CDM. In: ArXiv Astrophysics e-prints (1997), March.astro-ph/9703161

[86] Tegmark, Max et al.: Cosmological Constraints from the SDSS Luminous Red Galaxies. In: Phys.Rev. D74 (2006), S. 123507. astro-ph/0608632

[87] Riess, Adam G. et al.: New Hubble Space Telescope Discoveries of Type Ia Supernovae at z > 1:Narrowing Constraints on the Early Behavior of Dark Energy. In: Astrophys. J. 659 (2006), April, S.98–121. http://dx.doi.org/10.1086/510378. astro-ph/0611572. – DOI 10.1086/510378

[88] Ostriker, J. P. ; Steinhardt, P. J.: Cosmic Concordance. In: ArXiv Astrophysics e-prints (1995),May. astro-ph/9505066

[89] Ren, Jie ; Meng, Xin-He: Dark viscous fluid described by a unified equation of state in cosmology.In: ArXiv Astrophysics e-prints (2006). astro-ph/0605010

[90] Efstathiou, G. et al.: The Planck Bluebook. in press at the ESA Publication Division. PlanckConsortia, 2005. – The Scientific Program of Planck

[91] Springel, Volker ; Hernquist, Lars: The history of star formation in a LCDM universe. In: Mon.Not. Roy. Astron. Soc. 339 (2003), S. 312. astro-ph/0206395

33

Bibliography Bibliography

[92] Lemson, Gerard: Halo and Galaxy Formation Histories from the Millennium Simulation: Publicrelease of a VO-oriented and SQL- queryable database for studying the evolution of galaxies in theLambdaCDM cosmogony. In: ArXiv Astrophysics e-prints (2006), aug. astro-ph/0608019

[93] Virgo Consortium. http://www.virgo.dur.ac.uk. Version: 2007

[94] Max Planck Institute for Astrophysics, Garching. http://www.mpa-garching.mpg.de. Version: 2007

[95] Springel, Volker et al.: Simulating the joint evolution of quasars, galaxies and their large-scaledistribution. In: Nature 435 (2005), S. 629–636. astro-ph/0504097

[96] Fan, Xiaohui et al.: A Survey of z > 5.7 Quasars in the Sloan Digital Sky Survey II: Discovery ofThree Additional Quasars at z > 6. In: Astron. J. 125 (2003), S. 1649. astro-ph/0301135

[97] Cole, Shaun et al.: The 2dF Galaxy Redshift Survey: Power-spectrum analysis of the final dataset andcosmological implications. In: Mon. Not. Roy. Astron. Soc. 362 (2005), S. 505–534. astro-ph/0501174

[98] Springel, Volker ; Frenk, Carlos S. ; White, Simon D. M.: The large-scale structure of theUniverse. In: Nature 440 (2006). astro-ph/0604561

[99] Cosmological Evolution Survey (COSMOS). http://cosmos.astro.caltech.edu. Version: 2007

[100] York, Donald G. et al.: The Sloan Digital Sky Survey: technical summary. In: Astron. J. 120 (2000),S. 1579–1587. astro-ph/0006396

[101] Weinberg, David H. et al.: The Fifth Data Release of the Sloan Digital Sky Survey. In: Astrophys.J. (submitted) (2006)

[102] Scoville, Nick et al.: The Cosmic Evolution Survey (COSMOS) – Overview. In: ArXiv Astrophysicse-prints (2006). astro-ph/0612305

[103] ESA European Space Agency: First 3D Map of the Universe’s Dark Matter Scaffolding.http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=40484. Version: 2007

[104] Scoville, Nick et al.: COSMOS : Hubble Space Telescope Observations. In: ArXiv Astrophysicse-prints (2006). astro-ph/0612306

34