Performance of the Electromagnetic Calorimeter of AMS-02 on the International Space Station ans measurement of the positronic fraction in the 1.5 – 350 GeV energy range.

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Performance of the Electromagnetic Calorimeter of AMS-02 on the International Space Station ans measurement of the positronic fraction in the 1.5 – 350

GeV energy range.

Laurent Basara

To cite this version:

Laurent Basara. Performance of the Electromagnetic Calorimeter of AMS-02 on the International

Space Station ans measurement of the positronic fraction in the 1.5 – 350 GeV energy range.. Other

[cond-mat.other]. Université de Grenoble, 2014. English. �NNT : 2014GRENY012�. �tel-01155127�

THÈSE

Pour obtenir le grade de

DOCTEUR DE L’UNIVERSITÉ DE GRENOBLE

Spécialité : Physique Subatomique et Astroparticules

Arrêté ministériel : 7 août 2006

Présentée par

Laurent Basara

Thèse dirigée par Sylvie Rosier-Lees

préparée au sein du Laboratoire d’Annecy-le-Vieux de Physique des Particules

et de l’Ecole Doctorale de Physique de Grenoble

Performance of the electromagnetic calorimeter of the AMS-02 experiment on the International Space Station and measurement of the positron fraction in the 1.5 – 350 GeV energy range

Thèse soutenue publiquement le 5 mai 2014, devant le jury composé de :

Yannis KARYOTAKIS

Directeur de recherche, LAPP - CNRS , Président

Andrei KOUNINE

Senior research scientist, MIT, Rapporteur

Pascal VINCENT

Professeur, LPNHE, Rapporteur

Fernando DI BARAO

Professeur, LIP - Lisbonne, Examinateur

Oscar ADRIANI

Professeur, Universita degli Studi di Firenze, Examinateur

Sylvie ROSIER-LEES

Directeur de recherche, LAPP - CNRS, Directeur de thèse

i

ii

À mon père.

À ma mère.

iii

iv

Contents

I. Physics of cosmic rays 3

1. Milestones of Physics in the early twentieth century 5

1.1. General introduction . . . . 5

1.2. Small clouds in a clear sky . . . . 6

1.2.1. The dawn of the twentieth century . . . . 6

1.2.2. Einstein’s Annus Mirabilis . . . . 6

1.2.3. The discharge of the electroscope . . . . 9

1.3. The discovery of cosmic rays . . . 11

1.3.1. Victor Hess and the “radiation from above” . . . 11

1.3.2. The nature of Cosmic Rays . . . 12

1.3.3. Thematic displacement . . . 14

1.4. From General Relativity to the Big-Bang . . . 14

1.4.1. Relativities . . . 14

1.4.2. Friedmann-Lemaître-Robertson-Walker metrics . . . 15

1.4.3. Hubble observations and the Big-Bang . . . 16

2. Modern cosmology and the problem of Dark Matter 19 2.1. Thermal history of the Universe . . . 19

2.1.1. A little bit of thermodynamics . . . 19

2.1.2. Parameterization of FLRW . . . 21

2.1.3. Interactions decoupling . . . 22

2.1.4. Calculation of relic densities and decouplings . . . 23

v

2.1.5. Planck results . . . 26

2.2. Evidence of dark matter . . . 27

2.2.1. Spiral galaxies rotation curves . . . 27

2.2.2. Bullet cluster . . . 30

2.3. Candidates and detection . . . 31

2.3.1. First candidates . . . 31

2.3.2. The standard model of particle physics and its limitations . . . . 31

2.3.3. Supersymmetry . . . 33

2.3.4. Other candidates . . . 36

2.3.5. Detection . . . 37

3. Cosmic rays 39 3.1. Spectrum . . . 39

3.1.1. Introduction . . . 39

3.1.2. Overall spectrum . . . 40

3.1.3. Nuclei . . . 44

3.1.4. Gamma rays . . . 44

3.2. Interaction of particles with matter . . . 47

3.2.1. High-energy photons . . . 48

3.2.2. Electrons and positrons . . . 48

3.2.3. Electromagnetic showers development . . . 52

3.2.4. Hadronic behavior . . . 54

3.3. Recent results from cosmic-ray detectors . . . 58

II. AMS-02: description and performances on the ISS 61 4. The AMS-02 experiment 63 4.1. Introduction and objectives . . . 63

4.2. Permanent magnet . . . 67

4.3. Transition Radiation Detector (TRD) . . . 68

4.4. Time-of-Flight System (ToF) . . . 71

4.5. Silicon Tracker . . . 72

4.6. Ring-Imaging Čerenkov Detector (RICH) . . . 76

4.7. Electromagnetic Calorimeter (ECAL) . . . 79

4.7.1. Layout . . . 79

4.7.2. Electronic readout . . . 80

vi

4.7.3. Correlation to temperatures . . . 84

4.7.4. Performance . . . 94

4.7.5. Conclusion on the Electronic CALorimeter . . . 95

4.8. Anti-Coincidence Counter (ACC) . . . 95

4.9. Star Tracker and GPS . . . 96

4.10. Electronics . . . 96

5. Commissionning with minimum ionizing particles 99 5.1. Corrections applied to the signal . . . 99

5.1.1. Motivation . . . 99

5.1.2. General selection . . . 101

5.1.3. Span . . . 103

5.1.4. Interlude: interaction length determination . . . 103

5.1.5. MIPs theoretical energy distribution . . . 105

5.2. Previous corrections applied to the signal . . . 106

5.2.1. Influence of the statistics on Most Probable value (MPV ) and Maximum (Max) . . . 107

5.2.2. Attenuation corrections . . . 109

5.2.3. Intercells equalization . . . 110

5.2.4. New weights after attenuation . . . 112

5.3. Layer-by-layer attenuation corrections . . . 112

5.3.1. Motivations . . . 112

5.3.2. Cells binning . . . 115

5.3.3. Results for X and Y planes . . . 117

5.3.4. Superlayer-by-superlayer corrections . . . 118

5.4. Cross-checks with KSC cosmics . . . 121

5.4.1. Attenuation profiles . . . 121

5.4.2. Shaping time measurements . . . 121

5.4.3. Conclusion . . . 122

5.5. Aging effects measurements . . . 124

5.5.1. Objectives . . . 124

5.5.2. MIP selection . . . 125

5.5.3. Temperature correlations . . . 125

5.5.4. Aging effect measurements . . . 132

5.6. Conclusion . . . 136

vii

6. The ECAL-standalone Charge Estimator (EQE) 137

6.1. Going beyond the cell level . . . 137

6.1.1. Cell position correction . . . 137

6.1.2. Fibers . . . 138

6.1.3. First 2D model . . . 139

6.1.4. Drawbacks of the 2D model . . . 140

6.2. Results for the 3D model . . . 141

6.2.1. The model . . . 141

6.2.2. C++ implementation . . . 143

6.2.3. Path length on Monte-Carlo . . . 143

6.2.4. Interlude: the Bertrand paradox and the notion of mean path length145 6.2.5. Estimation of mean path length using actual data . . . 147

6.2.6. From geometrical to physical model . . . 148

6.3. Motivation and feasibility . . . 149

6.4. Bethe-Bloch curves . . . 151

6.4.1. Selection using the Tracker charge . . . 151

6.4.2. Results . . . 151

6.4.3. Relativistic rising estimation . . . 155

6.5. Buiding the estimator . . . 155

6.5.1. Maximum Likelihood Estimators . . . 155

6.5.2. Application to EQE . . . 157

6.6. Performances of the estimator . . . 158

6.6.1. Influence of the number of layers . . . 158

6.6.2. Comparisons with the tracker charge . . . 158

6.6.3. Linearity checks . . . 160

6.7. Higher charges . . . 161

6.7.1. Quenching effect . . . 161

6.7.2. Splines . . . 163

III. Electron/proton rejection and positron fraction determination167 7. Proton/electron rejection: a first estimator 169 7.1. Motivations . . . 169

7.2. Electromagnetic/hadronic discrimination: a first approach . . . 170

7.2.1. E/P ratio . . . 170

7.2.2. Quantifying the misinterpreted events . . . 171

viii

7.2.3. Simple cuts approach limitations . . . 172

7.3. Implementation of a multi-variate analysis . . . 175

7.3.1. Datasets . . . 175

7.3.2. Variables selection . . . 177

7.3.3. Implementation . . . 179

7.4. A first Ecal Standalone Estimator . . . 180

7.4.1. Datasets preselection . . . 180

7.4.2. Variables selection and normalization . . . 181

7.4.3. Energy binning . . . 183

7.4.4. Energy bins results renormalization . . . 184

7.4.5. Final rejection results and prospects . . . 185

8. New ECAL estimator using Monte-Carlo 189 8.1. Need for Monte-Carlo boost . . . 189

8.1.1. Motivations . . . 189

8.1.2. Data/MC comparisons . . . 191

8.2. Smearing . . . 194

8.2.1. General principle and simple cases . . . 194

8.2.2. Shape-related variables . . . 194

8.2.3. S3/S5 ratio . . . 195

8.2.4. Data/Monte-Carlo comparison for energy density variables . . . . 195

8.3. Estimator performance . . . 201

8.3.1. Rankings . . . 201

8.3.2. Binned estimator distributions . . . 203

8.3.3. Overfitting check . . . 207

8.3.4. Comparison between Monte-Carlo samples . . . 210

8.3.5. Comparison between Monte-Carlo and beam test . . . 213

8.4. Rejection factor . . . 216

9. Positronic fraction measurement 219 9.1. Event selection using templates and first estimation of the positronic ratio 220 9.1.1. Introduction and general selections . . . 220

9.1.2. Selections relying on the ESE and TRD likelihood estimator . . . 221

9.1.3. Definition of the signal and background templates from the ISS Data225 9.1.4. Performance of the fit templates . . . 228

9.1.5. Optimal cuts on the selectors . . . 234 9.1.6. Positronic ratio measurement before the charge confusion corrections237

ix

9.2. Charge confusion corrections . . . 240

9.2.1. Charge confusion definition . . . 240

9.2.2. Tracker layer pattern and data weighting . . . 242

9.2.3. E/P selection . . . 247

9.2.4. Charge confusion estimation . . . 248

9.3. Final result . . . 252

9.3.1. Systematic uncertainties . . . 252

9.3.2. Statistical error . . . 254

9.3.3. Positron ratio . . . 258

9.3.4. Conclusions and prospectives . . . 261

IV. Appendices 263 A. Temperature sensors mapping check 265 B. Commissioning Datasets 271 B.1. Introduction . . . 271

B.2. Beam test events . . . 272

B.3. KSC Cosmics . . . 273

B.4. Taiwan Monte-Carlo . . . 275

B.5. Spectrum Monte-Carlo . . . 277

C. Variables used in ESEv3 279 C.1. Introduction . . . 279

C.2. Shape-based variables . . . 279

C.2.1. S3S5x, S3S5y, S3S5

. . . 279

C.2.2. Number of layers at MIP . . . 281

C.2.3. Shower Footprint . . . 281

C.2.4. Shower longitudinal dispersion . . . 282

C.2.5. Shower lateral dispersion . . . 283

C.2.6. Longitudinal profile fit . . . 284

C.3. Energy-deposited related variables . . . 285

C.3.1. Rear leakages . . . 285

C.3.2. Rear leakage . . . 285

C.3.3. Deviation from expected EM energy . . . 286

C.3.4. Energy density distributions . . . 286

x

C.3.5. Compatibility with fit . . . 287 C.3.6. N5ZProfile . . . 287 C.3.7. N5ZProfileMaxRatio . . . 290 D. Implementation of the proton-electron estimator 291 D.1. Motivations . . . 291 D.2. Complementary cumulative distribution function . . . 292

List of figures 301

List of tables 313

xi

xii

“ When I heard the learn’d astronomer,

When the proofs, the figures, were ranged in columns before me,

When I was shown the charts and diagrams, to add, divide, and measure them, When I sitting heard the astronomer where he lectured with much applause in the lecture-room,

How soon unaccountable I became tired and sick, Till rising and gliding out I wander’d off by myself, In the mystical moist night-air, and from time to time, Look’d up in perfect silence at the stars. ”

— Walt Whitman, Leaves of Grass (1855)

1

2

Part I.

Physics of cosmic rays

3

1

Milestones of Physics in the early twentieth century

Ce premier chapitre introductif présente les principaux jalons parcourant l’astrophysique de la première moitié du vingtième siècle. Quoiqu’étant sans lien direct avec l’objet particulier de notre étude, il permet de la recontextualiser historiquement. Une première partie décrit l’état de la physique fondamentale à l’aube du vingtième siècle, la seconde se concentre sur la découverte des rayons cosmiques, l’explication de leur nature, et leur impact sur la physique des particules, une dernière Section traitant quant à elle de cosmologie, des théories de la relativité à celle du Big Bang.

—

1.1. General introduction

This first Part aims at a general introduction to the history, theories and phenomena at stake in the physics of cosmic rays. The current Chapter reviews some of the discoveries of the first half of the Twentieth Century that gave birth to some of the branches of Physics which the rest of this work deals with, without of course pretending to stand among them. Chapter 2 uses the tools of modern cosmology to exhibit some evidences of the existence of dark matter, along with some physical mechanisms which could lead to its eventual indirect detection by particle detectors in space. The hasty reader can skip without qualms both Chapters, since their content won’t be directly used in the core

5

6 Chapter 1. Milestones of Physics in the early twentieth century

results of this work. On the contrary, Chapter 3, finally, reviews physical results directly relevant to the rest of this study.

1.2. Small clouds in a clear sky

1.2.1. The dawn of the twentieth century

The end of the nineteenth century is often considered as a time where many physicists thought their discipline was near to its end [Wró06]. We can quote the French chemist Marcellin Berthelot, who wrote in 1885 [Ber85] “Le monde est aujourd’hui sans mystère”

(the world is now without mystery) or Albert A. Michelson, who said in 1903 [Mic03]:

“The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote [...] Our future discoveries must be looked for in the sixth place of decimals”

The most famous citation, however, probably comes from the introduction of a lecture made by Lord Kelvin in 1900 [Kel01], reproduced in Figure 1.1. The extensive quote reads:

“The beauty and clearness of the dynamical theory, which asserts heat and light to be modes of motion, is at present obscured by two clouds. I. The first came into existence with the undulatory theory of light, and was dealt with by Fresnel and Dr. Thomas Young; it involved the question, how could the earth move through an elastic solid, such as essentially is the luminiferous ether?

II. The second is the Maxwell-Boltzmann doctrine regarding the partition of energy.”

1.2.2. Einstein’s Annus Mirabilis

The two clouds he is referring to will respectively lead to the development of the special

relativity and the quantum mechanics, mostly triggered by two articles of Albert Einstein

1.2. Small clouds in a clear sky 7

Figure 1.1.: The address of Lord Kelvin during the First International Congress of Physics,

6-12 August 1900, later reproduced in [Kel01]

8 Chapter 1. Milestones of Physics in the early twentieth century

Figure 1.2.: Albert Einstein around 1905

in 1905. That year, often designed as its Annus Mirabilis (“year of wonders”), he published four articles in Annalen der Physics

:

• On a Heuristic Point of View Concerning the Production and Transformation of Light [Ein05c], in which he asserts that energy can only be exchanged trough quanta of finite energy (later called photons) For the photoelectric effect, considering h the Boltzmann constant, ν the frequency of the incident light, and P the work performed by the electron before leaving the cathode, the energy E exchanged is derived by:

E = hν (1.2.1)

This will solve the second cloud and lead to the development of quantum physics.

• On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat [Ein05b], on the Brownian movement, will play an important role in the acceptance of the kinetic theory of matter, and a theoretical proof of the existence of atoms.

• On the electrodynamics of moving bodies [Ein05d] considers the Maxwell equations from a symmetric point of view (independent of the observer), which leads to break

1

the liberty to translate the original titles in English was taken; the original titles are available in the

Bibliography

1.2. Small clouds in a clear sky 9

the concepts of absolute time and space, and discredit the idea of a “luminiferous ether”.

• Does the Inertia of a Body Depend Upon Its Energy Content? [Ein05a] states that

“If a body emits the energy L in the form of radiation, its mass decreases by L/V

”, primer form of the formula on equivalence of energy and matter

E = mc

(1.2.2)

1.2.3. The discharge of the electroscope

Some other small “clouds”, however, proved to have great further development. Among the phenomena which were not explained, was the progressive loss of charge in the electroscope, a device used to measure the rate of ion production inside a hermetically sealed container. About that time, electroscopes are constituted by two thin conductive foils suspended at an electrode. When the electrode is charged, the foils carry a charge of same sign and are pushed apart; the greater the charge, the further they go, so that a measurement of the distance between them gives a direct estimation of electrode’s charge.

However, although isolated from ground, the electroscopes loss their charge through time, which tends to indicate that the discharge is made through the air. On the other hand, air is known to be an excellent electric insulator, keeping open to the question of the discharge.

A first element of answer came in 1896 (before Einstein’s articles). At that time, the recent equations of Maxwell seemed to indicate that light was rather made of electromagnetic waves than particles. However, that year, the french physicist Antoine Henri Becquerel discovered, by studying phosphorescent elements, that uranium salts emitted radiations able to discharge electrical materials far from the source [Béc96]. He established that those emitted rays increased the conductivity of gas through the creation of new charge carriers he named ions. Pierre and Marie Curie discovered that polonium and radium emitted these radiations as well, and coined the term radioactivity.

In 1897, Sir Joseph John Thomson proved that the electric charge is carried by

particles, that he named corpuscles [Tho97]. These new particles had the same charge

as the hydrogen atom, but a e/m ratio 1700 times greater. This will be a great success

both for the understanding of electricity and, along with the discovery of radioactivity,

10 Chapter 1. Milestones of Physics in the early twentieth century

for the atomic model. A few years later, in 1911, Robert Millikan would identify these new particles with the electrons and measure their absolute charge.

The explanation of the discharge of the electroscope wass finally given in 1900 by two German physicists: Julius Elster and Hans Geitel [EG00]. They showed that the discharge was significantly more important at an altitude of 3 km than on ground. By proving that the ionization of the air was the same phenomenon as the ionization from gas having been in contact with radioactive substances, they could conclude that a sort of natural radioactivity constituted the source of ionization in the atmosphere and of the electric discharge of charged conductors.

Figure 1.3.: Wulf’s Strahlunsapparat

In 1909, Theodor Wulf created a new electroscope, the Strahlungsapparat, by replacing

the foils by two conducting wires, attached at the top and the bottom, see Figure 1.3. A

microscope combined with a micrometer allowed to measure the distance between the two

wires in the middle with an unmatched precision. Despite Elster and Geitel observations,

it was widely believed that the natural radiation from ground was the source of the

electroscope discharge. Wulf predicted that if he moved far enough from those sources,

less radiation would be detected, advising to do balloon-bourne experiments [Wul09],

and tested himself the hypothesis in 1910 by comparing the radiation at the top and

bottom of the Eiffel Tower. Knowing the absorption coefficient of the air, it was expected,

at a 80 m height, half the discharge value with respect to that on ground, and nearly

no radiation on top (330 m) of the Eiffel tower. He found that the ionization fell from

1.3. The discovery of cosmic rays 11

6 ions cm

s

on ground to 3.5 ions cm

s

on top of the Eiffel tower, in contrast with the almost-null expected value [Wul10]. He concluded that one had either to expect an extra source of gamma-rays in the atmosphere, or a smaller absorption coefficient in air.

1.3. The discovery of cosmic rays

1.3.1. Victor Hess and the “radiation from above”

In 1910 and 1911, two flights between 300 and 2800 m were made by the German physicist Albert Gockel; he found that the radiation seemed to increase at high altitudes, but stated his conclusions with great doubts. In 1911 Domenico Pacini observed simultaneous variations of the rate of ionization over a lake, over the sea, and at a depth of 3 m from the surface. Pacini concluded from the decrease of radioactivity underwater that a certain part of the ionization had to be due to sources other than the radioactivity of the Earth [Pac12].

The real pushing forward was made by Victor Hess. Before 1911, the absorption coefficient in air had never been measured, what he did by using radioactive substances placed at various distances of a Wulf electrometer, finding 4.47 × 10

cm

(the density law predicted 4.4 × 10

cm

).

To test the other hypothesis, Hess decided to do a series of flights in balloons. He disposed of a Strahlungsapparat specifically developed by Wulf for measurements at high altitude and underwater. He made a total of nine flights, the seven last of them taking place between April and August 1912, to an altitude up to 5300 m. The last, and most famous, of them, took place the seventh of August, 1912, during a near-total eclipse to rule out the Sun as radiation source. Figure 1.4 shows him a few weeks before this flight.

The conclusions were the following: the ionization radiation decreases steady up to an altitude of about 1600 m, after which in increases up to 5300 m, where the amount of radiation has doubled, showing that penetrating radiation is entering the atmosphere from a source other than the Sun. Hess called the radiation Höhenstrahlung, radiation from above.

Those results were confirmed in 1913 and 1914 by Werner Kolhörster, who noted an

increase in the ionization up to an altitude of 9300 m.

12 Chapter 1. Milestones of Physics in the early twentieth century

Figure 1.4.: Victor Hess in his balloon, a few weeks before his most famous flight.

1.3.2. The nature of Cosmic Rays

In 1926, the term cosmic rays was coined by Robert Millikan who had made measurements of ionization due to cosmic rays from deep under water to high altitudes and around the globe. Millikan believed his measurements proved that the primary cosmic rays were energetic photons, and proposed a theory explaining their formation in interstellar space as by-products of the fusion of hydrogen atoms into heavier elements.

On the other hand, scientists like Compton believed that the cosmic rays were charged particles. A famous controversy opposed the two scientists, doing the front page of newspaper, see Figure 1.5. The evidence supporting the latter theory was brought by J.

Clay, who found evidence in 1927 of a variation of cosmic ray intensity with latitude, which indicated that the primary cosmic rays are deflected by the geomagnetic field and should therefore be charged particles, not photons [Cla27].

In 1928, Johannes Geiger and his student Walter Müller invented the Geiger-Müller

counter who could measure all sort of ionizing radiation. Walter Bothe, who had assisted

Geiger in 1924, got interested in the cosmic rays. Using two Geiger-Müller counters to

1.3. The discovery of cosmic rays 13

Figure 1.5.: First page of the New-York Times from December 31, 1932

probe for coincidences, they proved that the gamma rays were not the main source of cosmic rays.

In 1934, Bruno Rossi found that the intensity of cosmic rays is greater from the West, proving that most primaries are positive. Observing that the rate of near-simultaneous discharges of two widely-separated Geiger counters was larger than the expected accidental rate, he wrote in his report on the experiment [Ros34]:

“it seems that once in a while the recording equipment is struck by very extensive showers of particles, which causes coincidences between the counters, even placed at large distances from one another.”

In 1938, the French physicist Pierre Auger, unaware of Rossi’s earlier report, detected the same phenomenon and investigated it in some detail. He made coincidence studies using two groups of counters separated by distances going from 2 to 300 m, both at the ground level, and at altitudes of 3500 m (Jungfraujoch) and 2900 m (Pic du Midi).

[Aug+39]. He showed the existence of cosmic rays showers of great surface, and made

an evaluation of their energy to estimate the energy of the primary particle, greater

14 Chapter 1. Milestones of Physics in the early twentieth century

than 10

eV. He concluded that high-energy primary cosmic-ray particles interact with air nuclei high in the atmosphere, initiating a cascade of secondary interactions that ultimately yields a shower of electrons, and photons that reach ground level.

1.3.3. Thematic displacement

Before the invention of particle accelerators, cosmic rays became an effective way to discover and study high-energy particles. The fog chamber, invented by Charles Wilson in 1911, was combined with a magnet by Anderson and Millikan, allowing to test for the composition of those cosmic rays. In 1932, Anderson discovered particles behaving like electrons but of opposite curvature sign, therefore of opposite charge [And33]. They were identified with positrons (e

), the antiparticle of electron predicted by Paul Dirac a few years earlier [Dir28]. For this discovery, Anderson received in 1936 the Nobel Prize in Physics, a prize he shared with Hess for the discovery of the extraterrestrial origin of the cosmic rays.

In the following decades, and with the progress of fog chambers, many particles would be discovered, among them the muon in 1936 [Mil39], the pion in 1947 [LOP47], and, the same year, the kaons [RB47]. Starting from 1953, the particle physicists moved from the study of cosmic rays to the newly-created particle accelerators, and then colliders. The cosmic rays became a research field by themselves, aimed at studying the composition and origin of these astroparticles.

1.4. From General Relativity to the Big-Bang

1.4.1. Relativities

The four articles published by Albert Einstein in 1905 (see Section 1.2.2), through the introduction of new symmetries (of energy exchanges, of independence of the observer), changed deeply the way the world was perceived, by redefining concepts such as time and space.

However, the special relativity was not compatible with the Newtonian theory of

gravitation, which described the gravitational interaction as instantaneous, whereas

Einstein’s special relativity stated that nothing could go faster than the speed of light in

1.4. From General Relativity to the Big-Bang 15

vacuum. Moreover, the underlying concept of simultaneity of the two events, depending on the observer, was not compatible with the global vision of Einstein, based on the symmetry of the observer.

Einstein began working on the subject in 1907 and presented his theory of General Relativity in 1915 [Ein15]. The general concept is that the Universe is in a non-euclidian space-time, whose local curvature, G

, is relied to the mass-energy contents of the universe, T

, through:

G

= 8πGT

(1.4.1)

It correctly described the precession of Mercury apside, which was by then unexplained, providing a strong argument in favor of the theory.

1.4.2. Friedmann-Lemaître-Robertson-Walker metrics

Current observations lead to assume the cosmological principle, asserting that the universe is isotropic around each of its points, which leads in return to homogeneity. Of course, due to the existing structures of the universe, it is only a so-called SHI (Statistical Homogeneity and Isotropy).

In the 1920s and 1930s, four authors, Alexander Friedmann, Georges Lemaître, Howard Percy Robertson and Arthur Geoffrey Walker, independently developed what would be known as the Friedmann-Lemaître-Robertson-Walker, or FLRW, metrics. They showed that the generic metric which meets the SHI assumption takes, under spherical coordinates, the form:

ds

= c

dt

− a(t)

^! dχ

+ S

(χ)dΩ

^" (1.4.2) Where:

• ds is the element of space-time length.

• a(t) is the scale factor of the universe: in a non-static universe, the distance R(t) between two points varies through time; if t

is the present moment, for which a(t

) = 1, a(t) asserts the ratio R(t)/R(t

).

• χ, on the contrary, determines the comobile distance, unchanged through time.

16 Chapter 1. Milestones of Physics in the early twentieth century

• dΩ

= dθ

+ sin(θ)

dφ

is the solid angle

• t is the cosmic time.

• S

(χ) depends on the space-time curvature of the Universe, and has for values R(t

) sinh(χ/R(t

)), χ and R(t

) sin(χ/R(t

)) respectively for a hyperbolic, flat and elliptical geometry.

The metric, combined with Einstein’s field equation, which allows to assess a(t), gives the two following equations:

⎧

⎪ ⎪

⎨

⎪ ⎪

⎩ ' a ˙

a

(

= 8πG 3 ρ − K

a

¨ a

a = − 4πG

3 (ρ + 3p)

(1.4.3)

Where ρ and p are the energy density and pressure of the energy, which will be calculated in the next Section. The combination of those two equations gives the equation of conservation of energy-impulsion:

˙

ρ + 3H

(ρ + p) = 0 (1.4.4)

Where H

≡ a/a. This equation was derived by Georges Lemaître in 1927. He noted ˙ that, if the quantity H

was different from zero, the Universe would either be expanding or crunching [Lem27].

1.4.3. Hubble observations and the Big-Bang

In 1929, Edwin Hubble is the first to observe precisely that Galaxies recede from us with velocities, v, proportional to their distance, R, another way to assess the value of H

[Hub29]:

v = H

R (1.4.5)

and he gives a numerical value to what is now known as the Hubble constant:

H

= (71 ± 4) km/s/Mpc (1.4.6)

where 1 pc ≡ 3.2 ly.

1.4. From General Relativity to the Big-Bang 17

This result means that the Universe is in expansion. If the arrow of time is reversed, it contracts in direction of a single hot point, defining an time origin for the universe, approximately given by H

≈ 13.8 × 10

yr.

The phenomenon is now called Big-Bang, a term coined by Fred Hoyle on BBC’s radio Third Programme broadcast on 28 March 1949. Hoyle remained a strong opponent to the Big-Bang Theory all of his life but, despite what is often said, the term had no pejorative intention but was intended to give a striking image. Fred Hoyle later co-authored, along with Geoffrey and Margaret Burbidge, and William Fowler, a landmark paper now referred, from their initials, as B

FH [Bur+57], which laid the foundations of stellar nucleosynthesis, explaining the elementary processes allowing the synthesis of all elements in stars, from Lithium to Uranium, and of their relative abundances.

Ironically, along with the recession from galaxies, the paper co-authored by one of its

strongest opponents was, after the recession of galaxies, the second of the three great

arguments which established the Big-Bang as the standard theory. The third one, the

CMB, comes from the consequences of the thermal history of the Universe.

18

2

Modern cosmology and the problem of Dark Matter

Le chapitre précédent a introduit les outils et perspectives de la physique de la première moitié du siècle dernier. Ce chapitre actualise ces connaissances en mettant en oeuvre les outils en vigueur en cosmologie. Il revient notamment sur l’histoire thermique de l’Univers, en s’attardant sur les calculs de densités reliques, qui permettent de placer des contraintes sur les densités actuelles des différents types d’énergie. Les récents résultats de l’expérience Planck permettent de préciser numériquement ces contraintes, et de fournir un premier indice de l’existence de matière sombre. En analysant les autres indices plaidant en faveur de cette existence, nous détaillons quelques candidats plausibles, en particulier les WIMPs supersymétriques, avant de préciser la manière dont ils pourraient être détectés.

—

2.1. Thermal history of the Universe

2.1.1. A little bit of thermodynamics

For a particle i being at thermodynamical equilibrium at temperature T , the phase-space distribution can be written:

f

(⃗ p) = g

' L 2π

(

1

exp[(E

− µ

)/T ] + c (2.1.1)

19

20 Chapter 2. Modern cosmology and the problem of Dark Matter

where:

• g

is the number of spin states of the species;

• χ

is the chemical potential;

• T is the temperature;

• E

is the energy of the state;

• c = − 1 for bosons, +1 for fermions;

•

' L 2π

(

is the number of particles in a cubic volume of edge length L.

The number density n

, the energy density ρ

and the partial pressure p

are determined by f

:

⎧

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

n

= g

(2π)

) d

pf

(⃗ p) ρ

= g

(2π)

) d

pE

f

(⃗ p) p

= g

(2π)

) d

p | ⃗ p |

3E

f

(⃗ p)

(2.1.2)

Two limits are important in cosmology: the ultra-relativistic limit (T ≫ µ or T ≫ m) and the non-relativistic dilute gas (T ≪ m). In the first case, the number and energy density n and ρ depend upon the presence of bosonic or fermionic particles:

⎧

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎩

n

= ζ(3)

π

g

T

for bosons n

= 3

4 ζ (3)

π

g

T

for fermions

(2.1.3)

⎧

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎩

ρ

= π

30 g

T

for bosons ρ

= 7

8 π

30 g

T

for fermions

(2.1.4)

2.1. Thermal history of the Universe 21

with ζ the Riemann zeta function, such that ζ(3) ≈ 1.202. In the non-relativistic limit, these quantities are independent of the particle type:

⎧

⎪ ⎪

⎨

⎪ ⎪

⎩

n

= g

' m

T 2π

(

e

ρ

= m

n

(2.1.5)

Where V ≡ L

, the pressure, in both cases, is given by the isentropic equation of state:

p = − ∂U

∂V

*

*

*

*

*

(2.1.6) and which gives p ≈ 0 if T ≪ m and, if T ≫ m:

p = 1 3V

+ p

E f(⃗ p)d⃗ p

= ρ

3 (2.1.7)

2.1.2. Parameterization of FLRW

If the previous quantities are injected in equation (1.4.4), the results depend on the equation of state of the Universe, ρ = wp. Under a Universe dominated by matter, which is non-relativistic:

˙ ρ

ρ

= − 3 a ˙

a ⇒ ρ

∝ 1

a

(2.1.8)

and under the ultra-relativistic case of a universe dominated by radiation:

˙ ρ

ρ

= − 4 a ˙

a ⇒ ρ

∝ 1

a

(2.1.9)

The factor 3 for the matter case can be intuitively seen as the dilution factor sus- tained by an unchanged energy in an expanding three-dimensional universe. As seen in equation (1.2.1), the energy of photons of frequency ν (or wavelength λ) is given by E = hν = hc/λ. In an expanding Universe (it is the definition of redshift), the wavelength (and energy) increase proportionally to a, which explains the total factor of 4 for the

radiative domination case.

22 Chapter 2. Modern cosmology and the problem of Dark Matter

We also consider the case of a vacuum energy for which ρ

is a constant. The FRLW equation can then be parameterized:

H

= 8πG 3

,

ρ

+ ρ

+ ρ

− K

a

-

(2.1.10)

We want to check the evolution of this equation between a time t of covariant distance a and today (parameters t

, a

). To simplify the equations, by analogy with the other terms, are defined an energy density of curvature such that ρ

≡ − H

/a

, and the critical energy:

ρ

≡ 3H

8πG (2.1.11)

It allows to normalize the equation by defining the terms Ω

, Ω

, Ω

and Ω

such that Ω

= ρ

/ρ

, leading to Ω

+ Ω

+ Ω

+ Ω

= 1.

Finally:

H

= ρ

.

Ω

+ Ω

' a

a

(

+ Ω

' a

a

(

+ Ω

' a

a

(

/

(2.1.12)

2.1.3. Interactions decoupling

In a universe close to thermal equilibrium, and for relativistic particles, the total number of particle density can be written:

n

= ⁰

i∈particle types

n

= ⁰

i∈bosons

n

+ ⁰

j∈fermions

n

= 1 π

1

g

(T ) + 3 4 g

(T )

2

ζ(3)T

(2.1.13)

where g

and g

are the sums of degenerescence factors of, respectively, bosonic and fermionic species. Similarly for the density energy, with g

the effective degenerescence factor:

ρ

= π

30

1

g

(T ) + 7 8 g

(T )

2

T

= π

30 g

(T )T

(2.1.14)

2.1. Thermal history of the Universe 23

Putting the previous expression into Friedmann equation, and with the Planck mass M

≡ G

≈ 1.2209 × 10

GeV c

, it finally comes:

H

= 8πG 3

π

30 g

T

⇒ H ≈ 1.66 √ g

T

M

(2.1.15)

2.1.4. Calculation of relic densities and decouplings

Those relations will help us to derive the thermal history of the universe, which in return will give us information and constraints for the following of our study. The basic procedure is the following: by reversing the time, the temperature, thus the energy gradually increases. The equation (1.2.2), E = mc

indicates that, above the temperature corresponding to E, particles of mass greater than m become pure energy. Reversely, with the normal time arrow, the energy progressively becomes particles, depending on the mass of the particles and whether they are relativistic or not. The next Sections will go through the main processes involved.

The general idea is that interactions between particles occur as long as the interaction rate of a particle per degree of freedom at redshift z, Γ(z), is much greater than the expansion rate of the Universe, H(z). To the first order:

Γ(z) ≫ H(z) ⇒ n ⟨ σv ⟩ ≫ T

M

(2.1.16) where σ depends on the interaction, and v = 1 if the particles are relativistic and (T /m)

otherwise.

The first second of the Universe

The thermal history of the universe starts at 10 TeV (10

s), a temperature at which all interactions of the standard model are at equilibrium.

At 100 GeV (10

s), comes the electroweak phase transition, where the particles acquire their mass. The “soup” is, then, made of heavy quarks, the bosons become non-relativistic, leading to their drop of equilibrium and decay, since they are unstable.

At 100 MeV (10

s) , happens the QCD phase transition, where quarks and glu-

ons become hadrons. We have neutrons, protons, electrons, positrons, six families of

neutrinos/antineutrinos, and photons.

24 Chapter 2. Modern cosmology and the problem of Dark Matter

Neutrinos decoupling

The neutrinos are the only species which interacts only weakly, and will be the first to decouple, when they are still relativistic (v = 1). The cross-section σ for such phenomena is, for T ≪ M

, in the order of G

T

, with the Fermi constant G

≈ 1.1664 × 10

GeV

. The condition for equilibrium arises at:

T

× G

T

≫ T

/M

(2.1.17)

T ∼ (G

M

)

∼ (10

GeV × 10

GeV

)

= 1 MeV (2.1.18) A precise calculus gives 1.5 MeV. It has been proven that neutrinos have a mass, m

; it could be sufficient for them to become non-relativistic after some time, and give a significant contribution Ω

to the equation (2.1.12). In that case:

Ω

= ρ

ρ

= m

n

≈ 0.2 m

10 eV (2.1.19)

It gives us a strong constraint: in order to make a substantial contribution to the total mass of the Universe (for example, 20 %), neutrinos should have a mass of at least 10 eV, when the Planck collaboration reports the combined mass of the three neutrino varieties to be less than 0.23 eV [Ade+13]. Note that all other particles coupled only through weak interactions decouple in the same way: such particles should have a mass of several eV to give a significant contribution to the total mass of the Universe.

Electrons freeze-out

After the decoupling of weak interactions, the only two-body interaction changing the number of electrons is:

e

+ e

⇀ ↽ γ + γ (2.1.20)

2.1. Thermal history of the Universe 25

The annihilation rate is σ ∼ α

/m

, with the fine-structure constant α ≈ 1/137. The number of electrons is frozen if

(m

T )

e

× α

/m

× (T /m

)

≫ T

/M

(2.1.21) α

/m

e

M

≫ 1 (2.1.22) Which leads to

T ∼ m

/40 ∼ 12 keV (2.1.23)

Similarly to what was done with neutrinos, constraints can be put on particles decoupling in a non-relativistic way to give a contribution to equation (2.1.12). With m

the mass of such particles (which would typically be of 1 GeV), and T

( ∼ 100 MeV) their freeze-out temperature, the resulting critical density, Ω

can be estimated to:

Ω

= 5 × 10

cm

⟨ σv ⟩

m

T

(2.1.24)

A substantial contribution (0.1 < Ω

< 1) gives ⟨ σv ⟩

∼ 10

cm

, which is a typical weak-interaction cross-section. The general-class of particles having those characteristics is called Weakly-Interacting Massive Particles, or WIMPs.

Primordial nucleosynthesis

The next step is the decoupling of nuclear interactions, allowing the calculation of the relic densities of nuclei. Despite their interest, these computations won’t be of much use in the following of this study, and won’t be detailed. Neutrons (which, free, have a lifetime of (885.7 ± 0.8) s), are bound into deuterium nuclei, whats is called the deuterium bottleneck, since no further reaction can happen before equilibrium of this reaction is reached (which happens at 60 keV). Deuterium nuclei lead to the creation of

H and

He, allowing in turn the creation of

He. The time is not sufficient to create more heavier nuclei than in “trace” densities, and the Coulomb repulsion stops the reactions at about 30 keV, signing the end of nucleosynthesis.

At the end of this reaction, the degrees of freedom in g

are the sum of the two spin

states of the photons, electrons, positrons, neutrinos and antineutrinos. By noting N

26 Chapter 2. Modern cosmology and the problem of Dark Matter

the number of neutrinos and antineutrino families:

g

= 11 2 + 7

4 N

(2.1.25)

It led to the first estimation of N

= 3 ± 0.05, in consistency with further measurements [Ber+12].

Recombination and decoupling

After nucleosynthesis, the Universe is constituted of protons, helium, photons, and electrons in equilibrium. Compton scattering allows the species to be in equilibrium through γ + p ⇀ ↽ γ + p and γ + e

⇀ ↽ γ + e

. The universe is opaque (dark ages ), since the mean free path of photons in the electron-baryon plasma is very small. They are practically embedded into matter, and interact with it through pressure waves (baryonic acoustic oscillations, or BAO).

Around 1 eV electrons and protons begin to form a significant amount of hydrogen, causing the photons do drop out of equilibrium. Those decoupled photons, with a mean free path now infinite, continue their trajectory until today, and form the Cosmic Microwave Background, or CMB, which can be detected, due to space dilatation, at a lowered temperature of 2.71 K.

The matter, on its side, aggregates into large-scale structures. The pressure waves have stopped at the same radius; in terms of angular distributions, the statistical distance between two structures will reflect this radius. Similarly, the CMB has inhomogeneities, and the statistical distance between two of them, along with the radius the BAO reached at the time of recombinaison, allow us to fix all the parameters previously discussed.

2.1.5. Planck results

The Planck experiment published, a few months before the redaction of this manuscript,

the most precise values of the parameters relevant to our study. They are detailed in

Table 2.1. The values come from [Ade+13] and combine the results of Planck with

the prior results of WMAP (also on the CMB), and the BAO measurements; personal

calculations were made to normalize the densities.

2.2. Evidence of dark matter 27

Table 2.1.: Main Planck results in cosmology

Parameter Symbol Value Unit

Age of the Universe t

13.80 ± 0.04 Gy

Hubble’s constant H

67.80 ± 0.77 (km/s)/Mpc

Dark energy density Ω

69.2 ± 1.0 %

Baryon density Ω

4.82 ± 0.05 %

Matter density Ω

30.6 ± 0.4 %

Curvature of space Ω

− 0.5 ± 6.5 !

Number of neutrinos families N

3.3 ± 0.5 −

Today, the radiation contribution is negligible, along with the contribution for cur- vature (meaning that the Universe is most probably flat). The main component of the energy is the “dark energy” which is beyond the scope of this study. The interesting point is that the baryons compose only a small fraction ( about 15 %) of the total matter density; the missing matter is called “Dark Matter”. From our previous results, in can not be made of neutrinos or other relativistic species (hot dark-matter model, excluded by the fact that such particles would be seen, and by considerations on the formation of large structures) but most probably of WIMPs (cold dark matter model ). The standard model in cosmology consists in the massive presence of Dark energy (Λ), Cold Dark Matter (CDM ) and no curvature; it is therefore called the ΛCDM model.

The next Sections review the other evidences of Dark Matter presence and detail some of the main candidates to it.

2.2. Evidence of dark matter

2.2.1. Spiral galaxies rotation curves

Many observations tend to indicate that there is far more matter in the universe that

the one visible from the luminous part of the galaxies. The idea was born in the spirit of

Fritz Zwicky, who, in the 1933, measured the speed of galaxies in the Coma cluster. By

applying the virial theorem to those speeds he inferred that the total mass should be 400

times greater that what was visible [Zwi33]. This was long known as the “missing mass

theorem”.

28 Chapter 2. Modern cosmology and the problem of Dark Matter

In 1939, Horace Babcock measured the rotation curve for Andromeda, suggesting an increasing mass-to-luminosity ratio [Bab39]. In 1959, Louise Volders demonstrated that the spiral galaxy M33 does not spin as expected according to Keplerian dynamics [Vol59].

The main progress on that subject were made by Vera Rubin, a young astronomer which, in 1975, discovered that most stars in spiral galaxies orbit at the same speed, implying a mass density uniform beyond the galactic bulge. The results were presented in a famous 1980 paper [RFT80].

Practically, assuming most of the mass of the galaxies is in the galactic bulge, the stars should have a rotation curve following (like stars around the sun) the Newton’s version of Kepler third’s law which states that:

ρ(r) = 3[v(r)]

4πGr

(2.2.1)

where ρ(r) is the radial density profile, v(r) the radial orbital velocity profile and G the gravitational constant.

If v(r) is approximately constant one should see a density ρ ∼ r

when the distance from core increases. Therefore Navarro et al. wrote that

“a correlation would be expected between the luminosity of binary galaxies and the relative velocity of their components. Similarly, there should be a correlation between the velocity of a satellite galaxy relative to its primary and the rotation velocity of the primary’s disk. No such correlations are apparent in existing data [NFW95]”.

They derived the Navarro-Frenk-White profile which is consistent with observations:

ρ(r) = ρ

r R

'

1 + r R

(

(2.2.2)

where the central density ρ

and the scale radius R

vary from halo to halo.

Figure 2.1 shows similar (and more visible) results of the curves from which the relation was built [BBS91]. We can see that the data differ from theoretical “naive”

curves.

2.2. Evidence of dark matter 29

Figure 2.1.: Three parameter (mass-to-light ratio of the disk, halo core radius, halo asymptotic

circular velocity) dark-halo fits (solid curves) to the rotation curves of sample

galaxies, along with individual components (dashed curves: visible components,

dotted curves: gas, dash-dot curves: dark halo) [BBS91]

30 Chapter 2. Modern cosmology and the problem of Dark Matter

2.2.2. Bullet cluster

The 1E 0657-56, as known as bullet cluster, results from the collision of two galaxy clusters. Its study began in 2003 [Mar+03], and was completed, many by the same team, in 2006 [Clo+06]. The cluster has two visible components. Hot gases, which can be observed through X-rays, account for most of the baryonic mass of the cluster. Their interaction in the collision slows them down. The other component of ordinary matter are the stars, whose light make them visible. Contrary to the gases, they interact mostly gravitationally, and are less slowed down than gases. The total mass distribution of the cluster can be estimated through the effect of gravitational lensing. Figure 2.2 shows a Chandra image of the bullet cluster along with weak lensing contours. The greates part of the mass contained within the cluster is not in the baryonic distribution, and did not interact as much as the gas. It is a evidence of dark matter made of (weakly-interacting) particles, which can not be fully accounted by alternative missing mass models such as MOND (MOdified Newton Dynamics).

Figure 2.2.: 500 ks Chandra image of the bullet cluster. The green contours show the weak

lensing reconstruction [Clo+06].

2.3. Candidates and detection 31

2.3. Candidates and detection

2.3.1. First candidates

Dark Matter constitutes a large fraction of the energy in the Universe. So far, thanks to Planck results, it is only known that it is not baryonic matter. This Section reviews the main candidates to Dark Matter.

The first candidate that can be thought of is a class of exotic objects, like planets or stars, but not made of ordinary matter, and too dark or small to be seen. This class is referred as MAssive Compact Halo Objects, or MACHOs. Primordial Black Holes (PBHs), formed by some violent epochs in Big Bang, could be good candidates. For PMHs created during the radiation-dominated universe, causality considerations show that their mass must be lighter than 1000 solar masses, and, on the other hand, heavier than 10

solar masses, otherwise they would emit a visible Hawking radiation [Mur07].

The MACHO and EROS collaborations have estimated the possible amount of MACHOs in our galactic halo through the study of microlensing effects in the Large Magellan Cloud, concluding that MACHOs of 10

to 30 solar masses can not account for the Dark Matter in our galactic halo [Tis+06].

Other primer candidates have been neutrinos, which were already ruled out, CHAMPs (CHArged Massive Particles), and Strongly Interacting Massive Particles (SIMPs), the two latter being excluded as shown, respectively, in [Dim+90] and [Sta+90]. That brings back to the study of WIMPs, not made of baryonic matter. The next Section, after a review of the standard model, will try to show how the supersymmetry could provide a valuable candidate to those WIMPs.

2.3.2. The standard model of particle physics and its limitations

The standard model was mainly built upon the study of the particles discovered in the

cosmic rays, as explained in the Section 1.3.2. The hadrons (like protons or kaons)

were ultimately proven not to be elementary particles, but constituted of quarks. A

third generation of leptons was discovered, along with the corresponding neutrinos and

antineutrinos. Finally, there aree six quarks, six leptons and their associated antiparticles,

and six bosons. The last particle of the standard model is the Higgs boson. In 2012,

both Atlas and CMS experiments presented data on the “observation of a new particle”,

32 Chapter 2. Modern cosmology and the problem of Dark Matter

of mass respectively of 126.0 ± 0.4 (stat) ± 0.4 (sys) GeV/c

and 125.3 ± 0.4 (stat) ± 0.5 (sys) GeV/c

, for significance of 5.9 and 5 σ. On 14 March 2013 the CERN confirmed

that:

“CMS and ATLAS have compared a number of options for the spin-parity of this particle, and these all prefer no spin and positive parity [two fundamental criteria of a Higgs boson consistent with the Standard Model]. This, coupled with the measured interactions of the new particle with other particles, strongly indicates that it is a Higgs boson [Pra13].”

It was included in Figure 2.3.

2.3 MeV/c² 2/3

1/2

u

up

1.3 MeV/c² 2/3

1/2

c

charm

173 GeV/c² 2/3

1/2

t

top

0 0

1

g

gluon

126 GeV/c² 0

0

H

Higgs b.

4.8 MeV/c² -1/3

1/2

d

down

95 MeV/c² -1/3

1/2

s

strange

4.2 GeV/c² -1/3

1/2

b

bottom

0 0

1

γ

photon

0.51 MeV/c² -1

1/2

e

electron

105.7 MeV/c² -1

1/2

µ

muon

1.8 GeV/c² -1

1/2

τ

tau

91.2 GeV/c² 0

1

Z

Z boson

<2.2 eV/c² 0

1/2

ν _e

electron n.

<0.17 MeV/c² 0

1/2

ν _µ

muon n.

<15.5 MeV/c² 0

1/2

ν _τ

tau n.

80.4 GeV/c²

±11

W boson W

Figure 2.3.: The Standard Model of elementary particles. Each of the first three column stands for a generation of fermions (I, II, II). For each particle, the background is violet for quarks, green for leptons, red for gauge bosons, and yellow for the Higgs boson, and the three numbers are, respectively, the mass, charge, and spin of the particle.

However, there are several phenomena/theories that the standard model can not explain:

• Gravity: the standard model is incompatible with the General Relativity, which, in

2013, has proven to accurately describe the Universe above the quantum level.