Cover Composition VIII Kandinsky 1923, Scalar field models. dark energy, The current universe seems to have an accelerating expan. sion due to negative pressure known as dark energy This. master thesis surveys dynamical dark energy models such as. Quintessence and K essence which can solve the fine tuning. and the coincidence problem Since in these models quantum. fields are responsible for large scale effects it seems to be an. excellent subject to investigate unification of small and large. scale physics At first sight acceleration seems to be forbid. den in string theory due to a No Go theorem However this. No Go theorem can be circumvented by non compact inter. nal spaces Then it is shown that a dilaton field arising from. compactifications in supergravity can act as dynamical dark. University of Amsterdam, Institute for Theoretical Physics. 1018 XE Amsterdam, June 10 2008, Acknowledgements, I would like to express my gratitude towards my supervisor Kostas Skenderis. He suggested this subject and his critical reading encouraged me to study in. depth many articles and helped me to formulate precisely. voor Luisa Yael, Like many children I was wondering about nature As a young boy at primary. school physics had caught already my interest and I even wanted to write. a paper about the theory of General Relativity of Einstein Of course this. subject was inaccessible for me and I decided to write about Galileo Galile. My interests brought me to study theoretical physics and the choice for the. subject of my master thesis The study of the relation between the small. and the large scales in physics is very interesting and therefore I wanted to. write my master thesis around this problem This directly implied that my. thesis should be a broad work in which different theories should be discussed. In search of a suitable thesis subject I met dr Kostas Skenderis He gave. me an article 1 about dark energy and asked me to review this for master. students This article caught my interest because it is about a fundamen. tal problem related to the connection of the large and the small scales in. cluding many different fields in theoretical physics and has also a feed back. with observations in astrophysics Besides this article the articles found on. www arxiv org brought me an almost overwhelming amount of information. of dark energy This was great and exciting to do, I hope that the result of my endeavour is a self contained overview in which. many different topics of dark energy are reviewed sometimes improveda land. scape of 4 dimensional effective theories and related to each other. Boaz van der Plas, 1 Introduction 1, 2 Framework 5. 2 1 The FRW Universe 5, 2 1 1 Curved space time 5, 2 1 2 The Robertson Walker metric 7. 2 1 3 A universe filled with a perfect fluid 9, 2 2 Cosmology and dark energy 13. 2 2 1 Observations 13, 2 2 2 Standard candles and dark energy 14. 2 2 3 The standard picture of the universe 20, 2 2 4 Inflation 22. 2 3 Dark energy models 25, 2 3 1 A true cosmological constant 25. 2 3 2 Dark energy models 28, 3 Dynamical dark energy 31. 3 1 Scalar field models 31, 3 1 1 The Einstein Hilbert action 32. 3 1 2 Quintessence 35, 3 1 3 K essence 40, 3 1 4 Phantom field 43. 3 1 5 Consistency conditions 44, 3 2 String inspired models 48. 3 2 1 Dilatonic dark energy 48, 3 2 2 Dilatonic ghost condensate 50. 3 2 3 Tachyon field 52, 3 2 4 Chaplygin gas 54, 3 3 Scaling solutions 61. 3 3 1 A unified approach 61, 3 3 2 A unified approach to scaling solutions in a general. cosmological background 62, 3 3 3 Quintessence 69, 3 3 4 Tachyon 70. 3 3 5 Dilatonic ghost condensate 71, 3 3 6 K essence 72. 4 String theory and dark energy 73, 4 1 Compactification and acceleration 73. 4 1 1 String theory 74, 4 1 2 A simple compactification 76. 4 1 3 Superstring theory 77, 4 1 4 Supergravity 79. 4 1 5 No Go theorem 79, 4 1 6 Gauged supergravity 81. 4 2 Quintessence from M theory 85, 4 2 1 D dimensional Quintessence 85. 4 2 2 Quintessence in N 1 supergravity 89, 4 2 3 Quintessence in non compact SUGRA 90. 5 Conclusion and discussion 93, Introduction, The discovery in 1998 that the expansion of the universe accelerates was. astonishing How is it possible that the universe accelerates while the dom. inant force on large scales the attractive gravitation should at least slow. down the expansion The cosmological constant that Einstein originally had. introduced in his equations was placed back to create a negative pressure re. sulting in the current acceleration The energy stored in the cosmological. constant is called the dark energy But what is this dark energy and why. does it seems to have the value of the critical density 10 3 eV 4. Since it is associated with the vacuum energy density in quantum field the. ory we would expect that the cosmological constant would be much bigger. Then a remaining question is why has the cosmological constant its present. value If it is not the vacuum energy density what are the underlying phys. ical processes that cause this obscure dark energy. In this master thesis we will explore different scalar field theories that obey. an equation of state such that it creates a negative pressure that leads to an. accelerated expansion of the universe Our goal is to introduce theoretical. scalar field models that possibly give understanding about the nature of dark. In the last century physicist accomplished a tremendous step forward in. understanding nature For large scales Einstein gave a description of gravity. by curved space time in his theory of General Relativity 1917 Soon after. his discovery Friedmann Robertson and Walker accomplished a model of the. evolution of the universe, At the other extreme for small scales Bohr Einstein Dirac Heisenbeg and. 2 CHAPTER 1 INTRODUCTION, others developed quantum mechanics at the beginning of the 20th century. By these findings it was possible for astrophysicist to understand the light. emission of stars which resulted in the measurability of the velocities of stars. by the redshift, This knowledge together with General Relativity GR and the Friedmann. Robertson Walker FRW universe was a first step to model the evolution of. the universe The Big Bang theory was born The knowledge of quantum. field theory that started in the 50s with the resulting Standard Model made. it possible to count back until 10 6 s after the beginning of the Big Bang. Thereafter intensified attempts are made to unify the large and the small. scales in string theory Mid 1980s string theory had its break through with. the superstring theory in which five different theories have been formulated. Recently it was found that these theories are equivalent and should originate. from one single theory the so called M theory However string theory is not. verified experimentally yet Moreover although fundamentally string theory. is assumed to be unique there are many different theories in many dimen. sions and the conditions for a valid effective 4 dimensional theory are still. Since 1998 astrophysical observations suggest that the universe has an accel. erating instead of a decelerating expansion as expected from the gravitational. force by the total amount of mass in the universe Thus we need an energy. that works in the opposite way of gravity the so called dark energy. Astrophysicists measured that we only see about 5 percent of the energy of. the universe the baryonic or normal matter The matter we do not see. gives about 25 percent of the energy of the universe This is called the dark. matter and should be exotic matter that weakly interacts with atoms and. light However we do not know what this dark matter is although supersym. metric particles are a strong candidate The rest of the energy of the universe. about 70 percent is the so called dark energy To achieve a static universe. Einstein introduced in 1917 this dark energy as a cosmological constant. in his equations Later Einstein considered his introduction of 6 0 as his. biggest mistake after Hubble s discovery of the expanding universe in 1927. This made his constant superfluous and thus he set 0 After 1998 the. cosmological constant was reintroduced to explain the observed accelerating. expansion of the universe, In the last decade physicists have build numerous models in the search for. dark energy This master thesis gives a self contained story starting from. the GR and the FRW universe to the frontline of cosmological string theory. The focus is on physical dark energy models in which a weakly interacting. physical field generates the negative pressure Having said this we have to. keep in mind that it is impossible to treat all models and theories Never. theless this work connects different topics in physics in a self contained story. in order to get more understanding of dark energy, Apart from the Introduction this thesis is divided into four chapters. Chapter 2 Framework, The first section FRW universe introduces a general model of the universe. From GR we obtain a FRW universe with a radiation matter and dark. energy dominated era In section 2 2 Cosmology and dark energy arguments. for dark energy are given by astrophysical observations A possible standard. picture of the universe is formulated and Inflation is discussed. Section 2 3 Dark energy models gives a global picture of the current ideas. for dark energy We will introduce three types of dark energy models a true. cosmological constant physical and geometrical models. Chapter 3 Dynamical dark energy, The core of this thesis consists of section 3 1 Scalar field models and section. 3 2 String inspired models First we will introduce two fundamental different. scalar field models for dark energy namely Quintessence and K essence Sec. ondly we will survey different string inspired dynamical models and relate. these with Quintessence or K essence Using the fact that certain dynam. ical scalar field models admit scaling solutions we can obtain scenario s in. which dark energy naturally becomes dominant independent of the initial. conditions at late time We investigate these scaling solutions in section 3 3. Scaling solutions We set up a unified approach giving scaling solutions of the. different scalar field models arising from one general Lagrangian. Chaper 4 String theory and dark energy, Since scalar field particles if they exist should appear at very high energy. we will investigate in this chapter a cosmological model derived from a fun. damental and mathematical consistent theory which incorporates both the. small and the large scales In section 4 1 Compactification and acceleration. we give an overview of string theory and a No Go theorem is formulated. which forbids acceleration in the standard way of dimensional reduction in. string theory A way to avoid this No Go theorem will be introduced using. non compact gauged supergravity This chapter lays foundations for chap. ter 4 2 Quintessence from M theory in which we extend the 4 dimensional. Quintessence model of chapter 5 to D dimensions and using this derive a. 4 dimensional supergravity model admitting acceleration which is mathemat. Scalar eld models for dark energy Abstract The current universe seems to have an accelerating expan sion due to negative pressure known as dark energy This master thesis surve

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