Colliding Branes and Its Application to String Cosmology Yu-Ichi Takamizu
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THESIS Colliding branes and its application to string cosmology ブレイン衝突とストリング宇宙論への応用 Department of Physics, Waseda University Yu-ichi Takamizu 高水 裕一 A doctoral dissertation submitted to Department of Physics, Waseda University July, 2007 Contents 1 Introduction 5 2 Standard cosmological scenario 11 2.1 Big bang scenario ........................................ 11 2.1.1 Big bang scenario ..................................... 11 2.1.2 Observational evidences ................................. 18 2.1.3 Problems in Big bang scenario ............................. 21 2.2 Inflation cosmology and Reheating mechanism ........................ 22 2.2.1 BasicIdea......................................... 23 2.2.2 Dynamics of inflation .................................. 24 2.2.3 Reheating mechanism .................................. 25 2.2.4 Theoretical models and Observational constraints ................... 29 2.3 Cosmological inhomogeneity ................................... 31 2.3.1 Cosmological perturbation theory ............................ 32 2.3.2 Evolution of perturbations ................................ 35 2.3.3 Origin of cosmological structure on large scale .................... 43 2.3.4 CMB anisotropy ..................................... 47 2.4 Dark energy and dark matter .................................. 51 2.4.1 Observational evidences for dark energy ........................ 51 2.4.2 Dark energy ........................................ 53 2.4.3 Observational evidences for dark matter ........................ 55 2.4.4 Dark matter ....................................... 57 2.5 Summary ............................................. 59 2.5.1 The modern imaging of the universe .......................... 59 2.5.2 Problems in the standard cosmology .......................... 60 3 String theory and Brane-world scenario 63 3.1 Superstring theory ........................................ 63 3.1.1 Bosonic string spectrum ................................. 65 3.1.2 T-duality and D-brane .................................. 68 3.1.3 Hagedorn temperature ................................. 69 3.1.4 Supergravity....................................... 69 3.2 Brane-world ............................................ 71 3.2.1 Domainwallmodel.................................... 71 3.2.2 ADDmodel........................................ 72 3.2.3 RSmodel......................................... 73 3.2.4 DGPmodel........................................ 75 3.2.5 DD¯ -branes inflation model ............................... 75 3.2.6 Ekpyroticmodel..................................... 76 3.2.7 KKLTmodel....................................... 76 3 4 CONTENTS 3.3 Summary ............................................. 77 4 Two scenarios based on colliding branes 79 4.1 Colliding branes/Ekpyrotic universe scenario ......................... 79 4.1.1 Idea of ekpyrotic universe ................................ 79 4.1.2 Dynamic of the universe and Spectrum of fluctuations ................ 83 4.2 String (brane) gas cosmological scenario ............................ 85 4.3 Summary ............................................. 86 5 Colliding branes in Minkowski spacetime 87 5.1 Basic idea of colliding branes .................................. 87 5.2 Reheating mechanism in colliding two branes universe .................... 88 5.2.1 Time evolution of domain walls ............................ 88 5.2.2 Particle production on a moving domain wall ..................... 89 5.2.3 Summary ......................................... 96 5.2.4 Appendix ......................................... 96 5.3 Fermions on Colliding Brane .................................. 98 5.3.1 Fermions on moving branes ............................... 99 5.3.2 Initial setup and Outgoing states ............................ 101 5.3.3 Time evolution of fermion wave functions ....................... 102 5.3.4 Fermion numbers on domain walls after collision ................... 105 5.3.5 Summary ......................................... 105 6 Colliding branes in curved spacetime 107 6.1 Collision of two domain walls in asymptotic Anti de Sitter spacetime ........... 107 6.1.1 Basic equations and initial settings ........................... 107 6.1.2 Time evolution of scalar field .............................. 110 6.1.3 Time evolution of metric ................................. 113 6.1.4 Summary ......................................... 116 6.1.5 Appendix ......................................... 116 6.2 Dynamics of colliding branes and black brane production .................. 118 6.2.1 ModelI.......................................... 118 6.2.2 Horizon formation .................................... 121 6.2.3 ModelII..........................................122 6.2.4 Summary ......................................... 123 6.3 Dimensionality problem ..................................... 123 6.3.1 String gas in 10D dilaton gravity ............................ 125 6.3.2 Hagedorn regime ..................................... 126 6.3.3 Time evolution of universe and dilaton ......................... 126 6.3.4 Thermal equilibrium of string gas ........................... 129 6.3.5 Summary ......................................... 132 6.3.6 Appendix ......................................... 133 7 Conclusions 137 Chapter 1 Introduction Cosmology Cosmology represents a fundamental study on the large scale physics in the whole Universe, to investigate its origin and evolution. Here, the word, “Universe” is all existing matter and space considered as a whole, in other words, where the law of nature exits. Then cosmology is also a study of its all components, existing in which the universe, how they formed, how they have evolved 1. It has a long history to understand the Universe as the world around humanity’s place including of course science, philosophy, and religion. Cosmology firstly started as a simple questions in early history of humankind about 100 thousands years ago, such as ”What’s going on around me?” or ”How does the Universe (God) work?” It was based on local pray as respecting God of Nature who creates phenomena around the world. The earliest scientific observations for understanding the Universe were related to the epoch of four big civilizations in 5000 years ago, Egypt, Mesopotamia, India and China. Actually in the ancient cosmology of Egypt, the sun god, Ra, was seen to control the annual solar motion along the horizon. Modern cosmology as a study of science is usually known as a beginning with Albert Einstein who found the Special and General Relativity. The general relativity not only gives a precise description of gravitational force but also has completely changed the idea of space and time. Gravity is described as a geometrical object, and energy and momentum densities of matter fields deform a spacetime. As a result, a spacetime becomes a dynamical object, which allows us to study the universe as a whole. Therefore, the cosmology was born as one of physical science by this theory. Alexander Friedmann derived the Friedmann equations describing expansion of the universe in 1921. In 1927 Georges Lemaitre proposed the big bang theory and Edwin Hubble discovered the red shift by expanding of the universe in 1929 (the so-called Hubble law) [212, 211]. In 1964 Arno Penzias and Robert Woodrow Wilson detected the cosmic microwave background radiation [355, 466] as a consequence of Big bang theory. Thus cosmology has long history as old as human beings and may be perhaps one kind of thinking ourself as their birth? or, their history? or themselves? We don’t say it clearly, but the universe is of course, the largest special “nature” existing around us, which works as a some kind of “mirror” to human being, attracting humankind for a long time. Recently, the idea of Anthropic Principle seems to be related to such direction of discussion. All the physical laws of Nature involves particular physical constants such as the gravitational constant, the speed of light, the electric charge and Planck’s constant etc. Some are derived from physical laws, however, for most, their values are arbitrary. If all above constants take critical values in a possible narrow range, life like a humankind could be created, otherwise no life comes who understands the universe. The Anthropic Principle may be not scientific, but is one approach to explain the above fundamental question; what our world, the universe takes special values of physical constants? This idea is interesting to think in the point of view of landscape existing in String theory as you will see later, where there exist other different universes and this principle may give a necessary condition for the existence of our universe. In this sense, based on the Anthropic principle, the universe 1The word “Cosmology” is from the Greek: cosmologia,(cosmos) the universe seen as a well-ordered whole + (logos) word, reason, plan. 5 6 CHAPTER 1. INTRODUCTION is just a mirror to face ourselves. Cosmological hierarchy structures The cosmology is also studying of large scale structures, such as galaxy and cluster. Here, let us introduce briefly cosmological structures by looking up to large scales from the our planet, Earth. This is interesting in the view point of an ancient study, “Cosmology”, which ancient man asked, ”What’s going on around me?” The Earth is the planet whose radius is R ∼ 6378km and circuit 2πR ∼ 40.08 thousand km 2. The galaxy which the Sun and we live in, of course, is called the Milky Way or the Galaxy 3. Our Galaxy is made of a bulge of old stars, whose size is