THE MATHEMATICAL THEORY OF COSMIC STRINGS
CONTENTS CHAPTER 1 A THERMAL HISTORY OF THE UNIVERSE AND PRIMORDIAL NUCLEOSYNTHESIS 1 1 A QUICK OVERVIEW 1 1 1 A USEFUL REFRESHER OF THE FRIEDMANN–LEMAÎTRE MODEL 1 1 2 THE MAJOR EVENTS OF THE BIG BANG 1 2 GAMOW’S YLEM 1 2 1 INTRODUCTORY REMARKS 1 2 2 NUMERICAL DENSITY 1 2 3 ENERGY DENSITY AND PRESSURE 1 2 4 ENTROPY DENSITY 1 2 5 EFFECTIVE DEGREES OF FREEDOM OF THE YLEM 1 2 6 NUMERICAL DEVELOPMENTS 1 3 COOLING KINETICS 1 3 1 PLANCK MASS 1 3 2 RELATION BETWEEN SCALE FACTOR AND TEMPERATURE 1 3 3 RELATION BETWEEN COSMIC TIME AND TEMPERATURE 1 4 THERMAL DECOUPLING OF NEUTRINOS 1 4 1 THE CONCEPT OF CROSS-SECTION 1 4 2 THERMAL EQUILIBRIUM BREAKING 1 4 3 THE NEUTRINIC FOSSIL RADIATION 1 4 4 THE COWSIK AND MCCLELLAND LIMIT 1 5 PRIMORDIAL NUCLEOSYNTHESIS 1 5 1 TRANSMUTATION BETWEEN PROTONS AND NEUTRONS 1 5 2 BREAKING OF THE NEUTRON–PROTON EQUILIBRIUM 1 5 3 THE FORMATION OF DEUTERIUM 1 5 4 THE COOKING OF LIGHT ELEMENTS 1 6 CHEMICAL DECOUPLING OF HEAVY PARTICLES 1 6 1 EVOLUTIONARY EQUATION 1 6 2 APPROXIMATE SOLUTION 1 6 3 RELIC DENSITY 1 6 4 THE LEE AND WEINBERG LIMIT 1 7 DISSIPATION OF THE PRIMORDIAL FOG 1 7 1 SAHA’S EQUILIBRIUM 1 7 2 PHOTOIONIZATION AND RECOMBINATION OF NEUTRAL HYDROGEN 1 7 3 LYMAN-ALPHA LINE AND ATOMIC RESONANCE 1 7 4 THE THREE-LEVEL ATOM MODEL 1 7 5 RECOMBINATION OF THE PRIMORDIAL PLASMA AND RESIDUAL IONIZATION 1 8 REFERENCES CHAPTER 2 COSMOLOGICAL MICROWAVE BACKGROUND 2 1 INTRODUCTION 2 1 1 LAST SCATTERING SURFACE 2 1 2 CMB ANISOTROPIES 2 1 3 CONVENTIONS 2 2 REVISITING THE THOMSON SCATTERING 2 2 1 SCATTERING RATE 2 2 2 OPTICAL DEPTH 2 2 3 VISIBILITY FUNCTION 2 2 4 DIFFUSION LENGTH 2 3 LINEAR COSMOLOGICAL PERTURBATIONS 2 3 1 WHY LINEAR PERTURBATIONS? 2 3 2 CLASSI?CATION OF PERTURBATIONS 2 3 3 COMOVING FOURIER SPACE 2 3 4 LINEARIZED EINSTEIN EQUATION 2 3 5 SPECIES PRESENT IN THE STANDARD COSMOLOGICAL MODEL 2 3 6 GENERAL EQUATIONS OF MOTION 2 4 FORMAL DESCRIPTION OF TEMPERATURE ANISOTROPIES 2 4 1 PHOTON PROPAGATION 2 4 2 TEMPERATURE ANISOTROPY IN A GIVEN DIRECTION 2 5 STOCHASTIC THEORY OF COSMOLOGICAL PERTURBATIONS 2 5 1 INITIAL CONDITIONS 2 5 2 POWER SPECTRUM AND TRANSFER FUNCTIONS 2 5 3 SPECTRUM OF TEMPERATURE ANISOTROPIES 2 6 PHYSICS OF TEMPERATURE ANISOTROPIES 2 6 1 LINE-OF-SIGHT INTEGRAL IN FOURIER SPACE 2 6 2 PERTURBATION EVOLUTION 2 6 3 CONTRIBUTIONS TO THE TEMPERATURE SPECTRUM 2 6 4 DEPENDENCY ON COSMOLOGICAL PARAMETERS 2 7 OTHER CONTRIBUTIONS AND OBSERVABLES 2 7 1 OVERVIEW OF CMB POLARIZATION 2 7 2 CMB GRAVITATIONAL LENSING 2 7 3 OVERVIEW OF THE ROLE OF GRAVITATIONAL WAVES 2 7 4 ASTROPHYSICAL FOREGROUND 2 7 5 SPECTRAL DISTORTIONS 2 8 CBM OBSERVATIONS 2 8 1 CURRENT OBSERVATIONS 2 8 2 FUTURE OBSERVATIONS 2 9 REFERENCES CHAPTER 3 COSMOLOGICAL IN?ATION 3 1 OVERVIEW 3 2 INTRODUCTION 3 2 1 SHORTCOMINGS OF THE HOT BIG BANG MODEL 3 2 2 THE MECHANISM OF IN?ATION 3 3 SINGLE-?ELD SLOW-ROLL IN?ATION 3 3 1 HOMOGENEOUS DYNAMICS 3 3 2 DYNAMICS OF ?UCTUATIONS 3 4 THE PHYSICS OF IN?ATION BEYOND TOY MODELS 3 4 1 ULTRAVIOLET SENSITIVITY 3 4 2 MULTI?ELD EFFECTS 3 5 PRIMORDIAL NON-GAUSSIANITIES 3 5 1 OVERVIEW 3 5 2 DETERMINING PRIMORDIAL NON-GAUSSIANITIES 3 5 3 NON-GAUSSIAN SHAPES 3 6 CONCLUSION 3 7 REFERENCES CHAPTER 4 DARK MATTER 4 1 INTRODUCTION 4 1 1 SUBJECT PRESENTATION 4 1 2 BRIEF HISTORICAL OVERVIEW 4 1 3 CURRENT INTEREST IN THE SUBJECT 4 1 4 WARNING 4 1 5 SUMMARY OF EVIDENCE IN FAVOR OF DARK MATTER 4 2 AT THE GALAXY SCALE 4 2 1 INTRODUCTION 4 2 2 SPIRAL GALAXY ROTATION CURVES 4 2 3 VELOCITY DISPERSION IN ELLIPTICAL GALAXIES 4 2 4 APPLICATION TO THE GRAVITATIONAL CASE 4 2 5 DWARF SPHEROID GALAXIES 4 2 6 OUR GALAXY 4 2 7 DENSITY PRO?LES 4 3 AT THE GALAXY CLUSTER SCALE 4 3 1 VELOCITY DISPERSION 4 3 2 HOT GAS X-RAY 4 3 3 GRAVITATIONAL LENSES 4 3 4 THE SUNYAEV–ZELDOVICH EFFECT 4 4 COSMOLOGY 4 4 1 GEOMETRY OF THE UNIVERSE 4 4 2 PRIMORDIAL NUCLEOSYNTHESIS 4 4 3 THE FORMATION OF LARGE STRUCTURES 4 5 THE NATURE OF DARK MATTER 4 5 1 BARYONIC CANDIDATES 4 5 2 GRAVITY MICROLENSING 4 5 3 PARTICLES 4 6 DETECTION 4 6 1 DIRECT DETECTION 4 6 2 INDIRECT DETECTION 4 6 3 CLUMPS 4 7 MODI?ED DYNAMICS AND GRAVITY 4 7 1 MOND THEORY 4 7 2 THE BULLET CLUSTER 4 8 CONCLUSION 4 9 REFERENCES A THERMAL HISTORY OF THE UNIVERSE AND PRIMORDIAL NUCLEOSYNTHESIS THE UNIVERSE HAS BEEN EXPANDING AND EVOLVING SINCE IT WAS BORN BILLIONS OF YEARS AGO WHAT WAS ITS BIRTH LIKE? WHAT WERE THE MAJOR STAGES OF ITS INFANCY? ANY QUEST FOR THE ORIGINS OF THE WORLD MUST ANSWER THESE LEGITIMATE QUESTIONS AND THUS RECONSTRUCT THE PAST IN ORDER TO DEVELOP THIS WORK THAT COULD BE ASSIMILATED TO ARCHAEOLOGICAL INVESTIGATIONS, WE CAN PROBE THE UNIVERSE SINCE LIGHT PROPAGATES AT ?NITE SPEED, THE LIGHT THAT REACHES US ON EARTH TODAY WAS EMITTED IN A PAST ALL THE MORE REMOTE AS THE DISTANCE TO THE SOURCE INCREASES TO OBSERVE INTO GREAT DISTANCES IS THEREFORE TO OBSERVE EARLY JUST LIKE THE HUBBLE DEEP FIELD, THE DEEP SURVEYS ACHIEVED BY THE BEST TELESCOPES BRING FORWARD EVER EVOLVING GALAXIES THAT HAVE NOT YET REACHED THE FULL MATURITY WHICH THEY EXHIBIT TODAY COLLISIONS AND MERGERS BETWEEN SPIRAL GALAXIES ARE NUMEROUS AND GIVE BIRTH TO GIANT ELLIPTICAL GALAXIES IN SOME, ENTIRE POPULATIONS OF STARS ARE BEING BORN, GIVING RISE TO DAZZLING STARBURST ACTIVITY HOWEVER, THERE IS AN IMPASSABLE WALL, CALLED THE LAST INTERACTION SURFACE, FROM WHICH THE COSMIC MICROWAVE BACKGROUND (CMB) WAS RADIATED, DISCOVERED IN 1964 (PENZIAS AND WILSON 1965) AND INTERPRETED AS THE REMNANT OF THE PRIMORDIAL ?REBALL (DICKE ET AL 1965) BEYOND THIS WALL, MORE THAN 10 BILLION LIGHT YEARS AWAY, THE UNIVERSE IS IONIZED ATOMS ARE STRIPPED OF THEIR ELECTRONIC CLOUD TO FORM A PLASMA " [1 1] THE MATHEMATICAL THEORY OF COSMIC STRINGS A THERMAL HISTORY OF THE UNIVERSE AND PRIMORDIAL NUCLEOSYNTHESIS [1 5] 1 STRICTLY SPEAKING, THE DISTANCE BETWEEN THE CONSIDERED POINT AND THE CENTER OF THE COORDINATE SYSTEM IS IN THIS CASE EQUAL TO A(T)R0 R DT [1 10] 2 BARYONS CONSTITUTE A FAMILY OF ELEMENTARY PARTICLES UNDERGOING THE STRONG INTERACTION THIS TERM REFERS HERE TO PROTONS AND NEUTRONS AND, BY EXTENSION, TO ORDINARY MATTER THE MATHEMATICAL THEORY OF COSMIC STRINGS A THERMAL HISTORY OF THE UNIVERSE AND PRIMORDIAL NUCLEOSYNTHESIS BY DENOTING WITH U ??A3 THE ENERGY OF THE COSMIC ?UID INSIDE THE COVOLUME V = A3, THAT IS A VOLUME EXPANDING WITH THE EXPANDING UNIVERSE, WE ?ND THE ?RST LAW OF THERMODYNAMICS WITH: DU = ?P DV = ?P DV + T DS WHERE DS =