The Cosmic Spacetime

The growth of cosmology into a precision science represents one of the most remarkable stories of the past century. Much has been written chronicling this development, but rarely has any of it focused on the most critical element of this work–the cosmic spacetime itself. Addressing this lacuna is the principal focus of this book, documenting the growing body of evidence compelling us–not only to use this famous solution to Einstein's equations in order to refine the current paradigm, but–to probe its foundation at a much deeper level. Its excursion from the smallest to largest possible scales insightfully reveals an emerging link between the Universe we behold and the established tenets of our most fundamental physical theories. Key Features: Uncovers the critical link between the Local Flatness Theorem in general relativity and the symmetries informing the spacetime's metric coefficients Develops a physical explanation for some of the most unpalatable coincidences in cosmology Provides a sober assessment of the horizon problems precluding our full understanding of the early Universe Reveals a possible explanation for the origin of rest-mass energy in Einstein's theory In spite of its technical layout, this book does not shy away from introducing the principal players who have made the most enduring contributions to this field. Anyone with a graduate level foundation in physics and astronomy will be able to easily follow its contents. Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Dedication 6 Contents 8 Preface 14 Section I: Space and Time 16 Chapter 1: Introduction 18 1.1 HOT BIG BANG COSMOLOGY 21 1.2 LARGE-SCALE STRUCTURE 25 1.3 THE COSMOLOGICAL PRINCIPLE: HOMOGENEITY & ISOTROPY 27 Chapter 2: Theoretical Background 32 2.1 BASIC CONSIDERATIONS 32 2.2 SPECIAL RELATIVITY 33 2.3 FOUR-DIMENSIONAL SPACETIME 36 2.4 ACCELERATED FRAMES 42 2.5 RELATIVISTIC CONTINUUM MECHANICS 49 2.6 THE PRINCIPLE OF EQUIVALENCE 55 2.7 EINSTEIN’S FIELD EQUATIONS 59 Chapter 3: The Black-hole Spacetime 68 3.1 SCHWARZSCHILD METRIC 68 3.2 KERR METRIC 74 Chapter 4: The Cosmic Spacetime 82 4.1 THE METRIC IN COMOVING COORDINATES 82 4.2 FRIEDMANN, LEMAˆITRE, ROBERTSON & WALKER 88 4.3 PHYSICAL INTERPRETATION OF THE FRIEDMANN EQUATION 97 Section II: The Metric 100 Chapter 5: Special FLRW Solutions 102 5.1 MILNE UNIVERSE 102 5.2 DE SITTER SPACE 105 5.3 EINSTEIN-DE SITTER UNIVERSE 107 5.4 STATIC FLRW SOLUTIONS 108 5.4.1 Minkowski Spacetime 108 5.4.2 The Milne Universe (Revisited) 108 5.4.3 de Sitter Space 109 5.4.4 The Lanczos Universe 110 5.4.5 The Lanczos Universe with 111 5.4.6 Anti-de Sitter Space (Negative 111 Chapter 6: The Standard Model 114 6.1 STANDARD MODEL BASICS 116 6.2 THE EARLY UNIVERSE 120 6.3 NUCLEOSYNTHESIS 128 6.4 THE COSMIC MICROWAVE BACKGROUND 134 6.5 THE TEMPERATURE HORIZON PROBLEM 141 6.6 BASIC SLOW-ROLL INFLATION 149 6.7 QUANTUM FLUCTUATIONS IN THE INFLATON FIELD 156 Chapter 7: The Gravitational Horizon 168 7.1 BRIEF HISTORICAL BACKGROUND 169 7.2 THE BIRKHOFF-JEBSEN THEOREM 172 7.3 THE (APPARENT) GRAVITATIONAL HORIZON 176 7.4 GEODESICS IN THE FLRW SPACETIME 178 7.5 PROPER SIZE OF THE VISIBLE UNIVERSE 181 7.6 A COMPARISON OF THE GRAVITATIONAL, PARTICLE AND EVENT HORIZONS 184 7.7 GAUGE TRANSFORMATIONS WITH 185 7.7.1 de Sitter Space 186 7.7.2 A Cosmology with 187 7.7.3 Radiation Dominated Universe 188 7.7.4 Einstein-de Sitter Universe ( 189 7.7.5 The Standard ( 190 Chapter 8: Cosmological Redshift 192 8.1 DEFINITION 194 8.2 REDSHIFT IN STATIC SPACETIMES 196 8.2.1 Minkowski Spacetime 197 8.2.2 The Milne Universe 198 8.2.3 de Sitter Space 199 8.2.4 The Lanczos Universe 201 8.2.5 A Lanczos Universe with 202 8.2.6 Anti-de Sitter Space 203 8.3 EXPANSION OF SPACE 204 Section III: The Metric 206 Chapter 9: Emergence of Zero Active Mass 208 9.1 A MOST CURIOUS COINCIDENCE 208 9.2 AGE OF THE UNIVERSE VERSUS REDSHIFT 211 9.3 HIGH-REDSHIFT QUASARS 213 9.4 HIGH-REDSHIFT GALAXIES 224 9.5 CONCLUDING REMARKS 227 Chapter 10: Theoretical Basis for Zero Active Mass 228 10.1 THE LAPSE FUNCTION 228 10.2 THE COSMOLOGICAL ORIGIN OF 233 Chapter 11: The Horizon Problems 240 11.1 INFLATION: A STATUS UPDATE 241 11.2 VIABILITY OF SLOW-ROLL INFLATION 247 11.3 THE TEMPERATURE HORIZON PROBLEM 257 11.4 THE ELECTROWEAK HORIZON PROBLEM 259 Chapter 12: Observing Zero Active Mass 266 12.1 STATISTICAL METHODS USED IN MODEL SELECTION 266 12.1.1 Information Criteria 267 12.1.2 Two-point Diagnostics 270 12.2 REDSHIFT-DEPENDENT EXPANSION RATE 272 12.3 THE HUBBLE DIAGRAM 278 12.3.1 Active Galactic Nuclei 280 12.3.2 Type Ia Supernovae 284 12.4 THE ANGULAR-DIAMETER DISTANCE 294 12.5 STRONG GRAVITATIONAL LENSING 303 12.6 THE COSMIC EQUATION OF STATE 313 Chapter 13: Structure Formation 318 13.1 QUANTUM FLUCTUATIONS AT THE PLANCK SCALE 319 13.2 STRUCTURE FORMATION 326 13.2.1 Perturbed Boltzmann Equation, 329 13.2.2 Perturbed Boltzmann Equation, Zero Active Mass 332 13.2.3 The Observed Matter Power Spectrum 339 13.3 THE OBSERVED HALO MASS FUNCTION 344 Chapter 14: Future Prospects 352 14.1 A DIRECT TEST OF THE COSMIC SPACETIME 352 14.2 OBSERVATIONS OVER THE COMING DECADES 355 14.2.1 Dark Energy Surveys 355 14.2.2 B-mode Polarization in the CMB 357 14.2.3 Re-ionization History 358 14.2.4 The Redshift-time Relation 359 14.3 MAJOR UNSOLVED PROBLEMS IN 362 14.3.1 The Trans-Planckian Problem 362 14.3.2 The Horizon Problems 363 14.3.3 BBN and the 363 14.3.4 Timeline for SMBHs and Early Galaxies 364 14.3.5 The Halo Mass Function 365 14.3.6 The Hubble Constant 365 14.4 OPEN QUESTIONS IN FLRW WITH ZERO ACTIVE MASS 366 14.4.1 BBN 366 14.4.2 N-Body Simulations 368 14.4.3 CMB Power Spectrum Below One Degree 369 14.5 CONCLUDING REMARKS 371 Bibliography 374 Index 412 The growth of cosmology into a precision science represents one of the most remarkable stories of the past century. Much has been written chronicling this development, but rarely has any of it focused on the most critical element of this work the cosmic spacetime itself. Addressing this lacuna is the principal focus of this book, documenting the growing body of evidence compelling us not only to use this famous solution to Einstein's equations in order to refine the current paradigm, but to probe its foundation at a much deeper level. Its excursion from the smallest to largest possible scales insightfully reveals an emerging link between the Universe we behold and the established tenets of our most fundamental physical theories. Key Features: Uncovers the critical link between the Local Flatness Theorem in general relativity and the symmetries informing the spacetime's metric coefficients Develops a physical explanation for some of the most unpalatable coincidences in cosmology Provides a sober assessment of the horizon problems precluding our full understanding of the early Universe Reveals a possible explanation for the origin of rest-mass energy in Einstein's theory In spite of its technical layout, this book does not shy away from introducing the principal players who have made the most enduring contributions to this field. Anyone with a graduate level foundation in physics and astronomy will be able to easily follow its contents This book documents the growing body of evidence compelling us to probe its foundation at a much deeper level. Its excursion from the smallest to largest possible scales insightfully reveals an emerging link between the Universe we behold and the established tenets of our most fundamental physical theories.