Springer Tracts in Modern Physics Volume 210 ManagingEditor:G.Höhler,Karlsruhe Editors: J.Kühn,Karlsruhe Th.Müller,Karlsruhe A.Ruckenstein,NewJersey F.Steiner,Ulm J.Trümper,Garching P.Wölfle,Karlsruhe StartingwithVolume165,SpringerTractsinModernPhysicsispartofthe[SpringerLink]service. ForallcustomerswithstandingordersforSpringerTractsinModernPhysicsweofferthefulltext inelectronicformvia[SpringerLink]freeofcharge.Pleasecontactyourlibrarianwhocanreceive apasswordforfreeaccesstothefullarticlesbyregistrationat: springerlink.com Ifyoudonothaveastandingorderyoucanneverthelessbrowseonlinethroughthetableofcontents ofthevolumesandtheabstractsofeacharticleandperformafulltextsearch. Thereyouwillalsofindmoreinformationabouttheseries. Springer Tracts in Modern Physics SpringerTractsinModernPhysicsprovidescomprehensiveandcriticalreviewsoftopicsofcurrentin- terestinphysics.Thefollowingfieldsareemphasized:elementary particlephysics,solid-statephysics, complexsystems,andfundamentalastrophysics. 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Seealsospringeronline.com ManagingEditor Solid-StatePhysics,Editors GerhardHöhler AndreiRuckenstein EditorforTheAmericas InstitutfürTheoretischeTeilchenphysik UniversitätKarlsruhe DepartmentofPhysicsandAstronomy Postfach6980 Rutgers,TheStateUniversityofNewJersey 76128Karlsruhe,Germany 136FrelinghuysenRoad Phone:+49(721)6083375 Piscataway,NJ08854-8019,USA Fax:+49(721)370726 Phone:+1(732)4454329 Email:[email protected] Fax:+1(732)445-4343 www-ttp.physik.uni-karlsruhe.de/ Email:[email protected] www.physics.rutgers.edu/people/pips/ ElementaryParticlePhysics,Editors Ruckenstein.html PeterWölfle JohannH.Kühn InstitutfürTheoriederKondensiertenMaterie InstitutfürTheoretischeTeilchenphysik UniversitätKarlsruhe UniversitätKarlsruhe Postfach6980 Postfach6980 76128Karlsruhe,Germany 76128Karlsruhe,Germany Phone:+49(721)6083372 Phone:+49(721)6083590 Fax:+49(721)698150 Fax:+49(721)370726 Email:woelfl[email protected] Email:[email protected] www-ttp.physik.uni-karlsruhe.de/∼jk www-tkm.physik.uni-karlsruhe.de ThomasMüller ComplexSystems,Editor InstitutfürExperimentelleKernphysik FrankSteiner FakultätfürPhysik UniversitätKarlsruhe AbteilungTheoretischePhysik Postfach6980 UniversitätUlm 76128Karlsruhe,Germany Albert-Einstein-Allee11 Phone:+49(721)6083524 89069Ulm,Germany Fax:+49(721)6072621 Phone:+49(731)5022910 Email:[email protected] Fax:+49(731)5022924 www-ekp.physik.uni-karlsruhe.de Email:[email protected] www.physik.uni-ulm.de/theo/qc/group.html FundamentalAstrophysics,Editor JoachimTrümper Max-Planck-InstitutfürExtraterrestrischePhysik Postfach1312 85741Garching,Germany Phone:+49(89)30003559 Fax:+49(89)30003315 Email:[email protected] www.mpe-garching.mpg.de/index.html Dierck-Ekkehard Liebscher Cosmology With100 Figures ABC Dierck-EkkehardLiebscher AstrophysikalischesInstitutPotsdam AnderSternwarte16 14482Potsdam,Germany E-mail:[email protected] LibraryofCongressControlNumber:2005921920 PhysicsandAstronomyClassificationScheme(PACS): 98.80.-k,04.20.Cv,01.55.+b ISSNprintedition:0081-3869 ISSNelectronicedition:1615-0430 ISBN-10 3-540-23261-3 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-23261-2 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermitted onlyundertheprovisions oftheGermanCopyright LawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springeronline.com (cid:1)c Springer-VerlagBerlinHeidelberg2005 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorandTechBooksusingaSpringerLATEXmacropackage Coverconcept:eStudioCalamarSteinen Coverproduction:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11316787 56/3141/jl 543210 Preface Thisbookisintendedtoexplainthetheoreticalissuesofcosmology.Tosome extent, it is a complement to presentations that emphasise the progress and success of our observational power and the importance of determining phe- nomenological parameters. Since Einstein solved Newton’s paradox through use of the theory of general relativity, the universe has been described in that framework. The present standard model is the result of many observa- tions, which culminated in the celebrated determination of the structure in the microwave background. This book will acknowledge this development, of course, but it will emphasise the theoretical aspects of cosmology that can be easily forgotten in view of such a success. Just as stars and black holes have beenfound through meresolutions to theory (i.e. without observation), cosmologytooisascientifictopic,andwassoevenbeforeanyoneknewwhat to observe. The central issue of cosmology is the compatibility of different areas of physics.Thiscompatibilityoughtnotbeboundedorrestrictedbyhorizonsof any kind. In contrast, the realm of experience is always limited by the obser- vational or experimental means currently available or possible in principle. The compatibility, or universality, of physics is nonetheless indispensable for our trust in physics itself. Cosmologyisanexcitingscience,eventhoughandindeedbecauseneither its subject nor its epistemological status can be described in a way that is generally accepted. On the one hand, cosmology is the science of global extrapolation; on the other, it is understood as the science of the gigantic historyoftheevolvinguniverse.Thisevolutionisduetotheoverallexpansion, whichpushesmanysystemsawayfromequilibriumeventhoughtheexpansion itself is adiabatic. Modern cosmology is closely connected with the physics of elementary particles and their interactions. The energy scale of elementary particles, atoms andmolecules is reflectedin a timescale in the history of the universe. The expansion of the universe must have begun from a time when it was so hot, that any value of energy that we might consider at present (from the nuclearbindingenergytothegrandunificationscale)hasexistedasathermal energy in the distant past. In particular, the processes that are believed to occurattheenergiesofgrandunificationwillonlyhavetakenplaceinapast VI Preface so distant that their consequences will, at best, be observed as relics of that stage. Moderncosmologyiscloselyconnectedwiththetheoryofgeneralrelativ- ity,andthefundamentalquestionofitsrelationtoquantumtheoryisreflected in the question of the origin of our universe. Cosmological models reveal the abstract features of the singularities of general relativity and all their conse- quences.Generalrelativityhastounderstandotherexcitingobjects,too,but theuniverseisthesimplestobjectthatcanbeconsistentlydescribedonlyby curved space-time, which is the essential invention of general relativity. Nevertheless, the notions of cosmology can be explained without going too deeply into general relativity or quantum theory, because the simplicity of the overall motion allows us to reduce most of the arguments to phenom- enologicalthermodynamics,andbecausethegeneral-relativisticequationsfor the expansion of the universe can be reduced in the case of our cosmological models to equations that are formally equivalent to Newtonian ones. This book will introduce the reader to cosmology not only with general arguments,butalsowiththekeyformulae.Theseformulaewillbegivenwith their derivations. The reader not only should get an overall impression, but also should be enabled to read more advanced literature without too much trouble to put everything into place. The network of basic arguments and basic facts will be presented in a formal context, too. In this way, the book will be not only an introduction, but also a kind of revision course. Modern cosmology rests upon the continuing expansion of the universe, which was identified first by Hubble through an appropriate interpretation of the redshift that increased with the apparent distance. Friedmann had already found the correct geometrical form of the cosmic expansion as a so- lution of the field equations of Einstein’s theory of general relativity. The expansion triggers the decay of various kinds of equilibria. The slow forma- tion of structure by gravitational condensation and the fast nuclear binding reactions can be understood as the result of such decays. Two basic obser- vations support the picture of the Friedmann universe. First, we observe a comparativelylargeconcentrationofdeuterium,whichcanonlybeexplained by some suddenly halted nucleosynthesis in a universal neutron-rich envi- ronment. The result of primordial nucleosynthesis is explained by reaction kinetics that describe processes that start after cooling and are halted after rarefaction, both of which result from the expansion. Second, the heat bath of the universe, now decoupled and cooled by the expansion, forms the mi- crowavebackground.Thisbackgroundisastonishinglyisotropic.Thedensity perturbations that are reflected in its fluctuations are very weak and have a remarkably featureless power spectrum. After1980,aloopofargumentwasfoundthatmakestheargumentsabout the evolution of the universe very strong (Fig. 1). When applied to baryon– antibaryon annihilation in a charge-symmetric universe, the nucleosynthe- sis paradigm leads to a present baryon concentration that is 10 orders of Preface VII Monopole Symmetry Inflation problem breakdown Seed Baryon suppression asymmetry Structure Gamow’s evolution conjecture Microwave Friedmann Heliumand background expansion deuterium Hubble expansion Fig. 1. The cosmic loop magnitudelowerthantheobservedvalue.Aninitialbaryonasymmetrymust be assumed, and grand unified theories provide an explanation of how such anasymmetrycanresultfromasymmetricdecaysofGUTparticles.Ingrand unifiedtheories,symmetriesmaybespontaneouslybroken.Thisisfineforthe baryon asymmetry, but is bad for the present density of the universe, which would exceed the observed value by far since the breakdown of symmetry leaves too many topological quasi-particles (monopoles) behind. How do we get rid of these monopoles? We need a dilution in a phase of exponential increase of the size of the universe. By coincidence or not, the grand unified theories contain the concept of a high-temperature vacuum that drives such an inflation when the temperature falls below the temperature of symmetry breakdown.Theuniversecoolsdowntonearlyabsolutezerobeforethehigh- temperature vacuum decays into a low-temperature vacuum, releasing its energy density into the degrees of freedom of ordinary GUT particles. Now, the zero-point fluctuations that remain lose their coherence and turn into ordinary perturbations, which show correctly the power spectrum inferred from observations. Grand unified theories also keep ready exotic particles VIII Preface that may be candidates for the dark matter that is suspected to exist in galaxies, clusters and large-scale structures. The loop is closed by the phe- nomena in the microwave background and the evolution of structure in the transparent universe. This book will follow this cosmic loop. After a short introduction to the geometryofspace-timeingeneralandthegeometryofanexpandinguniverse inparticular,weexplainthenuclear-synthesisparadigm,followtheargument about inflation, describe the inflationary scenario, consider the origin of per- turbations and follow their evolution to the recombination time and after. This closes the loop. Finally, we return to the basic questions of consistency andthecosmologicalsingularity,i.e.weconsiderthebasicnotionsofquantum gravity and other extensions of the standard picture. Potsdam Dierck-Ekkehard Liebscher February 2005 Contents 1 Basics .................................................... 1 1.1 The Cosmos, the Universe and the Metagalaxy............. 1 1.2 Explanation and Evolution .............................. 3 1.3 The Cosmological Principle .............................. 5 1.4 Observation and Measurement ........................... 6 1.5 Newton’s Paradox ...................................... 8 1.6 Olbers’ Paradox........................................ 8 1.7 Lambert’s Solution ..................................... 10 1.8 Einstein’s Solution...................................... 11 1.9 Friedmann’s Solution and the Hubble Expansion ........... 12 1.10 Gamow’s Conjecture.................................... 16 1.11 Jeans’ Problem......................................... 18 1.12 Lemaˆıtre’s Problem..................................... 19 References ................................................. 22 2 Relativity................................................. 25 2.1 Relativity of Simultaneity: Solving Fresnel’s Paradox........ 25 2.2 The Equivalence Principle, Deflection of Light and Geometry .................................. 34 2.3 General Relativity: Solving Galileo’s Paradox .............. 42 2.4 Positive Curvature: Solving Newton’s Paradox ............. 46 References ................................................. 50 3 Expansion ................................................ 53 3.1 The Friedmann Equations ............................... 53 3.2 The Worlds of Constant Curvature ....................... 58 3.3 Worlds with Matter Only................................ 65 3.4 Barotropic Components of Matter ........................ 67 3.5 Friedmann’s Universe: Solving Olbers’ Paradox............. 69 3.6 The Cosmological Singularity ............................ 74 References ................................................. 77 4 Cosmometry.............................................. 79 4.1 The Past Light Cone.................................... 79 4.2 Horizons .............................................. 81 X Contents 4.3 The Cosmological Redshift .............................. 83 4.4 Distance Definitions in the Expanding Universe ............ 88 4.5 Distance Determinations ................................ 92 4.6 Determinations of the Expansion Rate .................... 98 4.7 Curvature Determinations ............................... 101 4.8 Gravitational Lenses .................................... 104 4.9 Quasar Absorption Forests............................... 111 References ................................................. 115 5 Matter and Radiation .................................... 119 5.1 The Average Mass Density .............................. 119 5.2 Counting Procedures.................................... 123 5.3 The Microwave Background Radiation .................... 125 5.4 The Photon Bath and the Notion of the Ether ............. 127 5.5 The Homogeneous Large-Scale Structure .................. 129 5.6 Correlations and Power Spectra .......................... 131 5.7 Fractal Structure ....................................... 137 References ................................................. 139 6 Standard Synthesis ....................................... 143 6.1 The Gamow Universe ................................... 143 6.2 The Standard Process and the Decay of Equilibrium ........ 146 6.3 The Primordial Nucleosynthesis .......................... 154 6.4 Weakly Interacting Particles ............................. 160 6.5 The Problem of the Baryons ............................. 161 References ................................................. 166 7 Inflation .................................................. 167 7.1 The Implications of the Monopole Problem ................ 167 7.2 The Vacuum........................................... 168 7.3 The Inflaton Field ...................................... 170 7.4 Homogeneous Inflation .................................. 173 7.5 Concomitant Solutions to Fundamental Problems........... 176 7.6 Inhomogeneities and Inflation ............................ 182 7.7 Variations ............................................. 184 References ................................................. 188 8 Structure Formation in the Opaque Universe ............. 189 8.1 Perturbations in General ................................ 189 8.2 The Relativistic Approach and the Evolution on Large Scales ........................ 194 8.3 Inhomogeneities and Inflation ............................ 199 8.4 The Evolution of Small Scales and the Microwave Background .......................... 205 References ................................................. 212