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8 Jun 2014 Dynamics of the Universe in Problems Dynamics of the Universe in Problems A thousand problems in Cosmology 1 1 1 Yu.L. Bolotin ∗ , V.A. Cherkaskiy , G.I. Ivashkevych , O.A. Lemets1, I.V. Tanatarov1,2, D.A. Yerokhin1 1A.I.Akhiezer Institute for Theoretical Physics, National Science Center ”Kharkov Institute of Physics and Technology”, Akademicheskaya Str. 1, 61108 Kharkov, Ukraine 2V.N. Karazin Kharkov National University, Svobody Sq. 4, 61077 Kharkov, Ukraine arXiv:0904.0382v9 [astro-ph.CO] 8 Jun 2014 ∗[email protected] 2 Contents 1 Cosmo-warm-up 11 1.1 Astronomy”beforetheCommonEra” . 11 1.2 Quantitieslargeandsmall ........................... 12 1.3 Geometricwarm-up............................... 14 1.4 Astrophysicalwarm-up ............................. 14 1.5 Planck scales and fundamental constants c, G, ~ ............... 15 1.6 Gravity ..................................... 17 1.7 Forestforthetrees ............................... 19 1.8 LifeonMars? .................................. 19 1.8.1 FinetuningoftheUniverse . 20 1.9 Thermowarm-up ................................ 20 1.10 PlaywithNumbersafterSivaram . 21 2 Dynamics of Expanding Universe 23 2.1 Homogeneous and isotropic Universe, Hubble’s law . 23 2.2 Equations of General Relativity . 24 2.3 Friedman–Lemaˆıtre–Robertson–Walker (FLRW) metric . ..... 27 2.4 Expanding Universe: ordinarity, difficulties and paradoxes . .... 29 2.4.1 Warm-up ................................ 29 2.4.2 Thetetheredgalaxyproblem . 30 2.4.3 Cosmologicalredshift. .. .. 31 2.5 FriedmanEquations .............................. 32 2.6 Newtoniancosmology.............................. 34 2.7 Energy balance in an expanding Universe . 34 2.8 Cosmography .................................. 35 3 Dynamics of the Universe in the Big Bang model 43 3.1 Generalquestions................................ 43 3.2 Solutions of Friedman equations in the Big Bang model . 43 3.3 The role of curvature in the dynamics of the Universe . ... 47 3.4 TheMilneUniverse............................... 48 3.5 Cosmologicalhorizons ............................. 48 3.6 Energy conditions and the Raychaudhuri equation . .... 50 3.6.1 Energyconditions............................ 50 3.6.2 Raychaudhuriequation. 51 3 4 CONTENTS 3.6.3 Sudden Future Singularities . 52 3.7 Influence of cosmological expansion on local systems . ..... 52 3.8 Dynamics of the Universe in terms of redshift and conformal time ..... 53 3.9 TimeEvolutionofCMB ............................ 54 4 Black Holes. Problems 57 4.1 Technicalwarm-up ............................... 57 4.1.1 Uniformly accelerated observer, Rindler metric . 57 4.1.2 Metricincurvedspacetime. 57 4.2 Schwarzschildblackhole ............................ 58 4.2.1 Simple problems . 59 4.2.2 Symmetries and integrals of motion of Schwarzschild metric . 59 4.2.3 Radial motion in Schwarzchild metric . 60 4.2.4 Blacknessofblackholes ........................ 60 4.2.5 Orbital motion, effective potential . 61 4.2.6 Miscellaneous problems . 61 4.2.7 Different coordinates, maximal extension . 62 4.3 Kerrblackhole ................................. 63 4.3.1 General axially symmetric metric . 63 4.3.2 Limiting cases . 64 4.3.3 Horizons and singularity . 64 4.3.4 Stationary limit . 65 4.3.5 ErgosphereandthePenroseprocess . 65 4.3.6 Integralsofmotion ........................... 66 4.3.7 The laws of mechanics of black holes . 66 4.3.8 Particles’ motion in the equatorial plane . 67 4.4 Physics in general black hole spacetimes . 68 4.4.1 Frames, time intervals and distances . 68 4.4.2 Fiducialobservers............................ 69 4.4.3 Collision of particles: general relationships . 69 4.5 Astrophysicalblackholes............................ 70 4.5.1 Preliminary . 70 4.5.2 Quantumeffects............................. 72 5 Cosmic Microwave Background (CMB) 73 5.1 Thermodynamics of Black-Body Radiation . 73 5.2 TimeEvolutionofCMB ............................ 73 5.3 StatisticalpropertiesofCMB . .. .. 75 5.4 PrimaryanisotropyofCMB . .. .. 75 5.5 CMBinteractionwithothercomponents . 76 5.6 Extras ...................................... 77 CONTENTS 5 6 Thermodynamics of Universe 79 6.1 Thermodynamical Properties of Elementary Particles . .... 79 6.2 Thermodynamics of Non-Relativistic Gas . 80 6.3 EntropyofExpandingUniverse . 81 6.4 Connection between Temperature and Redshift . ... 82 6.5 Peculiarities of Thermodynamics in Early Universe . 82 6.6 TheSahaequation ............................... 83 6.7 PrimaryNucleosynthesis ............................ 83 7 Perturbation Theory 85 7.1 Non relativistic small perturbation theory . 85 7.2 Introduction to relativistic theory of small perturbations . ...... 86 7.2.1 Perturbationsonflatbackground . 86 7.2.2 Metrics perturbations, coordinates transforms and perturbed energy– momentumtensor............................ 87 7.3 Expansion of cosmological perturbations in helicities . .. 87 7.3.1 Scalar perturbations in conformal Newtonian gauge . 88 7.3.2 Evolution of vector and tensor perturbations . 89 7.4 CMBanisotropy ................................ 90 7.5 Initial perturbations in the Universe . 90 7.5.1 Fluctuations power spectrum: non–relativistic approach . ... 90 7.5.2 Quantum fluctuations of fields in inflationary Universe . 91 8 Inflationary Universe 93 8.1 Problems of the Hot Universe (the Big Bang Model) . 93 8.1.1 TheHorizonProblem.......................... 93 8.1.2 TheFlatnessProblem ......................... 93 8.1.3 TheEntropyProblem ......................... 94 8.1.4 The Primary Inhomogeneities Problem . 94 8.2 Cosmological Inflation: The Canonic Theory . 95 8.2.1 Scalar Field In Cosmology . 95 8.2.2 InflationaryIntroduction . 97 8.2.3 Inflation in the Slow-Roll Regime . 97 8.2.4 Solution of the Hot Big Bang Theory Problems . 98 8.2.5 DifferentModelsofInflation . 99 9 Dark Energy 105 9.1 TheCosmologicalConstant . .105 9.2 GeometryandDestiny .............................106 9.3 Time-dependent Cosmological Constant . 107 9.4 Static Einstein’s Universe . 108 9.5 DynamicalFormsofDarkEnergy . .108 9.5.1 TheQuintessence ............................108 9.5.2 TrackerFields..............................112 9.5.3 TheK-essence..............................113 6 CONTENTS 9.5.4 PhantomEnergy ............................114 9.5.5 Disintegration of Bound Structures . 115 9.5.6 Big Rip, Pseudo Rip, Little Rip . 116 9.5.7 TheStatefinder .............................118 9.5.8 CrossingthePhantomDivide . .118 9.6 LostandFound .................................119 9.7 SingleScalarCosmology ............................119 9.7.1 Exact Solutions for the Single Scalar Cosmology . 121 9.8 BianchiIModel.................................122 9.9 HybridExpansionLaw .............................123 9.10 ThePower-LawCosmology. .124 10 Dark Matter 125 10.1 Observational Evidence of the Dark matter Existence . ......125 10.2ANONIMOUS..................................126 10.3DarkMatterHalo................................126 10.4 CandidatesforDarkMatterParticles . 127 10.4.1 Standard Model Particles as Dark Matter Candidates . 127 10.4.2 Supersymmetric Candidate Particles . 127 10.5 DarkMatterDetection .............................127 10.6 TheDarkMatterintheSolarSystem . 128 10.7TheDarkStars .................................128 11 Interaction in the dark sector 129 11.1 Physical mechanism of energy exchange . 129 11.2 Phenomenology of interacting models . 129 11.3 Simple linear models . 131 11.4 Cosmological models with a change of the direction of energy transfer . 132 11.5 Non-linear interaction in the dark sector . 132 11.6 Phase space structure of models with interaction . .133 11.7 Peculiarity of dynamics of scalar field coupled to dark matter . .135 11.7.1 Interacting quintessence model . 135 11.7.2 InteractingPhantom . .136 11.7.3 Tachyonic Interacting Scalar Field . 136 11.8 Realization of interaction in the dark sector . 137 11.8.1 Time-dependent cosmological ”constant” . 138 11.9 Statefinder parameters for interacting dark energy and cold dark matter. 139 11.10Interactingholographicdarkenergy . .139 11.11Transientacceleration . 141 12 Standard Cosmological Model 145 12.1 CharacteristicParametersandScales . 145 12.2 EvolutionofUniverse..............................146 CONTENTS 7 13 Weak field limit and gravitational waves 151 13.1 Motivation and symmetries / Introduction . 151 13.2 Linearized Einstein equations . 152 13.3 Gauge transformations and degrees of freedom . ......152 13.4Waveequation..................................153 13.5 Transversetracelessgauge . 154 13.6 Gravitationalwavesandmatter . 155 13.7 Energyofgravitationalwaves . 156 13.8Binarysystems .................................156 13.9 Gravitational Waves: scale of the phenomenon . .156 13.10Generation and detection of gravitational waves . ......156 14 Observational Cosmology 159 14.1 Point gravitational lenses . 159 14.2 Microlensing and Weak Lensing . 160 15 Holographic Universe 161 15.1 Information, Entropy and Holographic Screen . .162 16 Horizons 167 16.1SimpleEnglish .................................167 16.2SimpleMath...................................169 16.3Compositemodels................................171 16.4Causalstructure................................. 171 16.5 Conformal diagrams: stationary black holes . 173 16.5.1 Schwarzschild-Kruskal black hole solution . 173 16.5.2 Other spherically symmetric black holes . 174 16.6Hubblesphere..................................176 16.7Properhorizons .................................177 16.8Inflation .....................................177 16.9Holography ...................................178 17 Quantum Cosmology 181 17.1 Introductiontovacuumfluctuations.
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