Wandering in the Background: A Cosmic Microwave Background Explorer by Wayne T. Hu B.A. (Princeton University) 1990 M.A. (University of California at Berkeley) 1992 A thesis submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA at BERKELEY Committee in charge: Professor Joseph Silk, Chair Professor Marc Davis Professor Hyron Spinrad 1995 1 Wandering in the Background: A Cosmic Microwave Background Explorer by Wayne T. Hu Doctor of Philosophy in Physics University of California at Berkeley Professor Joseph Silk, Chair We develop and examine the principles governing the formation of distortions in the cos- mic microwave background. Perturbations in the frequency or spectral distribution of the background probe the thermal history of the universe, whereas those in the angular temper- ature distribution probe its dynamics and geometry. Stressing model independent results, we show how the microwave background can be used to extract information on the mass density, vacuum density, baryon content, radiation content, expansion rate and some as- pects of structure formation in the universe. To address these issues, we develop elements of relativistic kinetic and perturbation theory as they become necessary for the description of the particle and gravitational interactions of the photons. Subtle issues such as fluc- tuation representation, or gauge, normal mode analysis in an open geometry, and second order effects are considered in detail. Employing analytic and numerical results, we con- struct anisotropies in a critical, open, and cosmological constant universe with adiabatic and/or isocurvature initial conditions allowing for possible early reionization. We find that anisotropy formation is a simple process governed by the Compton scattering of photons off electrons and their gravitational coupling to the other particle species in the universe. Chair Date The thesis of Wayne T. Hu is approved: Chair Date Date Date University of California, Berkeley 1995 Wandering in the Background: A Cosmic Microwave Background Explorer c copyright 1995 by Wayne T. Hu iii To Chuang-tzu, Said the disciple, “After I heard your words, one year and I ran wild, two years and I was tame, three years and positions interchanged, four years and things settled down, five years and things came to me ...” –Chuang-tzu, 27 From Chuang-tzu, I hear that there is a sacred tortoise which has been dead for three thousand years. His Majesty keeps it wrapped up in a box at the top of the hall in the ancestral shrine. Would this tortoise rather be dead, to be honored as preserved bones, or would it rather be alive and dragging its tail in the mud... Away with you! I shall drag my tail in the mud. –Chuang-tzu, 17 iv Contents List of Figures viii List of Tables x Preface xi Acknowledgements xiii 1 Overview 1 1.1CosmologicalBackground............................ 1 1.1.1 PerfectionandItsImplications..................... 2 1.1.2 ImperfectionandItsApplications.................... 4 1.2AnisotropyFormation.............................. 6 1.2.1 Acoustic Oscillations . 8 1.2.2 BaryonDrag............................... 10 1.2.3 DopplerEffect.............................. 11 1.2.4 PotentialEvolution............................ 11 1.2.5 PhotonDiffusionDamping....................... 14 1.2.6 IntegratedSachs-WolfeEffect...................... 15 1.2.7 ProjectionEffects............................ 16 1.3AnisotropySpectrum............................... 18 1.4RobustnesstoInitialConditions........................ 20 1.5Reionization.................................... 22 2 The Boltzmann Equation 25 2.1GravitationalInteractions............................ 26 2.1.1 MetricFluctuations........................... 26 2.1.2 GravitationalRedshiftandDilation.................. 27 2.1.3 Collisionless Brightness Equation . 30 2.2ComptonScattering............................... 31 2.2.1 Collision Integral . 32 2.2.2 IndividualTerms............................. 35 2.2.3 GeneralizedKompaneetsEquation................... 40 2.2.4 Collisional Brightness Equation . 41 v 3 Thermalization and Spectral Distortions 43 3.1 Collision Equations . 44 3.1.1 ComptonScatteringRevisited...................... 44 3.1.2 ElectronTemperatureEvolution.................... 45 3.1.3 Bremsstrahlung and Double Compton Scattering . 46 3.2ThermalizationOpticalDepthsandRates................... 47 3.2.1 Comptonization............................. 48 3.2.2 ChemicalPotentialFormation...................... 53 3.2.3 BlackbodyFormation.......................... 55 3.3LowFrequencyEvolution............................ 57 3.3.1 ChemicalPotentialEra......................... 59 3.3.2 Chemical Potential Freeze Out . 61 3.3.3 NegativeChemicalPotentials...................... 64 3.3.4 BalancedInjection............................ 64 3.4HighFrequencyEvolution............................ 66 3.4.1 AnalyticApproximations........................ 66 3.4.2 NumericalResults............................ 69 3.5ComparisonsandConstraints.......................... 73 3.5.1 ObservationalData............................ 73 3.5.2 ConstraintsonDecayingParticles................... 74 3.5.3 DissipationofAcousticWaves...................... 76 4 Multifluid Perturbation Theory 81 4.1NormalModeDecomposition.......................... 82 4.1.1 LaplacianEigenfunctions........................ 82 4.1.2 RadialRepresentation.......................... 83 4.1.3 CompletenessandSuperCurvatureModes.............. 84 4.1.4 HigherAngularFunctions........................ 87 4.2NewtonianGaugeEvolution........................... 88 4.2.1 MetricFluctuations........................... 88 4.2.2 ConservationEquations......................... 89 4.2.3 TotalMatterandItsComponents................... 93 4.2.4 Radiation................................. 94 4.2.5 Matter................................... 95 4.2.6 EinsteinEquations............................ 96 4.3Gauge....................................... 97 4.3.1 GaugeTransformations......................... 99 4.3.2 NewtonianGauge............................ 100 4.3.3 SynchronousGauge........................... 101 4.3.4 TotalMatterGauge........................... 104 4.3.5 HybridFormulation........................... 105 vi 5 Perturbation Evolution 107 5.1SuperhorizonEvolution............................. 108 5.1.1 TotalMatterEquation.......................... 108 5.1.2 GeneralSolution............................. 109 5.1.3 InitialConditions............................. 112 5.1.4 ComponentEvolution.......................... 114 5.1.5 Discussion................................. 117 5.2 Subhorizon Evolution before Recombination . 117 5.2.1 AnalyticAcousticSolutions....................... 118 5.2.2 Driven Acoustic Oscillations . 121 5.2.3 Damped Acoustic Oscillations . 124 5.3MatterEvolutionafterRecombination..................... 127 5.3.1 ComptonDrag.............................. 127 5.3.2 ReionizationinIsocurvatureModels.................. 129 6 Primary Anisotropies 131 6.1Overview..................................... 131 6.1.1 AnisotropySources............................ 132 6.1.2 ProjectionandFreeStreaming..................... 133 6.1.3 MathematicalDescription........................ 134 6.2Sachs-WolfeEffect................................ 135 6.2.1 OrdinarySachs-WolfeEffect....................... 137 6.2.2 IntegratedSachs-WolfeEffect...................... 139 6.2.3 Adiabatic Ω0 =1models........................ 141 6.2.4 AdiabaticΛModels........................... 144 6.2.5 AdiabaticOpenModels......................... 147 6.2.6 IsocurvatureΛandOpenModels.................... 149 6.3AcousticPeaks.................................. 152 6.3.1 MathematicalDescription........................ 153 6.3.2 LocationofthePeaks.......................... 154 6.3.3 HeightsofthePeaks........................... 156 6.3.4 DiffusionDampingatRecombination.................. 158 7 Secondary Anisotropies 161 7.1LinearContributions............................... 162 7.1.1 ReionizationDamping.......................... 162 7.1.2 COBE ConstraintsonPIBModels................... 164 7.1.3 AnisotropyRegeneration......................... 168 7.1.4 CancellationDamping.......................... 172 7.1.5 MinimalPIBAnisotropies........................ 174 7.2SecondOrderContributions........................... 176 7.2.1 GeneralizedDopplerEffect....................... 176 7.2.2 VishniacEffect.............................. 178 7.2.3 OtherSecondOrderEffects....................... 181 7.3BeyondPerturbationTheory:ASurvey.................... 184 vii 7.4FinalThoughts.................................. 186 Bibliography 188 A Toward Higher Accuracy: A CDM Example 197 A.1RefiningtheGravitationalPotentials...................... 198 A.1.1NeutrinoAnisotropicStress....................... 198 A.1.2SmallScaleRadiationFeedback..................... 201 A.2AnalyticConstructionto5%Accuracy..................... 203 A.2.1ExplicitTightCouplingSolutions................... 203 A.2.2RecombinationRevisited........................ 205 A.2.3AnalyticResults............................. 207 A.3Toward1%Accuracy............................... 210 A.3.1PolarizationDamping.......................... 210 A.3.2HeliumRecombination.......................... 214 A.3.3GravityWaves.............................. 215 A.3.4MassiveNeutrinos............................ 216 B Useful Quantities and Relations 219 B.1FRWParameters................................. 219 B.2TimeVariables.................................. 221 B.2.1ScaleFactorandRedshift.......................