Probing Dark Energy with Theory and Observations by Roland de Putter A dissertation 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, Berkeley Committee in charge: Professor Eric V. Linder, Co-Chair Professor Saul Perlmutter, Co-Chair Professor Chung-Pei Ma Professor Martin White Fall 2010 Probing Dark Energy with Theory and Observations Copyright 2010 by Roland de Putter 1 Abstract Probing Dark Energy with Theory and Observations by Roland de Putter Doctor of Philosophy in Physics University of California, Berkeley Professor Eric V. Linder, Co-Chair Professor Saul Perlmutter, Co-Chair The discovery of cosmic acceleration twelve years ago implies that our universe is dominated by dark energy, which is either a tiny cosmological constant or a mysterious fluid with large negative pressure, or that Einstein’s successful theory of gravity needs to be modified at large scales/low energies. Since then, independent evidence of a number of cosmological probes has firmly established the picture of a universe where dark energy (or the effective contribution from a modification of gravity) makes up about 72% of the total energy density. Whichever of the options mentioned above will turn out to be the right one, a satisfying explanation for cosmic acceleration will likely lead to important new insights in fundamental physics. The question of the physics behind cosmic acceleration is thus one of the most intriguing open questions in modern physics. In this thesis, we calculate current constraints on dark energy and study how to optimally use the cosmological tools at our disposal to learn about its nature. We will first present constraints from a host of recent data on the dark energy sound speed and equation of state for different dark energy models including early dark energy. We then study the observational properties of purely kinetic k-essence models and show how they can in principle be straightforwardly distinguished from quintessence models by their equation of state behavior. We next consider a large, representative set of dark energy and modified gravity models and show that they can be divided into a small set of observationally distinct classes. We also find that all non-early dark energy models we consider can be modeled extremely well by a simple linear equation of state form. We will then go on to discuss a number of alternative, model independent parametrizations of dark energy properties. Among other things, we find that principal component analysis is not as model-independent as one would like it to be and that assuming a fixed value for the high redshift equation of state can lead to a dangerous bias in the determination of the equation of state at low redshift. Finally, we discuss using weak gravitational lensing of cosmic microwave background (CMB) anisotropies as a cosmological probe. We compare 2 different methods for extracting cosmological information from the lensed CMB and show that CMB lensing will in the future be a useful tool for constraining dark energy and neutrino mass. Whereas marginalizing over neutrino mass can degrade dark energy constraints, CMB lensing helps to break the degeneracy between the two and restores the dark energy constraints to the level of the fixed neutrino mass case. i In loving memory of my uncle Rienk (1952 - 2010) ii Contents List of Figures v List of Tables xv 1 Introduction 1 1.1 ObservationalEffectsofDarkEnergy . 3 1.2 Discovery ................................. 4 1.3 CurrentConstraints............................ 7 1.4 Explaining Cosmic Acceleration . 9 1.4.1 TheCosmologicalConstant . 9 1.4.2 TheMultiverseandthe“AWord” . 11 1.4.3 DynamicalDarkEnergy . 12 1.4.4 Motivation............................. 13 1.4.5 ModifiedGravity ......................... 14 1.5 Observational Techniques and Prospects . ... 14 1.5.1 Supernovae ............................ 15 1.5.2 BaryonAcousticOscillations. 15 1.5.3 WeakLensing ........................... 17 1.5.4 Clusters .............................. 17 1.5.5 The Cosmic Microwave Background . 18 1.6 OtherQuestionsinCosmology. 19 1.7 Outline................................... 19 2 Measuring the Speed of Dark: Detecting Dark Energy Perturba- tions 21 2.1 Introduction ............................... 21 2.2 DarkEnergyPerturbations . 22 2.3 DarkEnergyModels ........................... 24 2.4 Impact on Cosmological Observations . 26 2.4.1 AngularPowerSpectra. 27 2.4.2 Estimating Constraints in Constant w Model ......... 31 2.4.3 Estimating Constraints in cEDE Model . 33 iii 2.5 MeasuringtheSpeedofDarkness . 34 2.6 Conclusions ................................ 37 3 Kinetic K-Essence and Quintessence 40 3.1 Introduction................................ 40 3.2 Motivation................................. 41 3.3 Themodel................................. 43 3.4 Restrictions on w(a)fromStability ................... 45 3.5 Distinguishing Kinetic K-Essence from Quintessence . ........ 47 3.6 Examples ................................. 49 3.6.1 Constant w ............................ 49 3.6.2 Time Variation: w(a)= w + w (1 a)............. 52 0 a − 3.6.3 Thawing/Freezing Regions . 53 3.6.4 GeneralizedChaplyginGas . 54 3.7 Conclusions ................................ 56 4 Calibrating Dark Energy 59 4.1 Introduction ............................... 59 4.2 DarkEnergyDynamics ......................... 60 4.2.1 PNGBModel ........................... 60 4.2.2 LinearPotential ......................... 62 4.2.3 SUGRAModel .......................... 63 4.2.4 BraneworldGravityModel . 64 4.2.5 Albrecht-SkordisModel . 64 4.2.6 CrossComparison ........................ 65 4.3 StretchingDarkEnergy . .. .. 67 4.4 ObservingDarkEnergy ......................... 71 4.5 FiguresofMerit ............................. 73 4.6 Conclusions ................................ 75 5 To Bin or Not To Bin: Decorrelating the Cosmic Equation of State 76 5.1 Introduction ............................... 76 5.2 Cosmological Information and the Equation of State . ..... 77 5.2.1 CosmologicalVariables . 77 5.2.2 InformationLocalization . 79 5.2.3 ExtractingtheEquationofState . 82 5.3 PrincipalComponents .......................... 82 5.3.1 Eigenmodes ............................ 83 5.3.2 Number of Eigenmodes and their Uncertainties . 85 5.4 UncorrelatedBandpowers . 87 5.4.1 ModesandWeights........................ 87 5.4.2 Decorrelated Estimates of the Equation of State . .. 88 iv 5.4.3 ContinuumLimit ......................... 91 5.5 BinnedEquationofState . .. .. 93 5.5.1 UncertaintiesandCorrelations. 93 5.5.2 FiguresofMerit ......................... 96 5.6 HighRedshiftEquationofStateandBias . 97 5.7 Physical Constraints on Equation of State . ... 101 5.7.1 EigenmodeExpansion . 101 5.7.2 TimeVariation .......................... 102 5.7.3 TestingtheEquationofState . 103 5.8 Conclusions ................................ 104 6 Future CMB Lensing Constraints on Neutrinos and Dark Energy 107 6.1 Introduction ............................... 107 6.2 Power Spectra Modeling: Theory and Experiments . ... 108 6.2.1 Theory .............................. 108 6.2.2 DeflectionField ......................... 109 6.2.3 Experiments............................ 112 6.3 NeutrinoMassConstraintsinΛCDM . 114 6.4 AddingDarkEnergyDynamics . 118 6.5 ExploringEarlyDarkEnergy . 120 6.6 Shortcut forJoint DarkEnergy Constraints . ... 126 6.7 Progress in Near-term Experiments: PolarBear . .... 129 6.8 Conclusions ................................ 131 Bibliography 134 A Angular Power Spectra: Definitions 156 B Properties of Decorrelated Modes 159 B.1 BasisExpansion ............................. 159 B.2 Basis Dependence of Eigenmodes . 160 B.3 Coordinate Dependence of Eigenmodes . 160 C Model Dependence of Decorrelated Equation of State Modes 163 D Fisher Parameter Bias Formula 165 v List of Figures 1.1 The supernova data used by the Supernova Cosmology Project and the High-z Supernova team to discover cosmic acceleration. Figure taken from[179].................................. 5 1.2 Left: Current constraints on Ωm and ΩΛ when the curvature is allowed to vary. Right: Constraints on a flat model with constant dark energy equationofstate. Figurescourtesyof[14] . 7 1.3 Current constraints from CMB, SNe and BAO on variation in the dark energy equation of state in the form of the linear parametrization w(a)= w0 + wa(1 a). A flat universe is assumed. Figure courtesy of [14]. ....................................− 8 1.4 Detection of the BAO feature in the galaxy correlation function by [132]. 16 2.1 The deviation of the power spectrum of the matter density perturba- tions (Newtonian gauge) from the cs = 1 case is plotted vs. wavenum- ber k. Three regions – above the Hubble scale (small k), below the sound horizon (large k), and the transition in between – can clearly be seen. The models have w = 0.8 (deviations will be smaller for w − closer to 1) and constant sound speed as labeled. For the cs = 0.1 case, we− also show the result (dashed curve) in terms of the gauge invariant variable Dg as defined in [76] (in that work Φ is equal to mi- nus our φ). This illustrates that the low k behavior is strongly gauge dependent. ................................ 25 2.2 The equation of state (lower three curves) and sound speed (upper three curves) as a function of scale factor are illustrated for two models. The aether model takes s = 3 (solid curves) or s = 1 (dashed curves) and w0 = 0.99; the early dark energy density Ωe is determined from these parameters.− Note that