Cosmology OXFORD. Steven Weinberg. University of Texas at Austin UNIVERSITY PRESS

Transcription

1 Cosmology Steven Weinberg University of Texas at Austin OXFORD UNIVERSITY PRESS

2 1 THE EXPANSION OF THE UNIVERSE Spacetime geometry 2 Robertson Walker metric 0 Co-moving coordinates q Proper distances q Momentum decay q Spatial geodesics q Number conservation q Energy & momentum conservation q Cold matter, hot matter, vacuum energy q Global geometry & topology 1.2 The cosmological redshift 10 Emission time vs. radial coordinate q Redshifts blueshifts q Hubble constant q Discovery of expansion q Changing redshifts 1.3 Distances at small redshift: The Hubble constant 13 Trigonometric parallax q Proper motions q Apparent luminosity: Main sequence, red clump stars, RR Lyrae stars, eclipsing binaries, Cepheid variables q Tully Fisher relation q Faber Jackson relation q Fundamental Plane q Type Ia supernovae q Surface brightness fluctuations q Result for Hubble constant 1.4 Luminosity distances and angular diameter distances 31 Luminosity distance q Deceleration parameter q Jerk & snap q Angular diameter distance 1.5 Dynamics of expansion 34 Einstein field equations q Friedmann equation q Newtonian derivation q Critical density q Flatness problem q Matter-dominated expansion q Radiation-dominated expansion q Vacuum-dominated expansion q De Sitter model q S2m, nr, S2A q Age of expansion q Luminosity distance formula q Future expansion q Historical note: cosmological constant 0 Historical note: steady state model 1.6 Distances at large redshifts: Accelerated expansion 45 Discovery of accelerated expansion q Newtonian interpretation q Gray dust? q Discovery of early deceleration q Other effects q Equation of state parameter w q X-ray observations q The cosmological constant problems 1.7 Cosmic expansion or tired light? 57 Surface brightness test q Supernova decline slowdown 1.8 Ages 59 Heavy element abundance q Main sequence turn-off q Age vs redshift xi

3 1.9 Masses 65 Virialized clusters of galaxies: X-ray luminosity of clusters of galaxies: S2BIS2m Intergalactic absorption 75 Optical depth q Resonant absorption q 21 cm absorption q Lyman a absorption q Gunn Peterson trough q Alcock Paczyliski analysis 1.11 Number counts 82 Number vs. z and Evolution q Radio source surveys 1.12 Quintessence 89 Scalar field theories q Power-law potential q Tracker solution q Twoparameter models 1.13 Horizons 98 Particle horizon q Event horizon 2 THE COSMIC MICROWAVE RADIATION BACKGROUND Expectations and discovery of the microwave background 101 Black body radiation q Early suggestions q Discovery q Rayleigh Jeans formula q CN absorption lines q Balloons & rockets q COBE & FIRAS q Energy density q Number density q Effect an cosmic rays 2.2 The equilibrium era 109 Entropy per baryon q Radiation matter equality q Energy decoupling 2.3 Recombination and last scattering 113 Maxwell Boltzmann distribution q Saha formula q Delay of n = 2 to n= 1 q Peebles analysis q Lyman a escape probability q Rate equation q Fractional ionization q Opacity q Jones Wyse approximation 2.4 The dipole anisotropy 129 Angelar dependence of temperature q U2 discovery q COBE & WMAP measurements q Kinematic quadrupole 2.5 The Sunyaev Zel'dovich effect 132 Kompaneets equation q Spectrum shift q Use with X-ray luminosity 2.6 Primary fluctuations in the microwave background: A first look 135 Partial-wave coefficients Multipole coefficients Q q Cosmic variance q Sachs Wolfe effect q Harrison Zel'dovich spectrum q Doppler fluctuations q Intrinsic temperature fluctuations q Integrated Sachs Wolfe effect q COBE observations xii

4 3 THE EARLY UNIVERSE Thermal history 149 Entropy density q Fermi Dirac & Bose Einstein distributions q Time vs. temperature q Effective number of species q Neutrino decoupling q Heating by electron positron annihilation q Neutrino masses & chemical potentials 3.2 Cosmological nucleosynthesis 159 Neutron proton conversion q Equilibrium nuclear abundances q Deuterium bottleneck q Helium abundance q Deuterium abundance q Hei abundance q Lithium abundance q S2Bh2 3.3 Baryonsynthesis and Leptonsynthesis 173 Sakharov conditions q Delayed decay q Electroweak nonconservation q Leptogenesis q Affleck Dine mechanism q Equilibrium baryonsynthesis 3.4 Cold dark matter 185 The bullet cluster q Leftover WIMP abundance q Sparticles q WIMP searches q Annihilation y rays q Axions & axinos 4 INFLATION Three puzzles 202 Flatness q Horizons q Monopoles 4.2 Slow-roll inflation 208 Bubble formation q New inflation q Slow-roll conditions q Power-law potential q Exponential potential q Reheating 4.3 Chaotic inflation, eternal inflation 216 Condition for eternal inflation q Condition for chaotic inflation 5 GENERAL THEORY OF SMALL FLUCTUATIONS Field equations 219 Perturbed Ricci tensor q Perturbed energy-momentum tensor q Scalar modes q Vector modes q Tensor modes 5.2 Fourier decomposition and stochastic initial conditions 228 Plane wave solutions q Stochastic parameters q Correlation functions q Helicity decomposition 5.3 Choosing a gauge 235 Gauge transformations q Newtonian gauge q Synchronous gauge q Conversion q Other gauges

5 5.4 Conservation outside the horizon 245 The quantities 7.Z. and q A conservation theorem q Conservation for isolated components 6 EVOLUTION OF COSMOLOGICAL FLUCTUATIONS Scalar perturbations kinetic theory 258 Cold dark matter q Baryonic plasma q Photon number density matrix perturbation Photon dimensionless intensity matrix Photon Boltzmann equations q Photon source functions q Photon pressure, density, anisotropic inertia q Photon line-of-sight solutions q Neutrino number density perturbation 8n q Neutrino dimensionless intensity J q Neutrino Boltzmann equations q Neutrino pressure, density, anisotropic inertia q Neutrino line-of-sight solutions q Gravitational field equations q Initial conditions 6.2 Scalar perturbations the hydrodynamic limit 274 Hydrodynamic & field equations q Adiabatic initial conditions q Nonadiabatic modes q Long & short wavelengths 6.3 Scalar perturbations long wavelengths 282 Evolution far outside horizon q Evolution in matter-dominated era 6.4 Scalar perturbations short wavelengths 289 Evolution in radiation-dominated era q Evolution deep inside horizon q Fast & slow modes q Matching 6.5 Scalar perturbations interpolation & transfer functions 303 Exact solution for pb = 0 q Transfer functions q Baryon density & damping effects 6.6 Tensor perturbations 312 Gravitational field equations q Photon Boltzmann equations q Photon source functions q Photon anisotropic inertia q Photon line-of-sight solution q Neutrino Boltzmann equations q Neutrino anisotropic inertia q Neutrino line-of-sight solutions q Evolution without damping q Transfer functions q Effect of damping 7 ANISOTROPIES IN THE MICROWAVE SKY General formulas for the temperature fluctuations 329 Line-of-sight formula q Rearrangement of scalar temperature fluctuation q Integrated Sachs Wolfe effect q Sudden decoupling approximation q Re-derivation following photon trajectories q Gauge invariance xiv

6 7.2 Temperature multipole coefficients: Scalar modes 343 General formula q Large approximation q Calculation of form factors q Silk & Landau damping q Comparison with numerical codes q Balloon & ground-based observations q WMAP q Results for cosmological parameters 7.3 Temperature multipole coefficients: Tensor modes 362 General formula q Calculation of gravitational wave amplitude q Calculation of source function q Large approximation q Sudden decoupling approximation q Numerical results 7.4 Polarization 370 Stokes parameters q Spherical harmonics of spin ±2 q Space-inversion properties q E and B polarization q Scalar modes: general formula q Scalar modes: large approximation q Scalar modes: numerical results q Scalar modes: observations q Tensor modes: general formula q Tensor modes: large approximation q Tensor modes: numerical results q Correlation functions 8 THE GROWTH OF STRUCTURE Linear perturbations after recombination 403 Hydrodynamic and field equations q Factorization of perturbations q Effect of vacuum energy q Power spectral function P(k) q Correlation function q Direct measurement of P(k) q Rms fluctuation ar q Measurements of P(k) q Baryon acoustic oscillations q Cosmic variance in measuring P(k) 8.2 Nonlinear growth 421 Spherically symmetric collapse q Calculation of criz q Press Schechter mass function 8.3 Collapse of baryonic matter 427 Jeans mass q Continuity & Euler equations q Power-law solutions q Critical wave number for baryon collapse 9 GRAVITATIONAL LENSES Lens equation for point masses 433 Derivation of lens equation q Image separation q Einstein ring 9.2 Magnification: Strong lensing and microlensing 436 Image luminosity q Conservation of surface brightness q Effective radius for strong lensing q Number counts q De Sitter model q Einstein de Sitter model q Lens survey q Microlensing observations xv

7 9.3 Extended lenses 443 Isothermal spheres q Lens equation q Lens luminosity q Number counts q Surveys 9.4 Time delay 447 Geometrical delay q Potential delay q Observations 9.5 Weak lensing 452 Calculation of deflection q Shear matrix q Ellipse matrix q Mean shear matrix q Shear field IC q Multipole coefficients q Large approximation q Measurement of P(k) q Correlation functions q Shear surveys 9.6 Cosmic strings 467 Calculation of deflection q A string suspect 10 INFLATION AS THE ORIGIN OF COSMOLOGICAL FLUCTUATIONS Scalar fluctuations during Inflation 470 Scalar field action q Field, density, pressure, and velocity perturbations q Field equations q WKB early-time solution q Fourier decomposition q Commutation relations q Bunch Davies vacuum q Gaussian statistics q Curvature perturbation 7?, q Mukhanov Sasaki equation q Limit Rq outside horizon q Number of e-foldings after horizon exit q Exponential potential q Measurement of spectral index & fluctuation strength q Values of exponential potential parameters q Justification of simple action 10.2 Tensor fluctuations during inflation 485 Gravitational field equation q WKB early-time solution q Fourier decomposition q Commutation relations q Scalar/tensor ratio r q Observational bounds on r 10.3 Fluctuations during inflation: The slow-roll approximation 488 Parameters e and Slow-roll approximation q Spectral index and fluctuation strength q Observational constraints on potential q Number of e-foldings after horizon exit 10.4 Multifield inflation 497 Gaussian, adiabatic, scale-invariant, & weak fluctuations q Thermal equilibrium after inflation q Evolution equations q WKB early-time solution q Vielbeins q Commutation relations q Slow-roll conditions q R. after horizon exit q What we have learned about inflation xvi

8 APPENDICES A. Some Useful Numbers 509 B. Review of General Relativity 511 C. Energy Transfer between Radiation and Electrons 531 D. The Ergodic Theorem 537 E. Gaussian Distributions 541 F. Newtonian Cosmology 543 G. Photon Polarization 547 H. The Relativistic Boltzmann Equation 551 GLOSSARY OF SYMBOLS 565 ASSORTED PROBLEMS 569 AUTHOR INDEX 575 SUBJECT INDEX 587 xvii