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This book is unique in the detailed, self-contained, and comprehensive treatment that it gives to the ideas and formulas that are used and tested in modern cosmological research. It divides into two parts, each of which provides enough material for a one-semester graduate course. The first part deals chiefly with the isotropic and homogeneous average universe; the second part concentrates on the departures from the average universe. Throughout the book the author presents detailed analytic calculations of cosmological phenomena, rather than just report results obtained elsewhere by numerical computation. The book is up to date, and gives detailed accounts of topics such as recombination, microwave background polarization, leptogenesis, gravitational lensing, structure formation, and multifield inflation, that are often treated superficially if at all in treatises on cosmology. Copious references to current research literature are supplied. Appendices include a brief introduction to general relativity, and a detailed derivation of the Boltzmann equation for photons and neutrinos used in calculations of cosmological evolution. Also provided is an assortment of problems.
Steven Weinberg is a member of the Physics and Astronomy Departments of the University of Texas at Austin. His research has been honored with the Nobel Prize in Physics and the National Medal of Science, and election to the US National Academy of Sciences, Britain's Royal Society, the Royal Irish Academy, the American Philosophical Society and the American Academy of Arts and Sciences. The American Philosophical Society awarded him the Benjamin Franklin Medal, with a citation that said he is "considered by many to be the preeminent theoretical physicist alive in the world today." His books include the three volume treatise, The Quantum Theory of Fields. Educated at Cornell, Copenhagen, and Princeton, he also holds honorary doctoral degrees from sixteen other universities. He taught at Columbia, Berkeley, M.I.T., and Harvard, where he was Higgins Professor of Physics, before coming to Texas in 1982.
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CONTENTS
1 THE EXPANSION OF THE UNIVERSE 1
1.1 Spacetime geometry 2
Robertson-Walker metric □ Co-moving coordinates □ Proper distances □ Momentum decay □ Spatial geodesies □ Number conservation □ Energy & momentum conservation □ Cold matter, hot matter, vacuum energy □ Global geometry & topology
1.2 The cosmological redshift 10
Emission time vs. radial coordinate □ Redshifts & blueshifts □ Hubble constant □ Discovery of expansion □ Changing redshifts
1.3 Distances at small redshift: The Hubble constant 13
Trigonometric parallax □ Proper motions □ Apparent luminosity: Main sequence, red clump stars, RR Lyrae stars, eclipsing binaries, Cepheid variables □ Tully-Fisher relation □ Faber-Jackson relation □ Fundamental Plane □ Type la supernovae □ Surface brightness fluctuations □ Result for Hubble constant
1.4 Luminosity distances and angular diameter distances 31
Luminosity distance □ Deceleration parameter □ Jerk & snap □ Angular diameter distance
1.5 Dynamics of expansion 34
Einstein field equations □ Friedmann equation □ Newtonian derivation □ Critical density □ Flatness problem □ Matter-dominated expansion □ Radiation-dominated expansion □ Vacuum-dominated expansion □ De Sitter model □ QM, &A d Age of expansion □ Luminosity distance formula □ Future expansion □ Historical note: cosmological constant □ Historical note: steady state model
1.6 Distances at large redshifts: Accelerated expansion 45
Discovery of accelerated expansion □ Newtonian interpretation □ Gray dust? □ Discovery of early deceleration □ Other effects □ Equation of state parameter w □ X-ray observations □ The cosmological constant problems
1.7 Cosmic expansion or tired light? 57
Surface brightness test □ Supernova decline slowdown
1.8 Ages 59
Heavy element abundance □ Main sequence turn-off □ Age vs. redshift
1.9 Masses 65
Virialized clusters of galaxies: QM d X-ray luminosity of clusters of galaxies: QB/^M
1.10 Intergalactic absorption 75
Optical depth □ Resonant absorption □ 21 cm absorption □ Lyman a absorption □ Gunn-Peterson trough □ Alcock-Paczyhski analysis
1.11 Number counts 82
Number vs. z and I □ Evolution □ Radio source surveys
1.12 Quintessence 89
Scalar field theories □ Power-law potential □ Tracker solution □ Two-parameter models
1.13 Horizons 98
Particle horizon □ Event horizon
2 THE COSMIC MICROWAVE RADIATION BACKGROUND 101
2.1 Expectations and discovery of the microwave background 101
Black body radiation □ Early suggestions □ Discovery □ Rayleigh-Jeans formula □ CN absorption lines □ Balloons & rockets □ COBE & FIRAS □ Energy density □ Number density □ Effect on cosmic rays
2.2 The equilibrium era 109
Entropy per baryon □ Radiation-matter equality □ Energy decoupling
2.3 Recombination and last scattering 113
Maxwell-Boltzmann distribution □ Saha formula □ Delay of n = 2 to n = 1 □ Peebles analysis □ Lyman a escape probability □ Rate equation □ Fractional ionization □ Opacity □ Jones-Wyse approximation
2.4 The dipole anisotropy 129
Angular dependence of temperature □ U2 discovery □ COBE & WMAP measurements □ Kinematic quadrupole
2.5 The Sunyaev-ZePdovich effect 132
Kompaneets equation □ Spectrum shift □ Use with X-ray luminosity
2.6 Primary fluctuations in the microwave background: A first look 135
Partial-wave coefficients atm □ Multipole coefficients Q □ Cosmic variance □ Sachs-Wolfe effect □ Harrison-Zel'dovich spectrum □ Doppler fluctuations □ Intrinsic temperature fluctuations □ Integrated Sachs-Wolfe effect □ COBE observations
3 THE EARLY UNIVERSE 149
3.1 Thermal history 149
Entropy density □ Fermi-Dirac & Bose-Einstein distributions □ Time vs. temperature □ Effective number of species □ Neutrino decoupling □ Heating by electron-positron annihilation □ Neutrino masses & chemical potentials
3.2 Cosmological nucleosynthesis 159
Neutron-proton conversion □ Equilibrium nuclear abundances □ Deuterium bottleneck □ Helium abundance □ Deuterium abundance □ He3 abundance □ Lithium abundance □ f2#/z2
3.3 Baryonsynthesis and Leptonsynthesis 173
Sakharov conditions □ Delayed decay □ Electroweak nonconservation □ Leptogenesis □ Affleck-Dine mechanism □ Equilibrium baryonsynthesis
3.4 Cold dark matter 185
The bullet cluster □ Leftover WIMP abundance □ Sparticles □ WIMP
searches □ Annihilation y rays □ Axions & axinos
4 INFLATION 201
4.1 Three puzzles 202
Flatness □ Horizons □ Monopoles
4.2 Slow-roll inflation 208
Bubble formation □ New inflation □ Slow-roll conditions □ Power-law potential □ Exponential potential □ Reheating
4.3 Chaotic inflation, eternal inflation 216
Condition for eternal inflation □ Condition for chaotic inflation
5 GENERAL THEORY OF SMALL FLUCTUATIONS 219
5.1 Field equations 219
Perturbed Ricci tensor □ Perturbed energy-momentum tensor □ Scalar modes □ Vector modes □ Tensor modes
5.2 Fourier decomposition and stochastic initial conditions 228
Plane wave solutions □ Stochastic parameters □ Correlation functions □ Helicity decomposition
5.3 Choosing a gauge 235
Gauge transformations □ Newtonian gauge □ Synchronous gauge □ Conversion □ Other gauges
5.4 Conservation outside the horizon 245
The quantities 1Z and f □ A conservation theorem □ Conservation for isolated components
6 EVOLUTION OF COSMOLOGICAL FLUCTUATIONS 257
6.1 Scalar perturbations - kinetic theory 258
Cold dark matter □ Baryonic plasma □ Photon number density matrix perturbation 8n^ □ Photon dimensionless intensity matrix Jy □ Photon Boltzmann equations □ Photon source functions □ Photon pressure, density, anisotropic inertia □ Photon line-of-sight solutions □ Neutrino number density perturbation 8nv □ Neutrino dimensionless intensity / □ Neutrino Boltzmann equations □ Neutrino pressure, density, anisotropic inertia □ Neutrino line-of-sight solutions □ Gravitational field equations □ Initial conditions
6.2 Scalar perturbations - the hydrodynamic limit 274
Hydrodynamic & field equations □ Adiabatic initial conditions □ Non-adiabatic modes □ Long & short wavelengths
6.3 Scalar perturbations - long wavelengths 282
Evolution far outside horizon □ Evolution in matter-dominated era
6.4 Scalar perturbations - short wavelengths 289
Evolution in radiation-dominated era □ Evolution deep inside horizon □ Fast & slow modes □ Matching
6.5 Scalar perturbations - interpolation & transfer functions 303
Exact solution for ps — 0 □ Transfer functions □ Baryon density & damping effects
6.6 Tensor perturbations 312
Gravitational field equations □ Photon Boltzmann equations □ Photon source functions □ Photon anisotropic inertia □ Photon line-of-sight solution □ Neutrino Boltzmann equations □ Neutrino anisotropic inertia □ Neutrino line-of-sight solutions □ Evolution without damping □ Transfer functions □ Effect of damping
7 ANISOTROPIES IN THE MICROWAVE SKY 329
7.1 General formulas for the temperature fluctuations 329
Line-of-sight formula □ Rearrangement of scalar temperature fluctuation □ Integrated Sachs-Wolfe effect □ Sudden decoupling approximation □ Re-derivation following photon trajectories □ Gauge invariance
7.2 Temperature multipole coefficients: Scalar modes 343
General formula □ Large I approximation □ Calculation of form factors □ Silk & Landau damping □ Comparison with numerical codes □ Balloon & ground-based observations □ WMAP □ Results for cosmological parameters
7.3 Temperature multipole coefficients: Tensor modes 362
General formula □ Calculation of gravitational wave amplitude □ Calculation of source function □ Large I approximation □ Sudden decoupling approximation □ Numerical results
7.4 Polarization 370
Stokes parameters □ Spherical harmonics of spin ±2 □ Space-inversion properties □ E and B polarization □ Scalar modes: general formula □ Scalar modes: large I approximation □ Scalar modes: numerical results □ Scalar modes: observations □ Tensor modes: general formula □ Tensor modes: large £ approximation □ Tensor modes: numerical results □ Correlation functions
8 THE GROWTH OF STRUCTURE 403
8.1 Linear perturbations after recombination 403
Hydrodynamic and field equations □ Factorization of perturbations □ Effect of vacuum energy □ Power spectral function P(k) □ Correlation function □ Direct measurement of P(k) □ Rms fluctuation OR □ Measurements of P{k) □ Baryon acoustic oscillations □ Cosmic variance in measuring P(k)
8.2 Nonlinear growth 421
Spherically symmetric collapse □ Calculation of OR □ Press-Schechter mass function
8.3 Collapse of baryonic matter 427
Jeans mass □ Continuity & Euler equations □ Power-law solutions □ Critical wave number for baryon collapse
9 GRAVITATIONAL LENSES 433
9.1 Lens equation for point masses 433
Derivation of lens equation □ Image separation □ Einstein ring
9.2 Magnification: Strong lensing and microlensing 436
Image luminosity □ Conservation of surface brightness □ Effective radius for strong lensing □ Number counts □ De Sitter model □ Einstein-de Sitter model □ Lens survey □ Microlensing observations
9.3 Extended lenses 443
Isothermal spheres □ Lens equation □ Lens luminosity □ Number counts □ Surveys
9.4 Time delay 447
Geometrical delay □ Potential delay □ Observations
9.5 Weak lensing 452
Calculation of deflection □ Shear matrix □ Ellipse matrix □ Mean shear matrix □ Shear field K □ Multipole coefficients □ Large t approximation □ Measurement of P(k) □ Correlation functions □ Shear surveys
9.6 Cosmic strings 467
Calculation of deflection □ A string suspect
10 INFLATION AS THE ORIGIN OF COSMOLOGICAL FLUCTUATIONS 469
10.1 Scalar fluctuations during inflation 470
Scalar field action □ Field, density, pressure, and velocity perturbations □ Field equations □ WKB early-time solution □ Fourier decomposition □ Commutation relations □ Bunch-Davies vacuum □ Gaussian statistics □ Curvature perturbation 1Z □ Mukhanov-Sasaki equation □ Limit 1Z°Q outside horizon □ Number of e-foldings after horizon exit □ Exponential potential □ Measurement of spectral index & fluctuation strength □ Values of exponential potential parameters □ Justification of simple action
10.2 Tensor fluctuations during inflation 485
Gravitational field equation □ WKB early-time solution □ Fourier decomposition □ Commutation relations □ Scalar/tensor ratio r □ Observational bounds on r
10.3 Fluctuations during inflation: The slow-roll approximation 488
Parameters e and 8 □ Slow-roll approximation □ Spectral index and fluctuation strength □ Observational constraints on potential □ Number of e-foldings after horizon exit
10.4 Multifield inflation 497
Gaussian, adiabatic, scale-invariant, & weak fluctuations □ Thermal equilibrium after inflation □ Evolution equations □ WKB early time solution □ Vielbeins □ Commutation relations □ Slow-roll conditions □ 1Z after horizon exit □ What we have learned about inflation
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
