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CALTECH PH 136A - Cosmology

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Contents27 Cosmology 127.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.2 Homogeneity and Isotropy of the Universe;Robertson-Walker Line Element . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 The Stress-energy Tensor and the Einstein Field Equation . . . . . . . . . . 927.4 Evolution of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.4.1 Constituents of the universe: Cold matter, radiation, and dark energy 1327.4.2 The vacuum stress-energy tensor . . . . . . . . . . . . . . . . . . . . 1427.4.3 Evolution of the densities . . . . . . . . . . . . . . . . . . . . . . . . 1427.4.4 Evolution in time and redshift . . . . . . . . . . . . . . . . . . . . . . 1727.4.5 Physical processes in the expanding universe . . . . . . . . . . . . . . 2227.5 Observational Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2727.5.1 Parameters characterizing the universe . . . . . . . . . . . . . . . . . 2727.5.2 Local Lorentz frame of homogeneous observers near Earth . . . . . . 2827.5.3 Hubble expansion rate . . . . . . . . . . . . . . . . . . . . . . . . . . 2927.5.4 Primordial nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . 3027.5.5 Density of Cold Dark Matter . . . . . . . . . . . . . . . . . . . . . . 3027.5.6 Radiation Temperature and Density . . . . . . . . . . . . . . . . . . . 3127.5.7 Anisotropy of the CMB: Measurements of the DopplerPeaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3127.5.8 Age of the universe: Constraint on the dar k energy . . . . . . . . . . 3627.5.9 Magnitude-Redshift relation for type Ia supernovae:Confirmation that the universe is accelerating . . . . . . . . . . . . . 3727.6 The Big-Bang Singularity, Quantum Gravity, and the Initial Conditions ofthe Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4027.7 Inflationary Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4227.7.1 Amplification of Primordial Gr avitational Waves by Inflation . . . . . 4327.7.2 Search for Primordial Gravitational Waves by their Influence on theCMB; Probing the Inflationary Expansion Rate . . . . . . . . . . . . 430Chapter 27CosmologyVersion 0627.1.K.pdf, 23 May 2007.Please send comments, sugges tions, and errata via e mail to [email protected] ch.edu or onpaper to Kip Thorne, 130- 33 Caltech, Pasade na CA 9 1125[NOTE: I have done a quick and incomplete revision of this chapter in light of the observa-tional data from the WMAP satellite (Bennett et. al. 20 03). I have not yet had time to dothis carefully. — Kip]Box 27.1Reader’s Guide• This chapter relies significant ly on– The sp ecial relativity portions of Chap. 1.– Chapter 23, on t he transition from special relativity to general relativity.– Chapter 24, on t he fundamental concepts of general relativity.– Sec. 25.3.3 on local energy-momentum conservation for a perfect fluid and Sec.25.6 on the many-fingered nature of time.• In addition, Box 27.3 and Ex. 27.7 of this chapter rely on the Planckian distributionfunction for thermalized photons and its evolution (Liouville’s theorem or collision-less Boltzmann equation), as presented in Secs. 2.2.4, 2.3 , and sec:02EvolutionLawsof Chap. 2.27.1 Ove rviewGeneral Relativity is an indispensable foundation fo r understanding the large scale structureand evolution of the universe (cosmology), but it is only one foundation o ut of many. Thecrudest of understandings can be a chieved with general relativity a nd little else; but moredetailed and deeper understandings require combining general relativity with quantum field12theory, nuclear and atomic physics, thermodynamics, fluid mechanics, and large bodies o fastrophysical lore.In this chapter we shall explore aspects of cosmology which are sufficiently crude that gen-eral relativity, augmented by only bits and pieces of other physics, can provide an adequatefoundation. Our exploration will simultaneously illustrate key aspects of general r elativityand give the reader an overview of modern cosmology.We shall begin in Sec. 27.2 by discussing the observational data that suggest our universeis homogeneous and isotropic when averaged over regions of space huge compared to clustersof galaxies, and we then shall construct a spacetime metric for an idealized homogeneous,isotropic model for the universe. In Sec. 27.3 we shall construct a stress-energy tensor thatdescribes, approximately, the total, averaged energy and pressure of the universe’s matterand radiatio n; and we shall insert that stress-energy tensor and the metric of Sec. 27.2 intothe Einstein field equation, thereby deducing a set of equations that govern t he evolution ofthe universe. In Sec. 27.4 we shall study the predictions that those evolution equations makefor the rate of expansion o f the universe and the manner in which the expansion changeswith time, and we shall describe the most important physical processes that have occuredin the universe during its evolution into its present state. As we shall see, the details of theexpansion are determined by the values of seven parameters that can b e measured today—with the caveat that there may be some big surprises associated with the so-called darkenergy. In Sec. 27 .5 we shall describe the astronomical observations by which the universe’sseven parameters are being measured, and the multifaceted evidence fo r dark energy. InSec. 27.6 we shall discuss the big-bang singularity in which the universe probably began,and shall discuss the fact that this singularity, like singularities inside black holes, is a signalthat general relativity breaks down and must be replaced by a quantum theory of gravitywhich (hopefully) will not predict singular behavior. We shall also examine a few featuresthat the quantum theory of gravity is likely to exhibit. Finally, in Sec. 27.7 we shall discussthe “inflationary” epoch that the universe appears to have …


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CALTECH PH 136A - Cosmology

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