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The Time Evolution of the Cosmic Microwave Background Photosphere THE TIME EVOLUTION OF THE COSMIC MICROWAVE BACKGROUND PHOTOSPHERE Stuart R. Lange Advisor: Lyman Page Reader: Paul Steinhardt A senior thesis submitted in partial fulfillment of the Artium Baccalaureus in Physics Princeton University April 23, 2007 To Mom and Dad i PLEDGE This paper represents my own work in accordance with University regulations. Stuart R. Lange ii ACKNOWLEDGMENTS I would like to express my sincere gratitude to Professor Lyman Page, whose masterful teaching style in introductory Physics drew me into the field, and whose care and guidance made this project an incredibly rewarding experience. Professor Page deserves the credit for the initial concept for this project, and his vast knowledge of the field kept me on track as I worked through the theory and its implementation. My explanation of the background material (chapters 1-4) relies heavily on Scott Dodelson’s Modern Cosmology (2003). Although I have cited Dodelson’s work numerous times throughout this paper, I here acknowledge its influence on the overall organization and structure of this document and on the background derivations contained therein. This project required an extensive amount of programming, much of which was built upon pre- existing code. The central implementation in this paper is an extension of the existing CAMB (Code for Anisotropies in the Microwave Background) framework,1 by by Antony Lewis and Anthony Challinor (2000). CAMB, in turn, is based on CMBFAST,2 by Uros Seljak and Matias Zaldarriaga (1996). In addition, some of the results in this paper have been derived using the HEALPix3 (Górski et al., 2005) package and the CFITSIO4 (Hanisch et al., 2001) library. Finally, I would like to thank my many friends and family members who have supported me on this project and throughout my time at Princeton. Thank you for making my experience over the last four years so wonderful. Special thanks are due to the Princeton University Band, my true home away from home, for making it all worthwhile. Three cheers for Old Nassau! 1http://camb.info/ 2http://cfa-www.harvard.edu/ mzaldarr/CMBFAST/cmbfast.html 3http://healpix.jpl.nasa.gov/healpixSoftwareGetHealpix.shtml 4http://heasarc.nasa.gov/docs/software/fitsio/fitsio.html iii ABSTRACT The Time Evolution of the Cosmic Microwave Background Photosphere Stuart R. Lange The rapid advance in precision measurements of the cosmic microwave background (CMB), which is exemplified by the recent WMAP results (Spergel et al., 2006), gives us unprecedented ability to consider novel questions about the nature of the CMB. This thesis presents a method to model the evolution of the CMB in time from the point of view of a stationary observer. This model requires extensions both to the existing theory of CMB modeling and the existing implementation. The model has the ability to create representative visualizations of the evolution of the CMB over several time steps. Using the model, we are able to determine that the CMB power spectrum shifts to smaller scales as time progresses, that the late-time ISW effect causes the low-` tail to eventually overtake the first acoustic peak in a ΛCDM universe, and that the parameters required of an experiment designed to measure the time evolution effect cannot be attained in the near future. In addition to presenting the model and the theory behind it, this thesis reviews CMB cosmology from basic principles through the derivation of the line of sight integration approach for CMB modeling. iv CONTENTS 1 Introduction to CMB Cosmology 1 1.1 The Expanding Universe . 1 1.2 A Notational Interlude . 3 1.2.1 Conformal Time . 3 1.2.2 Redshift . 4 1.3 Implications of Expansion . 5 1.4 The Horizon Problem . 8 1.5 Anisotropies . 9 1.6 CMB Measurement . 12 1.7 The Next Step . 14 2 Spacetime, Geometry and Isotropy 16 2.1 Components of the Universe . 16 2.1.1 Photons . 17 2.1.2 Baryons . 19 2.1.3 Dark Matter . 19 2.1.4 Dark Energy . 20 2.2 The Metric . 21 2.2.1 Four-vectors and Special Relativity . 21 2.2.2 The Metric in an Expanding Universe . 22 2.2.3 A Word on Curvature . 23 2.3 A Mathematical Interlude . 23 2.3.1 Vectors, One-Forms, and Tensors . 23 2.3.2 Particle Motion . 25 2.3.3 Application to the FRW Metric . 26 2.4 The Einstein Equation . 27 2.4.1 The Stress-Energy Tensor . 27 2.4.2 The Field Equations . 28 2.4.3 Application to the FRW Universe . 28 2.5 Rates, Times and Distances in the FRW Universe . 31 2.5.1 Evolution of the Hubble Rate . 31 2.5.2 Time Calculations . 32 2.5.3 Distance Calculations . 33 3 The Perturbed Universe 35 3.1 Preliminaries . 35 3.1.1 The Metric . 35 3.1.2 The Christoffel Symbol . 36 v CONTENTS vi 3.2 The Boltzmann Equation for Photons . 37 3.2.1 Collisionless Form for an Arbitrary Distribution . 37 3.2.2 Application to a Perturbed Distribution Function . 39 3.2.3 Collision Terms . 41 3.2.4 Putting it All Together . 42 3.3 Additional Equations . 44 4 The CMB Spectrum 47 4.1 The Line of Sight Integration Approach . 47 4.2 Calculating the Power Spectrum . 51 4.3 The Power Spectrum and the Cosmological Parameters . 54 5 Modeling CMB Evolution 58 5.1 Extension of CMB Theory . 58 5.1.1 The C` Covariance Matrix . 58 5.1.2 Measuring the Effect . 60 5.1.3 From C` to a Representative a`m .......................... 62 5.2 Implementation . 63 5.2.1 The Original CAMB Framework . 63 5.2.2 Alterations to CAMB . 65 5.2.3 Additional Programs . 68 6 Results 69 6.1 Power Spectrum Analysis of Late-Time Effects . 70 6.2 Evolution of the Sky Map . 74 6.3 Difference Analysis of Near-Time Effects . 78 6.4 Sanity Checks . 80 6.4.1 The Covariance Matrix . 80 6.4.2 Recalculated Spectra . 82 6.5 A Theoretical Experiment . 84 CHAPTER 1 INTRODUCTION TO CMB COSMOLOGY Recent experiments researching the cosmic microwave background radiation (CMB), such as the Wilkinson Microwave Anisotropy Probe (WMAP),1 have increased our understanding of the uni- verse enormously. By studying this faint afterglow of the big bang, these experiments are able to determine important characteristics of the universe — such as its age, expansion rate, density, and composition — to high precision. This rapid increase in knowledge allows us to consider more novel theoretical questions regarding the CMB itself. In this thesis, I propose to tackle such a question — How does the visible CMB evolve in time from the point of view of a stationary observer? A complete answer to this question requires an understanding of the equations governing the evolution of the components of the universe. This chapter aims to begin the development of that understanding by introducing the basic principles of cosmological physics. 1.1 THE EXPANDING UNIVERSE Concepts such as “The Big Bang” and “The Expanding Universe” are part of common parlance today, but the scale and origins of the universe were hotly debated in the early and middle part of the twentieth century. In fact, the 1920 discussion between astronomers Harlow Shapley and Heber Curtis concerning the nature and scale of the Milky Way galaxy became known as the Great Debate.2 The subject of the debate was whether the object Andromeda and other similar “spiral nebulae”, as galaxies were then known, are part of our Milky Way galaxy or rather individual galaxies in their own right, separated from our own by incredible distances. Shapley argued the former point, adding that our Milky Way galaxy itself is of quite large scale, and that our solar system is not located near its center. Curtis countered, placing the solar system near the center of a relatively small Milky Way galaxy that was one of many galaxies in the larger universe. With the benefit of hindsight, we know that Shapley was right about the scale of the Milky Way and our location within it, while Curtis was right about the nature of the “spiral nebulae”. The Great Debate was not immediately resolved, but the case for a universe consisting of mul- tiple separate galaxies was greatly furthered by Edwin Hubble’s work of the mid-1920s, in which he measured the distances to several nearby “spiral nebulae”, including Andromeda, by observing Cepheid variable stars within the nebulae.3 His early measurements of Andromeda itself (Hubble, 1929b) revealed it to be much more distant than the maximum extent of the Milky Way galaxy proposed by Shapley, proving that the “spiral nebulae” are in fact distinct galaxies. 1http://lambda.gsfc.nasa.gov/product/map/current/ 2The proceedings of the debate were published a year later (Shapley and Curtis, 1921). For a historical description of the event, see Trimble (1995). 3A Cepheid variable star is a standard candle – an object that shines with a known luminosity. Combining this known luminosity with the observed luminosity of the object, the inverse square law gives the distance. 1 CHAPTER 1. INTRODUCTION TO CMB COSMOLOGY 2 The earlier work of Vesto Slipher (1915) documented the relative velocities of many nearby galaxies, via measurements of their spectral shift. Slipher found that eleven of the fifteen nebulae he studied had redshifted spectra, implying that they are moving away from us.4 Hubble com- bined this work with his own distance measurements (Hubble, 1929a), revealing a simple linear relationship: v = H0r (1.1) where v is the recessional velocity of galaxy, and r is its distance. The constant H0, which became known as Hubble’s constant, is traditionally measured in kilometers per second per megaparsec.
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