# Workbook

- Page ID
- 5098

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)It has been quite a century for our understanding of the cosmos. As I write these words at the beginning of 2017 it was just over 100 years ago, in November 1915, that Albert Einstein finished development of his general theory of relativity. Among many other things, this theory provided the proper context for interpreting Edwin Hubble's distance-redshift law, published in 1929, as due to the expansion of the Universe. In the 40s, Gamow and Alpher speculated that the dense conditions that must have existed earlier in this expanding universe could provide another site, in addition to the cores of stars, for the fusion of light elements to heavier ones. In fact, to avoid over-production of the elements, this earlier, denser phase would have to be very hot. In the 1960s Bell Labs scientists accidentally stumbled upon the thermal radiation left over from that heat, that exists in the current epoch as a nearly uniform microwave glow. With that discovery, the idea that the universe used to be hot, dense, and expanding very rapidly became the dominant cosmological paradigm known as the "Big Bang."

The subsequent fifty years of the past century saw much progress as well. We now know that we do *not* know what constitutes 95% of the mass/energy in the universe. Only 5% of the mass/energy is composed of constituents in the particle physicist's standard model. Most of the rest is "dark energy" which smoothly fills the universe and dilutes only slowly, if at all, as the universe expands. The rest is "dark matter" that, like George Lucas's mystical Force, "pervades us and binds the galaxy together." Measurements of light element abundances, combined with modern, precision, version's of Gamow and Alpher's big bang nucleosynthesis calculations, give us confidence we understand the expansion back to an epoch when the presently observable universe was \(10^{27}\) times smaller in volume than it is now. A speculative theory, known as cosmic inflation, has met with much empirical success, giving us some level of confidence we may understand something about events at yet higher densities and even earlier times.

In this quarter-long course we will at least touch upon all the topics in the above two paragraphs. We will learn how to think about the expanding universe using concepts from Einstein's theory of general relativity. We will use Newtonian gravity to derive the dynamical equations that relate the expansion rate to the matter content of the universe. Connecting the expansion dynamics to observables such as luminosity distances and redshifts, we will see how astronomers use observations to probe these dynamics, and thereby the contents of the cosmos, including the mysterious dark energy.

We will introduce some basic results of kinetic theory to understand why big bang nucleosynthesis leads to atomic matter that is, by mass, about 25% Hydrogen, 75% Helium with only trace amounts of heavier elements. We'll use this kinetic theory, applied to atomic rather than nuclear reactions, to explore perhaps the most informative cosmological observable: the cosmic microwave background. Finally, we will study how an early epoch of inflationary expansion, driven by an exotic material with negative pressure, can explain some of the otherwise puzzling features of the observed universe.

- Introduction
- We claim to know the composition of the universe at this early time, dominated almost entirely by thermal distributions of photons and subatomic particles called neutrinos. We know in detail many aspects of the evolutionary process that connects this early universe to the current one. Our models of this evolution have been highly predictive and enormously successful.

- Overview
- We provide an overview of our subject, broken into two parts. The first is focused on the discovery of the expansion of the universe in 1929, and the theoretical context for this discovery, which is given by Einstein's general theory of relativity (GR). The second is on the implications of this expansion for the early history of the universe, and relics from that period observable today such as the cosmic microwave background and the lightest chemical elements.

- 1: Euclidean Geometry
- This chapter is entirely focused on the Euclidean geometry that is familiar to you, but reviewed in a language that may be unfamiliar. The new language will help us journey into the foreign territory of Riemannian geometry. Our exploration of that territory will then help you to drop your pre-conceived notions about space and to begin to understand the broader possibilities -- possibilities that are not only mathematically beautiful, but that appear to be realized in nature.

- 2: Curvature
- We introduce the notion of "curvature'' in an attempt to loosen up your understanding of the nature of space, to have you better prepared to think about the expansion of space.

- 3: Galilean Relativity
- We now extend our discussion of spatial geometry to spacetime geometry. We begin with Galilean relativity, which we will then generalize in the next section to Einstein (or Lorentz) relativity.

- 4: Einstein Relativity
- The Maxwell equations are inconsistent with Galilean relativity. Here we review Einstein's solution to this problem, which preserves the principle of relativity, and replaces the Galilean transformation with a Lorentz transformation.

- 5: The Simplest Expanding Spacetime
- We begin our exploration of physics in an expanding spacetime with a spacetime with just one spatial dimension that is not expanding: a 1+1-dimensional Minkowski spacetime. We then generalize it slightly so that the spatial dimension is expanding. After introduction of notions of age and "past horizon," we go on to calculate these quantities for some special cases.

- 6: Redshifts
- We begin to work out observational consequences of living in an expanding spatially homogeneous and isotropic universe. In this and the next two chapters we derive Hubble's Law, \(v = H_0 d\), and a more general version of it valid for arbitrarily large distances.

- 7: Distances as Determined by Standard Candles
- The consequences of expansion are recorded in the relationship between distance and redshift. Here we introduce the so-called standard candle method of distance determination. We work out the theoretical relationship between flux, luminosity, curvature constant k, coordinate distance between observer and source, and the redshift of the source. This moves us one step closer to being able to infer the expansion history from observations.

- 8: The Distance-Redshift Relation
- We complete the work begun in the previous chapter of creating a framework for inferring the expansion history from observations of standard candles over a range of redshifts and distances. We do so by relating, for a given object, the coordinate distance, \(d\), and its redshift \(z\), to the curvature constant and the changing value of the expansion rate between the time the light left the object and our reception of it.

- 9: Dynamics of the Expansion
- In the following set of chapters we will derive the dynamical equations that relate the matter content in a homogeneous and isotropic universe to the evolution of the scale factor over time.

- 10: A Newtonian Homogeneous Expanding Universe
- Retreating to the use of Newtonian concepts, we show that for a universe to be filled with an expanding fluid that remains homogeneous over time, the flow must be what we call a Hubble flow, with relative velocities proportional to distance. Thus we derive Hubble's law using Newtonian concepts, setting ourselves up for the next chapter in which we use Newtonian dynamics to relate the expansion rate to the contents of the cosmos.

- 11: The Friedmann Equation
- Sticking with our Newtonian expanding universe, we will now derive the Friedmann equation that relates how the scale factor changes in time to the mass/energy density. We will proceed by using the Newtonian concept of energy conservation. (You may be surprised to hear me call this a Newtonian concept, but the fact is that energy conservation does not fully survive the transition from Newton to Einstein).

- 12: Particle Kinematics in an Expanding Universe - Newtonian Analysis
- We investigate how an observer, at rest in their local rest frame, will observe the evolution of peculiar velocities of free particles. In a Newtonian analysis, the local rest frame will be an inertial frame (one in which Newton's laws of motion apply) only if there is no acceleration of the scale factor (\ \ddot a = 0\). We discuss the difference with a relativistic analysis. This chapter can be skipped without harming preparation for subsequent chapters.

- 13: The Evolution of Mass-Energy Density and a First Glance at the Contents of the Cosmos
- We have seen that the rate of change of the scale factor depends on the mass density \( \rho \). In order to determine how the scale factor evolves with time, we thus need to know how the density evolves as the scale factor changes, a subject we investigate here.

- 14: Energy and Momentum Conservation
- The lack of energy conservation in an expanding universe is quite surprising to people with any training in physics and therefore merits some discussion, which we present here in this chapter. The student could skip this chapter and proceed to 15 without serious harm. If, subsequently, the lack of energy conservation becomes too troubling, know that this chapter is here for you.

- 15: Pressure and Energy Density Evolution
- We apply local conservation of energy, valid in general relativity, to infer how density changes in response to scale factor changes, a response that depends on the relationship between pressure and density.

- 16: Distance and Magnitude
- There are a bewildering array of different kinds of distances in cosmology. We catalog them here as a resource for you as needed. We also introduce and define other related astronomical technical terms: apparent and absolute magnitudes.

- 17: Parallax, Cepheid Variables, Supernovae, and Distance Measurement
- Key to observing the consequences of this expansion is the ability to measure distances to things that are very far away. Here we cover the basics of how that is done. We have to do it in steps, getting distances to nearby objects and then using those objects to calibrate other objects that can be used to get to even further distances. We refer to this sequence of distance determinations as the distance ladder.

- 18: Cosmological Data Analysis
- We introduce the reader to the exciting subject of generating new knowledge from data, and the process of Bayesian inference in particular. We apply it to the inference of cosmological parameters from data that were reduced from supernova observations.

- 19: The Early Universe
- To understand the "primordial soup" and its relics, we now turn our attention from a relativistic understanding of the curvature and expansion of space, to statistical mechanics. We begin with equilibrium statistical mechanics, before moving on to a discussion of departures from equilibrium. We will study the production in the big bang of helium, photons, other "hot" relics such as neutrinos, and "cold" relics such as the dark matter, and the relevant observations that test our understanding.

- 20: Equilibrium Statistical Mechanics
- Out of the early Universe we get the light elements, a lot of photons and, as it turns out, a bunch of neutrinos and other relics of our hot past as well. To understand the production of these particles we now turn to the subject of equilibrium statistical mechanics.

- 21: Equilibrium Particle Abundances
- At sufficiently high temperatures and densities, reactions that create and destroy particles can become sufficiently rapid that an equilibrium abundance is achieved. In this chapter we assume that such reaction rates are sufficiently high and work out the resulting abundances as a function of the key controlling parameter. We will thus see how equilibrium abundances change as the universe expands and cools.

- 22: Hot and Cold Relics of the Big Bang
- As the temperature and density drops, the reactions necessary to maintain chemical equilibrium can become too slow to continue to do so. This departure from equilibrium can occur while the particles are relativistic, in which case we have "hot relics," or when the particles are non-relativistic in which case we say we have "cold relics." The cosmic microwave and neutrino backgrounds are hot relics. The dark matter may be a cold relic.

- 23: Overview of Thermal History
- This chapter does not yet exist. We intend to include here a summary of the thermal history of the cosmos assuming the standard cosmological model.

- 24: Big Bang Nucleosynthesis - Predictions
- Big Bang Nucleosynthesis is the process by which light elements formed during the Big Bang. The agreement between predicted abundances and inferences from observations of primordial (pre-stellar) abundances is a major pillar of the theory of the hot big bang and reason we can speak with some confidence about events in the primordial plasma in the first few minutes of the expansion. Elements created at these very early times include Deuterium, Helium-3, Lithium-7, and, most abundantly, Helium-4.

- 26: Introduction to the Cosmic Microwave Background
- Predicted in the late 1940s, and discovered accidentally in the 1960s, the Cosmic Microwave Background (CMB) is a cornerstone of the edifice of modern cosmology. We review its discovery and then present the "surface of last scattering"; the thin shell around us, at a distance now of about 46 billion light years, where most of the CMB photons we see today last interacted with matter. We discuss its high degree of isotropy, reflecting the high degree of homogeneity in the early universe.

- 27: Cosmic Microwave Background Anisotropies
- Here we will introduce you to a physical system that is a beautiful gift of nature: the plasma that existed from the first fractions of a second of the Big Bang until it transitioned to a neutral gas 380,000 years later. Gently disturbed away from equilibrium by mysterious, very early-universe processes, the plasma is an unusually simple, natural system whose dynamics are calculable and also observable in maps of CMB intensity and polarization.

- 28: The Spectrum of the CMB
- No other natural source of radiation has ever been measured to be as consistent with black-body radiation as is the case with the CMB. Here we look at the measurements of the spectrum from a Nobel-prize winning instrument on the COBE satellite, before turning to the question of why the CMB is so near to being a black body and what we can learn from that.

- 29: Solving the Wave Equation with Fourier Transforms
- We use Fourier methods to solve for the evolution of \(\Psi(x,t)\) assuming it obeys a wave equation and that we are given appropriate initial conditions. Fourier methods have a broad range of applications in both experimental and theoretical physics, and other sciences as well. For the student of physics, time spent developing facility with the Fourier transform is time well spent.

- 30: The First Few Hundred Thousand Years: The Dynamics of the Primordial Plasma
- We explain the origin of the peaks in the CMB power spectrum as arising from acoustic dynamics in the primordial plasma.

- 31: Structure Formation
- Under the gravitational influence of dark matter, small fluctuations in the matter density field evolve. Particularly dense regions collapse into nonlinear, self-gravitating systems called dark matter halos, which form the nodes of the web of galaxies that cosmologists observe today.

- 32: Galaxy Formation
- After recombination, baryons fall into the gravitational potential wells provided by dark matter halos, beginning the process of star and galaxy formation. This Chapter explores how galaxies form and evolve, and how they relate to dark matter halos.

- A1: Spacetime Geometry
- We begin our exploration of physics in an expanding spacetime with a spacetime with just one spatial dimension that is not expanding: a 1+1-dimensional Minkowski spacetime. We then generalize it slightly so that the spatial dimension is expanding. After introduction of notions of age and "past horizon," we go on to calculate these quantities for some special cases.

- A2: Redshifts
- We begin to work out observational consequences of living in an expanding spatially homogeneous and isotropic universe. In this and the next two chapters we derive Hubble's Law, \(v = H_0 d\), and a more general version of it valid for arbitrarily large distances.

- A3: Spatially Homogeneous and Isotropic Spacetimes
- We begin to work out observational consequences of living in an expanding spatially homogeneous and isotropic universe. In this and the next two chapters we derive Hubble's Law, \(v = H_0 d\), and a more general version of it valid for arbitrarily large distances.

- A4: Euclidean Geometry
- This chapter is entirely focused on the Euclidean geometry that is familiar to you, but reviewed in a language that may be unfamiliar. The new language will help us journey into the foreign territory of Riemannian geometry. Our exploration of that territory will then help you to drop your pre-conceived notions about space and to begin to understand the broader possibilities -- possibilities that are not only mathematically beautiful, but that appear to be realized in nature.

Thumbnail: This is a modification of the Flammarion Woodcut is an enigmatic woodcut by an unknown artist. The woodcut depicts a man peering through the Earth's atmosphere as if it were a curtain to look at the inner workings of the universe. The original caption below the picture (not included here) translated to: "A medieval missionary tells that he has found the point where heaven and Earth meet...".

## Contributor

Lloyd Knox (UC Davis)