# A Cosmology Workbook

- Page ID
- 5098

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.

- 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: the cosmic microwave background and the lightest chemical elements. The consistency of such observations with

- 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
- While the principle of relativity holds, its specific implementation as Galilean relativity does not. As you know, because you have studied special relativity, this is indeed the correct solution to the puzzle of the Maxwell Equations lack of invariance under a Galilean transformation.

- 5: The Simplest Expanding Spacetime
- In this chapter we begin our exploration of physics in an expanding spacetime. We start with a spacetime with just one spatial dimension that is not expanding: a 1+1-dimensional Minkowski spacetime. We then generalize it slightly to describe a spacetime with one spatial dimension that is expanding. With additional assumptions we then calculate the age of this spacetime as well as the "past horizon."

- 6: Redshifts
- We begin to work out observational consequences of living in an expanding spatially homogeneous and isotropic universe.

- 7: Distances as Determined by Standard Candles
- Measuring the flux (energy/unit time/unit area) give us a way to figure out the distance to the object assuming we know its luminosity and that it is emitting isotropically. This is the so-called standard candle method of distance determination. Here we work out the theoretical relationship between flux, luminosity, curvature constant k, coordinate distance between observer and source, and the redshift of the source.

- 8: The Distance-Redshift Relation
- For a given scale factor history, a(t) , one can work out a relationship between luminosity distance and redshift. This will be useful to us because it shows how we can infer a(t) from measurements of luminosity distance and redshift, over a range of redshifts.

- 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
- Consider a homogeneous expanding universe in a Newtonian manner, where space is fixed, and the universe is filled with a fluid that is moving in such a manner as to keep the density spatially uniform. If homogeneity is to be preserved over time, then the motion must be such that the separation between any pair of fluid elements must scale up with time in the same way.

- 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 want to understand 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 acceleration is a constant.

- 13: The Evolution of Mass-Energy Density and a First Glance at the Contents of the Cosmos
- We've 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.

- 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
- There is a sense in which energy is conserved in general relativity. We say it is locally conserved, which effectively means that in a sufficiently small region of spacetime, the change in energy is equal to the flux of energy across the boundary of the region, including that via any work being done on the region.

- 16: Distance and Magnitude
- We have the invariant distance equation for a homogeneous and isotropic universe. We now introduce no fewer than five kinds of spatial distances: Coordinate distance, Physical distance, Comoving distance, Luminosity distance, and Comoving angular diameter distance.

- 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
- There are two distinct aspects of data analysis: model comparison and parameter estimation. In model comparison we try to determine which model is better than another. In parameter estimation, we have one assumed model and we are estimating the parameters of that model. We focus here on parameter estimation.

- 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 come to understand the production in the big bang of Helium, photons, other "hot" relics such as neutrinos, and "cold" relics such as the dark matter. We will also discuss the observations that test

- 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 abundance changes 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.

- 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.

- 25: The Cosmic Microwave Background
- Cosmic Microwave Background Radiation, or CMB radiation, is a prediction of Big Bang theory. This theory asserts that the early universe was occupied by a hot, dense plasma of photons, electrons and baryons that was opaque to electromagnetic radiation. The photons were constantly scattering off particles, particularly electrons, in the plasma. As the universe underwent adiabatic expansion, the plasma cooled until it became favorable for electrons to combine with protons and form hydrogen atoms.

- 26: The First Few Hundred Thousand Years: Observations of the Primordial Plasma
- 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 dynamical system. Due to its simplicity, we can calculate the evolution of these disturbances with high accuracy and use these calculations to predict

- 27: The Fourier Transform
- Here, we will focus on the use of Fourier methods to solve for the evolution of \( \Psi(x,t) \) assuming it obeys the above equation and that we are given the value of \(\Psi\) and its time derivative at some initial time for all values of \(x\). Fourier methods have a broad range of applications in physics. They have utility well beyond the dynamics of the wave equation in both experimental and theoretical physics. For the student of physics, time spent developing facility with Fourier transfor

- 28: 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.

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)