# Table of Contents

- Page ID
- 19007

In this quarter-long course, 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.

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 with1: 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 test20: 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