AST4220: Cosmology I Øystein Elgarøy 2 Contents 1 Cosmological models 1 1.1 Special relativity: space and time as a unity . 1 1.2 Curvedspacetime......................... 3 1.3 Curved spaces: the surface of a sphere . 4 1.4 The Robertson-Walker line element . 6 1.5 Redshifts and cosmological distances . 9 1.5.1 Thecosmicredshift . 9 1.5.2 Properdistance. 11 1.5.3 The luminosity distance . 13 1.5.4 The angular diameter distance . 14 1.5.5 The comoving coordinate r ............... 15 1.6 TheFriedmannequations . 15 1.6.1 Timetomemorize! . 20 1.7 Equationsofstate ........................ 21 1.7.1 Dust: non-relativistic matter . 21 1.7.2 Radiation: relativistic matter . 22 1.8 The evolution of the energy density . 22 1.9 The cosmological constant . 24 1.10 Some classic cosmological models . 26 1.10.1 Spatially flat, dust- or radiation-only models . 27 1.10.2 Spatially flat, empty universe with a cosmological con- stant............................ 29 1.10.3 Open and closed dust models with no cosmological constant.......................... 31 1.10.4 Models with more than one component . 34 1.10.5 Models with matter and radiation . 35 1.10.6 TheflatΛCDMmodel. 37 1.10.7 Models with matter, curvature and a cosmological con- stant............................ 40 1.11Horizons.............................. 42 1.11.1 Theeventhorizon . 44 1.11.2 Theparticlehorizon . 45 1.11.3 Examples ......................... 46 I II CONTENTS 1.12 The Steady State model . 48 1.13 Some observable quantities and how to calculate them . 50 1.14 Closingcomments . 52 1.15Exercises ............................. 53 2 The early, hot universe 61 2.1 Radiation temperature in the early universe . 61 2.2 Statisticalphysics: abriefreview . 62 2.3 An extremely short course on particle physics . 68 2.4 Entropy .............................. 75 2.5 TheBoltzmannequation. 78 2.6 Freeze-outofdarkmatter . 79 2.7 BigBangNucleosynthesis . 83 2.8 Recombination .......................... 90 2.9 Concludingremarks . 93 2.10Exercises ............................. 93 3 Inflation 99 3.1 PuzzlesintheBigBangmodel . 99 3.2 The idea of inflation: de Sitter-space to the rescue! . 101 3.3 Scalarfieldsandinflation . 104 3.3.1 Example: inflaction in a φ2 potential . 109 3.3.2 Reheating.........................112 3.4 Fluctuations............................ 112 3.4.1 Inflation and gravitational waves . 115 3.4.2 The connection to observations . 118 3.4.3 Optional material: the spectrum of density perturba- tions............................120 3.5 Exercises .............................121 4 Structure formation 125 4.1 Non-relativisticfluids. 126 4.2 TheJeanslength ......................... 131 4.3 The Jeans instability in an expanding medium . 133 4.4 Perturbationsinarelativisticgas . 134 4.5 A comment on the perturbations in the gravitational potential 135 4.6 TheMeszaroseffect ....................... 135 4.7 The statistical properties of density perturbations . 137 4.8 Fluctuations in the cosmic microwave background . 141 4.9 Exercises .............................145 Chapter 1 Cosmological models Cosmology is the study of the universe as a whole. We want to learn about its size, its shape and its age. Also, we want to understand the distribution of matter in the form of galaxies, clusters of galaxies and so on, and how this distribution arose. Even more ambitiously, we want to know how the universe started and how it will end. These are all bold questions to ask, and the fact that we are now getting closer to answering many of them is a testimony to the tremendous theoretical and, perhaps most important, observational effort invested over the past century. It is not totally inaccurate to say that modern cosmology started with Einstein’s theory of general relativity (from now on called GR for short). GR is the overarching framework for modern cosmology, and we cannot avoid starting this course with at least a brief account of some of the most important features of this theory. 1.1 Special relativity: space and time as a unity Special relativity, as you may recall, deals with inertial frames and how physical quantities measured by observers moving with constant velocity relative to each other are related. The two basic principles are: 1. The speed of light in empty space, c, is the same for all observers. 2. The laws of physics are the same in all inertial frames. From these principles the strange, but by now familiar, results of special relativity can be derived: the Lorentz transformations, length contraction, time dilation etc. The most common textbook approach is to start from the Lorentz transformations relating the position and time for an event as observed in two different inertial frames. However, all the familiar results can be obtained by focusing instead on the invariance of the spacetime interval (here given in Cartesian coordinates) ds2 = c2dt2 dx2 dy2 dz2 (1.1) − − − 1 2 CHAPTER 1. COSMOLOGICAL MODELS for two events separated by the time interval dt and by coordinate distances dx, dy, and dz. The invariance of this quantity for all inertial observers follows directly from the principles of relativity. To see how familiar results can be derived from this viewpoint, consider the phenomenon of length contraction: Imagine a long rod of length L as measured by an observer at rest in the frame S. Another observer is travelling at speed v relative to the frame S, at rest in the origin of his frame S′. When the observer in S′ passes the first end point of the rod, both observers start their clocks, and they both stop them when they see the observer in S′ pass the second end point of the rod. To the observer in S, this happens after a time dt = L/v. Since the observer in S′ is at rest in the origin of his frame, he measures no spatial coordinate difference between the two events, but a time difference dt′ = τ. Thus, from the invariance of the interval we have L 2 ds2 = c2 L2 = c2τ 2 02 v µ ¶ − − from which we find L v2 τ = 1 . v s − c2 Since the observer in S′ sees the first end point of the rod receding at a speed v, he therefore calculates that the length of the rod is v2 L′ = vτ = L 1 γL < L. (1.2) s − c2 ≡ Similarly, we can derive the usual time dilation result: moving clocks run at a slower rate (i.e. record a shorter time interval between two given events) than clocks at rest. Consider once again our two observers in S and S′ whose clocks are synchronized as the origin of S′ passes the origin of S at t = t′ = 0. This is the first event. A second event, happening at the origin of S′ is recorded by both observers after a time ∆t in S, ∆t′ in S′. From the invariance of the interval, we then have 2 2 2 2 2 2 c ∆t v ∆t = c ∆t′ − which gives ∆t′ ∆t′ ∆t = = > ∆t′. 1 v2/c2 γ − This approach to specialp relativity emphasizes the unity of space and time: in relating events as seen by observers in relative motion, both the time and the coordinate separation of the events enter. Also, the geometrical aspect of special relativity is emphasized: spacetime ‘distances’ (intervals) play the fundamental role in that they are the same for all observers. These 1.2. CURVED SPACETIME 3 features carry over into general relativity. General relativity is essential for describing physics in accelerated reference frames and gravitation. A novel feature is that acceleration and gravitation lead us to introduce the concept of curved spacetime. In the following section we will explore why this is so. 1.2 Curved spacetime In introductory mechanics we learned that in the Earth’s gravitational field all bodies fall with the same acceleration, which near the surface of the Earth is the familiar g = 9.81 m/s2. This result rests on the fact that the mass which appears in Newton’s law of gravitation is the same as that appearing in Newton’s second law F = ma. This equality of gravitational and inertial mass is called the equivalence principle of Newtonian physics. We will use this as a starting point for motivating the notion of curved spacetime and the equivalence of uniform acceleration and uniform gravitational fields. Consider a situation where you are situated on the floor of an elevator, resting on the Earth’s surface. The elevator has no windows and is in every way imaginable sealed off from its surroundings. Near the roof of the elevator there is a mechanism which can drop objects of various masses towards the floor. You carry out experiments and notice the usual things like, e.g. that two objects dropped at the same time also reach the floor at the same time, and that they all accelerate with the same acceleration g. Next we move the elevator into space, far away from the gravitational influence of the Earth and other massive objects, and provide it with an engine which keeps it moving with constant acceleration g. You carry out the same experiments. There is now no gravitational force on the objects, but since the floor of the elevator is accelerating towards the objects, you will see exactly the same things happen as you did when situated on the surface of the Earth: all objects accelerate towards the floor with constant acceleration g. There is no way you can distinguish between the two situations based on these experiments, and so they are completely equivalent: you cannot distinguish uniform acceleration from a uniform gravitational field! Einstein took this result one step further and formulated his version of the equivalence principle: You cannot make any experiment which will distinguish between a uniform gravitational field and being in a uniformly accelerated reference frame! This has the further effect that a light ray will be bent in a gravitational field.