1. Introduction

You have two weeks to complete this unit. Below I give a syllabus for the unit, together with guidance as to where you can find the relevant material. More detailed advice on how to approach the work is given in the introduction to Unit 1, and will not be repeated here.

This unit contains the theoretical core of the course, and the conceptual challenges which go with it. You should be prepared to spend about 15 hours on the unit, plus a further 4 hours for the assessed exercise.
 

2. Syllabus & sources

This unit covers the core theory needed to understand modern developments in observational cosmology. A full G.R. treatment would start from the Robertson-Walker metric, which is inserted into Einstein's field equations to derive the Friedmann and acceleration equations. This is unnecessarily complex for our purposes, since the dynamical equations can be derived and understood from a Newtonian perspective, and this is the route we will take. A course note on the subject is available below. The lecture for this module will also be used to discuss solutions to these equations, and the nature of the cosmological constant.

Despite this Newtonian approach, familiarity with the R-W metric and spatial curvature are essential to interpreting cosmological observations, so these will also be introduced. For this unit, I think that you will find Liddle a good deal clearer than Rowan-Robinson.
 

Topic	Sources	Comments
Dynamical equations of the Universe:     Assumptions - Newtonian treatment      The Friedmann equation      The acceleration equation	Course note (pdf) , RR(4.3, 5.1), L1(3.0-3.6)  L2(3.0-3.6)
Solutions to the equations with Lambda=0:      For matter (dust); radiation; matter + radiation       Flat, open and closed Universes       Behaviour  of the scale factor with time	L1(4), RR(4.6), unit 2 lecture L2(5)
Cosmological Parameters :      H,q,Omega,Lambda - meanings and definitions	L1(6), RR(4.7), unit 2 lecture L2(6)
The cosmological constant:      Friedmann equation with Lambda term      Physical interpretation      Dynamical solutions - static & accelerating Universes	L2(7), L1(6.4), RR(4.8), unit 2 lecture (& discussion class)
Curved space-time       Flat, spherical and hyperbolic geometries       Relation to the Friedmann models	L1(5), L2(4), RR(4.4, 4.5)
The Robertson-Walker metric      The form and meaning of the R-W metric       General Relativity also gives the Friedmann equation	RR(4.4, 4.5), L2(A1.1)
Cosmological Redshift      Relation between redshift and expansion  (& scale factor)	L1(4.2), RR(7.4) L2(5.2, A2.1)
Age of the Universe      Dependence on Ho, and Omega (for Lambda=0)      Comparison  with ages of stars       Effect of the cosmological constant	L1(7), RR(4.9 & p.163) L2(8)

Notes
1.  Key: RR=Rowan-Robinson (4th edition), L1=Liddle (1st edition) - relevant sections, or sometimes complete chapters, eg L(5), are given in brackets.  L2=Liddle, 2nd edition. Older editions of Rowan-Robinson give an out-of-date treatment of the cosmological constant.
2. The topics listed are not of equal size.
3. References given are not by any means the only ones (e.g. check out some of the links and references on the Home Page), but they should provide a reasonable treatment.
4. For the more complex topics it pays to consult several sources and to synthesize the results. This takes longer, but should result in a better understanding.

3. Self-test problems

Use these questions as you proceed through the unit, to judge whether your coverage of the material and level of understanding are adequate. Answers are just a click away, via the      button, but you will greatly reduce the diagnostic value of the questions if you look at the solutions before making a serious attempt to answer the question yourself.

1. Why is the Cosmological Principle valid for cosmological analysis, given the very large variations in density seen in the local universe? 
2. Why does the derivation of the Friedmann equation fail in the case of an "island universe", in which the matter density is uniform, but has an edge? (Hint: think about one of the very first assumptions in the derivation) 
3. In the absence of a cosmological constant, how do the expansion velocity (da/dt) and the Hubble parameter evolve  as a function of the scale factor  in (a) an empty universe, and (b) a flat (k=0) universe? (Hint:  look at the Friedmann equation) 
4. Three simple geometrical tests for the curvature of space are: (a) the sum of the angles in a triangle, (b) the convergence/divergence of initially parallel lines, and (c) the circumference of a circle of radius r.
Consider the cases of a sphere and a saddle to explain what happens in each case for spaces of positive and negative curvature.
5. The proper distance between two points P and Q is the distance which would be obtained by a chain of comoving observers laying rulers end-to-end along a geodesic between the two at one cosmic instant. It can be calculated by integrating over the metric at a given time (so that dt=0).
For the Robertson-Walker metric, show that at the present epoch (i.e. a=1) the proper distance from the origin to a galaxy at comoving radial coordinate r is greater than r if the space is positively curved. 


6. In the case of a flat, matter-dominated universe, with zero cosmological constant, how does the density evolve as a function of time? 
7. Sketch how the scale factor would evolve in a matter dominated universe with zero cosmological constant and  k<0, and explain the shape of the curve by considering the behaviour of the two terms on the RHS of the Friedmann equation. 
8. Why is it reasonable to set the pressure P=0 when considering a matter-dominated universe? 
9. Show that the cosmological constant Lambda must have dimensions of time^-2.
What is the physical significance of the timescale given by 1/sqrt(Lambda)?
Calculate the value of 1/sqrt(Lambda) for a universe in which the cosmological constant contributes 0.7 of the critical density. 
10. Calculate the age of a universe with Omega=0 and H₀ =70 km/s/Mpc.
If Omega>0 (and Lambda=0), is the age greater or less? Explain why. 
11. At what redshift was the mean density of matter in the Universe 1000 times its present value? 

 

a) Lecture:  The Dynamics of the Universe

In this lecture I will discuss the nature of the solutions to the dynamical equations for the Universe, the relationship to space-time curvature. 

b) Discussion class: Cosmological Concepts #2

We will first discuss the nature of the Cosmological Constant, and its inclusion in the Friedmann equations.
Then, continuing along the lines of the first discussion class, we will discuss more questions designed to help your understanding of important conceptual  issues in cosmology, such as : 

   What is the difference between a cosmological redshift and a Doppler shift?
   What is the resolution of Olbers paradox?

c) Assessed exercise:

The first assessed exercise for third year students is available here. It must be completed and returned to the Teaching Office by 4pm on Friday 7 November. Year 4 students are advised to look at this for practice, but not submit solutions.

The assessed exercise (worth 30% of  the module) for Year 4 students is available here. It must be completed by the end of term.