1. Introduction

You have two weeks to complete this final 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 less material than the others, but you will need to familiarise yourself with using the various formulae involved, and to think carefully about what they mean, in order to get a good grasp of them. You should aim to spend about 12 hours on the unit, and take the self-test problems very seriously.

2. Syllabus & sources

In this final unit, we make the connection between theory and observation. This involves the relationship between abstract parameters like comoving coordinates, and things we can observe, like redshift, flux, and size. You will find that Liddle (in his Advanced Topics) and Rowan-Robinson contain much of what you need. As usual, Liddle is clearer, whilst Rowan-Robinson gives more observation-related material. Since this topic is challenging, I will also devote lecture 5 to laying out some of the basic ideas and mathematical apparatus.

Finally, I suggest rounding the course off by reading the interesting and perceptive esssay by the Astronomer Royal, Martin Rees. This is mostly about cosmology, and will give you the "big picture". You should find that you can understand it a lot better than would have been the case three months ago. Take it home for Christmas!
 

Topic	Sources	Comments
Relationship between redshift & distance   Light rays and the R-W metric   Role of curvature and of H(z)   Low z approximation	L2(A2.1) and Unit 5 lecture	Note that rigorous solutions for r(z) in general cosmologies normally have to be calculated numerically.
Flux & luminosity distance   Surface area of a sphere    Effect of redshift on flux   Definition of DL   Application to high z SNe	Unit 5 lecture, RR(7.2,7.6) L2(A2.3)	Note that RR defines P to be the luminosity radiated into unit solid angle - i.e. P=L/(4.pi).
Angular size & angular-diameter distance   Definition of DA   Behaviour of angular size     with redshift	RR(7.8) L2(A2.4) Unit 5 lecture	Remember that RR uses Z=1+z.
Magnitude-redshift relation & K-correction	RR(7.7)
Space density and number counts   Volume element vs redshift   Number counts for a non-evolving population   Effects of source evolution on number counts	RR(7.9, 3.5) L2(A2.5)
Piecing it all together	Rees paper	Read this interesting essay on our understanding of the Universe by Martin Rees, for a perspective on where cosmology has now brought us.

Notes
1.  Key: RR=Rowan-Robinson,  L2=Liddle, 2nd edition - relevant sections are given in brackets.
2. The topics listed are not of equal size.

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 it that properties of objects (sizes, magnitudes etc.) at cosmological distances behave in rather unexpected ways even if space is flat? 
2. Starting from the acceleration equation, and other equations from your notes from Lecture 2, demonstrate that equation 5 from Lecture 5 is correct - i.e. q = 0.5xOmega_m + Omega_r + Omega_Lam. 
   
3. Show that D_L tends to the "common sense" result, D _L=cz/H₀, at low redshift, from the full matter-dominated solution (equation 7 from Lecture 5). (Note that this is obviously true for the aproximate result given by equation 8.)    
4. Are apparent bolometric magnitudes of sources brighter or fainter than naive expectations based on simple Hubble Law distances? What is the main effect causing this? 
5. What is the luminosity distance in Mpc of a galaxy at z=1 in a matter dominated Universe with zero cosmological constant, H₀ =70 km/s/Mpc and Omega₀=1 ? If this galaxy has an observed  bolometric flux  of  1x10^-18  W/m²  what is its intrinsic bolometric luminosity?
6. Use the approximate result from  the lectures (Lect.5 equation 4),  r=c/Ho (z - 1/2 (1+qo)z² ) , to derive a low redshift formula for the angular diameter of an object of proper size l at redshift z, in a universe with deceleration parameter q₀.

In the case of a matter dominated universe, can you see why objects should look larger if space is more positively curved (i.e. larger values of q ₀)?  
7. Assuming that the physical spatial scale of the highest amplitude fluctuations in the microwave background is independent of cosmology, use the results of the last section to explain why the angular scale size of these fluctuations would be smaller in an open universe than in a flat one. This is the crucial test of Omega which has been carried out recently. You may find it helpful to refer to the following MAP web page for this. 
8. Combining the flux and angular size relations, how should the surface brightness (i.e. flux per unit solid angle) of an extended object like a galaxy scale with redshift? 
9. Demonstrate that the K-correction, K(z), is equal to 2.5(alpha-1)log(1+z) for a power law spectrum. (Refer to RR(7.7) for the definition of alpha).
Explain physically why K(z) is larger for a steeper (i.e. "redder") spectrum.   
10. It is quite straightforward to show that N(S), the number of sources bighter than flux S, scales as S^-1.5 for any population uniformly distributed in a static Euclidean space. Rowan-Robinson comments in his section 7.9 that the log N:log S relation is predicted to be flatter than this (i.e. the slope is less than 1.5) for all sensible cosmological models. In other words fewer faint sources are observed than in the simple Euclidean case. Why do you think this should be? 
11. If a population is undergoing luminosity evolution with L^* (z)=L^*(0)x(1+z)³, show that the flux from systems with L=L^* increases with z at moderate redshifts - i.e. typical brightest members of the class look brighter at larger distances! This is what happens with quasars. 
12. If the currently emerging "consensus cosmology" is correct, what will the future evolution of the Universe look like? 

a) Lecture:  Connecting Theory and Observation

In this lecture I will present some of the formulae required to link the theoretical results (e.g. Robertson-Walker metric and Friedmann equation) to the observed properties of the Universe.

b) Discussion class: The Course etc.

We will finish our discussion of angular diameter distance, discuss the feedback from the course questionnaire and take a look at the third assessed exercise for Year 3 students. Be sure to bring your notes.