ObservationalCosmology

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 the following equation is correct: q = 0.5xOmega_m + Omega_r + Omega_Lam. 
   
3. Question removed... as it's not useful. Placeholder here so as not to renumber problems.
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 ,  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?