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 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?