Introduction to Cosmology Example Exam Question: Hints

Lecture 5: Single Component Universes

Solve the Friedmann equation for the case of a matter-dominated flat universe. [3]
This is a standard question: you should be able to do all the single-component solutions we discussed in this lecture. The method is the following:
  1. Write down the Friedmann euation with only the non-zero terms.
  2. Take square root and rearrange to get all the a-terms on one side.
  3. Integrate, and apply appropriate boundary condition (usually a = 0 at t = 0, but for the Λ-only universe it's a = 1 at t = t0).
  4. Consider the resulting equation for the case a = 1 to get an expression for t0.
  5. Substitute this back into the equation for a.
Hence show that, in such a universe, the proper distance of an object at redshift z is

dP = (2c/H0) (1 – 1/√(1+z)),

where H0 is the present value of the Hubble parameter. 		[3]
This is a setpiece derivation which we did in Homework 1. Take your solution to the Friedmann equation, and insert into the equation for the proper distance, dP = ∫ dt/a(t).
In a flat, matter-dominated universe with H0 = 50 km s-1 Mpc-1, a quasar is observed at redshift z = 4.0.
(i)  Calculate the proper distance of the quasar. [1]
Simply a case of putting in the numbers. The only thing that can go wrong is that you forget to convert H into sensible units!
(ii) Calculate the time at which the light from the quasar was emitted, and hence the look-back time for the quasar. [2]
First use the given value of H0 to calculate t0, then insert into your expression for a(t) to get te (remembering that 1 + z = 1/a). Finally, remember that what you are asked for is the look-back time t0 – te, not simply te!

(2008 Q5(b).)

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