| Show that, for a universe without a cosmological constant, the scale factor a(t) will | |
| (i) reach a finite maximum value and then recollapse, if k > 0; | [1] |
| (ii) increase at an ever-decreasing rate, i.e. ȧ → 0 as t → ∞, if k = 0; | [1] |
| (iii) increase at a rate which tends to a constant value as t → ∞, if k < 0. | [1] |
| Define the density parameter Ω, and show that the three cases above correspond to Ω > 1, Ω = 1 and Ω < 1 respectively. | [3] |
| In case (i), calculate the maximum value of a if Ωr0 ≈ 0 and Ωm0 = 1.1. | [3] |
(2006 Resit Q5(a).)