| 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] |
| The Friedmann equation for the case where Λ = 0 is given by | |
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| To solve the above cases, you need to (a) consider whether ȧ is ever zero (this corresponds to the expansion stopping, i.e. to a maximum value of a), and (b) if ȧ never goes to zero for finite a, think about what happens as a becomes very large (which is equivalent to t becoming very large). | |
| Define the density parameter Ω, and show that the three cases above correspond to Ω > 1, Ω = 1 and Ω < 1 respectively. | [3] |
| You should re-express the above Friedmann equation in terms of H and Ω(t). Then take all the terms in H over to the lefthand side. You should find that all your quantities are squared (therefore necessarily positive) except for k itself and (1 - Ω); therefore the sign of k directly gives you the sign of (1 - Ω). | |
| In case (i), calculate the maximum value of a if Ωr0 ≈ 0 and Ωm0 = 1.1. | [3] |
| We did this in Problems Class 2. It is simple if you recall that the maximum value of a will occur when ȧ = 0. |
(2006 Resit Q5(a).)