Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 58 / 1 / ii

Table of contents
- Solution ii

Solution

L = 8 π G ρ_{Λ} /3 > 0

. The Friedmann effective potential for matter, radiation and vacuum now reads

V (a) = - \frac{M}{a} - \frac{Q}{a ^{2}} - L a^{2}, a_{t}^{2} + V (a) = E, E = - k .

With nonzero nonnegative dust or radiation,

V

tends to

- \infty

at both ends. Its stationary point satisfies

2 L a_{E}^{4} - M a_{E} - 2 Q = 0.

Equivalently

V^{'} (a) = 0

when

2 L = M / a^{3} + 2 Q / a^{4}

; the right-hand side decreases strictly, so there is exactly one stationary point. It is a maximum because

V^{''} (a) = - \frac{2 M}{a ^{3}} - \frac{6 Q}{a ^{4}} - 2 L < 0.

The energy line in the preceding sketch therefore gives five broad types of history:

Monotonic expansion. For $E > V (a_{E})$ , the universe expands from $a = 0$ to arbitrarily large $a$ .
Monotonic contraction. Reversing cosmic time gives the corresponding contracting history.
Recollapse. For $E < V (a_{E})$ , the inner allowed interval ends at $a_{-}$ : the universe expands from zero to $a_{-}$ and contracts back to zero.
Bounce. The outer allowed interval starts at $a_{+}$ . A universe contracting from large $a$ reaches $a_{+}$ and re-expands. Here $V^{'} (a_{+}) < 0$ , so $a_{tt} > 0$ at the turning point.
Static equilibrium. For $E = V (a_{E})$ , $a = a_{E}$ and $a_{t} = 0$ give the Einstein static universe.

At the critical energy there are also marginal branches within the monotonic classes: expansion from zero can approach

a_{E}

asymptotically, or expansion can start asymptotically at

a_{E}

and proceed to infinity; contraction gives their reversals. They must not be mistaken for reaching the static solution in finite time. Indeed near the maximum,

E - V (a) ≃ - \frac{1}{2} V^{''} (a_{E}) (a - a_{E})^{2},

so

d t ≃ d a / [- V^{''} (a_{E}) /2 ∣ a - a_{E} ∣]

diverges logarithmically. The static equilibrium is unstable. Since

V (a_{E}) < 0

, the recollapsing, bouncing and static cases require positive spatial curvature

k > 0

.

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