 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ii/paper-2/15d/b/solution

The two scalings give

ρ = ρ_{r 0} a^{- 4} + ρ_{s 0} a^{- 2}

. Flatness and the Friedmann equation give

H^{2} = 8 π Gρ /3

, with

H_{0}^{2} = (8 π G /3) (ρ_{r 0} + ρ_{s 0})

. Thus

(\frac{a ˙}{a})^{2} = \frac{H _{0}^{2}}{a ^{4}} \frac{a ^{2} + β}{1 + β} .

On the expanding branch, putting the Big Bang at zero, separation gives

t = \frac{1 + β}{H _{0}} (a^{2} + β - β), a^{2} = \frac{H _{0}^{2} t ^{2}}{1 + β} + \frac{2 H _{0} t β}{1 + β} .

Its positive square root is the required solution. If

β > 0

, its early behaviour is

a \propto t^{1/2}

, as for radiation. The values

a = 1

and

a^{2} = β

give

t_{0} = \frac{1 + β}{H _{0}} (1 + β - β), t_{eq} = \frac{β 1 + β}{H _{0}} (2 - 1) .

For

β = 0

there is only the string component, with linear expansion and no radiation era.

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