Radiation--cosmic-string Friedmann solution

Consider a spatially flat universe containing radiation and a cosmic string network. If their densities are both

ρ_{eq}

at scale factor

a_{eq}

, then

ρ_{r} = ρ_{eq} (\frac{a _{eq}}{a})^{4}, ρ_{s} = ρ_{eq} (\frac{a _{eq}}{a})^{2} .

In conformal time

d τ = d t / a

, the flat Friedmann equation becomes

(\frac{d a}{d τ})^{2} = \frac{8 π G a _{eq}^{2}}{3 c ^{2}} ρ_{eq} (a^{2} + a_{eq}^{2}) .

Choosing the Big Bang at

τ = 0

, its expanding solution is

a (τ) = a_{eq} sinh (a_{eq} \frac{8 π G ρ _{eq}}{3 c ^{2}} τ) .

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 15D / Solution

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