Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 14D / b

Table of contents
- Solution b

Solution

Here

c_{s}^{2} = c^{2} \frac{d P}{d ρ} = \frac{4}{3} α c^{2} \overset{ρ}{ˉ}^{1/3} .

Since

\overset{ρ}{ˉ} \propto t^{- 2}

and

a \propto t^{2/3}

,

c_{s}^{2} / a^{2} = L_{0}^{2} / t^{2}

for a constant

L_{0}^{2}

. With

\overset{a}{˙} / a = 2/ (3 t)

and

4 π G \overset{ρ}{ˉ} / c^{2} = 2/ (3 t^{2})

, substitution of

δ = t^{n}

gives

n^{2} + \frac{1}{3} n + L_{0}^{2} k^{2} - \frac{2}{3} = 0,

so

n_{\pm} = - \frac{1}{6} \pm \frac{25}{36} - L_{0}^{2} k^{2} .

For

k ≪ k_{J} = 5/ (6 L_{0})

, the modes approach

t^{2/3}

and

t^{- 1}

. For

k ≫ k_{J}

, the exponents are complex with real part

- 1/6

, giving logarithmic oscillations with decaying envelope

t^{- 1/6}

. Taking

L_{0} ≃ 50 Mpc

would pressure-suppress structure on enormous, cosmologically important scales and is incompatible with the observed growth of smaller-scale structure.

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