Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 3 / c

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- Solution c

Solution

Neutrino free streaming gives

δ_{ν} ≃ 0

on the stated scales, so

δ_{m} = (1 - f_{ν}) δ_{c b}

. With derivatives with respect to

N = lo g a

, the growth equation becomes

δ_{c b, NN} + \frac{1}{2} δ_{c b, N} - \frac{3}{2} (1 - f_{ν}) δ_{c b} = 0.

For

δ_{c b} \propto a^{p}

,

p = \frac{- 1 + 25 - 24 f _{ν}}{4} = 1 - \frac{3}{5} f_{ν} + O (f_{ν}^{2}) .

Hence

X (f_{ν}) = 1 - \frac{3}{5} f_{ν}

to first order.

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