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

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- Solution d

Solution

Relative to massless-neutrino growth after

z_{ν}

,

\frac{δ _{c b}}{δ _{0}} = (\frac{a}{a _{ν}})^{- 3 f_{ν} /5} ≃ 1 - \frac{3}{5} f_{ν} lo g \frac{1 + z _{ν}}{1 + z} .

The total contrast has the additional factor

1 - f_{ν}

. Squaring its amplitude to obtain the cosmological density power spectrum gives

P_{f_{ν}} (k, z) ≃ [1 - 2 f_{ν} - \frac{6}{5} f_{ν} lo g \frac{1 + z _{ν}}{1 + z}] P_{0} (k, z) .

The two terms respectively encode the unclustered neutrino fraction and the accumulated slowing of cold-matter growth.

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