Solution

Write the matter density as a sum over normalized halo profiles,

ρ (x) = \sum_{i} M_{i} u (x - x_{i} ∣ M_{i}), \int d^{3} x u (x ∣ M) = 1.

(1)

For

k \neq = 0

, its density contrast is

δ (k) = \frac{1}{ρ ˉ} \sum_{i} M_{i} u (k ∣ M_{i}) e^{- i k \cdot x_{i}} .

(2)

If halo locations form an uncorrelated Poisson process, only equal-halo terms survive after subtracting the homogeneous contribution. Replacing the sum per unit volume by the halo abundance gives the one-halo term

P (k) = P_{1 h} (k) = \int_{0}^{\infty} d M \frac{d n}{d M} (\frac{M}{ρ ˉ})^{2} ∣ u (k ∣ M) ∣^{2} .

(3)

Solved by gpt-5.6-sol high.

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d
3
Paper 312
iii
2026
Past exam of the mathematics course of the University of Cambridge
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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 3 / d / Solution

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