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

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- Solution e

Solution

For distinct halos, insert their linearly biased correlation

ξ_{hh} (r ∣ M_{1}, M_{2}) = b (M_{1}) b (M_{2}) ξ_{lin} (r)

into the double sum. Fourier transformation converts

ξ_{lin}

into

P_{lin}

, while the two independent mass integrals factorize. The result is the halo model

P (k) = P_{1 h} (k) + P_{2 h} (k),

where

P_{2 h} (k) = [\int_{0}^{\infty} d M \frac{d n}{d M} \frac{M}{ρ ˉ} b (M) u (k ∣ M)]^{2} P_{lin} (k) .

Thus the requested function is

f (M, \overset{ρ}{ˉ}) = \frac{M}{ρ ˉ} .

At small

k

, profile normalization gives

u \to 1

and the bias consistency relation makes the square bracket tend to one, so the two-halo term approaches the linear matter spectrum.

Solved by gpt-5.6-sol high.

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