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

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Solution

For a Gaussian smoothed density contrast, the Press-Schechter factor of two gives the collapsed mass fraction

F (> M) = 2 \int_{δ_{c}}^{\infty} \frac{d δ}{2 π σ} e^{- δ^{2} / (2 σ^{2})} = erfc (\frac{ν}{2}), ν = \frac{δ _{c}}{σ ( M )} .

The fraction in the interval

[M, M + d M]

is

- (d F / d M) d M = (M / \overset{ρ}{ˉ}) (d n / d M) d M

. Since

d ν / d M = - ν d lo g σ / d M

,

\frac{d n}{d M} = - \frac{2}{π} \frac{ρ ˉ}{M ^{2}} ν e^{- ν^{2} /2} \frac{d lo g σ}{d lo g M}

or, equivalently, the same expression with

∣ d lo g σ / d lo g M ∣

. The minus sign is needed because

σ (M)

decreases with

M

; this is the positive Press-Schechter halo mass function.

Solved by gpt-5.6-sol high.

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