Solution

On wavelengths much larger than halos, all matter is partitioned among halos. The mass-weighted halo overdensity must therefore equal the matter overdensity. Since

δ_{h} (M) = b (M) δ_{m}

, mass conservation requires

\int_{0}^{\infty} d M \frac{d n}{d M} \frac{M}{ρ ˉ} b (M) = 1.

(1)

For the Press-Schechter formalism, the mass-fraction measure becomes

f (ν) d ν = \frac{2}{π} e^{- ν^{2} /2} d ν, ν \geq 0.

(2)

It is normalized and is a half-normal distribution, so

\int_{0}^{\infty} f (ν) d ν = 1, \int_{0}^{\infty} ν^{2} f (ν) d ν = 1.

(3)

Using the linear Eulerian halo bias

b = 1 + b_{L} = 1 + \frac{ν ^{2} - 1}{δ _{c}}

(4)

therefore gives

\int_{0}^{\infty} f (ν) b (ν) d ν = 1 + \frac{1 - 1}{δ _{c}} = 1,

(5)

which verifies the consistency relation.

Solved by gpt-5.6-sol high.

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

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