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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-41/3/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 41 / 3 / ii / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

With the specified Fourier transform convention, the inverse is

δ (x) = \int d^{3} k δ_{k} e^{- i k \cdot x} / (2 π)^{3}

. Inserting both inverse transforms into the cosmological two-point correlation function and applying the given Dirac delta covariance yields

ξ (r) = \int \frac{d ^{3} k d ^{3} k ^{'}}{( 2 π ) ^{6}} e^{- i k \cdot x - i k^{'} \cdot (x + r)} ⟨ δ_{k} δ_{k^{'}} ⟩ = \int \frac{d ^{3} k}{( 2 π ) ^{3}} P (k) e^{i k \cdot r} .

For an isotropic matter power spectrum, align the polar axis with

r

. The angular integral is

2 π \int_{- 1}^{1} e^{ik r μ} d μ = 4 π sin (k r) / (k r)

. Hence the isotropic cosmological correlation-power-spectrum relation is

ξ (r) = \frac{1}{2 π ^{2}} \int_{0}^{\infty} k^{2} P (k) \frac{sin ( k r )}{k r} d k .

The angular kernel is the order-zero Spherical Bessel function

j_{0} (k r)

. At

r = 0

, replace it by its limit one, giving the unsmoothed variance when the integral converges. The argument assumes an ordinary integrable spectrum, or an explicitly specified distributional Fourier transform interpretation otherwise.

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