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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-58/3/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 58 / 3 / b / Solution by

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

Define the matter power spectrum by the Fourier transform convention

δ (x, τ) = \int \frac{d ^{3} k}{( 2 π ) ^{3}} δ_{k} (τ) e^{i k \cdot x}, ⟨ δ_{k} δ_{k^{'}}^{*} ⟩ = (2 π)^{3} δ^{3} (k - k^{'}) P_{δ} (k, τ) .

Statistical isotropy then gives

⟨ δ^{2} ⟩ = \int d lo g k \frac{k ^{3} P _{δ} ( k , τ )}{2 π ^{2}} .

For the growing mode found above, write

P_{δ} (k, τ) = τ^{4} P_{A} (k)

. At cosmological horizon entry,

τ_{H} \propto k^{- 1}

, so the variance contribution per logarithmic interval is proportional to

k^{3} k^{- 4} P_{A} (k) = P_{A} (k) / k

. Making it independent of

k

requires

P_{A} (k) \propto k

. Hence

P_{δ} (k, τ) \propto τ^{4} k .

This is the Harrison-Zeldovich spectrum. Scale invariance applies at horizon entry, not to the unprocessed density variance on a common late-time slice. In the notation

δ_{k} \propto k^{n /2} τ^{2}

, it means $n = 1$ , with random Fourier coefficients supplying the ensemble statistics.

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