Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 4 / a

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Solution

The creation and annihilation operators obey the canonical bosonic commutation relations

[\overset{a}{^}_{k}, \overset{a}{^}_{k^{'}}^{†}] = (2 π)^{3} δ^{(3)} (k - k^{'}),

[\overset{a}{^}_{k}, \overset{a}{^}_{k^{'}}] = [\overset{a}{^}_{k}^{†}, \overset{a}{^}_{k^{'}}^{†}] = 0, \overset{a}{^}_{k} ∣0 ⟩ = 0.

Since

δ ϕ = \hat{f} / a

, the vacuum two-point correlation function is

⟨ 0∣ δ ϕ (τ, x) δ ϕ (τ, x + r) ∣0 ⟩ = \frac{1}{a ^{2}} \int \frac{d ^{3} k}{( 2 π ) ^{3}} ∣ f_{k} (τ) ∣^{2} e^{- i k \cdot r} .

For

f_{k} = \frac{e ^{- ik τ}}{2 k} (1 - \frac{i}{k τ}),

one has

∣ f_{k} ∣^{2} = (1 + 1/ (k^{2} τ^{2})) / (2 k)

. Comparison with the definition of the dimensionless power spectrum gives

Δ_{δ ϕ}^{2} = \frac{k ^{3}}{2 π ^{2} a ^{2}} ∣ f_{k} ∣^{2} = \frac{H ^{2}}{4 π ^{2}} (1 + k^{2} τ^{2}),

where

a = - 1/ (H τ)

. On superhorizon scales,

k ≪ a H

or

∣ k τ ∣ ≪ 1

, and therefore

Δ_{δ ϕ}^{2} = (\frac{H}{2 π})^{2} .

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