Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 312 / 2 / iv

Expand the quantized field using the normalized mode

u_{k} (η) = A_{k} \frac{1 + ik η}{η ^{3}} e^{- ik η} .

M_{Pl}^{2} z^{2} (u_{k} u_{k}^{*'} - u_{k}^{*} u_{k}^{'}) = i

x = k η

, the primed correlators are

⟨ RR ⟩^{'} = \frac{∣ A _{k} ∣ ^{2}}{η ^{6}} (1 + x^{2}),

⟨ R \dot{R} ⟩^{'} = \frac{H ∣ A _{k} ∣ ^{2}}{η ^{6}} (3 + 2 x^{2} - i x^{3}),

⟨ \dot{R} \dot{R} ⟩^{'} = \frac{H ^{2} ∣ A _{k} ∣ ^{2}}{η ^{6}} (9 + 3 x^{2} + x^{4}) .

Each unprimed expression includes

(2 π)^{3} δ^{(3)} (k + k^{'})

. The imaginary part of the mixed correlator gives

∣ ⟨[R, \dot{R}] ⟩^{'} ∣ = \frac{2 H ∣ A _{k} ∣ ^{2}}{∣ η ∣ ^{6}} ∣ x ∣^{3} .

C (k, η) = \frac{2∣ k η ∣ ^{3}}{( 1 + k ^{2} η ^{2} ) ( 9 + 3 k ^{2} η ^{2} + k ^{4} η ^{4} )} ≃ \frac{2}{3} ∣ k η ∣^{3} .

It tends to zero on superhorizon scales, so the perturbation becomes effectively classical even while its amplitude grows as

a^{3}

. In ordinary slow-roll inflation,

R

instead freezes to a constant; its analogous normalized commutator also vanishes, but behaves as

∣ k η ∣

for the leading de Sitter mode rather than

∣ k η ∣^{3}

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