Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 1 / c

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- Solution c

Solution

Expanding the free field as

φ_{k} (τ) = f_{k} (τ) a_{k} + f_{k}^{*} (τ) a_{- k}^{†}

and using the vacuum creation and annihilation operators algebra gives

⟨ φ_{k} (τ) φ_{k^{'}} (τ)⟩ = (2 π)^{3} δ^{(3)} (k + k^{'}) \frac{H ^{2} τ ^{2}}{2 c k} .

Thus the dimensional power spectrum is

P_{φ} (k, τ) = H^{2} τ^{2} / (2 c k)

. Unlike a minimally coupled massless inflationary fluctuation, this conformally coupled field decays as

a^{- 1}

and does not freeze at late time.

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