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

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- Solution c

Solution

At horizon exit, the primordial tensor power spectrum is proportional to

H^{2}

. Therefore

n_{T} = \frac{d l o g Δ _{t}^{2}}{d l o g k} = \frac{- 2 ϵ}{1 - ϵ} = - 2 ϵ + O (ϵ^{2}) .

Thus

C = 2 .

The ratio of the stated tensor and scalar spectra is

r = \frac{8 M _{Pl}^{- 2} ( H /2 π ) ^{2}}{( 2 ϵ M _{Pl}^{2} ) ^{- 1} ( H /2 π ) ^{2}} = 16 ϵ .

Eliminating

ϵ

gives the single-field inflation consistency relation

r = - 8 n_{T} .

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