Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 2 / e

Table of contents
- Solution e

Solution

Comparing the stated curvature power spectrum with

P_{R} = 2 π^{2} A_{s} / k^{3}

at the pivot scale gives the primordial scalar amplitude

A_{s} = \frac{H ^{2}}{8 π ^{2} ϵ _{V} M _{Pl}^{2}} ≃ \frac{Λ ^{12}}{12 π ^{2} M _{Pl}^{6} m ^{4} ϕ ^{2}} .

The scalar spectral index is

n_{s} - 1 ≃ - 6 ϵ_{V} + 2 η_{V} ≃ - \frac{3 M _{Pl}^{2} m ^{4} ϕ ^{2}}{Λ ^{8}} - \frac{2 M _{Pl}^{2} m ^{2}}{Λ ^{4}} .

Sufficiently close to the hilltop,

ϵ_{V}

is suppressed by

ϕ^{2}

and the

η_{V}

term controls the spectral tilt.

Solved by gpt-5.6-sol high.

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