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

Table of contents
- Solution f

Solution

Neglecting the

ϵ_{V}

contribution to the tilt, the observed value gives

- \frac{1}{30} ≃ - \frac{2 M _{Pl}^{2} m ^{2}}{Λ ^{4}}, \frac{M _{Pl}^{2} m ^{2}}{Λ ^{4}} ≃ \frac{1}{60} .

Part (d) then gives

ϕ_{60} = e^{- 1} ϕ_{e}

. Using the formal end estimate from part (b),

ϵ_{V} (ϕ_{60}) ≃ e^{- 2}

. The amplitude relation therefore yields

\frac{H}{M _{Pl}} = 8 π^{2} A_{s} ϵ_{V} (ϕ_{60}) ≃ \frac{8 π ^{2} ( 2 \times 1 0 ^{- 9} )}{e} ≃ 1.5 \times 1 0^{- 4} .

Thus the requested order-of-magnitude estimate is

H / M_{Pl} \sim 1 0^{- 4} .

The simultaneous approximations make

ϵ_{V} (ϕ_{60})

only moderately small, so this numerical result should be read as the order-of-magnitude estimate requested rather than a precision fit.

Solved by gpt-5.6-sol high.

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