Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 4 / c

Table of contents
- Solution c

Solution

For the monomial inflation potential

V = λ M_{Pl}^{4} (ϕ / M_{Pl})^{α}

,

ϵ ≃ \frac{M _{Pl}^{2}}{2} (\frac{V _{, ϕ}}{V})^{2} = \frac{α ^{2} M _{Pl}^{2}}{2 ϕ ^{2}},

and

η = 4 ϵ - 2 M_{Pl}^{2} \frac{V _{, ϕϕ}}{V} = \frac{2 α M _{Pl}^{2}}{ϕ ^{2}} .

Slow-roll inflation is possible when both are much smaller than one:

\frac{ϕ ^{2}}{M _{Pl}^{2}} ≫ max (\frac{α ^{2}}{2}, 2 α) .

Inflation ends once one of these conditions fails, commonly near

∣ ϕ ∣ ≃ α M_{Pl} / 2

when

ϵ ≃ 1

.

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