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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 14A / c / Solution

The Friedmann acceleration equation gives

\frac{a ¨}{a} = - \frac{4 π G}{3 c ^{2}} (ρ + 3 P) .

For the scalar field,

ρ + 3 P = 2 \dot{ϕ}^{2} - 2 V (ϕ),

and hence

\frac{a ¨}{a} = \frac{8 π G}{3 c ^{2}} [V (ϕ) - \dot{ϕ}^{2}] .

An inflationary phase occurs precisely when potential energy dominates enough that

V (ϕ) > \dot{ϕ}^{2} .

Solved by gpt-5.6-sol high.

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