Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 15B / b

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- Solution b

Solution

When

V

is approximately constant,

d V / d ϕ ≃ 0

, and exponential expansion gives the constant Hubble parameter

H = H_{inf}

. Take the spatial Fourier transform

ϕ (x, t) = \int \frac{d ^{3} k}{( 2 π ) ^{3}} ϕ_{k} (t) e^{i k \cdot x} .

Since

\nabla^{2} e^{i k \cdot x} = - k^{2} e^{i k \cdot x},

each independent inflationary scalar Fourier mode satisfies

\ddot{ϕ}_{k} + 3 H_{inf} \dot{ϕ}_{k} + \frac{c ^{2} k ^{2}}{a ^{2}} ϕ_{k} = 0 .

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