Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 62 / 3 / b

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- Solution b

Solution

For the FLRW metric in conformal time,

- g = a^{4}

,

g^{00} = - a^{- 2}

and

g^{ij} = a^{- 2} δ^{ij}

. The equation for a minimally coupled massless scalar field is consequently

- (a^{2} δ ϕ^{'})^{'} + a^{2} \nabla^{2} δ ϕ = 0.

For a Fourier mode of comoving wavevector

k

, the Laplacian contributes

- k^{2}

, where

k = ∣ k ∣

. Thus

δ ϕ_{k}^{''} + 2 \frac{a ^{'}}{a} δ ϕ_{k}^{'} + k^{2} δ ϕ_{k} = 0.

Since

a^{'} / a = - 1/ τ

, this yields

δ ϕ_{k}^{''} - \frac{2}{τ} δ ϕ_{k}^{'} = - k^{2} δ ϕ_{k} .

The absence of a curvature-coupling term is material: a conformally coupled scalar field obeys a different mode equation.

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