Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 312 / 1 / i / Solution

Varying the scalar action and Fourier transforming the spatial coordinates gives the Klein-Gordon equation

ϕ_{k}^{''} + 2 \frac{a ^{'}}{a} ϕ_{k}^{'} + (k^{2} + a^{2} m^{2}) ϕ_{k} = 0.

In de Sitter spacetime,

a = - 1/ (Hη)

and

m^{2} = 2 H^{2}

, so

ϕ_{k}^{''} - \frac{2}{η} ϕ_{k}^{'} + (k^{2} + \frac{2}{η ^{2}}) ϕ_{k} = 0 .

Direct substitution shows that

η e^{- ik η}

and

η e^{ik η}

are independent solutions. Hence

ϕ (η, k) = η [C_{+} (k) e^{- ik η} + C_{-} (k) e^{ik η}] .

The mass value is the one that gives a conformally coupled scalar field in four-dimensional de Sitter spacetime.

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