Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 62 / 2 / a

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- Solution a

Solution

Vary the scalar field and integrate by parts, taking the variation to vanish at the boundary:

δ S = \int d^{4} x - g {\nabla_{μ} \nabla^{μ} ϕ - V_{, ϕ}} δ ϕ .

Thus the curved-spacetime Klein-Gordon equation is

□ ϕ - V_{, ϕ} = 0

. In the flat FRW metric,

- g = a^{3}

and

g^{00} = - 1

. A spatially homogeneous field therefore has

□ ϕ = \frac{1}{a ^{3}} \partial_{t} (- a^{3} \dot{ϕ}) = - \ddot{ϕ} - 3 \frac{a ˙}{a} \dot{ϕ} .

Consequently

\ddot{ϕ} + 3 H \dot{ϕ} + V_{, ϕ} = 0, H = \frac{a ˙}{a} .

Dots now denote cosmic time derivatives. The

3 H \dot{ϕ}

term is Hubble friction caused by expansion of the comoving volume.

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