Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 312 / 2 / i

Table of contents
- Solution i

Solution

The massless field equation is

ϕ^{''} + 2 H ϕ^{'} - \nabla^{2} ϕ = 0, H = \frac{a ^{'}}{a} .

For a homogeneous solution this becomes

(a^{2} \overset{ˉ}{ϕ}^{'})^{'} = 0.

Therefore

\overset{ˉ}{ϕ}^{'} \propto a^{- 2}, p = - 2 .

Since

\dot{\overset{ˉ}{ϕ}} = \overset{ˉ}{ϕ}^{'} / a

, the cosmic-time velocity decays as

a^{- 3}

.

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