Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 4 / c

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- Solution c

Solution

When

H ≪ m_{σ}

over an oscillation, the background equation reduces to

\ddot{\overset{σ}{ˉ}} + m_{σ}^{2} \overset{σ}{ˉ} = 0, \overset{σ}{ˉ} = Σ cos (m_{σ} t + φ) .

Time averaging gives

⟨ \dot{\overset{σ}{ˉ}}^{2} ⟩ = m_{σ}^{2} ⟨ \overset{σ}{ˉ}^{2} ⟩

, hence

⟨ P_{σ} ⟩ = 0, ⟨ ρ_{σ} ⟩ = \frac{1}{2} m_{σ}^{2} Σ^{2} .

The oscillating quadratic scalar is therefore pressureless matter. Its continuity equation gives

ρ_{σ} \propto a^{- 3},

with the slowly varying amplitude obeying

Σ \propto a^{- 3/2}

.

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