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

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- Solution b

Solution

Linearizing the curvaton equation and Fourier transforming gives

δ σ_{k}^{''} + 2 \frac{a ^{'}}{a} δ σ_{k}^{'} + (k^{2} + a^{2} m_{σ}^{2}) δ σ_{k} = 0.

f_{k}^{σ} = a δ σ_{k}

yields

(f_{k}^{σ})^{''} + (k^{2} + a^{2} m_{σ}^{2} - \frac{a ^{''}}{a}) f_{k}^{σ} = 0.

Since

m_{σ} ≪ H

, the mass term is negligible and this is the equation stated. It has the same Bunch-Davies mode as the inflaton perturbation, so after horizon exit

Δ_{δ σ}^{2} (k) = (\frac{H}{2 π})^{2}

up to a small mass-induced spectral tilt.

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