= Solution
Linearizing the <curvaton> equation and Fourier transforming gives
$$
\delta\sigma_k''+2\frac{a'}a\delta\sigma_k'
+(k^2+a^2m_\sigma^2)\delta\sigma_k=0.
$$
Writing $f_k^\sigma=a\delta\sigma_k$ yields
$$
(f_k^\sigma)''+\left(k^2+a^2m_\sigma^2-\frac{a''}{a}\right)f_k^\sigma=0.
$$
Since $m_\sigma\ll H$, 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
$$
\boxed{\Delta_{\delta\sigma}^2(k)=\left(\frac H{2\pi}\right)^2}
$$
up to a small mass-induced spectral tilt.
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