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For either scalar field $f$ with mass $m_f$, the supplied covariant equation in the FLRW metric is the Klein-Gordon equationSince $\dot{f}=f^{\prime }/a$, $\ddot{f}=(f^{\prime \prime }-\mathcal{H}{f}^{\prime })/a^2$ and $H=\mathcal{H}/a$, its conformal-time form isThe homogeneous scalar field backgrounds satisfy the same equations without spatial derivatives. After subtracting them, let $\delta \varphi =u/a$ and $\delta \chi =v/a$. The key derivatives areBecause $a^{\prime \prime }/a={\mathcal{H}}^{\prime }+{\mathcal{H}}^2$, the expansion-friction term cancels after this canonical rescaling of a scalar cosmological perturbation. The fluctuation equations areThe quadratic potential has no mixed derivative, and the stipulated neglect of metric perturbations removes gravitational mixing. Thus these two rescaled fluctuations evolve independently in this approximation.

The given symmetric Fourier transform convention sends $-{\nabla }^2$ to $k^2$, where $k=|\mathbf{k}|$. Hence the Fourier-mode equations areThey are harmonic-oscillator equations with time-dependent squared frequencies; a negative effective squared frequency for a rescaled field does not itself imply growth of the physical field $v/a$.