= Solution
For either <scalar field> $f$ with mass $m_f$, the supplied covariant equation in the <FLRW metric> is the <Klein-Gordon equation>
$$
\ddot f+3H\dot f-a^{-2}\nabla^2f+m_f^2f=0.
$$
Since $\dot f=f'/a$, $\ddot f=(f''-\mathcal Hf')/a^2$ and $H=\mathcal H/a$, its \b[conformal-time form] is
$$
\boxed{\phi''+2\mathcal H\phi'-\nabla^2\phi+a^2m_\phi^2\phi=0,
\qquad
\chi''+2\mathcal H\chi'-\nabla^2\chi+a^2m_\chi^2\chi=0.}
$$
The homogeneous <scalar field> backgrounds satisfy the same equations without spatial derivatives. After subtracting them, let $\delta\phi=u/a$ and $\delta\chi=v/a$. The key derivatives are
$$
(u/a)'=\frac{u'-\mathcal Hu}{a},\qquad
(u/a)''=\frac{u''-2\mathcal Hu'+(\mathcal H^2-\mathcal H')u}{a}.
$$
Because $a''/a=\mathcal H'+\mathcal H^2$, the expansion-friction term cancels after this <canonical rescaling of a scalar cosmological perturbation>. The \b[fluctuation equations] are
$$
\boxed{u''-\nabla^2u+\left(a^2m_\phi^2-\frac{a''}a\right)u=0,
\qquad
v''-\nabla^2v+\left(a^2m_\chi^2-\frac{a''}a\right)v=0.}
$$
The 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 \b[Fourier-mode equations] are
$$
\boxed{u_{\mathbf k}''+\left(k^2+a^2m_\phi^2-\frac{a''}a\right)u_{\mathbf k}=0,
\qquad
v_{\mathbf k}''+\left(k^2+a^2m_\chi^2-\frac{a''}a\right)v_{\mathbf k}=0.}
$$
They 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 $v/a$.
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