= Solution
Subtract the homogeneous background equation and linearize the <inflaton equation of motion>. The field perturbation satisfies
$$
\delta\phi''+2\frac{a'}a\delta\phi'-\nabla^2\delta\phi+a^2V_{,\phi\phi}(\bar\phi)\delta\phi=0.
$$
With $f=a\delta\phi$, direct differentiation cancels the first-derivative term and gives
$$
f''-\nabla^2f+\left(a^2V_{,\phi\phi}-\frac{a''}a\right)f=0.
$$
The <conformal-time quadratic action for an inflaton perturbation>, after a boundary term is discarded, is
$$
S^{(2)}=\frac12\int d\tau\,d^3x\left\{(f')^2-(\nabla f)^2+\left(\frac{a''}a-a^2V_{,\phi\phi}\right)f^2\right\}.
$$
For the specified <de Sitter spacetime> background, $a''/a=2/\tau^2=2a^2H^2$. The light-field approximation $|V_{,\phi\phi}|\ll2H^2$ therefore permits dropping the mass term. Taking a spatial <Fourier transform> yields the <Canonically rescaled de Sitter scalar mode> equation
$$
\boxed{f_{\mathbf k}''+\left(k^2-\frac{a''}a\right)f_{\mathbf k}=0,\qquad\frac{a''}a=\frac2{\tau^2}.}
$$
Its kinetic term is canonical, with momentum $\pi_f=f'$. Each independent real Fourier component consequently has the canonical coordinate/momentum algebra of a <quantum harmonic oscillator>, with time-dependent squared frequency $\omega_k^2=k^2-a''/a$. This is why <canonical quantization of a real scalar field> proceeds by oscillator creation and annihilation operators. On <superhorizon scales> the squared frequency is negative: the system is then an inverted time-dependent oscillator, not a stationary oscillator with a positive frequency. The canonical commutation relations remain valid.
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