= Solution
Use $\hbar=c=1$. For a canonical light <scalar field> with effective mass $m^2\ll H^2$, define the canonical Fourier mode $u_k=a\,\delta\Phi_k$. In the nearly de Sitter approximation it obeys
$$
u_k''+\left(k^2-\frac{a''}{a}\right)u_k=0,\qquad
a\simeq-\frac1{H\eta},\qquad \frac{a''}{a}\simeq\frac2{\eta^2}.
$$
Deep inside the <Hubble radius>, $k\gg aH$, the mode behaves like a Minkowski oscillator. Choosing the <Bunch-Davies vacuum> sets $u_k\to e^{-ik\eta}/\sqrt{2k}$. The normalized de Sitter solution is
$$
u_k=\frac{e^{-ik\eta}}{\sqrt{2k}}\left(1-\frac{i}{k\eta}\right).
$$
Inflation stretches the physical wavelength until it exits the Hubble radius. For $|k\eta|\ll1$, the growing scalar solution freezes with $|\delta\Phi_k|^2=|u_k/a|^2\to H^2/(2k^3)$. The <dimensionless power spectrum> is therefore
$$
\boxed{\mathcal P_{\delta\Phi}(k)=\frac{k^3}{2\pi^2}|\delta\Phi_k|^2
\simeq\left(\frac{H_k}{2\pi}\right)^2}.
$$
Here $H_k$ is evaluated when $k=aH$, since slow evolution near horizon exit fixes the frozen amplitude. Nearly constant $H$ and a light field make this spectrum almost scale-invariant; slow background evolution and small mass corrections produce a small tilt. The stretched vacuum modes supply the primordial stochastic fluctuations.
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