= Solution
In the far past, $|ck\tau|\gg1$, the term $2/\tau^2$ is negligible and the rescaled mode is a <quantum harmonic oscillator> of angular frequency $ck$. The stated choice of its ground state is the <Bunch-Davies vacuum>.
At late times, after the mode is far outside the Hubble scale, $|ck\tau|\ll1$. Its <variance> becomes
$$
P_k
=\frac{\hbar H_{\rm inf}^2}{2c^3k^3}
\left(1+c^2k^2\tau^2\right)
\longrightarrow
\frac{\hbar H_{\rm inf}^2}{2c^3k^3}.
$$
Since $P_k$ has dimensions of length cubed, the dimensionless power is proportional to $k^3P_k$. More precisely, with the standard isotropic Fourier convention,
$$
\Delta^2(k)=\frac{k^3P_k}{2\pi^2}
\longrightarrow
\frac{\hbar H_{\rm inf}^2}{4\pi^2c^3},
$$
which is independent of the <wavenumber> $k$. Thus equal logarithmic intervals in $k$ carry equal late-time power. This is the sense in which inflation naturally generates a <scale-invariant inflationary power spectrum>.
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