= Solution
Expanding the free field as
$$
\varphi_{\mathbf k}(\tau)=f_k(\tau)a_{\mathbf k}
+f_k^*(\tau)a_{-\mathbf k}^\dagger
$$
and using the vacuum <creation and annihilation operators> algebra gives
$$
\boxed{
\langle\varphi_{\mathbf k}(\tau)\varphi_{\mathbf k'}(\tau)\rangle
=(2\pi)^3\delta^{(3)}(\mathbf k+\mathbf k')
\frac{H^2\tau^2}{2ck}}.
$$
Thus the dimensional <power spectrum> is $P_\varphi(k,\tau)=H^2\tau^2/(2ck)$. Unlike a minimally coupled massless inflationary fluctuation, this conformally coupled field decays as $a^{-1}$ and does not freeze at late time.
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