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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-312/1/c/solution