The creation and annihilation operators obey the <canonical commutation relations>
$$
[\widehat a_{\mathbf k},\widehat a_{\mathbf k'}^\dagger]
=(2\pi)^3\delta^{(3)}(\mathbf k-\mathbf k'),
\qquad
[\widehat a_{\mathbf k},\widehat a_{\mathbf k'}]
=[\widehat a_{\mathbf k}^\dagger,\widehat a_{\mathbf k'}^\dagger]=0.
$$
Since $\delta\widehat\phi=\widehat f/a$, the vacuum two-point function identifies
$$
\Delta_{\delta\phi}^2(k,\tau)
=\frac{k^3}{2\pi^2}\frac{|f_k(\tau)|^2}{a^2}
=\frac{H^2}{4\pi^2}(1+k^2\tau^2).
$$
On <superhorizon scales>, $|k\tau|=k/(aH)\ll1$, so
$$
\boxed{\Delta_{\delta\phi}^2=\left(\frac H{2\pi}\right)^2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-310/4/a/solution