The <creation and annihilation operators> obey
$$
[\hat a_{\mathbf k},\hat a_{\mathbf k'}^\dagger]
=(2\pi)^3\delta^{(3)}(\mathbf k-\mathbf k'),
\qquad
[\hat a_{\mathbf k},\hat a_{\mathbf k'}]
=[\hat a_{\mathbf k}^\dagger,\hat a_{\mathbf k'}^\dagger]=0.
$$
The <two-point correlation function> in their vacuum has Fourier amplitude $|f_k|^2/a^2$. After <cosmological horizon exit>, $|k\tau|\ll1$, and the stated <Bunch-Davies vacuum> mode and $a=-1/(H\tau)$ give
$$
|f_k|^2\simeq\frac1{2k^3\tau^2},
\qquad
\frac{|f_k|^2}{a^2}=\frac{H^2}{2k^3}.
$$
The dimensionless <power spectrum> is consequently
$$
\boxed{\Delta_{\delta\phi}^2(k)
=\frac{k^3}{2\pi^2}\frac{|f_k|^2}{a^2}
=\left(\frac{H}{2\pi}\right)^2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-310/4/a/solution