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

The creation and annihilation operators obey the canonical commutation relationsSince $\delta \hat{\varphi }=\hat{f}/a$, the vacuum two-point function identifiesOn superhorizon scales, $|k\tau |=k/(aH)\ll 1$, so

For each mode, the right side ofis evaluated at cosmological horizon exit, $k=aH$. Sincefirst order in the Hubble slow-roll parameters gives

For the monomial inflation potential $V=\lambda M_{\mathrm{Pl}}^4(\varphi /M_{\mathrm{Pl}}{)}^{\alpha }$,andSlow-roll inflation is possible when both are much smaller than one:Inflation ends once one of these conditions fails, commonly near $|\varphi |\simeq \alpha M_{\mathrm{Pl}}/\sqrt{2}$ when $ϵ\simeq 1$.

The expressions in part c implySince the tensor-to-scalar ratio is $r=16ϵ$,For positive $\alpha$, this class predicts a red scalar tilt andA blue tilt with positive $r$, or $r\ge 8(1-n_s)$, is incompatible. An allowed pair fixes $\alpha =2r/[8(1-n_s)-r]$, subject to slow roll and observational bounds.