A finite duration of cosmic inflation can result in features ${\mathcal{P}}_{\mathcal{R}}(k)=|{\alpha }_k-{\beta }_k{\mathrm{e}}^{\mathrm{i}{\delta }_k}{|}^2{\mathcal{P}}_{\mathcal{R}}^{(0)}(k)$ in the primordial power spectrum that carry information about a quantum gravity phase before inflation. While the almost scale-invariant power spectrum ${\mathcal{P}}_{\mathcal{R}}^{(0)}$ for the quasi-Bunch-Davies vacuum is fully determined by the inflationary background dynamics, the Bogoliubov coefficients ${\alpha }_k$ and ${\beta }_k$ for the squeezed vacuum depend on new physics beyond inflation and have been used to produce phenomenological templates for the features. The phase ${\delta }_k$ vanishes in de Sitter space and therefore is often neglected, but it results in nontrivial effects in quasi–de Sitter inflationary geometries. Here we consider a large class of effective theories of inflation and provide a closed-form expression for ${\delta }_k$ and for the fully expanded power spectrum up to next-to-next-leading order (N2LO) in the Hubble flow expansion. In particular, for the Starobinsky model of inflation we find that this relative phase can be expressed in terms of the scalar tilt $n_{\mathrm{s}}$ as ${\delta }_{k*}=\frac{\pi }{2}(n_{\mathrm{s}}-1)-\frac{\pi }{4}(n_{\mathrm{s}}-1{)}^2\ln (k/k_{*})$. The relative phase results in a negative shift and a running frequency that have been considered in the most studied phenomenological templates for primordial features, thus providing precise theoretical predictions for upcoming cosmological observations.

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• Inflation
• Quantum fields in curved spacetime
• Quantum gravity