Squeezed vacua and primordial features in effective theories of inflation at N2LO

TL;DR

This paper derives a detailed theoretical expression for phase features in the primordial power spectrum during inflation, incorporating effects beyond leading order in effective theories, with implications for interpreting cosmological data.

Contribution

It provides a closed-form, next-to-next-leading order (N2LO) expression for the phase in primordial features within effective inflationary theories, connecting it to observable parameters like the scalar tilt.

Findings

Derived a closed-form expression for the phase _k in inflationary models.

Connected the phase _k to the scalar tilt n_s in the Starobinsky model.

Predicted a negative shift and running frequency in primordial features.

Abstract

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 non-trivial 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 $\delta_{k\ast}=\frac{\pi}{2}(n_\mathrm{s}-1)-\frac{\pi}{4}(n_\mathrm{s}-1)^2\,\ln(k/k_*)$. 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.