Choice of Quantum Vacuum for Inflation Observables

Abstract

We investigate the modifications to inflationary observables that arise when adopting an $α$-vacuum instead of the standard Bunch--Davies vacuum for quantum fluctuations during inflation. Within the Starobinsky inflationary model, we compute and compare the scalar spectral index, its running, and the running of the running arising from different choices of the initial vacuum state. We further examine the energy scales associated with $α$-vacua and argue that, for any number of extra spatial dimensions, the relevant scale can be truncated at the Hubble scale, $\sim$$\mathcal{O}(10^{13})\,\mathrm{GeV}$, without conflict with current Cavendish-type experimental bounds on sub-millimeter gravity ($\sim$$250\,μ\mathrm{m}$). Our analysis demonstrates that the $α$-vacuum is subject to stringent constraints as a viable de~Sitter-invariant alternative to the Euclidean (Bunch--Davies) vacuum, with the corrections that it induces in the inflationary observables being strongly limited by the latest Planck data.

Figures (4)

• Figure S1: Comparison between the $\alpha$-vacuum prediction for the power spectrum given by Equation (\ref{['Pr']}) and its BD limit, ${\cal P}_{\cal R}^{\rm (BD)}\equiv (H^4/(4 \pi^2 \dot \phi^2)$, as a function of $\Lambda/H$. The BD result is recovered for $\Lambda\gg H$.
• Figure S2: Range of values that satisfy $R < r_{cav}$ in terms of the number of extra spatial dimensions $n$ and the energy scale of gravity on the bulk $M_{*}$.
• Figure S3: Normalization of the Starobinsky potential as a function of $\lambda$, computed using the full power spectrum result Equation (\ref{['Pr']}). The dashed line shows the BD limit.
• Figure S4: Predictions for (a) $n_s$, (b) $\alpha_s$, and (c) $\beta_s$ using Equations (\ref{['nsalpha']})--(\ref{['d2nsalpha']}), with $N_*=60$. Solid: $\alpha$-vacuum with $\lambda$-dependent corrections. Dashed: BD limit.