Corrections to $n_s$ and $n_t$ from high scale physics

Abstract

When modelling inflaton fluctuations as a free quantum scalar field, the initial vacuum is conventionally imposed at the infinite past. This is called the Bunch-Davies (BD) vacuum. If however an asymptotically Minkowskian past does not exist, this requires modifications. We derive corrections to the scalar spectral index $n_s$ and the tensor tilt $n_t$ descending from arbitrary mixed states or from explicit non-BD initial conditions. The former may stem from some pre-inflationary background and can redshift away whereas the latter are induced by a timelike hypersurface parametrising a physical cut-off. In both cases, we find that corrections scale in parts or fully as $\mathcal O(\epsilon)$ where $\epsilon$ is the first slow-roll parameter. The precise observational footprint is hence dependent on the model driving inflation. Further, we show how the inflationary consistency relation is altered. We thus provide an analytic handle on possible high scale or pre-inflationary physics.