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