  The next generation of cosmological observations will be sensitive to small
deviations from a pure power law in the primordial power spectrum of the
curvature perturbations. In the context of slow-roll inflation, these
deviations are expected and correspond to the so-called running of the spectral
index. Their measurement would bring as much information as the discovery of
deviations from scale invariance. However, robust parameter inference requires
to marginalize over any possible higher order uncertainties, which have been,
up to now, not fully determined. We tackle this issue by deriving the
inflationary scalar and tensor slow-roll power spectra at next-to-next-to-next
to leading order (N3LO), fully expanded around an observable pivot wavenumber,
for all single field inflationary models having minimal and non-minimal kinetic
terms. Our result therefore encompasses string-inspired inflationary models
having a varying speed of sound.
