## Author(s)

Martin, Jerome, Ringeval, Christophe, Vennin, Vincent## Abstract

Making observable predictions for cosmic inflation requires determining when the wavenumbers of astrophysical interest today exited the Hubble radius during the inflationary epoch. These instants are commonly evaluated using the slow-roll approximation and measured in e-folds $\Delta N=N-N_\mathrm{end}$, in reference to the e-fold $N_\mathrm{end}$ at which inflation ended. Slow roll being necessarily violated towards the end of inflation, both the approximated trajectory and $N_\mathrm{end}$ are determined at, typically, one or two e-folds precision. Up to now, such an uncertainty has been innocuous, but this will no longer be the case with the forthcoming cosmological measurements. In this work, we introduce a new and simple analytical method, on top of the usual slow-roll approximation, that reduces uncertainties on $\Delta N$ to less than a tenth of an e-fold.

## Figures

Absolute error, in {\efolds}, of the slow-roll approximated trajectory (in red) with respect to the exact value of $\Delta N(\phi)$ for various prototypical models of inflation. The blue curve shows $\Delta\Nsree(\phi) - \Delta N(\phi)$ where $\Delta\Nsree = \Nsr(\phi) - \Nsr(\phiend)$, $\phiend$ being the \emph{exact} field value at which inflation stops. The differences between the red and blue curves are the errors induced by using $\phiendsr$ instead of $\phiend$ (see text). Let us notice that the Pseudo Natural Inflationary model (lower right) is an extreme case as it has its parameters purposely chosen to be in a slow-roll violating regime (incompatible with current data).

Absolute error, in {\efolds}, of the velocity-corrected trajectory $\Delta\Nsrvc - \Delta N$ (blue curve), of the velocity plus end-point corrected trajectories $\Delta\Nsrvcm - \Delta N$ (green curve) and $\Delta\Nsrvcp - \Delta N$ (magenta curve), with respect to the exact value of $\Delta N(\phi)$ for various prototypical models of inflation. The red curve is the error associated with the traditional slow-roll approximation, same as in \cref{fig:srtrajs}.

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