Clocking the end of cosmic inflation

Author(s)

Auclair, Pierre, Blachier, Baptiste, Ringeval, Christophe

Abstract

Making observable predictions for cosmic inflation requiresdetermining when the wavenumbers of astrophysical interest todayexited the Hubble radius during the inflationary epoch. Theseinstants are commonly evaluated using the slow-roll approximation andmeasured in e-folds Δ N=N - N$_{end}$, in reference to the e-fold N$_{end}$ at which inflation ended. Slow roll being necessarilyviolated towards the end of inflation, both the approximatedtrajectory and N$_{end}$ are determined at, typically, one or twoe-folds precision. Up to now, such an uncertainty has beeninnocuous, but this will no longer be the case with the forthcomingcosmological measurements. In this work, we introduce a new andsimple analytical method, on top of the usual slow-rollapproximation, that reduces uncertainties on ΔN to less thana tenth of an e-fold.

Figures

Caption

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).

Caption

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}.