Sharp inflaton potentials and bi-spectra: Effects of smoothening the discontinuity

TL;DR

This paper investigates how smoothing the sharp features in inflaton potentials, specifically in the Starobinsky model, affects the scalar bi-spectrum, showing that smoothing restores scale invariance at high wavenumbers.

Contribution

It analytically and numerically demonstrates that smoothing discontinuities in the inflaton potential leads to a scale-invariant bi-spectrum at large wavenumbers, resolving unphysical growth issues.

Findings

Smoothing the potential removes the unphysical linear growth of the bi-spectrum.

Analytical and numerical results agree on the scale invariance restoration.

The developed BINGO code efficiently computes the bi-spectrum for smooth potentials.

Abstract

Sharp shapes in the inflaton potentials often lead to short departures from slow roll which, in turn, result in deviations from scale invariance in the scalar power spectrum. Typically, in such situations, the scalar power spectrum exhibits a burst of features associated with modes that leave the Hubble radius either immediately before or during the epoch of fast roll. Moreover, one also finds that the power spectrum turns scale invariant at smaller scales corresponding to modes that leave the Hubble radius at later stages, when slow roll has been restored. In other words, the imprints of brief departures from slow roll, arising out of sharp shapes in the inflaton potential, are usually of a finite width in the scalar power spectrum. Intuitively, one may imagine that the scalar bi-spectrum too may exhibit a similar behavior, i.e. a restoration of scale invariance at small scales, when slow roll has been reestablished. However, in the case of the Starobinsky model (viz. the model described by a linear inflaton potential with a sudden change in its slope) involving the canonical scalar field, it has been found that, a rather sharp, though short, departure from slow roll can leave a lasting and significant imprint on the bi-spectrum. The bi-spectrum in this case is found to grow linearly with the wavenumber at small scales, a behavior which is clearly unphysical. In this work, we study the effects of smoothening the discontinuity in the Starobinsky model on the scalar bi-spectrum. Focusing on the equilateral limit, we analytically show that, for smoother potentials, the bi-spectrum indeed turns scale invariant at suitably large wavenumbers. We also confirm the analytical results numerically using our newly developed code BINGO. We conclude with a few comments on certain related points.