Comparing sharp and smooth transitions of the second slow-roll parameter in single-field inflation

TL;DR

This paper compares sharp and smooth transitions of the second slow-roll parameter in single-field inflation, analyzing their effects on curvature perturbations and non-Gaussianities, with implications for the validity of the Wands duality.

Contribution

It introduces a detailed comparison of sharp versus smooth transitions of the second slow-roll parameter and their impact on the bispectrum and loop corrections in inflation.

Findings

Wands duality condition determines vanishing of certain corrections.

Sharp transitions cause large one-loop corrections at the transition.

Smooth transitions maintain smaller, controlled corrections.

Abstract

In single-field inflation, violation of the slow-roll approximation can lead to growth of curvature perturbation outside the horizon. This violation is characterized by a period with a large negative value of the second slow-roll parameter. At an early time, inflation must satisfy the slow-roll approximation, so the large-scale curvature perturbation can explain the cosmic microwave background fluctuations. At intermediate time, it is viable to have a theory that violates the slow-roll approximation, which implies amplification of the curvature perturbation on small scales. Specifically, we consider ultraslow-roll inflation as the intermediate period. At late time, inflation should go back to the slow roll period so that it can end. This means that there are two transitions of the second slow-roll parameter. In this paper, we compare two different possibilities for the second transition: sharp and smooth transitions. Focusing on effects generated by the relevant cubic self-interaction of the curvature perturbation, we find that the bispectrum and one-loop correction to the power spectrum due to the change of the second slow-roll parameter vanish if and only if the Mukhanov-Sasaki equation for perturbation satisfies a specific condition called Wands duality. We also find in the case of sharp transition that, even though this duality is satisfied in the ultraslow-roll and slow-roll phases, it is severely violated at the transition so that the resultant one-loop correction is extremely large inversely proportional to the duration of the transition.