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

Understand

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.