An exact model for enhancing/suppressing primordial fluctuations

Enhancements of primordial curvature fluctuations in single field inflation often involve departures from attractor trajectories in the phase space. We study enhancement/suppression of primordial fluctuations in one of the simplest models with exact background solutions for arbitrary initial conditions: a single field inflationary model with a piecewise exponential potential. We then present close to exact analytical solutions for primordial fluctuations in a general transition between two slow-roll attractors, valid whether the first slow parameter increases or decreases. The main features in the primordial spectrum are determined by the ratio of exponents of the potential. We also discuss the imprint of such features in the induced GW spectrum. Lastly, we apply the $\delta N$ formalism to discuss non-Gaussianities and the tail of the probability distribution. We find that while non-Gaussianities are at most ${\cal O}(1)$ in the case of enhancement, they can be very large in the case of suppression. Our work can be easily generalized to multiple piecewise exponential potentials.