Evolution of Linear Perturbations under Time-Dependent Hubble Friction I: SR-USR-SR Inflation

TL;DR

This paper analyzes the evolution of linear perturbations during SR-USR-SR inflation with instantaneous transitions, deriving accurate formulas for the power spectrum and revealing new cancellation effects influencing observable features.

Contribution

It introduces a systematic method to derive asymptotic expressions for mode functions during inflation transitions, clarifying the origin of power spectrum dips and oscillations.

Findings

Finite dip in power spectrum caused by cancellation between two growing modes.

Derived analytical formulas match numerical results for amplitude and oscillations.

Formulas provide testable predictions for future CMB observations.

Abstract

In this paper, we revisit the linear perturbation (including the comoving curvature perturbation and field perturbation) dynamics in the SR-USR-SR inflation with instantaneous transitions. Using the junction method and asymptotic expansions of Hankel functions, we derive accurate asymptotic expressions for the time evolution of mode functions and the resulting power spectrum, based on three systematic rules for identifying the dominant terms across transitions. Our results reveal that a finite dip of the final power spectrum arises from the cancellation between two growing modes within the linear perturbation theory, rather than between constant and growing terms as previously suggested. We also provide analytical descriptions of the amplitude enhancement and oscillatory features in the linear power spectrum, in agreement with numerical computations. These simple, tractable formulas not only facilitate theoretical calculations but also yield testable predictions for future CMB observations.