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

Abstract

The origin of the finite dip in the curvature power spectrum of instantaneous Slow-Roll (SR)-Ulta-Slow-Roll (USR)-SR inflation remains controversial at linear order, and its full spectral features still lack a complete asymptotic analytical description. We revisit linear perturbation dynamics in this framework. Using the junction method and asymptotic expansions of Hankel functions, we for the first time derive accurate and simple asymptotic expressions for mode evolution and the resulting power spectrum, based on three systematic rules for dominant-term identification across transitions. We find the finite dip arises from cancellation between two growing modes within linear perturbation theory, rather than between constant and growing terms as previously suggested. We also provide analytical descriptions of the amplitude enhancement and the two oscillatory patterns observed in the spectrum. All asypmtotic analytical results are validated by numerical calculations with the reconstructed SR-USR-SR inflationary potential.

37 pages, 9 figures