Absence of one-loop effects on large scales from small scales in non-slow-roll dynamics II: Quartic interactions and consistency relations

TL;DR

This paper demonstrates that in transient non-slow-roll inflationary phases, one-loop quantum corrections do not affect large-scale primordial fluctuations, confirming the robustness of these scales against small-scale physics.

Contribution

The authors explicitly prove the absence of one-loop corrections to large scales from small scales in non-slow-roll dynamics, including the role of quartic interactions and consistency relations.

Findings

One-loop corrections to large scales are independent of small-scale physics.

Quartic interactions are crucial in expressing one-loop diagrams as three-point functions.

Consistency relations hold for arbitrary transient non-slow-roll phases.

Abstract

We prove explicitly the absence of one-loop corrections to large scales from small scales in transient non-slow-roll dynamics. Specifically, we address loop corrections to the primordial power spectrum, relative to tree-level, that are independent of the ratio between the two scales. We review all the necessary components, adapted to our context, to express one-loop diagrams as three-point functions, emphasizing the crucial role played by quartic interactions. Notably, we include the quartic Hamiltonian induced by the cubic Lagrangian and quartic interactions that are ensured by diffeomorphism invariance. We then explicitly prove consistency relations for an arbitrary transient non-slow-roll phase involving operators with (time) derivatives. Finally, we calculate one-loop corrections by including contributions from the relevant cubic and quartic interactions, and express the final result as a total derivative term over comoving momenta, utilizing the consistency relations we established. This leads us to conclude that one-loop corrections to long-wavelength modes are unaffected by the physics of short and enhanced modes in non-slow-roll dynamics.