[Paper Review] Absence of one-loop effects on large scales from small scales in non-slow-roll dynamics

Jacopo Fumagalli|arXiv (Cornell University)|May 30, 2023

Spectroscopy and Quantum Chemical Studies27 citations

TL;DR

The paper proves that one-loop corrections to the large-scale power spectrum from small-scale non-slow-roll dynamics are always negligible due to volume suppression, with exact cancellations when all relevant interactions are included. This holds across two equivalent Hamiltonian formalisms and includes boundary-term considerations.

ABSTRACT

We question the existence of one-loop corrections to the large-scale power spectrum from small-scale modes in non-slow-roll dynamics which are not volume suppressed by the ratio of the short to long distance scales. One-loop contributions proportional to the long wavelength tree-level power spectrum, and not sharing this suppression, have appeared in studies involving interactions singled out by the non-slow-roll dynamics. In this context, we show the relevance of seemingly irrelevant interactions terms, such as the one provided by total derivative terms (boundary terms), and how they equally lead to non-volume suppressed contributions and exact cancellations.

Motivation & Objective

• Motivate the study of non-slow-roll phases during inflation to generate enhanced small-scale fluctuations, as in primordial black hole scenarios.
• Investigate whether one-loop corrections to the large-scale power spectrum from enhanced small scales are significant.
• Clarify the role of various interaction terms, including boundary (total derivative) terms, in equal-time correlators.
• Demonstrate the cancellations that lead to negligible one-loop contributions when all relevant interactions are accounted for.

Proposed method

• Compute one-loop corrections to the equal-time two-point function using the in-in formalism.
• Use two equivalent forms of the third-order action to derive interaction Hamiltonians and check cancellations.
• Split the cubic action into bulk and boundary pieces and evaluate their contributions to the one-loop power spectrum.
• Apply approximations appropriate for transient non-slow-roll phases with a top-hat eta profile and p << k, focusing on long versus short scale mode interactions.
• Perform nested-commutator evaluations of two insertions of the cubic Hamiltonian to obtain the one-loop corrections.
• Demonstrate exact cancellations between contributions proportional to the long-wavelength tree-level power spectrum and boundary-term effects.

Experimental results

Research questions

• RQ1 Do one-loop corrections to the large-scale power spectrum from small-scale non-slow-roll dynamics remain significant after accounting for all relevant interactions?
• RQ2 Are boundary terms and different equivalent Hamiltonian representations sufficient to establish cancellations of these one-loop contributions?
• RQ3 How do bulk versus boundary contributions compare in generating or canceling loop corrections in transient non-slow-roll inflation?
• RQ4 Is the leading one-loop correction to the large-scale power spectrum volume suppressed relative to the short-scale cutoff, effectively negligible for p << k?

Key findings

• One-loop corrections to the large-scale power spectrum from small-scale non-slow-roll dynamics are negligible due to volume suppression (~p^3/k^3).
• Contributions proportional to the long-wavelength tree-level power spectrum cancel exactly when all relevant interactions are included.
• Boundary terms (total time derivatives) are crucial in one method but their effects cancel in the final result, confirming robustness across formalisms.
• Two equivalent approaches (one emphasizing boundary terms, the other reformulating to minimize boundary effects) yield the same cancellation and c = 0 in the leading Eq. (1) form.
• A residual, volume-suppressed term arising from a loop convolution is also present but negligible for p << k.