Decoupling of large-scale, adiabatic inflationary perturbations from enhanced small-scale modes at one-loop

TL;DR

This paper analyzes how large-scale inflationary perturbations are unaffected by small-scale modes at one-loop order, showing that the back-reaction decouples and is generally unobservable, using the separate universe approach and soft theorems.

Contribution

It demonstrates that one-loop back-reaction from small-scale modes decouples from large-scale perturbations and is scale-invariant, extending the separate universe method with a $ abla N$ approach.

Findings

Back-reaction at 1-loop is due to non-linearity or initial condition corrections.

The loop correction decouples from detailed properties of small-scale modes.

Back-reaction is scale-invariant and generally unobservable.

Abstract

We reconsider back-reaction from large amplitude, short-scale perturbations onto a long wavelength adiabatic mode. In a loop expansion of the long-mode power spectrum, this back-reaction appears first at 1-loop. Due to the separation between the long and short scales, the separate universe method provides a simple and efficient framework for this computation. In this paper, building on our earlier work, we employ a $\delta N$ formula for the long mode, which captures the effect of short scales. We show that back-reaction at 1-loop is due to either (i) non-linearity of the $\delta N$ formula, or (ii) 1-loop corrections to the initial conditions. We argue that contributions of type (ii) cannot themselves be described within the separate universe framework, but their properties can be constrained using soft theorems and a ''multi-point propagator'' expansion. When applied to a band of enhanced short-scale perturbations that crossed the horizon during inflation, our result shows that the loop correction decouples from their detailed properties. Furthermore, the back-reaction we obtain is scale-invariant. Its magnitude is model-dependent, but is degenerate with effects from modes that were still sub-horizon at the end of inflation. In this scenario (but not necessarily in all scenarios), we conclude that the effect is not observable.