Cancellation of one-loop correction to soft tensor power spectrum

TL;DR

This paper addresses whether enhanced small-scale scalar perturbations can induce scale-invariant one-loop corrections to the large-scale tensor power spectrum. It develops a soft tensor EFT by integrating out short-wavelength scalars on the Schwinger–Keldysh contour and proves that no such corrections arise in the soft limit $k \to 0$ for general time-dependent backgrounds. The cancellation relies on a central identity for Green's functions and is rooted in the diffeomorphism invariance of general relativity, with the remaining allowed operator $h_a^{ij} h_{aij}$ obeying the usual causality-suppressed $k^3$ scaling. The result strengthens the decoupling of small-scale scalar dynamics from large-scale tensor modes and suggests analogous all-order cancellations, with potential extensions to multi-loop and scalar-sector analyses.

Abstract

We demonstrate that there are no scale-invariant one-loop corrections to the superhorizon tensor perturbations from small-scale (potentially enhanced) scalar perturbations, irrespective of the details of inflationary background time evolution. For this purpose we derive a soft tensor effective field theory at leading order in the gradient expansion by integrating out small-scale scalar fluctuations in a general time-dependent background over the Schwinger-Keldysh contour, i.e., we perform loop calculations in the soft limit of external momentum. The absence of scale-invariant corrections originates from the diffeomorphism invariance of general relativity and is therefore unavoidable.

Figures (1)

• Figure 1: The path integral contour of the in-in formalism for the cosmological correlators. We evaluate the correlation functions of gravitons at the future infinity, which corresponds to the end of inflation.