Inflationary Correlation Functions without Infrared Divergences

TL;DR

The paper addresses infrared (IR) divergences in inflationary correlators by introducing IR-safe correlation functions defined on the reheating surface via an invariant distance. IR corrections from long-wavelength scalar and tensor modes are shown to factorize into background averages and sum to all orders, yielding closed-form, all-orders expressions for the IR-safe power spectrum and higher-point functions. In slow-roll, single-field inflation, the leading log-enhanced corrections to the power spectrum arise with scalar and tensor contributions, with tensor modes often dominating the $f_{\rm NL}$ corrections; these results are connected to, and reproduce, standard perturbative results in the appropriate limits. The framework unifies IR effects, provides practical all-orders resummations, and extends to a generalized $\delta N$-formalism, offering a path to multi-field extensions and a rigorous basis for understanding IR behavior in inflationary observables.

Abstract

Inflationary correlation functions are potentially affected by infrared divergences. For example, the two-point correlator of curvature perturbation at momentum k receives corrections ~ln(kL), where L is the size of the region in which the measurement is performed. We define infrared-safe correlation functions which have no sensitivity to the size L of the box used for the observation. The conventional correlators with their familiar log-enhanced corrections (both from scalar and tensor long-wavelength modes) are easily recovered from our IR-safe correlation functions. Among other examples, we illustrate this by calculating the corrections to the non-Gaussianity parameter f_NL coming from long-wavelength tensor modes. In our approach, the IR corrections automatically emerge in a resummed, all-orders form. For the scalar corrections, the resulting all-orders expression can be evaluated explicitly.