One-loop corrections to the curvature perturbation from inflation

TL;DR

This work analyzes one-loop corrections to the primordial curvature perturbation ζ in single-field slow-roll inflation, showing that corrections split into a quantum part from horizon-crossing interference and a classical part from superhorizon mode accumulation, with large logarithms potentially threatening naive perturbation theory across huge scale separations. Using the non-linear δN formalism, the authors derive a renormalized expression for the one-loop power spectrum that incorporates both q-loop and c-loop contributions, and they discuss infrared regularization via a finite box and the freedom to choose the initial slicing to maintain ζ conservation. They apply constant-tilt and monomial-potential analyses to estimate loop magnitudes, finding that for standard inflation (N ≈ 60) loops are typically small, but if inflation lasts exponentially long, loop corrections can become substantial, reflecting backreaction effects between Planck-scale theories and observable CMB scales. The results underscore the importance of accounting for loop corrections in connecting high-energy inflationary theories to data, and they point to necessary extensions to multi-field scenarios where isocurvature dynamics can further modify ζ loops and predictions.

Abstract

An estimate of the one-loop correction to the power spectrum of the primordial curvature perturbation is given, assuming it is generated during a phase of single-field, slow-roll inflation. The loop correction splits into two parts, which can be calculated separately: a purely quantum-mechanical contribution which is generated from the interference among quantized field modes around the time when they cross the horizon, and a classical contribution which comes from integrating the effect of field modes which have already passed far beyond the horizon. The loop correction contains logarithms which may invalidate the use of naive perturbation theory for cosmic microwave background (CMB) predictions when the scale associated with the CMB is exponentially different from the scale at which the fundamental theory which governs inflation is formulated.

Figures (2)

• Figure 1: Diagrammatic representation of the $\delta N$ expansion. To represent an $n$-point correlator one draws all $m$-valent vertices attached to $n$ external legs. Internal lines terminate at special vertices marked by a cross '$\times$'. One then draws all possible $j$-gons which connect the $\times$-vertices, of which two possible examples are displayed. 2-gons are treated specially for notational convenience (see Fig. \ref{['fig:one-two-gons']}). Each $m$-valent vertex contributes a "coupling constant" $\partial^{m-1} N/\partial \phi_\ast^{m-1}$.
• Figure 2: Special representation for 2-gon diagrams. In the left-hand diagram the two $\times$-vertices should be connected by a 2-gon, but this is difficult to draw. Instead, we adopt the notation shown in the right-hand diagram, where the 2-gon is represented by two parallel lines transverse to the propagators in which they are inserted. In principle there should also be 1-gon initial conditions, where a single $\times$-vertex is tied off at a one-point function $\langle \delta\phi \rangle$, but because we are neglecting the one-point function on large scales such 1-gons do not appear in these diagrams.