Density perturbations arising from multiple field slow-roll inflation

TL;DR

The paper develops a comprehensive framework for density perturbations in slow-roll inflation with multiple scalar fields residing on a (possibly curved) field manifold. By introducing vectorized slow-roll functions and projecting perturbations along and orthogonal to the background field velocity, the authors show that the gravitational potential decouples from perpendicular field perturbations to leading order, yielding a single-field-like adiabatic mode described by a generalized Mukhanov–Sasaki variable $u$. They quantify the quantum correlations of the Newtonian potential, justify the vacuum initial state for CMB-relevant scales, and provide explicit results for both flat and curved field spaces using the $F_n$ and $C_n$ formalisms. The work delivers a practical framework to compute the CMB power spectrum in multi-field inflation and highlights the crucial roles of field-space geometry and isocurvature modes in shaping observable perturbations.

Abstract

In this paper we analyze scalar gravitational perturbations on a Robertson-Walker background in the presence of multiple scalar fields that take values on a (geometrically non-trivial) field manifold during slow-roll inflation. For this purpose modified and generalized slow-roll functions are introduced and their properties examined. These functions make it possible to estimate to what extent the gravitational potential decouples from the scalar field perturbations. The correlation function of the gravitational potential is calculated in an arbitrary state. We argue that using the vacuum state seems a reasonable assumption for those perturbations that can be observed in the CMBR. Various aspects are illustrated by examples with multiple scalar fields that take values on flat and curved manifolds.