Cosmological Perturbations with Multiple Scalar Fields

TL;DR

This work develops a gauge-ready formalism to study scalar-type cosmological perturbations in models with multiple minimally coupled scalar fields within Einstein gravity. It derives a complete set of scalar perturbation equations, including gauge-invariant combinations such as $\delta \phi^I_\varphi$ and $\varphi_v$, and presents closed-form second-order evolution equations for the adiabatic mode and isocurvature perturbations $\delta \phi_{IJ}$, with explicit large-scale behavior. Under slow-roll, the adiabatic and isocurvature modes decouple and the adiabatic variable $\varphi_v$ is generally conserved on super-horizon scales, while special potentials yield conserved isocurvature combinations and, in two-field systems, a conserved isocurvature quantity for general potentials. These results provide practical tools for analyzing multi-field inflation and its imprint on large-scale structure, facilitating inflationary potential reconstruction and improved modeling of early-universe perturbations.

Abstract

We consider the evolution of perturbed cosmological spacetime with multiple scalar fields in Einstein gravity. A complete set of scalar-type perturbation equations is presented in a gauge-ready form, and we derived the closed set of second-order differential equations in several useful forms. Conserved behaviors of the perturbed three-space curvature in the comoving gauge, $φ_v$, under several conditions are clarified. Under the slow-roll conditions, the adiabatic and isocurvature modes decouple from each other, and in the large-scale limit we have (i) the adiabatic mode is generally conserved, (ii) for a couple of special potential the isocurvature modes decouple from each other and are described by conserved quantities, (iii) in the two field system, the isocurvature mode is described by a conserved quantity for the general potential.