Super-Horizon Scale Dynamics of Multi-Scalar Inflation

TL;DR

The paper addresses the evolution of super-horizon perturbations in multi-field inflation by developing a linear long-wavelength framework that links perturbations to derivatives of spatially homogeneous background solutions. It shows that, in a convenient gauge with the e-folding number $N$ unperturbed, the scalar-field perturbations ${\boldsymbol \chi}$ correspond to derivatives ${\partial{\boldsymbol \phi}}/{\partial\lambda^\alpha}$ with phase-space parameters $\lambda^\alpha=(N,\lambda^a)$, and provides a closed equation for ${\boldsymbol \chi}$ along with a formula for the curvature perturbation ${\cal R}_c$ on the comoving hypersurface. In a slow-roll limit, the results reduce to the Sasaki–Stewart expression, while a quasi-nonlinear extension via a gradient- (anti-Newtonian) expansion demonstrates how nonlinear scalar-field dynamics can be incorporated and how ${\cal R}_c$ can be estimated using background data, with potential implications for non-Gaussian statistics. Altogether, the approach offers a broad, background-solution–based method to analyze multi-field inflation perturbations, including nonlinear effects, without restricting to slow-roll.

Abstract

We consider the dynamics of a multi-component scalar field on super-horizon scales in the context of inflationary cosmology. We present a method to solve the perturbation equations on super- horizon scales, i.e., in the long wavelength limit, by using only the knowledge of spatially homogeneous background solutions. In doing so, we clarify the relation between the perturbation equations in the long wavelength limit and the background equations. Then as a natural extension of our formalism, we provide a strategy to study super-horizon scale perturbations beyond the standard linear perturbation theory. Namely we reformulate our method so as to take into account the nonlinear dynamics of the scalar field.