A General Analytic Formula for the Spectral Index of the Density Perturbations produced during Inflation

TL;DR

This paper addresses how inflationary density perturbations arise when multiple dynamical fields are active. It develops a general analytic framework based on the delta-N formalism, expressing the curvature perturbation spectrum $P_{\,\cal R}$ and its spectral index $n_{\cal R}$ in terms of derivatives of the expansion count $N$ with respect to the scalar fields and the geometry of field space, including its curvature. A compact result is $P_{\,\cal R} = (H/2\pi)^2 \; h^{ab} N_{;a} N_{;b}$ with the spectral index $n_{\cal R}-1$ given by a formula involving $(\ln V)^{;a}$ and the Riemann tensor $R^a_{\;bcd}$ of the field-space, while the tensor spectrum is $P_g=(H/2\pi)^2$ with $n_g = 2\dot H/H^2$. The work highlights that multi-field inflation generally yields non-unique trajectories, enabling potential observational signatures in the CMB and large-scale structure beyond single-field predictions.

Abstract

The standard calculation of the spectrum of density perturbations produced during inflation assumes that there is only one real dynamical degree of freedom during inflation. However, there is no reason to believe that this is actually the case. In this paper we derive general analytic formulae for the spectrum and spectral index of the density perturbations produced during inflation.