Multiple-field inflation and the CMB

TL;DR

The paper develops a comprehensive multi-field slow-roll framework with a potentially curved field space metric and derives explicit expressions for the CMB power-spectrum amplitudes and spectral indices for adiabatic, isocurvature, mixing, and tensor modes. It extends previous work by including derivatives of the spectral indices and by detailing the evolution of the total entropy perturbation ${\tilde S}$, offering generalized observational consistency relations that distinguish one-, two-, and multi-field scenarios through quantities like $U_{P e}$, $V_e$, and ${\tilde{\eta}}^{\perp}$. A key finding is that multi-field effects are governed by ${\tilde{\eta}}^{\perp}$; when this perpendicular-velocity component vanishes near the end of inflation, isocurvature and mixing effects are suppressed, though end-of-inflation and reheating can complicate the adiabatic amplitude. Tensor perturbations, in contrast, do not depend on the number of fields and provide a clean probe of the inflationary energy scale via $H_{\rm H}$, while the total entropy perturbation determines post-inflationary sourcing of adiabatic perturbations. Collectively, these results supply a practical framework to test the presence and number of active fields with upcoming CMB data by exploiting consistency relations, amplitude ratios, and the behavior of ${\tilde S}$ during radiation and matter domination.

Abstract

In this paper, we investigate some consequences of multiple-field inflation for the cosmic microwave background radiation (CMB). We derive expressions for the amplitudes, the spectral indices and the derivatives of the indices of the CMB power spectrum in the context of a very general multiple-field theory of slow-roll inflation, where the field metric can be non-trivial. Both scalar (adiabatic, isocurvature and mixing) and tensor perturbations are treated and the differences with single-field inflation are discussed. From these expressions, several relations are derived that can be used to determine the importance of multiple-field effects observationally from the CMB. We also study the evolution of the total entropy perturbation during radiation and matter domination and the influence of this on the isocurvature spectral quantities.