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Inflationary perturbations with multiple scalar fields

B. J. W. van Tent, S. Groot Nibbelink

TL;DR

The paper addresses how to compute inflationary perturbations in a multi-field setting with non-minimal kinetic terms by formulating an induced field-space basis and generalized slow-roll functions. It extends the Mukhanov-Sasaki framework to a vector-valued perturbation ${\mathbf{q}}$ and a scalar ${u}$, deriving coupled equations that couple adiabatic and entropy perturbations through ${\tilde{\eta}}^{\perp}$ and solving them across sub-horizon, transition, and super-horizon regimes; horizon crossing is treated with a Hankel function and a consistent matching yields a first-order analytic expression for the gravitational-potential correlator at recombination. A key finding is that entropy perturbations can contribute at leading order through the particular solution driven by ${\tilde{\eta}}^{\perp}$ and the projection $U_{P\,e}$, as shown in a quadratic two-field example where the particular part contributes substantially to both amplitude and spectral index $n-1$. The work generalizes prior results and provides a practical framework for predicting multi-field inflation signatures in the CMB and large-scale structure, applicable to string-inspired models with curved field manifolds.

Abstract

The calculation of scalar gravitational and matter perturbations during multiple-field inflation valid to first order in slow roll is discussed. These fields may be the coordinates of a non-trivial field manifold and hence have non-minimal kinetic terms. A basis for these perturbations determined by the background dynamics is introduced, and the slow-roll functions are generalized to the multiple-field case. Solutions for a perturbation mode in its three different behavioural regimes are combined, leading to an analytic expression for the correlator of the gravitational potential. Multiple-field effects caused by the coupling to the field perturbation perpendicular to the field velocity can even contribute at leading order. This is illustrated numerically with an example of a quadratic potential. (The material here is based on previous work by the authors presented in hep-ph/0107272.)

Inflationary perturbations with multiple scalar fields

TL;DR

The paper addresses how to compute inflationary perturbations in a multi-field setting with non-minimal kinetic terms by formulating an induced field-space basis and generalized slow-roll functions. It extends the Mukhanov-Sasaki framework to a vector-valued perturbation and a scalar , deriving coupled equations that couple adiabatic and entropy perturbations through and solving them across sub-horizon, transition, and super-horizon regimes; horizon crossing is treated with a Hankel function and a consistent matching yields a first-order analytic expression for the gravitational-potential correlator at recombination. A key finding is that entropy perturbations can contribute at leading order through the particular solution driven by and the projection , as shown in a quadratic two-field example where the particular part contributes substantially to both amplitude and spectral index . The work generalizes prior results and provides a practical framework for predicting multi-field inflation signatures in the CMB and large-scale structure, applicable to string-inspired models with curved field manifolds.

Abstract

The calculation of scalar gravitational and matter perturbations during multiple-field inflation valid to first order in slow roll is discussed. These fields may be the coordinates of a non-trivial field manifold and hence have non-minimal kinetic terms. A basis for these perturbations determined by the background dynamics is introduced, and the slow-roll functions are generalized to the multiple-field case. Solutions for a perturbation mode in its three different behavioural regimes are combined, leading to an analytic expression for the correlator of the gravitational potential. Multiple-field effects caused by the coupling to the field perturbation perpendicular to the field velocity can even contribute at leading order. This is illustrated numerically with an example of a quadratic potential. (The material here is based on previous work by the authors presented in hep-ph/0107272.)

Paper Structure

This paper contains 5 sections, 15 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Perturbations
  4. Example
  5. Conclusions

Figures (1)

  • Figure 1: a) The slow-roll functions and b) the particular contribution $U_{P}$ to the gravitational correlator as a function of the number of e-folds $N$ in the model with two fields on a flat manifold with a quadratic potential with masses $m_1 = 1 \cdot 10^{-5}$, $m_2 = 2.5 \cdot 10^{-5}$ and initial conditions $\phi_1 = 20$, $\phi_2 = 25$.