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Scalar perturbations during multiple-field slow-roll inflation

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

TL;DR

The paper addresses scalar perturbations during slow-roll inflation with multiple real fields on a curved field manifold. It develops a covariant, field-trajectory–driven framework that uses a background-dynamics–induced basis and generalized slow-roll functions to analyze perturbations in three regimes around horizon crossing, deriving first-order analytic solutions for the coupled perturbations and a nontrivial particular solution that sources the adiabatic mode. The authors provide explicit expressions for adiabatic, isocurvature, and cross-correlators of the gravitational potential at recombination, and validate the theory with a quadratic-potential two-field example showing that multi-field effects can be substantial when ${\tilde{\eta}}^{\perp}$ is large. This framework is broadly applicable to high-energy theories with many scalar fields and non-minimal kinetic terms, enabling robust predictions for CMB and large-scale structure and clarifying when multi-field effects matter.

Abstract

We calculate the scalar gravitational and matter perturbations in the context of slow-roll inflation with multiple scalar fields, that take values on a (curved) manifold, to first order in slow roll. For that purpose a basis for these perturbations determined by the background dynamics is introduced and multiple-field slow-roll functions are defined. To obtain analytic solutions to first order, the scalar perturbation modes have to be treated in three different regimes. Matching is performed by analytically identifying leading order asymptotic expansions in different regions. Possible sources for multiple-field effects in the gravitational potential are the particular solution caused by the coupling to the field perturbation perpendicular to the field velocity, and the rotation of the basis. The former can contribute even to leading order if the corresponding multiple-field slow-roll function is sizable during the last 60 e-folds. Making some simplifying assumptions, the evolution of adiabatic and isocurvature perturbations after inflation is discussed. The analytical results are illustrated and checked numerically with the example of a quadratic potential.

Scalar perturbations during multiple-field slow-roll inflation

TL;DR

The paper addresses scalar perturbations during slow-roll inflation with multiple real fields on a curved field manifold. It develops a covariant, field-trajectory–driven framework that uses a background-dynamics–induced basis and generalized slow-roll functions to analyze perturbations in three regimes around horizon crossing, deriving first-order analytic solutions for the coupled perturbations and a nontrivial particular solution that sources the adiabatic mode. The authors provide explicit expressions for adiabatic, isocurvature, and cross-correlators of the gravitational potential at recombination, and validate the theory with a quadratic-potential two-field example showing that multi-field effects can be substantial when is large. This framework is broadly applicable to high-energy theories with many scalar fields and non-minimal kinetic terms, enabling robust predictions for CMB and large-scale structure and clarifying when multi-field effects matter.

Abstract

We calculate the scalar gravitational and matter perturbations in the context of slow-roll inflation with multiple scalar fields, that take values on a (curved) manifold, to first order in slow roll. For that purpose a basis for these perturbations determined by the background dynamics is introduced and multiple-field slow-roll functions are defined. To obtain analytic solutions to first order, the scalar perturbation modes have to be treated in three different regimes. Matching is performed by analytically identifying leading order asymptotic expansions in different regions. Possible sources for multiple-field effects in the gravitational potential are the particular solution caused by the coupling to the field perturbation perpendicular to the field velocity, and the rotation of the basis. The former can contribute even to leading order if the corresponding multiple-field slow-roll function is sizable during the last 60 e-folds. Making some simplifying assumptions, the evolution of adiabatic and isocurvature perturbations after inflation is discussed. The analytical results are illustrated and checked numerically with the example of a quadratic potential.

Paper Structure

This paper contains 16 sections, 93 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Slow-roll background in multiple-field inflation
  3. Equations of motion for the background
  4. Slow roll
  5. Perturbations in multiple-field inflation
  6. Equations of motion for the perturbations
  7. Quantization of the perturbations
  8. Solutions of the perturbation equations to first order: setup
  9. Slow roll for the perturbations
  10. Solutions of the perturbation equations to first order: calculation
  11. Adiabatic and isocurvature perturbations after inflation
  12. Illustration: scalar fields with a quadratic potential
  13. Analytical expressions for background and perturbations
  14. Numerical example
  15. Conclusions and discussion
  16. ...and 1 more sections

Figures (2)

  • Figure 1: a) Background fields and b) slow-roll functions as a function of the number of e-folds 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 $\phi_1 = 20$, $\phi_2 = 25$.
  • Figure 2: a) The particular contribution $U_{P}$ to the gravitational correlator during the super-horizon region as a function of the number of e-folds. To show the relation with the behaviour of ${\tilde{\eta}}^\perp$, this slow-roll function has been plotted again in figure b), on the same horizontal scale as figure a).