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Non-Gaussianity beyond slow roll in multi-field inflation

Christian T. Byrnes, Gianmassimo Tasinato

TL;DR

This paper develops an exact, beyond-slow-roll framework for non-Gaussianity in multi-field inflation by recasting the dynamics with a separable Hubble rate H(φ,χ) = H^(1)(φ) + H^(2)(χ) within the Hamilton-Jacobi formalism and the δN approach. It derives compact, exact expressions for the spectral index n_ζ and the bispectrum parameter f_NL^(4) in terms of trajectory-specific quantities, highlighting how large isocurvature-to-adiabatic conversion (quantified by γ) can boost f_NL at the end of inflation even when slow-roll breaks down. Two exact two-field solutions illustrate the outcomes: a quadratic potential yields negligible non-Gaussianity, while an exponential potential can produce observably large f_NL near the end of inflation, with the trispectrum (τ_NL and g_NL) potentially becoming the dominant signal in some regimes. These results enable analytical exploration of regions in field space inaccessible to slow-roll approximations and invite extensions to post-inflation dynamics and non-canonical kinetic terms, broadening the phenomenological tests of multi-field inflation models.

Abstract

We study the non-Gaussianity generated during multiple-field inflation. We provide an exact expression for the bispectrum parameter f_NL which is valid beyond the slow-roll regime, valid for certain classes of inflationary models. We then study a new, exact multi-field inflationary model considering a case where the bispectrum grows to observable values at the end of inflation. We show that in this case the trispectrum is also large and may even provide the dominant signal of non-Gaussianity.

Non-Gaussianity beyond slow roll in multi-field inflation

TL;DR

This paper develops an exact, beyond-slow-roll framework for non-Gaussianity in multi-field inflation by recasting the dynamics with a separable Hubble rate H(φ,χ) = H^(1)(φ) + H^(2)(χ) within the Hamilton-Jacobi formalism and the δN approach. It derives compact, exact expressions for the spectral index n_ζ and the bispectrum parameter f_NL^(4) in terms of trajectory-specific quantities, highlighting how large isocurvature-to-adiabatic conversion (quantified by γ) can boost f_NL at the end of inflation even when slow-roll breaks down. Two exact two-field solutions illustrate the outcomes: a quadratic potential yields negligible non-Gaussianity, while an exponential potential can produce observably large f_NL near the end of inflation, with the trispectrum (τ_NL and g_NL) potentially becoming the dominant signal in some regimes. These results enable analytical exploration of regions in field space inaccessible to slow-roll approximations and invite extensions to post-inflation dynamics and non-canonical kinetic terms, broadening the phenomenological tests of multi-field inflation models.

Abstract

We study the non-Gaussianity generated during multiple-field inflation. We provide an exact expression for the bispectrum parameter f_NL which is valid beyond the slow-roll regime, valid for certain classes of inflationary models. We then study a new, exact multi-field inflationary model considering a case where the bispectrum grows to observable values at the end of inflation. We show that in this case the trispectrum is also large and may even provide the dominant signal of non-Gaussianity.

Paper Structure

This paper contains 10 sections, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Brief recap of $\delta N$ approach to evolution of perturbations
  3. The method
  4. The homogeneous equations
  5. Evolution of perturbations
  6. Curvature perturbation and non-Gaussianities during inflation
  7. Exact solutions
  8. Quadratic potential: no enhancement of $f_{\rm NL}$
  9. Exponential potential: enhancement of $f_{\rm NL}$ towards the end of inflation
  10. Conclusions and Outlook

Figures (2)

  • Figure 1: Plot showing $f_{NL}$ as given by (\ref{['fNLevolve']}) as a function of $\epsilon_H$ towards the end of inflation, for the values of the parameters given in the text. Inflation ends when $\epsilon_H=1$; for this example $f_{NL}\simeq-59$ at that time.
  • Figure 2: Left plot shows the trajectory considered for the parameters given after eq. (\ref{['fNLconcrete2']}) superimposed on a contour plot of the potential. The square on the trajectory indicates a point along the trajectory one $e$-folding before inflation ends as $\phi$ and $\chi$ roll towards zero. This shows that the fields roll much more quickly during the final stage of inflation, and the trajectory curves at the end as discussed in the text after (\ref{['straight']}). The right plot shows the potential for the same parameter values. Notice that inflation ends on the plateau long before the potential becomes negative.