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Non-Gaussianities in two-field inflation

Filippo Vernizzi, David Wands

Abstract

We study the bispectrum of the curvature perturbation on uniform energy density hypersurfaces in models of inflation with two scalar fields evolving simultaneously. In the case of a separable potential, it is possible to compute the curvature perturbation up to second order in the perturbations, generated on large scales due to the presence of non-adiabatic perturbations, by employing the $δN$-formalism, in the slow-roll approximation. In this case, we provide an analytic formula for the nonlinear parameter $f_{NL}$. We apply this formula to double inflation with two massive fields, showing that it does not generate significant non-Gaussianity; the nonlinear parameter at the end of inflation is slow-roll suppressed. Finally, we develop a numerical method for generic two-field models of inflation, which allows us to go beyond the slow-roll approximation and confirms our analytic results for double inflation.

Non-Gaussianities in two-field inflation

Abstract

We study the bispectrum of the curvature perturbation on uniform energy density hypersurfaces in models of inflation with two scalar fields evolving simultaneously. In the case of a separable potential, it is possible to compute the curvature perturbation up to second order in the perturbations, generated on large scales due to the presence of non-adiabatic perturbations, by employing the -formalism, in the slow-roll approximation. In this case, we provide an analytic formula for the nonlinear parameter . We apply this formula to double inflation with two massive fields, showing that it does not generate significant non-Gaussianity; the nonlinear parameter at the end of inflation is slow-roll suppressed. Finally, we develop a numerical method for generic two-field models of inflation, which allows us to go beyond the slow-roll approximation and confirms our analytic results for double inflation.

Paper Structure

This paper contains 13 sections, 75 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Non-Gaus-sia-n perturbations in multi-field inflation
  3. Two-point statistics
  4. Three-point statistics
  5. Two-field inflation with separable potential
  6. Slow-roll dynamics of background fields
  7. Perturbed expansion during slow-roll
  8. Curvature perturbation and non-Gaussianity during inflation
  9. Perturbations after inflation
  10. Double quadratic inflation
  11. Slow-roll analysis
  12. Numerical analysis
  13. Conclusion

Figures (6)

  • Figure 1: Examples of trajectories in field space are shown from Hubble exit, starting $60$$e$-foldings before the end of inflation at $\epsilon_e=1$, for $R=1/9$. The thick (orange) line represents the trajectory starting from $\phi=\chi=13 m_{\rm P}$, and shown from Hubble exit, $(\phi_*=8.2,\chi_*=12.9)$, to the end of inflation, $(\phi_e=0.0,\chi_e=1.4)$. The grey shading represents the space of all possible trajectories.
  • Figure 2: The values of the fields $\phi$ (solid line) and $\chi$ (dashed red line) during the inflationary trajectory of Fig. \ref{['fig:1']}, are shown as a function of the Hubble rate $H$, from $N\simeq 42$ to $N \simeq 4$$e$-foldings from the end of inflation. Note that time increases from right to left.
  • Figure 3: The energy densities of the fields $\rho_\phi$ (solid line) and $\rho_\chi$ (dashed red line) normalized to $(m_{\rm P} m_\phi)^2$, are shown as for Fig. \ref{['fig:2']}.
  • Figure 4: The power spectrum ${\cal P}_\zeta$ of the large-scale uniform density perturbation $\zeta$ during the inflationary trajectory of Fig. \ref{['fig:1']}, is shown as a function of the Hubble rate $H$, from $N\simeq 42$ to $N \simeq 4$$e$-foldings from the end of inflation ($m_{\rm P}=1$). The solid and the dashed lines represent the analytic and numerical calculations, respectively.
  • Figure 5: The spectral index $n_\zeta$ is shown as for Fig. \ref{['fig:4']}.
  • ...and 1 more figures