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Finite volume effects for non-Gaussian multi-field inflationary models

Francis Bernardeau, Jean-Philippe Uzan

TL;DR

This work analyzes finite-volume effects on non-Gaussian signatures in multi-field inflation with a quartically self-interacting auxiliary field $\chi$. Using a Langevin/random-walk framework for the super-Hubble modes, it shows that the finite survey scale induces a nonzero mean $\bar{\chi}$ that skews the observed PDF of the filtered field $\delta\chi_{\rm S}$ and generates a nonzero bispectrum, even if the ensemble distribution is symmetric. The authors derive how second and fourth moments survive finite-volume filtering and demonstrate that higher-order cumulants, notably the three-point function, acquire finite-volume contributions with amplitudes tied to $\bar{\chi}$ and the number of e-folds $N_e$. Their results highlight a third observational parameter in addition to the intrinsic non-Gaussianity strength and its PDF, with implications for interpreting CMB and LSS data in multi-field inflation models.

Abstract

Models of multi-field inflation exhibiting primordial non-Gaussianity have recently been introduced. This is the case in particular if the fluctuations of a light field scalar field, transverse to the inflaton direction, with quartic coupling can be transferred to the metric fluctuations. So far in those calculations only the ensemble statistical properties have been considered. We explore here how finite volume effects could affect those properties. We show that the expected non-Gaussian properties survive at a similar level when the finite volume effects are taken into account and also find that they can skew the metric distribution even though the ensemble distribution is symmetric.

Finite volume effects for non-Gaussian multi-field inflationary models

TL;DR

This work analyzes finite-volume effects on non-Gaussian signatures in multi-field inflation with a quartically self-interacting auxiliary field . Using a Langevin/random-walk framework for the super-Hubble modes, it shows that the finite survey scale induces a nonzero mean that skews the observed PDF of the filtered field and generates a nonzero bispectrum, even if the ensemble distribution is symmetric. The authors derive how second and fourth moments survive finite-volume filtering and demonstrate that higher-order cumulants, notably the three-point function, acquire finite-volume contributions with amplitudes tied to and the number of e-folds . Their results highlight a third observational parameter in addition to the intrinsic non-Gaussianity strength and its PDF, with implications for interpreting CMB and LSS data in multi-field inflation models.

Abstract

Models of multi-field inflation exhibiting primordial non-Gaussianity have recently been introduced. This is the case in particular if the fluctuations of a light field scalar field, transverse to the inflaton direction, with quartic coupling can be transferred to the metric fluctuations. So far in those calculations only the ensemble statistical properties have been considered. We explore here how finite volume effects could affect those properties. We show that the expected non-Gaussian properties survive at a similar level when the finite volume effects are taken into account and also find that they can skew the metric distribution even though the ensemble distribution is symmetric.

Paper Structure

This paper contains 13 sections, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. An overview of finite volume effects
  3. Models
  4. Self-interacting scalar fields in de Sitter space
  5. Second and fourth moment
  6. Lessons from a random walk approach
  7. Scales intervening in the problem
  8. The late time PDF of $\chi_{_{\rm H}}(\eta)$.
  9. Consequences on the shape of the PDF of $\delta\chi_{_{\rm S}}$
  10. Finite volume effects on high order cumulants
  11. The bi-spectrum
  12. The three-point cumulant
  13. Conclusions

Figures (2)

  • Figure 1: The different filtered quantities and scales entering our problem. $\chi_{_{\rm H}}$ follows a stochastic dynamics that can be described by a Langevin equation; it has a time dependent smoothing scale so that more and more modes contribute to the filtered field. The field values ${\bar{\chi}}$ and $\chi_{_{\rm S}}$ evolve according a classical Klein-Gordon equation; their smoothing scale is time independent; they coincide with $\chi_{_{\rm H}}$ at horizon crossing times.
  • Figure 2: PDF of $\delta\chi_{_{\rm S}}=\delta\chi_{_{\rm S}}-{\bar{\chi}}$ for different values of ${\bar{\chi}}$. The dashed line corresponds to a Gaussian distribution; the dot-dashed line to the deformed distribution of $\delta\chi_{_{\rm S}}$ when $\lambda N_e/H^2=1$ and ${\bar{\chi}}=0$ and the solid lines when the latter equals $0.5$ and $1$.