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Non-Gaussianity and constraints for the variance of perturbations in the curvaton model

Jussi Valiviita, Misao Sasaki, David Wands

TL;DR

This paper extends the curvaton model analysis of primordial non-Gaussianity beyond the sudden-decay approximation by applying the $\delta N$ formalism to non-instantaneous decay. It derives the exact nonlinear curvature perturbation and provides a numerical procedure to compute the transfer efficiency $r$, showing that discrepancies between sudden-decay and exact results exceed 1% only for a narrow range of $|f_{\rm NL}|$; however, Planck-level constraints may require the fully numerical treatment. It confirms that the sudden-decay approximation is adequate for WMAP3 constraints but could be insufficient for Planck-level bounds. It also discusses how UV-scale variance can influence non-Gaussianity and places an upper bound on small-scale variance, $\Delta^2_{UV} < 90$, with implications for curvaton parameter constraints.

Abstract

Recently, the primordial non-Gaussianity in the curvaton model has been predicted assuming sudden decay of the curvaton. We extend the calculation to non-instantaneous decay by employing delta N -formalism. The difference between the sudden-decay approximation and our numerical result is larger than 1% only if the non-linearity parameter is small, -1.16 < f_NL < 60. Thus it is safe to use the sudden-decay approximation when deriving constraints for the curvaton model from WMAP3 (f_NL < 114), but with the Planck forecast |f_NL| <5 one should employ the fully numerical result. Often, the curvaton perturbations $δσ$ have been assumed to be small compared to the background value of the curvaton field $σ_0$. Consequently, the variance $Δ^2 = <δσ^2> / σ_0^2$ has been assumed to be negligible. However, the measurements of CMB or large-scale structure perturbation amplitude do not constrain the variance if the main contribution to it comes from the ultraviolet (UV) scales, i.e., from smaller than observable scales. We discuss how, even in this case, observational constraints on non-Gaussianity set an upper bound to the small scale variance, Delta^2_UV < 90.

Non-Gaussianity and constraints for the variance of perturbations in the curvaton model

TL;DR

This paper extends the curvaton model analysis of primordial non-Gaussianity beyond the sudden-decay approximation by applying the formalism to non-instantaneous decay. It derives the exact nonlinear curvature perturbation and provides a numerical procedure to compute the transfer efficiency , showing that discrepancies between sudden-decay and exact results exceed 1% only for a narrow range of ; however, Planck-level constraints may require the fully numerical treatment. It confirms that the sudden-decay approximation is adequate for WMAP3 constraints but could be insufficient for Planck-level bounds. It also discusses how UV-scale variance can influence non-Gaussianity and places an upper bound on small-scale variance, , with implications for curvaton parameter constraints.

Abstract

Recently, the primordial non-Gaussianity in the curvaton model has been predicted assuming sudden decay of the curvaton. We extend the calculation to non-instantaneous decay by employing delta N -formalism. The difference between the sudden-decay approximation and our numerical result is larger than 1% only if the non-linearity parameter is small, -1.16 < f_NL < 60. Thus it is safe to use the sudden-decay approximation when deriving constraints for the curvaton model from WMAP3 (f_NL < 114), but with the Planck forecast |f_NL| <5 one should employ the fully numerical result. Often, the curvaton perturbations have been assumed to be small compared to the background value of the curvaton field . Consequently, the variance has been assumed to be negligible. However, the measurements of CMB or large-scale structure perturbation amplitude do not constrain the variance if the main contribution to it comes from the ultraviolet (UV) scales, i.e., from smaller than observable scales. We discuss how, even in this case, observational constraints on non-Gaussianity set an upper bound to the small scale variance, Delta^2_UV < 90.

Paper Structure

This paper contains 4 sections, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sudden-decay approximation
  3. Non-instantaneous decay
  4. Large variance on small scales

Figures (1)

  • Figure 1: (a) The non-linearity parameter $f_{\rm{NL}}$ as a function of curvature perturbation transfer efficiency $r = \zeta_{r, \rm out} / \zeta_{\sigma, \rm in}$. The analytical approximative, i.e., sudden-decay result ( red dashed line) crosses zero at $r = 0.58$ and is negative for $r>0.58$. The exact numerical result ( black solid line) crosses zero at $r=0.53$. (b) $r$ vs $p_{\rm in}$.