Non-Gaussianity of the primordial perturbation in the curvaton model

TL;DR

This study applies the $\delta N$-formalism to the curvaton model to quantify primordial non-Gaussianity, deriving the full non-linear mapping from initial Gaussian curvaton perturbations to the final curvature perturbation and computing the complete pdf. It provides both analytic results in the sudden-decay limit and a fully non-linear numerical treatment that accounts for gradual curvaton decay, yielding leading bispectrum and trispectrum contributions, including $g_{\rm NL}$. The authors demonstrate that the sudden-decay approximation is accurate for current observational bounds but offer a robust numerical framework for future precision tests and for constraining small-scale variance in the curvaton field. They also show how the non-Gaussian features encode information about the small-scale variance and how CMB-scale constraints limit this variance, enabling multiple consistency tests of the curvaton scenario.

Abstract

We use the delta N -formalism to investigate the non-Gaussianity of the primordial curvature perturbation in the curvaton scenario for the origin of structure. We numerically calculate the full probability distribution function allowing for the non-instantaneous decay of the curvaton and compare this with analytic results derived in the sudden-decay approximation. We also present results for the leading-order contribution to the primordial bispectrum and trispectrum. In the sudden-decay approximation we derive a fully non-linear expression relating the primordial perturbation to the initial curvaton perturbation. As an example of how non-Gaussianity provides additional constraints on model parameters, we show how the primordial bispectrum on CMB scales can be used to constrain variance on much smaller scales in the curvaton field. Our analytical and numerical results allow for multiple tests of primordial non-Gaussianity, and thus they can offer consistency tests of the curvaton scenario.

Figures (6)

• Figure 1: To achieve the same curvature perturbation transfer efficiency $r$ one needs to start from slightly different initial value of $p$ in the sudden-decay case ( red dashed line) than in the non-instantaneous decay case ( black solid line).
• Figure 2: The non-linearity parameter $f_{\rm{NL}}$ as a function of curvature perturbation transfer efficiency $r = \zeta_1 / \zeta_{{\chi} 1}$. 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) is negative for $r>0.53$. Here we assume that $g({\chi}_\ast)$ is linear.
• Figure 3: The non-linearity parameter $g_{\rm{NL}}$ as a function of curvature perturbation transfer efficiency $r = \zeta_1 / \zeta_{{\chi} 1}$. The analytical approximative, i.e., sudden-decay result ( red dashed line) crosses zero at $r = 0.83$ and is positive for $r>0.83$. The exact numerical result ( black solid line) is positive for $r>0.79$. Here we assume that $g({\chi}_\ast)$ is linear.
• Figure 4: (a) Pdfs at $r=0.00028$ ($p=0.00030$, $f_{\rm{NL}}=4432$). Red dashed line is for the sudden decay, $\tilde{f}_{\rm SD}(\zeta)$, and black solid line for the non-instantaneous decay, $\tilde{f}(\zeta)$. The solid green/grey line is the Gaussian "reference", $f_g(\zeta_1)$. (b) The ratio of non-Gaussian pdfs to the Gaussian one.
• Figure 5: (a) Pdfs as in Fig. \ref{['Fig:pdfcomparisonLARGEfnl']} but now at $r=0.010758$ ($p=0.011560$, $f_{\rm{NL}}=114$). In this figure the Gaussian reference (solid green/grey line) is completely indistinguishable from the fully non-linear ("non-Gaussian") pdfs. (b) The ratio of non-Gaussian pdfs to the Gaussian one.