Non-gaussianity from the second-order cosmological perturbation

TL;DR

This work provides a unified second-order treatment of primordial non-Gaussianity across inflationary and post-inflationary scenarios, focusing on the normalization parameter $f_{\rm NL}$ for the curvature perturbation $\zeta$. By clarifying multiple definitions of $\zeta$ and employing a consistent slow-roll framework, the authors show that single-component slow-roll inflation yields $|f_{\rm NL}|\ll 1$, while multi-component inflation and post-inflationary mechanisms (curvaton and inhomogeneous reheating) can produce a wider range of $f_{\rm NL}$ values, including $\pm\mathcal{O}(1)$ and beyond under specific conditions. They correct prior expectations in two-field models, emphasizing the role of trajectory curvature and the timing of non-Gaussianity generation, and highlight that preheating generally does not generate large non-Gaussianity unless light non-inflaton fields are involved. The paper underscores the importance of second-order calculations for interpreting current and forthcoming observations (e.g., Planck) and constraining early-Universe physics. Overall, it maps the landscape of second-order non-Gaussianity across canonical scenarios and provides a framework for translating theoretical predictions into observational limits.

Abstract

Several conserved and/or gauge invariant quantities described as the second-order curvature perturbation have been given in the literature. We revisit various scenarios for the generation of second-order non-gaussianity in the primordial curvature perturbation ζ, employing for the first time a unified notation and focusing on the normalisation f_{NL} of the bispectrum. When the classical curvature perturbation first appears a few Hubble times after horizon exit, |f_{NL}| is much less than 1 and is, therefore, negligible. Thereafter ζ(and hence f_{NL}) is conserved as long as the pressure is a unique function of energy density (adiabatic pressure). Non-adiabatic pressure comes presumably only from the effect of fields, other than the one pointing along the inflationary trajectory, which are light during inflation (`light non-inflaton fields'). During single-component inflation f_{NL} is constant, but multi-component inflation might generate |f_{NL}| \sim 1 or bigger. Preheating can affect f_{NL} only in atypical scenarios where it involves light non-inflaton fields. The curvaton scenario typically gives f_{NL} \ll -1 or f_{NL} = +5/4. The inhomogeneous reheating scenario can give a wide range of values for f_{NL}. Unless there is a detection, observation can eventually provide a limit |f_{NL}| \lsim 1, at which level it will be crucial to calculate the precise observational limit using second order theory.

Figures (1)

• Figure 1: Two different procedures for defining the fields in two-component inflation. The fields are denoted by $\varphi$ and $\sigma$. (a) The field $\varphi$ parameterises the distance along the inflaton trajectories, with uniform $\varphi$ corresponding to the equipotential lines. The field $\sigma$ parameterises the distance along the equipotentials. (b) The fields $\varphi$ and $\sigma$ are the components in a fixed orthonormal basis, aligned with the inflationary trajectory at a certain point in field space. The value of $\varphi$ is now the displacement along the tangent vector and the value of $\sigma$ is the displacement along the orthogonal vector. Working to second order in these displacements, the equipotentials no longer coincide with the lines of uniform $\varphi$.