Non-Gaussianity from the second-order cosmological perturbation

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 $\ensuremath{\zeta}$, employing for the first time a unified notation and focusing on the normalization ${f}_{\mathrm{NL}}$ of the bispectrum. When $\ensuremath{\zeta}$ first appears a few Hubble times after horizon exit, $|{f}_{\mathrm{NL}}|$ is much less than 1 and is, therefore, negligible. Thereafter $\ensuremath{\zeta}$ (and hence ${f}_{\mathrm{NL}}$) is conserved as long as the pressure is a unique function of energy density (adiabatic pressure). Nonadiabatic pressure comes presumably only from the effect of fields, other than the one pointing along the inflationary trajectory, which are light during inflation (``light noninflaton fields''). During single-component inflation ${f}_{\mathrm{NL}}$ is constant, but multicomponent inflation might generate $|{f}_{\mathrm{NL}}|\ensuremath{\sim}1$ or bigger. Preheating can affect ${f}_{\mathrm{NL}}$ only in atypical scenarios where it involves light noninflaton fields. The simplest curvaton scenario typically gives ${f}_{\mathrm{NL}}\ensuremath{\ll}\ensuremath{-}1$ or ${f}_{\mathrm{NL}}=+5/4$. The inhomogeneous reheating scenario can give a wide range of values for ${f}_{\mathrm{NL}}$. Unless there is a detection, observation can eventually provide a limit $|{f}_{\mathrm{NL}}|\ensuremath{\lesssim}1$, at which level it will be crucial to calculate the precise observational limit using second-order theory.