We extend the deltaN formalism so that it gives all of the stochastic properties of the primordial curvature perturbation zeta if the initial field perturbations are Gaussian. The calculation requires only the knowledge of some family of unperturbed universes. A formula is given for the normalization f(NL) of the bispectrum of zeta, which is the main signal of non-Gaussianity. Examples of the use of the formula are given, and its relation to cosmological perturbation theory is explained.

The δN formula for the primordial curvature perturbation ζ is extended to include vector as well as scalar fields. Formulas for the tree-level contributions to the spectrum and bispectrum of ζ are given, exhibiting statistical anisotropy. The one-loop contribution to the spectrum of ζ is also worked out. We then consider the generation of vector field perturbations from the vacuum, including the longitudinal component that will be present if there is no gauge invariance. Finally, the δN formula is applied to the vector curvaton and vector inflation models with the tensor perturbation also evaluated in the latter case.

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 fNL of the bispectrum. When ζ first appears a few Hubble times after horizon exit, |fNL| is much less than 1 and is, therefore, negligible. Thereafter ζ (and hence fNL) 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 fNL is constant, but multi-component inflation might generate |fNL| ∼ 1 or bigger. Preheating can affect fNL only in atypical scenarios where it involves light non-inflaton fields. The simplest curvaton scenario typically gives fNL ≪ −1 or fNL = +5/4. The inhomogeneous reheating scenario can give a wide range of values for fNL. Unless there is a detection, observation can eventually provide a limit |fNL| < ∼ 1, at which level it will be crucial to calculate the precise observational limit using second order theory.