A bound concerning primordial non-gaussianity

Seery and Lidsey have calculated the three-point correlator of the light scalar fields, a few Hubble times after horizon exit during inflation. Lyth and Rodriguez have calculated the contribution of this correlator to the three-point correlator of the primordial curvature perturbation. We calculate an upper bound on that contribution, showing that it is too small ever to be observable. Introduction. The study of non-gaussianity features in the primordial curvature perturbation ζ has become a subject of growing interest, because they provide a valuable discriminator between different models for its origin [1, 2, 3, 4]. While the relevant scales are outside the horizon, the curvature perturbation is given by [5, 6, 7] ζ(x, t) = δN(φi(x), ρ(t)) (1) = ∑ i Ni(t)δφi(x) + 1 2 ∑ ij Nij(t)δφi(x)δφj(x) + · · · . (2) In this expression, N is the number of e-folds of expansion, from a flat slice of spacetime on which the light fields during inflation have values φi(x) = φi + δφi(x), and ending on a slice which has uniform energy density ρ. The initial slice is taken to be a few e-folds after the relevant scales have left the horizon. The final slice can be any time after ζ has settled down to the time-independent value which provides an initial condition for the evolution of perturbations after horizon entry, and is constrained by observation. We use the notation Ni ≡ ∂N/∂φi and Nij ≡ ∂2N/∂φi∂φj , the derivatives being evaluated with the fields at their unperturbed values φi. According to first-order cosmological perturbation theory, the field perturbations δφi(x) are gaussian, with spectrum (H/2π) 2 where H is the Hubble parameter during inflation. (In this paper we ignore the scale dependence of the spectrum.) Using this result, Eq. (2) gives the evolution of ζ without any further use of cosmological perturbation theory. The first term is gaussian, and higher terms are responsible for any non-gaussianity.