The angular dependence of the three-point correlation function of the cosmic microwave background radiation as predicted by inflationary cosmologies

Abstract

Inflationary models predict a definite, model independent, angular dependence for the three-point correlation function of ∆T /T at large angles ( > ∼ 1 • ) which we calculate. The overall amplitude is model dependent and generically unobservably small, but may be large in some specific models. We compare our results with other models of nongaussian fluctuations.

Gauge-invariant second-order perturbations and non-Gaussianity from inflation

“…In this section we shall derive the behaviour on large scales of the comoving curvature perturbation at second order R (2) introduced in Eq. (21).…”

“…As in Ref. [8], but at the second-order level in the perturbations, an equation linking δ (2) ϕ and φ (2) holds, so that it is possible to obtain an explicit expression for δ (2) ϕ and hence for the curvature R (2) , once the equation for φ (2) has been solved. Indeed there are many differences with respect to the first-order case, as it will be evident to the reader from the details of the calculations which follow.…”

“…Indeed there are many differences with respect to the first-order case, as it will be evident to the reader from the details of the calculations which follow. The main difficulties arise due to the fact that -contrary to the first-order perturbation theory -also vector and tensor contributions are present, and to the fact that the two scalar potential φ (2) and ψ (2) differ even in the longitudinal gauge for the presence of source terms which are quadratic in the first-order perturbations.…”

“…(36) and (40) we have made use of the background relations a ′′ /a = H 2 + H ′ and (κ 2 /2) ϕ ′ 0 2 = H 2 − H ′ . We shall now describe how to isolate the equation for the potential φ (2) . We use the (0 − 0)-component of Einstein equations, the divergence of the (i − 0)-component and the trace of the (i − j)-component, both performed with the background metric δ ij .…”

“…So far, the problem of calculating the bispectrum of perturbations produced during inflation has been addressed by either looking at the effect of inflaton self-interactions (which necessarily generate non-linearities in its quantum fluctuations) in a fixed de Sitter background [2], or by using the so-called stochastic approach to inflation [3] 1 , where back-reaction effects of field fluctuations on the background metric are partially taken into account. An intriguing result of the stochastic approach is that the dominant source of non-Gaussianity actually comes from non-linear gravitational perturbations, rather than by inflaton self-interactions.…”