The Three--Point Correlation Function of the Cosmic Microwave Background in Inflationary Models

TL;DR

The paper develops a general, inflationary framework to quantify non-Gaussian signatures in the CMB via the temperature three-point function and skewness on large angular scales. Using stochastic inflation, it derives universal expressions for the ensemble-averaged inflaton bispectrum and the induced gravitational-potential bispectrum, connecting them to CMB multipoles through a hierarchical bispectrum structure. Specialization to the Sachs–Wolfe regime and several inflaton potentials shows that non-Gaussian effects are small, with mean skewness well below the Gaussian rms skewness set by cosmic variance, implying limited observability with COBE-scale data. The results provide a robust methodology and quantitative predictions for testing inflationary models and offer a framework applicable to other large-scale anisotropies with hierarchical bispectra.

Abstract

We analyze the temperature three--point correlation function and the skewness of the Cosmic Microwave Background (CMB), providing general relations in terms of multipole coefficients. We then focus on applications to large angular scale anisotropies, such as those measured by the {\em COBE} DMR, calculating the contribution to these quantities from primordial, inflation generated, scalar perturbations, via the Sachs--Wolfe effect. Using the techniques of stochastic inflation we are able to provide a {\it universal} expression for the ensemble averaged three--point function and for the corresponding skewness, which accounts for all primordial second--order effects. These general expressions would moreover apply to any situation where the bispectrum of the primordial gravitational potential has a {\em hierarchical} form. Our results are then specialized to a number of relevant models: power--law inflation driven by an exponential potential, chaotic inflation with a quartic and quadratic potential and a particular case of hybrid inflation. In all these cases non--Gaussian effects are small: as an example, the {\em mean} skewness is much smaller than the cosmic {\em rms} skewness implied by a Gaussian temperature fluctuation field.