The three-point correlation function of the cosmic microwave background in inflationary models

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 s anisotropies, such as those measured by the 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 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 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 mean skewness is much smaller than the cosmic rms skewness implied by a Gaussian temperature fluctuation field.