The Cosmic Microwave Background Bispectrum and Inflation

We derive an expression for the non-Gaussian cosmic-microwave-background (CMB) statistic $I_l^3$ defined recently by Ferreira, Magueijo, and G\'orski in terms of the slow-roll-inflation parameters $\epsilon$ and $\eta$. This result shows that a nonzero value of $I_l^3$ in COBE would rule out single-field slow-roll inflation. A sharp change in the slope of the inflaton potential could increase the predicted value of $I_l^3$, but not significantly. This further suggests that it will be difficult to account for such a detection in multiple-field models in which density perturbations are produced by quantum fluctuations in the scalar field driving inflation. An Appendix shows how to evaluate an integral that is needed in our calculation as well as in more general calculations of CMB bispectra.