The Cosmic Microwave Background Bispectrum and Inflation

TL;DR

The paper develops a practical framework to connect a CMB non-Gaussian statistic $I_l^3$ with inflationary dynamics by deriving it in terms of slow-roll parameters $ε$ and $η$ for single-field slow-roll inflation. It shows analytically that $I_l^3$ is of order the density-perturbation amplitude times a linear combination of $ε$ and $η$, specifically yielding $\sqrt{l(l+1)} I_l^3 = \frac{2}{m_{Pl}^2} \sqrt{\frac{3V}{ε}} (3ε-2η)$, which is far from unity given $10^{-5}$ perturbations, thus disfavoring a large non-Gaussian signal from such models. The authors also explore a single-field model with a feature (a discontinuity in the slope) and find only modest enhancement, with peak $I_l^3$ still substantially below unity; they note that while multi-field scenarios could, in principle, generate non-Gaussianity, they generally struggle to reach the COBE-level signal. A general, recursive method to compute the CMB bispectrum from an arbitrary spatial bispectrum is presented, along with an Appendix detailing recursion relations for the triple-Bessel integral, enabling efficient evaluation for broader analyses.

Abstract

We derive an expression for the non-Gaussian cosmic-microwave-background (CMB) statistic $I_l^3$ defined recently by Ferreira, Magueijo, and Górski in terms of the slow-roll-inflation parameters $ε$ and $η$. 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.