Cosmic microwave background bispectrum and slow roll inflation

TL;DR

This work analyzes non-Gaussian features in the CMB arising from slow-roll inflation, deriving the full angular bispectrum $C_{l1 l2 l3}$ in terms of slow-roll parameters $\epsilon$ and $\eta$ and relating it to the observable power spectra. It introduces an unbiased bispectrum estimator and computes its variance under a mildly non-Gaussian regime, enabling a quantitative assessment of the signal-to-noise for primordial non-Gaussianity. The authors show that the intrinsic slow-roll bispectrum is too small to explain tentative COBE-DMR non-Gaussian detections, with $(S/N)_{3,l}$ decreasing at higher $l$ in the small-$\ell$ regime, indicating that a significant non-primordial contribution would be required for such observations. Overall, the paper provides explicit slow-roll predictions for the CMB bispectrum and highlights the observational challenges in detecting primordial non-Gaussianity with current data.

Abstract

Recent tentative findings of non-Gaussian structure in the COBE-DMR dataset have triggered renewed attention to candidate models from which such intrinsic signature could arise. In the framework of slow roll inflation with built-in non linearities in the inflaton field evolution we present expressions for both the cosmic microwave background (CMB) skewness and the full angular bispectrum ${\cal C}_{\ell_1 \ell_2 \ell_3}$ in terms of the slow roll parameters. We use an estimator for the angular bispectrum recently proposed in the literature and calculate its variance for an arbitrary $\ell_i$ multipole combination. We stress that a real detection of non-Gaussianity in the CMB would imply that an important component of the anisotropies arises from processes {\it other} than primordial quantum fluctuations. We further investigate the behavior of the signal-to-(theoretical) noise ratio and demonstrate for generic inflationary models that it decreases in the limited range of small-$\ell$'s considered for increasing multipole $\ell$ while the opposite applies for the standard ${\cal C}_{\ell}$'s.

Figures (3)

• Figure 1: Normalised angular bispectrum ${\cal C}_{\ell \ell \ell}$ as predicted by a generic slow roll inflation model, in units of ${\cal C}_{2}^2$, with ${\cal C}_{2}=1.18\times 10^{-10}$ related to the quadrupole power spectrum normalization $Q_{\rm rms-PS} = 18.7 \mu K$, as a function of the multipole index $\ell$ for all even values up to $\ell = 20$ (white dots in main plot). Grey band corresponds to the normalized variance $\sigma_{\hat{f}_{\ell\ell\ell}}$ (also in units of ${\cal C}_{2}^2$) associated to the estimator of Eq. (\ref{['estim']}). In the inset we zoom up $\ell^2(\ell+1)^2{\cal C}_{\ell \ell \ell}$ in the same units, which permits to see the alternating sign of the normalised bispectrum and its actual smooth increase in amplitude with increasing $\ell$.
• Figure 2: Absolute value of the signal-to-noise ratio for the bispectrum $(S/N)_{3,\ell}$ versus the multipole index $\ell$.
• Figure 3: Absolute value of the product $\ell \left(^{\ell~~\ell~~\ell}_{0~~0~~0}\right)$ for different $\ell$'s. Note that the product vanish for all $\ell = odd$.